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Curriculum Vitae Agostino Prástaro
Edition April 2014, 1-60
C U R R I C U L U M
V I T A E
CURRICULUM VITAE
AGOSTINO PRÁSTARO
Fig. 1. Agostino Prástaro
UNIVERSITY OF ROMA LA SAPIENZA - ROMA - ITALY
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CURRICULUM VITAE AGOSTINO PRÁSTARO – APRIL 2014
University address:
Department of Methods and Mathematical Models for Applied Sciences(MEMOMAT)
University of Roma La Sapienza, Via A.Scarpa, 16-00161 - Roma, Italy.
Phone: +39-06-49766723; Fax: +39-06-4957647
E-mails: [email protected]; [email protected].
Home page: http://www.dmmm.uniroma1.it/ agostino.prastaro/HOMEPAGEPRAS.htm
Fig. 2. University of Roma La Sapienza Pictures.
Home address:
Fig. 3. Porta Maggiore.
Via L’Aquila, 29-00176 - Roma - Italy. (Phone & Fax: +39-06-7023432 )
CURRICULUM VITAE AGOSTINO PRÁSTARO – APRIL 2014
Acknowledgments
Fig. 4. Prástaro’s mathematicians poster 2013-2014.
“...A World Of Mathematicians...For A World With Mathematics... ”
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1. GENERAL INFORMATIONS
• Birth-place and birth-date: Civitavecchia (Roma - Italy), 01-11-1942.
• Diploma di maturità scientifica: Trieste - Italy, July 1960.
• Doctor degree in Theoretical Physics: University of Torino, Torino - Italy, July
1967.
• Research grant in Theoretical Physics: University of Torino, Torino - Italy,
1967–1968.
• Researcher in Mathematical Models and Processes of Materials: Montedison
Research Center, Ferrara - Italy, 1968–1975.
• School on Tensors and Group Theory Applied to Crystallography - Brooklyn
Crystallographic Laboratory, Cambridge, UK, 1970.
• Professor (incaricato) in Mathematical Physics, University of Lecce and University of Calabria, Cosenza - Italy, 1975–1976.
• Professor (incaricato) in Mathematical Physics, University of Calabria, Cosenza
- Italy, 1976-1980.
• Professor (associate) in Mathematical Physics, University of Calabria, Cosenza
- Italy 1980-1990.
• Professor (associate) in Mathematical Physics, University of Roma La Sapienza,
Roma - Italy, 1990–2013.
• Courses taught: Rational Mechanics, Institutions of Mathematical Physics,
Mathematical Physics, Mathematical Analysis, Continuum Mechanics, Differential
Geometry.
• Member in: Italian Mathematical Union (UMI), European Mathematical Society (EMS), National Group of Mathematical Physics (GNFM/INDAM), American Mathematical Society (AMS), International Federation of Nonlinear Analysts
(IFNA), Mathematical Association of America (MAA), International Mathematical
Union (IMU).
• Reviewer for: Mathematical Reviews, Zentralblatt Mathematics and some
other mathematical journals.
• Head of the research-team MIUR-Faculty of Engineering-University of Roma
La Sapienza: “Geometry of PDEs and Applications”.
• Member in National Project in Algebraic Topology and Differential Geometry
(1986–2004).
• Member of PhD Committee: “Models and Mathematical Methods for Technology and Sociology”, University of Roma La Sapienza (1999–2003).
• Fields of Research: Geometry of PDE’s (Differential Geometry, Algebraic Geometry and Algebraic Topology); (Co)bordism in PDE’s and quantum PDE’s; Geometry of PDE’s in Continuum Mechanics; Geometry of PDE’s in Quantum Field
Theory and Quantum Supergravity; Geometry of PDE’s in Mathematical Physics.
• Organizer of many international conferences. The last ones from 2000: WCNA2000 Organized Session Functional Equations Geometric Analysis, Catania, Italy
(2000); Joint Meeting AMS-UMI Special Session Advances in Differential Geometry of PDE’s and Applications, Pisa, Italy (2002); ICM-2006 Satellite Conference
Advances in PDE’s Geometry, Madrid, Spain (2006); WCNA 2008 - Organized
Session: Advances Geometric Analysis, Orlando, FL - USA (2008).
CURRICULUM VITAE AGOSTINO PRÁSTARO – APRIL 2014
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• (Some) Invited Speaker or Partecipant and Visiting Professor: Department
of Mathematics, Univ. P. Sabatier, Toulouse, France (1986); I.H.E.S., Buressur-Yvette, France (1988); Department of Mathematics, University of California,
Berkely, USA, (1990); University of Torun, Poland (1996); University of Budapest,
Hungary (1996); Technical University of Athens, Greece (1996); Third World Congress of Nonlinear Analysts (WCNA2000), Catania, Italy (2000); University of Debrecen, Hungary (2000); Department of Mathematics, University of Opava, Czech
Republic (2000); University of Bilbao, Spain (2000); Department of Mathematics,
University of Atlanta, USA (2003); Department of Mathematics, Florida Institute
of Technology, Melbourne, USA (2005); ICM-2006 and Facultad de Matematicas,
Universidad Complutense de Madrid, Madrid - Spain (2006); Department of Mathematics, Morehouse College, Atlanta, USA (2007); WCNA 2008 - Orlando, FL USA (2008).
•
Award Sapienza Ricerca 2010. (Awarded publications [58, 59, 60].)
• Member in IFNA Board of Global Advisors World Congress of IFNA, 2012
- Athens, Greece, June 25-July 1, 2012.
• Membership in Roma Sapienza Foundation (from 2014 ).
• Publications: 96 scientific publications including 3 monographs as author,
3 as editor and coauthor and 2 patents. (See pages 7, 21 and 27.)
[Metric informations from Googlescholar: ”author:Prastaro author:A.”: (i) Worksnumber: 116; (ii) Citations-number: 912; (iii) H-index: 17.]1
1Please visit Wikipedia for criticism about H-index.
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Fig. 5. School of Engineering, University of Roma La Sapienza.
CURRICULUM VITAE AGOSTINO PRÁSTARO – APRIL 2014
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2. SUMMARY OF PRINCIPAL RESULTS
• [1]. This work is inserted in the problematic of the Regge’s trajectories in
strong interactions. The paper shows a relation between Regge’s trajectories of schannel with the ones in the t-channel that allows us to obtain general behaviours
of diffusion amplitudes.
• [2, 3, 4, 5, 95, 96]. These are mathematical models for elongational flows,
extrusion and flash-spinning of polymers. These models allow us to obtain optimum
industrial conditions of manufacturing.
• [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 23, 25]. These works
are devoted to build new mathematics that allow us to describe mechanics in an
intrinsic and completely covariant way. This is obtained by introducing new spaces,
derivative spaces, that are the natural universal spaces for differential calculus and
PDEs. This point of view generalizes previous one introduced by Ehresmann and
allows us to treat all the differential objects in algebraic way, i.e., as generalized
tensor-like objects. In this context a generalized form of the Noether theorem
for any PDE, even if of non-Lagrangian type, is obtained. Then this mathematical
machinery is applied also to describe the physics of continuum media, gauge theories
and supergravity.
• [22, 24, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 50]. These works extend the
geometric formulation of PDEs, developed in the previous papers, in order to consider also singular solutions and to obtain a geometric formulation of quantization
of PDEs. This geometric approach gives an intrinsic formulation of the canonical
quantization of PDEs. All this theory is extended also to super PDEs, i.e., PDEs
defined in the category of supermanifolds. In particular, a geometric formulation
of the canonical quantization of PDEs in supergravity is obtained.
• [37, 38, 39, 42, 45, 46, 51, 55, 56, 63, 70, 89]. These works present a general
theory of integral (co)bordism in PDEs and develop general methods to calculate
the corresponding (co)bordism groups. Such results allow us to characterize also
global solutions by means of algebraic topologic methods, and to recognize tunnel
effects in such solutions. In particular, in [55] some improvements are given emphasizing the role played by singular and weak-solutions and their relations with the
bordism groups for smooth solutions. Tricomi equation, heat equation, Ricci-flow
equation and d’Alembert equation on finite dimensional smooth manifolds are considered there as interesting examples where to apply the general theory. Variational
PDE’s, and their global solutions characterizations, by means of integral bordism
groups, are given in [56].
The methods proposed are constructive, as they give us suitable tools to build
solutions. In particular in [39, 42, 44, 45, 46, 63] the Navier-Stokes equation it is
also carefully considered and it is proved for such an equation existence of global
(smooth) solutions. In this way it is solved an old well known problem in the
theory of PDEs, remained open for many decades. (See also [65] where some further
improvements are given.)2
2First results on the geometric theory of the Euler equation and the Navier-Stokes equation
were obtained in [15, 16, 17, 19].
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In [51] some applications of previous results on integral (co)bordism groups and
canonical quantization of PDEs are considered. In particular fully covariant canonical quantizations of elementary particles interacting with the gravitational field,
also for massive neutrinos, are obtained. (See also [89].)
• [32, 33, 34, 40, 41]. Interesting applications of the geometric theory of PDE’s,
and integral bordism group theory for PDE’s, as formulated by A.Prástaro, are
considered to characterize local and global solutions of the d’Alembert equation
and its generalizations on Rn . For such equations integral bordism groups are
explicitly calculated and global solutions built. Sophisticated solutions with tunnel
effects are recognized.
• [37, 43, 44, 49, 54, 57, 58, 59, 60, 76, 78]. Here the new concepts of quantum manifolds and quantum PDEs are introduced, and for such structures it is
formulated a new noncommutative mathematics, (differential geometry, algebraic
geometry, geometric theory of quantum PDEs and determination of their integral
(co)bordism groups), that extends previous one for commutative PDEs. Applications to many interesting quantum PDEs are given. In particular, in [44] the concept
of category of quantum quaternionic manifolds is introduced and for PDEs built
in such a category theorems of existence of local and global solutions are proved.
Extensions to the quantum Navier-Stokes equation of previous results, obtained in
[39, 42, 44, 45, 46, 63] for the Navier-Stokes equation, are given in [49]. Moreover,
in [54, 57, 58, 59, 60, 69] the theory of quantum PDEs is extended to quantum
super PDEs and applied to quantum Yang-Mills PDEs and quantum supergravity
PDEs. These papers extend previous results of the integral (co)bordism groups theory for quantum PDE’s, to super quantum PDE’s. Furthermore, quantized (super)
PDE’s are identified with quantum (super) PDE’s in a canonical way. A proof that
quantum (super) PDE’s are a more general approach in the description of quantum
physics, that goes beyond the point of view of the quantization of (super) classical
theories (PDE’s), is given. Similarly to the commutative case, integral (co)bordism
groups for quantum (super) PDE’s, are related to weak, singular and smooth solutions, showing algebraic relations between such groups. Such a theory is applied to
many interesting PDE’s, and in particular, to quantum super Yang-Mills equations.
Characterizations of important and very sophisticated solutions, as quantum tunnel
effects and quantum black holes evaporation processes, are obtained by means of
integral bordism groups.
The new algebraic topologic methods introduced allowed also to solve some important problems in Mathematics. In particular, in [54, 60] the open problem to
prove existence of global smooth solutions for quantum (super)Yang-Mills equations
with mass-gap is solved.
• [47, 48]. Here some interesting applications of the geometric theory of quantum
PDE’s are considered. In fact, generalizations in the category of quantum manifolds
of previous results on d’Alembert equation [32, 33, 34], are given.
• [89, 91]. These monographs present, for the first time, a systematic formulation
of the geometric theory of noncommutative PDE’s which is suitable enough to
be used for a mathematical description of quantum dynamics and quantum field
theory. A geometric theory of supersymmetric quantum PDE’s is also considered,
in order to describe quantum supergravity. Covariant and canonical quantizations
of (super) PDE’s are shown to be founded on the geometric theory of PDE’s and
to produce quantum (super) PDE’s by means of functors from the category of
CURRICULUM VITAE AGOSTINO PRÁSTARO – APRIL 2014
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commutative (super) PDE’s to the category of quantum (super) PDE’s. Global
properties of solutions to (super) (commutative) PDE’s are obtained by means of
their integral bordism groups. In particular the (quantum) Navier-Stokes problem
and the (quantum)Yang-Mills problem are considered showing that their solutions
can be obtained in the framework of the integral bordism groups for such equations.
• [61, 62]. In these two papers the theory on the integral bordism groups for
PDE’s, formulated by Prástaro, is applied to some interesting PDE’s. In particular, in the first part a new general theory is developed that recognizes natural webs
structures on PDE’s. These structures are important to solve (generalized) Cauchy
problems. Some applications interesting PDE’s of the Mathematical Physics are
also given (wave equation, Korteweg de Vries equation). In the second part applications of the general theory of webs on PDE’s is applied to some problems concerning
the Riemannian geometry. In particular are considered generalized Yamabe problems and the proof of the Poincaré conjecture for 3-dimensional compact Riemannian manifolds, via the Ricci-flow equation. The new algebraic topologic methods
introduced by A.Prástaro in the theory of PDE’s, allowed to give a new proof that
the Poincaré conjecture is true.
• [64, 65, 66, 67, 68]. In these papers an unified geometric theory of stability
for PDE’s and solutions of PDE’s, is formulated in the framework of Prástaro’s
geometric theory of partial differential equations, i.e., by using integral bordism
groups. Relations with the Ljapunov’s stability theory, and the classic Ulam problem for approximate homomorphisms are stressed. (See also [52, 53].) Examples
for some important PDE’s are given. In particular the theory is applied to new
anisotropic MHD-PDE’s encoding dynamics for incompressible plasma fluids with
nuclear energy production. Such PDE’s are related, on the ground of their integrability properties, to crystallographic groups, (extended crystal PDE’s).
• [69, 70, 71]. In these papers one further extends the theory of quantum (super) PDE’s, previously developed in [37, 43, 44, 49, 54, 57, 58, 59, 60, 76, 78] to
systematically adapt the algebraic topologic surgery techniques to quantum supermanifolds and to solutions of quantum super manifolds. In particular in [70], by
using also the Prástaro’s geometric theory of quantum super PDE’s, one formulates
and proves the quantum Poincaré conjecture. This generalizes to the category of
quantum super PDE’s, the well known Poincaré conjecture for 3-dimensional closed
Riemannian manifolds. (See also [62] for a proof of this conjecture that is different by one by G. Perelman.) Furthermore in [71] one generalizes to the category
of quantum supermanifolds QS , the variational calculus for variational problems,
constrained by PDE’s, as formulated in [56]. In this way one formulates also a new
quantum gravity theory that extends to the category QS the Einstein’s General
Relativity. There, and in [69] solutions of quantum gravity equations, interpreting
nuclides and quantum black-holes, are considered in details. In particular quantum
black-holes are solutions that describe very high energy level production of particles, where the effects of strong-quantum-gravity become dominant. A new concept
of quantum propagator is introduced that allows us to recognize Green kernels as
“linear approximations” of such non-linear quantum propagators. More precisely
in [60] we have seen how to identify solutions of quantum super Yang-Mills equations that come from the Dirac quantization of super classic counterpart of such
equations, i.e. (classic Dirac solutions). Here we go beyond classic Dirac solutions
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CURRICULUM VITAE AGOSTINO PRÁSTARO – APRIL 2014
and recognize solutions related to a new nonlinear geometric definition of quantum
super Yang-Mills equation propagators.
• [72, 73]. In these papers one gives a geometric formulation of singular PDE’s
theory on the ground of the PDE’s geometric theory by A.Prástaro, i.e. by considering integral bordism groups and surgery techniques. Many interesting applications are given by explicitly solving (singular) boundary value problems of physical
relevance. A particular attention is reserved to singular ODE’s, where a general
characterization of their singular points is obtained. Many examples of (singular)
boundary value problems for ODE’s are given too. Relations with the stability
theory for PDE’s and PDE’s solutions are given by utilizing the recent geometric
stability theory given by A.Prástaro in [64, 65, 66, 67, 68].
• [74]. In this paper we show that between PDE’s and crystallographic groups
there is an unforeseen relation. In fact we prove that integral bordism groups of
PDE’s can be considered extensions of crystallographic subgroups. In this respect
we can consider PDE’s as extended crystals. Then an algebraic-topological obstruction (crystal obstruction), characterizing existence of global smooth solutions for
smooth boundary value problems, is obtained. These results, are also extended to
singular PDE’s, introducing (extended crystal singular PDE’s). An application to
singular MHD-PDE’s, is given extending some our previous results on such equations, and showing existence of (finite times stable smooth) global solutions crossing
critical zone nuclear energy production
• [75, 76]. Aim of the second paper (partially announced in [75]) is to specialize our study to quantum supergravity Yang-Mills PDE’s (quantum SG-Yang-Mills
PDE’s). This type of equations have been previously introduced by us in some
recent works and appears very useful to encode quantum dynamics unifying, just
at quantum level, gravity with the other fundamental forces of Nature, i.e., electromagnetic, weak and strong forces. These equations extend, at quantum level,
some superclassical ones, well known in literature about supergravity. In fact supergravity, as has been usually considered, is a classical field theory, that, in some
sense comes from a generalization of Charles Ehresmann and Èlie Cartan’s differential geometry. Then classical supergravity requires to be quantized. But in this
way one discards nonlinear phenomena. In fact this quantization is obtained by
means of so-called quantum propagators, that are just associated to linearizations
of classical PDE’s. Our formulation, instead, works directly on noncommutative
manifolds (quantum supermanifolds), and the quantization is not more necessary.
In fact, whether it is performed in this noncommutative framework, it can bee seen
as a linear approximation of a more general nonlinear integration. In some previous papers this important aspect has been carefully proved. Here we characterize
quantum super PDE’s like extended crystals, in the sense that their integral bordism groups can be considered as extensions of crystallographic subgroups. This
approach generalizes our previous one for commutative PDE’s, and allows us to
identify an algebraic topologic obstruction to the existence of global smooth solutions for PDE’s in the category QS . Furthermore, for such solutions we study
their stability properties from a geometric point of view. Section 3. Here we consider “quantum gravity” in the category QS , and encoded by suitable quantum
\
Yang-Mills equations (quantum SG-Yang-Mills PDE’s), say (Y
M ). In this way we
are able to characterize quantum (super)gravity like a secondary object, associ\
ated to some geometric fundamental objects (fields), solutions of (Y
M ). Then the
CURRICULUM VITAE AGOSTINO PRÁSTARO – APRIL 2014
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mass properties of such solutions are directly pointed-out, without the necessity to
directly assume symmetry breaking Higgs-mechanisms. However, we recognize a
\
constraint in (Y
M ), that gives a pure quantum geometrodynamic mechanism able
\
to justify mass acquisition (or loss) to a quantum solution of (Y
M ). Furthermore,
\
nuclear particles and nuclides can be seen as suitable p-chain solution of (Y
M ),
and their energetic, thermodynamic and stability properties characterized.
• [77, 78, 79, 80]. The famous Poincaré’s conjecture is about n-dimensional
manifolds, with n = 3, (R. S. Hamilton, G. Perelman, A. Prástaro [74, 62]), but
there are also generalizations of this conjecture for higher dimension manifolds. For
n = 4 the Poincaré conjecture has been proved by M. Freedman and for n ≥ 5, by
S. Smale. More recently has been given a generalization for quantum supermanifolds by A. Prástaro, that has also proved it in [70]). Nowadays, one can state
that a generalized Poincaré conjecture can be proved, or disproved, depending on
the particular category C in which it is formulated. This problem aroses in the
framework of the geometric topology, but in order to be solved it was necessary to
go outside that framework and recast the problem in a theory of PDE’s. But the
more recent results by A. Prástaro [70, 74, 62]), have proved that it was necessary
to return inside the algebraic topology framework, applied to the PDE’s geometric
theory. Really, it was soon evident that remaining in a pure algebraic topologic
approach it was not enough to solve this conjecture. In fact, a fundamental idea
to solve this problem is to ask whether it is possible find a smooth manifold, V ,
that without singular points bords a 3-dimensional compact, closed, smooth, simply connected manifold N with S 3 , when N is homotopy equivalent to S 3 .∪The
bordism theory is able to state that a smooth manifold V such that ∂V = N S 3 ,
there exists, since the nonoriented and oriented 3-dimensional bordism groups Ω3
and + Ω3 respectively, are both trivial: Ω3 = + Ω3 = 0. However, by simply looking
to the above bordism groups it is impossible to state if V has singular points (i.e.,
has holes) or it is a cylinder. By the way, more informations can be obtained by
the h-cobordism theory. More precisely the h-cobordism theorem in a category C of
manifolds, states that if the compact manifold V has ∂V = N0 ⊔ N1 , such that
the inclusion maps Ni ,→ V , i = 0, 1, are homotopy equivalences, (i.e., V is a hcobordism), and π1 (Ni ) = 0, then V ∼
=C N0 × [0, 1]. This theorem holds for n ≥ 5
in the category of smooth manifolds (S. Smale) and for n = 4 in the category of
topological manifolds (M. Freedman). But it does not work for n = 3 !
A very important angular stone, in the long history about the solution of the
Poincaré conjecture, has been the introduction, by R. S. Hamilton, of a new approach recasting the problem in to solving a PDE, the Ricci flow equation, and
asking for nonsingular solutions there, that starting from a Riemannian manifold
(N, γ) arrive to the 3-dimensional sphere S 3 , respectively identified with initial and
final Cauchy manifolds in the Ricci flow equation. In that occasion the Mathematical Analysis, or more precisely the Functional Analysis, entered in the Poincaré
conjecture problem. This approach has had many improvements until the papers
by G. Perelman. More recently, A. Prástaro, by using his algebraic topologic theory of PDE’s was able to give a pure geometric proof of the Poincaré conjecture.
Let us emphasize that the usual geometric methods for PDE’s (Spencer, Cartan),
were able to formulate for nonlinear PDE’s, local existence theorems only, until
the introduction, by A. Prástaro, of the algebraic topologic methods in the PDE’s
geometric theory. These give suitable tools to calculate integral bordism groups
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CURRICULUM VITAE AGOSTINO PRÁSTARO – APRIL 2014
in PDE’s, and to characterize global solutions. Then, on the ground of integral
bordism groups, a new geometric theory of stability for PDE’s and solutions of
PDE’s has been built. These general methodologies allowed to A. Prástaro to solve
fundamental mathematical problems too, other than the Poincaré conjecture and
some of its generalizations, like characterization of global smooth solutions for the
Navier-Stokes equation and global smooth solutions with mass-gap for the quantum
Yang-Mills superequation. (See [38, 45, 46, 54, 60, 63, 70, 71, 74, 91, 81, 62].)
The main purpose of this paper is to emphasize some problems related to exotic
heat PDE’s recently focused.3
The following theorem is a direct issue from results contained in [74].
Theorem 2.1. Any 3-dimensional compact, closed simply connected smooth manifold, homotopy equivalent to S 3 , is diffeomorphic to S 3 .4
The main purpose of [78] is to show how, by using the PDE’s algebraic topology,
introduced by A. Prástaro, one can prove the Poincaré conjecture in any dimension
for the category of smooth manifolds, but also to identify exotic spheres. In the
framework of the PDE’s algebraic topology, the identification of exotic spheres
is possible thanks to an interaction between integral bordism groups of PDE’s,
conservation laws, surgery and geometric topology of manifolds. With this respect
we shall enter in some details on these subjects, in order to well understand and
explain the meaning of such interactions. So the paper splits in three sections other
than this Introduction. 2. Integral bordism groups in Ricci flow PDE’s. 3. Morse
theory in Ricci flow PDE’s. 4. h-Cobordism in Ricci flow PDE’s. The main result
is contained just in this last section and it is Theorem 2.2 that by utilizing the
previously considered results states (and proves) the following.5
Theorem 2.2. The generalized Poincaré conjecture, for any dimension n ≥ 1 is
true, i.e., any n-dimensional homotopy sphere M is homeomorphic to S n : M ≈ S n .
For 1 ≤ n ≤ 6, n ̸= 4, one has also that M is diffeomorphic to S n : M ∼
= Sn.
n
But for n > 6, it does not necessitate that M is diffeomorphic to S . This happens
when the Ricci flow equation, under the homotopy equivalence full admissibility
hypothesis, (see below for definition), becomes a 0-crystal.
Moreover, under the sphere full admissibility hypothesis, the Ricci flow equation
becomes a 0-crystal in any dimension n ≥ 1.
3The Ricci flow equation can be considered a generalization of the classical Fourier’s heat
equation ut − κuxx = 0. In this paper we call exotic heat equations, PDE’s that, like the Ricci
flow equation, are of the type F j ≡ ujt − f j (uik ) = 0, 1 ≤ i, j ≤ m, with the length |k| of the multiindex k ∈ {1, · · · , n}, given by 0 ≤ |k| ≤ s ≤ r, where F j : J r (W ) → R are analytic functions of
order r ≥ 0 on a fiber bundle π : W ≡ R × E(M ) → R × M , where E(M ) is a vector bundle over
M , with M an analytic manifold of dimension n. Let us emphasize that the structure of exotic
heat equation is the more suitable to use in order to prove (generalized) Poincaré conjectures.
In fact, the idea to use PDE’s to solve the Poincaré’s conjecture, was the initial motivation to
introduce and study the well-known Yamabe equation. But that road did not turn out a lucky
choice to prove the conjecture, even if the Yambe equation is a very important equation to study
conformal problems in Riemannian geometry. (See, e.g., [62].)
4This last result agrees with the Hauptvermuntung conjecture that was proved for (n = 2, 3)dimensional manifolds and disproved for (n ≥ 4)-dimensional manifolds. (See A. Casson, J.
Milnor, E. Moise, T. Radó, D. Sullivan, W. Tuschmann.)
5
Results of this paper agree with previous ones by J. Cerf, M. Freedman, M. A. Kervaire and
J. W. Milnor, E. Moise and S. Smale, and with the recent proofs of the Poincaré conjecture by R.
S. Hamilton, G. Perelman, and A. Prástaro.
CURRICULUM VITAE AGOSTINO PRÁSTARO – APRIL 2014
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In [79] a theorem characterizing global solutions of exotic 8-d’Alembert equation
is given. This theorem allows us to state that two diffeomorphic exotic 7-sphere,
identified with two Cauchy manifolds in (d′ A)8 over R8 , bound singular solutions
only, but they cannot bound smooth solutions. (Compare with the situation in
the Ricci flow equation on compact, simply connected 7-dimensional Rimennian
manifolds [78].)
In [80] are generalized results of the previous three papers to any PDE and relating them to general algebraic topological properties of PDE’s, like integral bordism
groups, conservation laws, spectral sequences, algebraic topological spectra. In particular it is emphasized their relations with exotic Cauchy manifolds, that motivate
the name “exotic” given to these equations. Theorem 2.2 is generalized to any PDE.
The main results in this paper is just the generalization to dimension n = 4, of such
theorem, obtained by means the proof of the smooth Poincaré conjecture. This is
the generalization of the Poincaré conjecture in the category of smooth manifolds
in dimension n = 4. This was a very important open problem, remained unsolved
also after the solution of the famous Poincaré conjecture.
• [81]. This paper aims to further develop the A. Prástaro’s geometric theory
of quantum PDE’s, by considering three different (even if related) subjects in this
theory. The first is a way to characterize quantum PDE’s by means of suitable
Heyting algebras. Nowadays these algebraic structures are considered important in
order to characterize quantum logics and quantum topoi. Really we prove that to
any quantum PDE can be associated a Heyting algebra, naturally arising from the
algebraic topologic structure of the PDE’s and that encodes its integral bordism
group.
Another aspect that we shall consider is the extension of the category Q of quantum manifolds, or QS of quantum supermanifolds, to the ones Qhyper of quantum
hypercomplex manifolds. These generalizations are obtained by extending a quantum algebra A, in the sense of A. Prástaro, by means of Cayley-Dickson algebras.
In this way one obtains a new category of noncommutative manifolds, that are useful in some geometric and physical applications. In fact there are some fashioned
research lines, concerning classical superstrings and classical super-2-branes, where
one handles with algebras belonging just to some term in the Cayley-Dickson construction. Thus it is interesting to emphasize that the Prástaro’s geometric theory
of quantum PDE’s can be directly applied also to PDE’s for such quantum hypercomplex manifolds. This allows us to encode quantum micro-worlds, by a general
theory that goes beyond the classical simple description of classical extended objects, and solves also the problem of their quantization.
Let us emphasize that in some previous works we have formulated a geometric theory of quantum manifolds that are noncommutative manifolds, where the
fundamental algebra is a suitable associative noncommutative topological algebra,
there called quantum algebra. Extensions to quantum supermanifolds and quantumquaternionic manifolds are also considered too. Furthermore, we have built a geometric theory of quantum PDE’s in these categories of noncommutative manifolds,
that allows us to obtain theorems of existence of local and global solutions, and
constructive methods to build such solutions too.
On the other hand it is well known that the sequences R ⊂ C ⊂ H ⊂ O ⊂ S,
where S is the sedenionic algebra, fit in the so-called Cayley-Dikson construction,
R ⊂ C ⊂ H ⊂ O ⊂ S ⊂ · · · ⊂ Ar ⊂ · · · , where Ar is a Cayley algebra of
14
CURRICULUM VITAE AGOSTINO PRÁSTARO – APRIL 2014
r
dimension 2r : Ar ∼
, r ≥ 0, A0 = R. Thus we can also consider quantum
= R2 ⊗
hypercomplex algebras A Ar , 1 ≤ r ≤ ∞, where A is⊗
a quantum algebra
in the
⊗
sense of Prástaro, obtaining the natural inclusions A R Ar ,→ A R Ar+1 . It
is important to note that the Cayley algebras Ar are not associative for r ≥ 3,
(even if they are exponential associative6) hence also the corresponding quantum
hypercomplex algebras are non-associative for r ≥ 3, despite the associativity of the
quantum algebra A. This fact introduces some particularity in the theory of PDE’s
on such algebras. The purpose of this paper is just to study which new behaviours
have PDE’s on the category Qhyper of quantum hypercomplex manifolds.
Let us emphasize that a first justification to use the category of quantum (super)
manifolds to formulate PDE’s that encode quantum physical phenomena, arises
from the fact that quantized PDE’s can be identified just with quantum (super)
PDE’s, i.e., PDE’s for such noncommutative manifolds. However, to formulate
equations just in the category Q (or QS ) allows us to go beyond the point of view
of quantization of classical systems, and capture more general nonlinear phenomena
in quantum worlds, that should be impossible to characterize by some quantization
process. (See Refs. [58, 59, 60, 70, 71, 75, 76].) In fact the concept of quantum
algebra (or quantum superalgebra) is the first important brick to put in order to
build a theory on quantum physical phenomena. In other words it is necessary to
extend the fundamental algebra of numbers, R, to a noncommutative algebra A,
just called quantum algebra. The general request on such a type of algebra can be
obtained on the ground of the mathematical logic. (See [89, 91].) In fact, we have
shown that the meaning of quantization of a classical theory, encoded by a PDE Ek ,
in the category of commutative manifolds, is a representation of the logic L(Ek ) of
the classic theory, into a quantum logic Lq . More precisely, L(Ek ) is the Boolean
algebra of subsets of the set Ω(Ek )c of solutions of Ek : L(Ek ) ≡ P(Ω(Ek )c ). (The
infinite dimensional manifold Ω(Ek )c is called also the classic limit of the quantum
situs of Ek .) Furthermore, Lq is an algebra A of (self-adjoint) operators on a
locally convex (or Hilbert) space H: Lq ≡ A ⊂ L(H). Then to quantize a PDE
Ek , means to define a map L(Ek ) → Lq , or an homomorphism of Boolean algebras
q : P(Ω(Ek )c ) → Pr (H), where Pr (H) is a Boolean algebra of projections on H.
This construction allowed us also to prove that a quantization of a classical theory,
can be identified by a functor relating the category of differential equations for
commutative (super)manifolds, with the category of quantum (super) PDE’s.
We can also extend a quantum algebra A, when the particular mathematical (or
physical) problem requires it useful. Then the extended algebra does not necessitate
to be associative. This is, for example, the case when the extension is made by
means of some algebra in the Cayley-Dickson construction, obtaining a quantumCayley-Dickson construction:

/ Q1 
/ Q2 
/ · · · 
/ Qr 
/ ···
Q0 ⊗
⊗
⊗
where Qr ≡ A R Ar , hence Q0 = A, Q1 = A R C and Q2 = A R H, etc.
The main of this paper is just to show that the Prástaro’s algebraic topologic
theory of quantum PDE’s, formulated starting from 80s, directly applies to these
non-associative quantum algebras arising in the above quantum-Cayley-Dickson
construction.
6i.e., z n+m = z n z m , ∀z ∈ A and n, m ∈ N.
r
CURRICULUM VITAE AGOSTINO PRÁSTARO – APRIL 2014
15
Finally, the last purpose of this paper is to extend the concept of exotic PDE’s,
recently introduced by A. Prástaro, for PDE’s in the category of commutative
manifolds, also for the ones in the category of quantum PDE’s. In particular, we
prove also smooth versions of quantum generalized Poincaré conjectures.
In the following we list the main results of this paper, assembled for sections. 2. It
is shown that to any quantum hypercomplex PDE Êk ⊂ Jˆnk (W ), can be associated
a topological spectrum (integral spectrum) and a Heyting algebra (integral Heyting
algebra) encoding some algebraic topologic properties of such a PDE. This allows us
to give a new constructive point of view to the actual approach to consider quantum
logic in field theory by means of topoi. 3. Here the Prástaro’s formal geometric
theory of PDE’s is extended from the category of quantum (super)manifolds to
the ones for quantum hypercomplex manifolds. Global solutions of PDE’s in the
category Qhyper , are characterized by means of suitable bordism groups. 4. An
algebraic topologic characterization of singular PDE’s in the category Qhyper is
given. 5. The concept of exotic PDE’s, perviously introduced by A. Prástaro
for PDE’s in the category of commutative manifolds, is extended to the category
Qhyper . Global solutions for exotic PDE’s in the category Qhyper , that allow to
classify smooth solutions starting from quantum homotopy spheres are classified.
In particular, an integral h-cobordism theorem in quantum Ricci flow PDE’s is
proved.
• [82, 83, 84]. In order to encode strong reactions of the high energy physics,
by means of quantum nonlinear propagators in the Prástaro’s geometric theory
of quantum super PDE’s, some related geometric structures are further developed
and characterized. In particular super-bundles of geometric objects in the category
QS of quantum supermanifolds are considered and quantum Lie derivative of sections of super bundle of geometric objects are calculated. Quantum supermanifolds
with classic limit are classified with respect to the holonomy groups of these last
commutative manifolds. A theorem characterizing quantum super manifolds with
structured classic limit as super bundles of geometric objects is obtained. A theorem on the characterization of chi-flow on suitable quantum manifolds is proved.
This solves a previous conjecture too. Quantum instantons and quantum solitons
are defined are useful generalizations of the previous ones, well-known in the literature. Quantum conservation laws for quantum super PDEs are characterized.
Quantum conservation laws are proved work for evaporating quantum black holes
too. Characterization of observed quantum nonlinear propagators, in the observed
quantum super Yang-Mills PDE, by means of conservation laws and observed energy is obtained. Some previous results by A. Prástaro about generalized Poincaré
conjecture and quantum exotic spheres, are generalized to the category Qhyper,S of
hypercomplex quantum supermanifolds. (This is the first part of a work divided in
two parts. For part II see [83].)
In the second part decomposition theorems of integral bordisms in quantum
super PDEs are obtained. In particular such theorems allow us to obtain representations of quantum nonlinear propagators in quantum super PDE’s, by means
of elementary ones (quantum handle decompositions of quantum nonlinear propagators). These are useful to encode nuclear and subnuclear reactions in quantum physics. Prástaro’s geometric theory of quantum PDE’s allows us to obtain
constructive and dynamically justified answers to some important open problems
in high energy physics. In fact a Regge-type relation between reduced quantum
16
CURRICULUM VITAE AGOSTINO PRÁSTARO – APRIL 2014
mass and quantum phenomenological spin is obtained. A dynamical quantum GellMann-Nishijima formula is given. An existence theorem of observed local and
global solutions with electric-charge-gap, is obtained for quantum super Yang-Mills
\
\
\
PDE’s, (Y
M )[i], by identifying a suitable constraint, (Y
M )[i]w ⊂ (Y
M )[i], quantum electromagnetic-Higgs PDE, bounded by a quantum super partial differential
\
\
relation (Goldstone)[i]
w ⊂ (Y M )[i], quantum electromagnetic Goldstone-boundary.
An electric neutral, connected, simply connected observed quantum particle, iden\
\
tified with a Cauchy data of (Y
M )[i], it is proved do not belong to (Y
M )[i]w .
\
Existence of Q-exotic quantum nonlinear propagators of (Y M )[i], i.e., quantum
nonlinear propagators that do not respect the quantum electric-charge conservation
is obtained. By using integral bordism groups of quantum super PDE’s, a quantum
crossing symmetry theorem is proved. As a by-product existence of massive photons and massive neutrinos are obtained. A dynamical proof that quarks can be
broken-down is given too. A quantum time, related to the observation of any quantum nonlinear propagator, is calculated. Then an apparent quantum time estimate
for any reaction is recognized. A criterion to identify solutions of the quantum super
Yang-Mills PDE encoding (de)confined quantum systems is given. Supersymmetric particles and supersymmetric reactions are classified on the ground of integral
\
bordism groups of the quantum super Yang-Mills PDE (Y
M ). Finally, existence
of the quantum Majorana neutrino is proved. As a by-product, the existence of a
new quasi-particle, that we call quantum Majorana neutralino, is recognized made
by means of two quantum Majorana neutrinos, a couple (νee , νēe ), supersymmetric
partner of (νe , ν̄e ), and two Higgsinos. (Part I and Part II are unified in arXiv.)
in the third part quantum nonlinear propagators in the observed quantum super
\
Yang-Mills PDE, (Y
M )[i], are further characterized. In particular, a criterion that
assures the zero lost quantum electric-charge is obtained. In a previous work [?],
we have characterized observed quantum nonlinear propagators V of the observed
\
quantum super Yang-Mills PDE, (Y
M )[i], proving that the total quantum electriccharge of incident particles in quantum reactions does not necessitate to be the
same of the total quantum electric-charge of outgoing particles. This allowed us
to define Q-exotic quantum nonlinear propagators ones where there is a non-zero
lost quantum electric-charge, Q[V ] ∈ A, in the corresponding encoded reactions.
\
(A is the fundamental quantum superalgebra in (Y
M )[i]) This important phenom\
ena, that is related to the gauge invariance of (Y M )[i], was non-well previously
understood, since the gauge invariance was wrongly interpreted. Really just the
gauge invariance is the main origin of such phenomenon, beside the structure of the
quantum nonlinear propagator. This fundamental aspect of quantum reactions in
\
(Y
M )[i], gives strong theoretical support to the guess about existence of quantum
reactions where the “electric-charge” is not conserved. This was quasi a dogma
in particle physics. However, there are in the world many heretical experimental
efforts to prove existence of decays like the following e− → γ + ν, i.e. electron
decay into a photon and neutrino. In this direction some first weak experimental
evidences were recently obtained. Some other exotic decays were also investigated,
as for example the exotic neutron’s decay: n → p + ν + ν̄. (See References quoted
in the paper.) With this respect, one cannot remark the singular role played, in the
history of the science in these last 120 years, by the electron, a very small and light
CURRICULUM VITAE AGOSTINO PRÁSTARO – APRIL 2014
17
particle. In fact, at the beginning of the last century was just the electron to produce a break-down in the Maxwell and Lorentz physical picture of the world, until
to produce a completely new point of view, i.e. the quantum physics. Now, after
120 years the electron appears to continue do not accept the place that physicists
have reserved to it in the world-puzzle.
Aim of the third paper is to further characterize a criterion to recognize under which constraints quantum nonlinear propagators preserve quantum electric
charges between incoming and outgoing particles. The main result of this third
^
\
\
part is the existence of a sub-equation (Y
M )[i]• ⊂ (Y
M )[i], were live solutions
strictly respecting the conservation of the quantum electric charge. Instead, so^
\
\
lutions bording Cauchy data contained in the sub-equation (Y
M )[i]• ⊂ (Y
M )[i],
^
\
that are globally outside (Y
M )[i]• , can violate the conservation of the quantum
electric charge. This effect is interpreted caused by the quantum supergravity. The
action of the quantum supergravity is able to guarantee existence of such more general quantum nonlinear propagators in quantum super Yang-Mills PDEs. In fact
quantum supergravity can deform quantum nonlinear propagators in order that
\
they can produce such exotic solutions of (Y
M )[i]. In other words, quantum exotic strong reactions exist as a by-product of quantum supergravity that produces
non-flat quantum nonlinear propagators. In the standard model quantum supergravity is completely forgotten. Without quantum supergravity, exotic quantum
propagators cannot be realized ! The main results are the following. • A criterion
to recognize under which constraints quantum nonlinear propagators have zero lost
quantum electric-charge. Our main result is the identification of a sub-equation
^
\
\
(Y
M )[i]• ⊂ (Y
M )[i], that is formally integrable and completely integrable, such
that all quantum reactions encoded there, are characterized by non-Q-exotic quantum nonlinear propagators. • A theorem proving that Q-exotic quantum nonlinear
\
propagators of (Y
M )[i] are solutions with exotic-quantum supergravity, i.e., havejα . • A justification of the
ing non zero observed quantum curvature components R
K
so-called quantum entanglement phenomenon on the algebraic topologic structure
of quantum nonlinear propagators. We prove that the EPR paradox is completely
solved in the framework of the Algebraic Topology of quantum PDE’s, as formulated by A. Prástaro. In fact, EPR paradox is related to a macroscopic model
of physical world (Einstein’s General Relativity (GR)). In order to reconcile this
model with quantum mechanics, it is necessary to extend GR to a noncommutative geometry, as made by the Prástaro’s Algebraic Topology of quantum (super)
PDE’s. In fact the logic of microworlds is not commutative, hence it is natural
that macroscopic mathematical models cannot justify quantum dynamics. In other
words, the incompleteness of quantum mechanics (QM) is the complementary incompleteness of the GR, since they talk at different levels: microscopic the first
(QM), macroscopic the second (GR). These different points of view can be reconciled by introducing the noncommutative logic of the QM in the geometric point of
view of the GR. But this must be made at the dynamic level ! It is not enough to
formulate some noncommutative geometry to encode microworlds ! Therefore the
necessity to formulated a geometric theory of quantum PDE’s as has been realized
\
by A. Prástaro. • Existence of solutions of (Y
M )[i] admitting negative (absolute)
temperature and their relations with quantum entanglement is given.
18
CURRICULUM VITAE AGOSTINO PRÁSTARO – APRIL 2014
• [85]. The well known Goldbach’s conjecture in number theory, remained unsolved up to now, was one of the most famous example of the Gödel ’s incompleteness theorem. In this paper we give a direct proof of this conjecture. Some useful
applications regarding geometry and quantum algebra are also obtained.
Our proof is founded on the experimental observation that fixed an even integer,
say 2n, n ≥ 1, and considered the highest prime number p1 ∈ P , that does not
exceed 2n, the difference 2n − p1 is often a prime number, or if not, we can pass
(1)
(1)
to consider the next prime number, say p1 < p1 , and find that 2n − p1 is just
a prime number. (We denote by P the set of prime numbers.) Otherwise, we can
(s)
(s)
continue this process, and after a finite number of steps, obtain that 2n−p1 = p2 ,
(s)
(s)
where p2 ∈ P . This process gives us a practical way to find two primes p1 and
(s)
(s)
(s)
p2 , such that 2n = p1 + p2 , hence satisfy the Goldbach’s conjecture. Of course
the question is ”Does this phenomenon is a law and why ?” The main result of this
paper is to prove that our criterion, is mathematically justified.
• [86]. In this paper we consider some problems in Number Theory called Landau’s problems listed by Edmund Landau at the 1912 International Congress of
Mathematicians. These problems are the following. 1. Goldbach’s conjecture. 2.
Twin prime conjecture. 3. Legendre’s conjecture. 4. Are there infinitely many
primes p such that p − 1 is a perfect square ? In [85] the proof of the Goldbach’s
conjecture has been already given. In this paper we show that by utilizing some
algebraic topologic methods introduced in [85], some Landau’s problems can be
proved too. Furthermore, for the above fourth Landau’s problem a Euler-Riemann
zeta function estimate is given and settled the problem negatively by evaluating the
cardinality of the set of solutions of a suitable Diophantine equation of RamanujanNagell-Lebesgue type.
• [87]. The Riemann hypothesis is the conjecture concerning the zeta Riemann
function ζ(s), given by B. Riemann (1892). The difficulty to prove this conjecture is
related to the fact that ζ(s) has been formulated in a some cryptic way as complex
continuation of hyperharmonic series and characterized by means of a functional
equation that in a sense caches its properties about the identifications of zeros. Our
approach to solve this conjecture has been to recast the zeta Riemann function ζ(s)
to a quantum mapping between quantum-complex 1-spheres, i.e., working in the
category Q of quantum manifolds as introduced by A. Prástaro. (See on this subject References [70, 81] and related works by the same author quoted therein.) More
precisely the fundamental quantum algebra is just A = C, and quantum-complex
manifolds are complex manifolds, where the quantum class of differentiability is
the holomorphic class. In this way one can reinterpret all the theory on complex
manifolds∪as a theory on quantum-complex manifolds. In particular the Riemann
sphere C {∞} can be identified with the quantum-complex 1-sphere Ŝ 1 , as considered in [70, 81]. The paper splits in two more sections. In Section 2 we resume
some fundamental definitions and results about the Riemann zeta function ζ(s).
In Section 3 the main result, i.e., the prof that the Riemann hypothesis is true, is
contained in Theorem 3.1. This is made splitting the proof in some steps (lemmas).
It is important to emphasize the central role played by Lemma 3.7. This focuses
the attention on the completed Riemann zeta function, ζ̃(s), that symmetrizes the
role between poles, with respect to the critical line of C, and between zeros, with
respect to the x = ℜ(s)-axis. Finally the conclusion can be obtained by extending
ζ̃(s) to a quantum-complex mapping ζ̂(s), between quantum-complex 1-spheres.
CURRICULUM VITAE AGOSTINO PRÁSTARO – APRIL 2014
19
Then by utilizing the properties of meromorphic functions between compact Riemann spheres, identified with quantum-complex 1-spheres, we arrive to prove that
all (non-trivial) zeros of ζ(s) must necessarily be on the critical line. In fact, the
extension of ζ̃(s) to ζ̂(s), reduces zeros of this last meromorphic function to have
have two simple zeros, symmetric with respect to the equator, and two simple poles,
symmetric with respect to the critical line. For symmetry properties, this implies
that also ζ̃(s) cannot have zeros outside the critical line, hence the same must
happen for ζ(s) for non-trivial zeros.
• [88]. Aim of this paper is to utilize previous Prástaro’s results on quantum
supergravity to prove that the geometric structure of quantum propagator encoding Universe at the Planck epoch is the cause of the Universe’s expansion.7 This
expansion is not caused by a strange exoteric force, but it is the boundary-effect of
the quantum nonlinear propagator encoding the Universe. In fact this propagator
has a boundary with thermodynamic quantum exotic components. The presence of
such exotic components produces an increasing of energy contents in the Universe,
seen as transversal sections of such a quantum nonlinear propagator. To the increasing of energy corresponds an increasing in the expansion of the Universe. Such
expansion has produced the passage of the Universe from the Higgs-universe, the
first massive universe at the Planck epoch, to the actual macroscopic one. However, yet in such a macroscopic age the expansion of the Universe can be justified
by using the same philosophy. This will be illustrated by adopting the Einstein’s
General Relativity equations, but taking into account the effect of its quantum
origin (Planck-epoch-legacy). With this respect one can state that the so-called
dark-energy-matter, is nothing else than the increasing in energy produced by the
thermodynamic exotic boundary encoding the Universe. Therefore it is a pure geometrodynamic bordism effect that produces an expansion of our Universe also at
the Einstein epoch. Paradoxically this is a consequence of the energy conservation
law that continues to work whether at the Planck epoch or at the Einstein age.
This Prástaro theory gives also a precise mathematical support to some early conjectures on the continuous creation of matter. (See, e.g., P. A. Dirac (1974), F.
Hoyle (1949) and F. Hoyle and J. V. Narlikar (1963).
7Georges Lematre (1927) and Edwin Hubble (1929) first proposed that the Universe is expanding. Lemaitre used Einstein’s General Relativity equations and Hubble estimated value of
the rate of expansion by observed red-shifts. The most precise measurement of the rate of the
Universe’s expansion, has been obtained by NASA’s Spitzer Space Telescope and published in
October 2012. Very recently (March 17, 2014) some scientists of the Harvard-Smithsonian Center
for Astrophysics, announced that observations with the telescope Bicep2 (Background Imaging of
Cosmic Extragalactic Polarization), located at the South Pole, allowed to give an experimental
proof of the Big Bang. (See the following link First Direct Evidence of Cosmic Inflation.)
20
CURRICULUM VITAE AGOSTINO PRÁSTARO – APRIL 2014
Fig. 6. Florida Institute of Technology - Melbourne, FL, USA - 2005.
CURRICULUM VITAE AGOSTINO PRÁSTARO – APRIL 2014
21
3. PUBLICATIONS
References
PAPERS
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of spinning, Tex. Res. J. 15(1975), 118–127. DOI: 10.1177/004051757504500206.
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Polymerer und für die Spinnbedingungen, Colloid & Polymer Science 255(2)(1977), 624–
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3(2)(1979), 15–31. MR0556645(81b:76014); Zbl 0427.76003.
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Torino Suppl. 114(1980/81), 289–292. MR0670263(83h:58013).
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Unione Mat. Ital. (5)S.-FM(1981), 107–129. MR0641761(83c:73002b); Zbl 0478.58005.
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(5)18-A(1981), 411–416. MR0633674(842:58049); Zbl 0471.58012.
[12] A. Prástaro, Dynamic conservation laws and the Korteweg-De Vries equation, Atti convegno
su onde e stabilità nei mezzi continui, Catania 1981, Quaderni CNR-GNFM, Catania (1982),
272–274.
[13] A. Prástaro, Spinor super bundles of geometric objects on spinG space-time structures, Boll.
Unione Mat. Ital. (6)1-B(1982), 1015–1028. MR0683489(84c:53036); Zbl 0501.53023.
[14] A. Prástaro, Gauge geometrodynamics, Riv. Nuovo Cimento 5(4) (1982), 1–122. DOI:
10.1007/BF02740593. MR0693882(84e:83045); Zbl 0695.58028.
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MR0823718(87g:58034).
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Geometrodynamics of non-relativistic continuous media.I: Spacetime structures, Rend. Sem. Mat. Univ. Politec. Torino40(2)(1982), 89–117.
MR0724201(85e:53095); Zbl 0525.53036.
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constitutive structures, Rend. Sem. Mat. Univ. Politec. Torino 43(1)(1985), 91–116.
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[18] A. Prástaro, A geometric point of view for the quantization of non-linear field theories, Atti
VI Convegno Nazionale di Relatività Generale e Fisica della Gravitazione, Firenze 1984, R.
Fabbri and M. Modugno (eds.), Pitagora Ed., Bologna (1986), 289–292.
[19] A. Prástaro, Dynamic conservation laws, Geometrodynamics Proceedings 1985, A. Prástaro
(ed.), World Scientific Publishing, Singapore (1985), 283–420. MR0825801(87g:53109);
Zbl 0645.58038. [About “Geometrodynamics” see also Wikipedia.]
[20] A. Prástaro & T. Regge, The group structure of supergravity, Ann. Inst. H. Poincaré Phys.
Théor. 44(1)(1986), 39–89. MR0834019(87i:83104); Zbl 0588.53066.
[21] V. Marino & A. Prástaro, On the geometric generalization of the Noether theorem, Lecture
Notes in Math. 1209, Springer-Verlag, Berlin (1986), 222–234. DOI: 10.1007/BFb0076634.
MR0863759(88j:58142); Zbl 0603.53058.
22
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[22] A. Prástaro, Quantum gravity and group model gauge theory, Journées Relativistes,
Toulouse, France 1986, A. Crumeyrolle (ed.), Univ. Paul Sabatier Toulouse (1986), 213–222.
[Recent results on the quantum geometrodynamics. Proceedings General Relativity and Gravitation, vol.1. July, 4-9, 1983, Padova - Italy. (B. Bertotti, F. de Felice & A. Pascolini eds.)
CNR - Roma (1983), 1145.]
[23] V. Marino & A. Prástaro, On the conservation laws of PDE’s, Rep. Math. Phys.
26(2)(1987/8), 211–225. DOI: 10.1016/0034-4877(88)90024-9. MR0991720(91b:58097);
Zbl 0695.58029.
[24] A. Prástaro, On the quantization of Newton equation, Atti IX Congresso AIMETA, Bari
1988, AIMETA(1988), 13–16.
[25] A. Prástaro, Wholly cohomological PDE’s, International Conference on Differential Geometry
and Applications, Dubrovnick (Yu), 1988, Univ. Beograd & Univ. Novi Sad (1989), 305–314.
MR1040078(91c:58151); Zbl 0695.58030.
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Janyska & D. Krupka (eds.), World Scientific Publishing, Teanek, NJ, (1990), 392–404.
MR1062046(91m:58175); Zbl 0796.35006.
[27] A. Prástaro, On the singular solutions of PDE’s, Atti X Congresso Nazionale AIMETA, Pisa
1990, AIMETA(1990), 17–20.
[28] A. Prástaro, Cobordism of PDE’s, Boll. Unione Mat. Ital. (7)5-B(1991), 977–1001.
MR1146763(93a:57037); Zbl 0746.57015.
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[30] A. Prástaro, Geometry of super PDE’s, Geometry of Partial Differential Equations, A.
Prástaro & Th. M. Rassias (eds.), World Scientific Publishing, River Edge, NJ, (1994), 259–
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and integral cobordism, Differential Geom. Appl. 4(3)(1994), 283–300. DOI: 10.1016/09262245(94)00017-4. MR1299399(96b:58122); Zbl 0808.58039.
[32] A. Prástaro & Th. M. Rassias, On a geometric approach to an equation of J. D’Alembert, Geometry in Partial Differential Equations, A. Prástaro & Th. M. Rassias (eds.), World Scientific
Publishing, River Edge, NJ, (1994), 316–322. MR1340223(96g:35131); Zbl 0879.35038.
[33] A. Prástaro, Th. M. Rassias & J. Šimša, Geometry of the J. D’Alembert equation, in Finite
Sums Decompositions in Mathematical Analysis, Th. M. Rassias & J. Šimša (eds.), J. Wiley
(1995), 133–159. MR96k:26006; Zbl 0859.26005.
[34] A. Prástaro & Th. M. Rassias, A geometric approach to an equation of J.
D’Alembert, Proc. Amer. Math. Soc. 123(5)(1995), 1597–1606. DOI: 10.2307/2161153.
MR1232143(95f:58007); Zbl 0839.58068.
[35] A. Prástaro, Geometry of quantized super PDE’s, The Interplay Between Differential Geometry and Differential Equations, V. Lychagin (ed.), Amer. Math. Soc. Transl. 2/167(1995),
165–192. MR1343988(96d:58159); Zbl 0844.58012.
[36] A. Prástaro, Quantum geometry of super PDE’s, Rep. Math. Phys. 37(1)(1996), 23–140.
DOI: 10.1016/0034-4877(96)88921-X. MR1394861(97e:58235); Zbl 0887.58064.
[37] A. Prástaro, (Co)bordism in PDEs and quantum PDEs, Rep. Math. Phys. 38(3)(1996), 443–
455. DOI: 10.1016/S0034-4877(97)84894-X. MR1437641(97m:58004); Zbl 0885.58094.
[38] A. Prástaro, Quantum and integral (co)bordisms in partial differential equations, Acta Appl.
Math. 51(3) (1998), 243–302. DOI: 10.1023/A:1005986024130. MR1437641(99d:58183);
Zbl 0924.58103.
[39] A. Prástaro, Quantum and integral bordism groups in the Navier-Stokes equation, New Developments in Differential Geometry, Budapest 1996, J. Szenthe (ed.), Kluwer Academic
Publishers, Dordrecht (1998), 343–360. MR1670467(2000h:58065); Zbl 0937.35133.
[40] A. Prástaro & Th. M. Rassias, On the set of solutions of the generalized d’Alembert equation, C. R. Acad. Sci. Paris 328(I-5)(1999), 389–394. DOI: 10.1016/S0764-4442(99)80177-3.
MR1678135(2000a:3508); Zbl 0931.35031.
[41] A. Prástaro & Th. M. Rassias, A geometric approach of the generalized d’Alembert equation, J. Comput. Appl. Math. 113(1-2)(2000), 93–122. DOI: 10.1016/S0377-0427(99)002472. MR1735816(2001c:58079); Zbl 0936.35011.
[42] A. Prástaro, (Co)bordism groups in PDEs, Acta Appl. Math. 59(2) (1999), 111–201. DOI:
10.1023/A:1006346916360. MR1741657(2001m:58046); Zbl 0949.35011.
CURRICULUM VITAE AGOSTINO PRÁSTARO – APRIL 2014
23
[43] A. Prástaro, (Co)bordism groups in quantum PDEs, Acta Appl. Math. 64(2)(2000), 111–217.
DOI: 10.1023/A:1010685903329. MR1826643(2002e:58037); Zbl 0978.58016.
[44] A. Prástaro, Theorems of existence of local and global solutions of PDEs in the category
of noncommutative quaternionic manifolds, Quaternionic Structures in Mathematics and
Physics, S. Marchiafava, P. Piccinni & M. Pontecorvo (eds.), World Scientific Publishing,
Singapore (2001), 329–337. MR1848873(2002f:58007); Zbl 0978.81038.
[45] A. Prástaro, Local and global solutions of the Navier-Stokes equation, Steps in Differential
Geometry, Proceedings of the Colloquium on Differential Geometry, 25–30 July, 2000, Debrecen, Hungary, L. Kozma, P. T. Nagy & L. Tomassy (eds.), Univ. Debrecen (2001), 263–271.
MR1859305(2002d:53008); Zbl 0983.35105.
[46] A. Prástaro, Navier-Stokes equation: Global existence and uniqueness. (A geometric way
to solve the “(N S)-problem”.), published as: Addendum I: Bordism Groups and the (NS)Problem, in Quantized. Partial Differential Equations, World Scientific Publishing, Singapore,
(2004), 377–434.
[47] A. Prástaro & Th. M. Rassias, A geometric approach to a noncommutative generalized
d’Alembert equation, C. R. Acad. Sc. Paris 330(I-7)(2000), 545–550. DOI: 10.1016/S07644442(00)00238-X. MR1760436(2001d:58026); Zbl 0966.35105.
[48] A. Prástaro & Th. M. Rassias, Results on the J. d’Alembert equation, Ann. Acad. Paed.
Cracoviensis. Studia Math. 1(2001)117–128. Zbl 1137.58308.
[49] A. Prástaro, Quantum manifolds and integral (co)bordism groups in quantum partial differential equations, Nonlinear Anal. Theory Methods Appl. 47/4(2001), 2609–2620. DOI:
10.1016/S0362-546X(01)00382-0. MR1972386(2004c:35343); Zbl 1042.35610.
[50] A. Prástaro, Dirac quantization, Encyclopaedia Math. Suppl.III., M. Hazwinkel (ed.), Kluwer
Academic Publishers, Dordrecht (2002), 127–129. DOI: 10.1007/978-0-306-48373-8.
[51] A. Prástaro, Integral bordisms and Green kernels in PDEs, Cubo 4(2)(2002), 316–370.
MR1928829(2003g:58056).
[52] A. Prástaro & Th. M. Rassias, On the Ulam stability in geometry of PDE’s , Functional
Equations Inequality and Applications, Th. M. Rassias (ed.), Kluwer Academic Publishers,
Dordrecht (2003), 139–147. MR2042561(2004k:58031); Zbl 1059.39024.
[53] A. Prástaro & Th. M. Rassias, Ulam stability in geometry of PDE’s, Nonlinear Funct. Anal.
Appl. 8(2)(2003), 259–278. MR1994707(2004g:35179); Zbl 1096.39028.
[54] A. Prástaro, Quantum super Yang-Mills equations: Global existence and mass-gap, Dynamic
Syst. Appl. 4(2004), 227–232. (Eds. G. S. Ladde, N. G. Madhin and M. Sambandham), Dynamic Publishers, Inc., Atlanta, USA. ISBN:1-890888-00-1. MR2117787(2005m:81203);
Zbl 1067.81097.
[55] A. Prástaro, Geometry of PDE’s.I: Integral bordism groups in PDE’s, J. Math. Anal. Appl.
319(2)(2006), 547–566. DOI: 10.1016/j.jmaa.2005.06.044. MR2227923(2007d:58031);
Zbl 1100.35007.
[56] A. Prástaro, Geometry of PDE’s.II: Variational PDE’s and integral bordism groups,
J. Math. Anal. Appl. 321(2)(2006), 930–948. DOI: 10.1016/j.jmaa.2005.08.037.
MR2241487(2007d:58032); Zbl 1160.58301.
[57] A. Prástaro, Conservation laws in quantum super PDE’s, Proceedings of the Conference
on Differential & Difference Equations and Applications (eds. R. P. Agarwal & K. Perera),
Hindawi Publishing Corporation, New York (2006), 943–952. MR23049427(2008b:58041);
Zbl 1131.35381.
[58] A. Prástaro, (Co)bordism groups in quantum super PDE’s.I: Quantum supermanifolds, Nonlinear Anal. Real World Appl. 8(2)(2007), 505–538. DOI: 10.1016/j.nonrwa.2005.12.008.
MR2289563(2008j:58052); Zbl 1152.58313.
[59] A. Prástaro, (Co)bordism groups in quantum super PDE’s.II: Quantum super PDE’s, Nonlinear Anal. Real World Appl. 8(2)(2007), 480–504. DOI: 10.1016/j.nonrwa.2005.12.007.
MR2289562(2008j:58053); Zbl 1152.58312.
[60] A. Prástaro, (Co)bordism groups in quantum super PDE’s.III: Quantum super YangMills equations, Nonlinear Anal. Real World Appl. 8(2)(2007), 447–479. DOI:
10.1016/j.nonrwa.2005.12.006. MR2289561(2009b:58082); Zbl 1152.58311.
[61] R. Agarwal & A. Prástaro, Geometry of PDE’s.III(I): Webs on PDE’s and integral bordism groups. The general theory, Adv. Math. Sci. Appl. 17(1)(2007), 239–266.
MR237378(2009j:58026); Zbl 1143.53017.
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CURRICULUM VITAE AGOSTINO PRÁSTARO – APRIL 2014
[62] R. Agarwal & A. Prástaro, Geometry of PDE’s.III(II): Webs on PDE’s and integral bordism
groups. Applications to Riemannian geometry PDE’s, Adv. Math. Sci. Appl. 17(1)(2007),
267–285. MR2337379(2009j:58027); Zbl 1140.53005.
[63] A. Prástaro, Geometry of PDE’s.IV: Navier-Stokes equation and integral bordism
groups, J. Math. Anal. Appl. 338(2)(2008), 1140–1151. DOI:10.1016/j.jmaa.2007.06.009.
MR2386488(2009j:58028); Zbl 1135.35064.
[64] A. Prástaro, (Un)stability and bordism groups in PDE’s, Banach J. Math. Anal. 1(1)(2007),
139–147. MR2350203(2009e:58036); Zbl 1130.58014.
[65] A. Prástaro, Extended crystal PDE’s stability.I: The general theory, Math. Comput.
Modelling. (2008). DOI: 10.1016/j.mcm.2008.07.020. MR2532085(2011b:58041); Zbl
1171.35322.
[66] A. Prástaro, Extended crystal PDE’s stability.II: The extended crystal MHD-PDE’s, Math.
Comput. Modelling. (2008). DOI: 10.1016/j.mcm.2008.07.021. MR2532086(2011b:58042);
Zbl 1171.35323.
[67] A. Prástaro, On the extended crystal PDE’s stability.I: The n-d’Alembert extended crystal PDE’s, Appl. Math. Comput. 204(1)(2008), 63–69. DOI: 10.1016/j.amc.2008.05.141.
MR2458340(2010h:58058); Zbl 1161.35054.
[68] A. Prástaro, On the extended crystal PDE’s stability.II: Entropy-regular-solutions in
MHD-PDE’s, Appl. Math. Comput. 204(1)(2008), 82–89. DOI: 10.1016/j.amc.2008.05.142.
MR2458342(2010h:58059); Zbl 1161.35462.
[69] A. Prástaro, On quantum black-hole solutions of quantum super Yang-Mills equations,
Dynamic Syst. Appl. 5(2008), 407–414. (Eds. G. S. Ladde, N. G. Madhin C. Peng
& M. Sambandham), Dynamic Publishers, Inc., Atlanta, USA. ISBN: 1-890888-01-6.
MR2468173(2010g:83040).
[70] A. Prástaro, Surgery and bordism groups in quantum partial differential equations.I: The
quantum Poincaré conjecture, Nonlinear Anal. Theory Methods Appl. 71(12)(2009), 502–
525. DOI: 10.1016/j.na.2008.11.077. MR2671857(2012b:58057); Zbl 1238.58025.
[71] A. Prástaro, Surgery and bordism groups in quantum partial differential equations.II: Variational quantum PDE’s, Nonlinear Anal. Theory Methods Appl. 71(12)(2009), 526–549. DOI:
10.1016/j.na.2008.10.063. MR2671858(2012b:58058); Zbl 1238.58026.
[72] R. P. Agarwal & A. Prástaro, Singular PDE’s geometry and boundary value problems, J. Nonlinear Conv. Anal. 9(3)(2008), 417–460. MR2478974(2010b:58030); Zbl 1171.35006.
[73] R. P. Agarwal & A. Prástaro, On singular PDE’s geometry and boundary value
problems, Appl. Anal. 88(8)(2009), 1115–1131. DOI: 10.1080/00036810902943612.
MR2568427(2010k:58033); Zbl 1180.35012.
[74] A. Prástaro, Extended crystal PDE’s. Mathematics Without Boundaries: Surveys in Pure
Mathematics. P. M. Pardalos and Th. M. Rassias (Eds.) Springer-Heidelberg New York Dordrecht London, (to appear). arXiv:0811.3693[math.AT].
[75] A. Prástaro, Quantum extended crystal PDE’s, Nonlinear Studies 18(3)(2011), 447–485.
arXiv:1105.0166[math.AT]. MR2012k:57043; Zbl 1253.35135.
[76] A. Prástaro, Quantum extended crystal super PDE’s. Nonlinear Anal. Real World Appl.
13(6)(2012), 2491–2529. DOI: 10.1016/j.nonrwa.2012.02.014. arXiv:0906.1363[math.AT].
MR2927202; Zbl 1258.81064.
[77] A. Prástaro, Exotic heat PDE’s, Commun. Math. Anal. 10(1)(2011), 64–81.
arXiv:1006.4483[math.GT]. MR2825954; Zbl 06008771.
[78] A. Prástaro, Exotic heat PDE’s.II. Essays in Mathematics and its Applications. In Honor of
Stephen Smale’s 80th Birthday. P. M. Pardalos and Th. M. Rassias (Eds.) Springer-Heidelberg
New York Dordrecht London (2012), 369–419. ISBN 978-3-642-28820-3 (Print) 978-3-28821-0
(Online). DOI: 10.1007/978-3-642-28821-0. arXiv: 1009.1176[math.AT]. MR2975595.
[79] A. Prástaro, Exotic n-d’Alembert PDE’s and stability. Nonlinear Analysis. Stability, Approximation and Inequalities. Series: Springer Optimization and its Applications Vol 68. P. M.
Pardalos, P. G. Georgiev and H. M. Srivastava (Eds.). Springer Optimization and its Applications Volume 68 (2012), 571–586. ISBN 978-1-4614-3498-6. arXiv:1011.0081[math.AT].
Zbl 06073130.
[80] A. Prástaro, Exotic PDE’s. Mathematics Without Boundaries: Surveys in Interdisciplinary
Research. P. M. Pardalos and Th. M. Rassias (Eds.) Springer-Heidelberg New York Dordrecht
London, (to appear). arXiv:1101.0283[math.AT].
CURRICULUM VITAE AGOSTINO PRÁSTARO – APRIL 2014
25
[81] A. Prástaro, Quantum exotic PDE’s. Nonlinear Anal. Real World Appl. 14(2)(2013), 893–
928. DOI: 10.1016/j.nonrwa.2012.04.001. arXiv:1106.0862[math.AT]. MR2991123. ; Zbl
06142846.
[82] A. Prástaro, Strong reactions in quantum super PDE’s. I: Quantum hypercomplex exotic
super PDE’s.
arXiv:1205.2984[math.AT]. (Part I and Part II are unified in arXiv.)
[83] A. Prástaro, Strong reactions in quantum super PDE’s. II: Nonlinear quantum propagators.
arXiv:1205.2984[math.AT].
[84] A. Prástaro, Strong reactions in quantum super PDE’s. III: Exotic quantum supergravity.
arXiv:1206.4956[math.AT].
[85] A. Prástaro, The Landau’s problems.I: The Goldbach’s conjecture proved.
arXiv:1208.2473[math.GM].
[86] A. Prástaro, The Landau’s problems.II: Landau’s problems solved.
arXiv:1208.2473[math.GM]. (Part I and Part II are unified in arXiv.)
[87] A. Prástaro, The Riemann hypothesis proved. arXiv:1305.6845[math.GM].
[88] A. Prástaro, Quantum Geometrodynamic Cosmology.
(Submitted for publication.)
MONOGRAPHS AND TEXTS
[89] A. Prástaro, Geometry of PDEs and Mechanics, World Scientific Publishing, River Edge,
NJ, 1996, 760 pp. ISBN 9810225202. MR1412798(98e:55182); Zbl 0866.35007.
[90] A. Prástaro, Elementi di Meccanica Razionale, Edizione 2010, Aracne Editrice, Roma, 2010,
446 pp. ISBN 978-88-548-3601-3.
[91] A. Prástaro, Quantized Partial Differential Equations, World Scientific Publishing,
River Edge, NJ, 2004, 500 pp. ISBN 981-238-764-1. MR2086084(2005f:58036); Zbl
1067.58022.
BOOKS (EDITOR AND COAUTHOR)
[92] A. Prástaro, Geometrodynamics Proceedings 1983 , Pitagora Ed., Bologna 1984. ISBN 88371-0286-0. MR0823711(86m:58007).
[93] A. Prástaro, Geometrodynamics Proceedings 1985, World Scientific Publishing, Singapore 1985. ISBN 9971-978-63-6. BookSG-Contents. MR0825784(86m:58008); Zbl
0637.00006.
[94] A. Prástaro & Th. M. Rassias, Geometry in Partial Differential Equations, World Scientific
Publishing, River Edge, NJ, 1994. ISBN 978-981-02-14-07-4. MR1340208(96a:58003); Zbl
0867.00017.
PATENTS
[95] A. Prástaro et al., 28/5/1975 - No 23.797 A/75. Process of production of fibrous structures
with high degree of birefringence.
[96] A. Prástaro et al., 11/7/1975 - No 25.334 A/75. Process of production of plexus-filament of
synthetic polymers by means of flash-spinning of polymers solutions.
26
CURRICULUM VITAE AGOSTINO PRÁSTARO – APRIL 2014
FIG. 7. Poster: ICM 2006 Satellite Conference “Advances PDE’s
Geometry”, August 31 - September 2, 2006, Madrid - Spain.
CURRICULUM VITAE AGOSTINO PRÁSTARO – APRIL 2014
27
4. ABSTRACTS OF PUBLICATIONS
ABSTRACTS OF PAPERS
(1) L. Corgnier, A. D’Adda & A. Prástaro, Regge pole vs. resonance duality
and boostrap calculations, Nuovo Cimento 57 A(4)(1968), 881–885.
Abstract. In this note we use the Regge pole vs. resonance duality to
make bootstrap calulations by comparing the behaviour at fixed scattering
angle of Regge like amplitude with that of a sum of resonance contributions.
(2) A. Prástaro & P. Parrini, A mathematical model for spinning molten polymer and conditions of spinning, Tex. Res. J. 45(1975), 118–127.
Abstract. The equation of spinning of molten polymers in the stationary
non-isothermal state have been solved by an analytical numerical method so
as to obtain temperature T (s) and section A(z) profiles along the spinning
axis z. T (z) and A(z) are thus correlated with the molecular parameters
of the molten polymer: viscosity, density, and extrusion temperature, and
polymer mass-flow rate. Furthermore, the correlation has been obtained
between fiber quality and steady-state solutions. A critical collection rate
and the critical extrusion output have been deduced from such correlations.
Above such critical values, breakage of the molten polymer takes place.
(3) A. Prástaro & P. Parrini, Ein mathematisches Model für das Verspinnen
geschmolzener Polymerer und für die Spinnbedingungen, Colloid & Polymer
Science 255(2)(1977), 624–633.
Abstract. This is a translated German version of the previous article.
(4) A. Prástaro, A mathematical model for spinning viscoelastic molten polymers, Riv. Mat. Univ. Parma (4)(1976), 295–313.
Abstract. A mathematical model for spinning viscoelastic materials is
proposed. This work can be considered the continuation of the papers
[2, 3] which treated the case of newtonian materials. The viscoelastic system, as more differs from the newtonian as the elastic component is present;
thus the viscoelastic mathematical model can be not be inferred from the
analysis of the previous paper; on the contrary the viscoelastic model includes, as particular case, the newtonian model. The spinning process was
analysed by adding the rheological equation for viscoelastic materials to
the set of simultaneous partial differential equations describing a general
molten spinning process. We gave the steady-state numerical solutions,
i.e. the filament croos-section A(z), filament temperature T (z) and filament tension F (z), as function of position z and we related them to the
parameters which influence the process of spinning: material parameters
and spinning conditions parameters. We related the yarn birefringence ∆n
to the same parameters also. Moreover, we proposed to investigate which
bounds impose the spinnability criterion on the viscoelastic paramaters and
which conditions realize the maximum yarn production with fixed denier
and section.
(5) A. Prástaro, Modello matematico sulla formazione della melt-fracture nei
polimeri fusi, Quad. Ing. Chim. Ital. Suppl. 13(3-4)(1977), 37–44.
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(6)
(7)
(8)
(9)
(10)
Abstract. The melt-fracture is one of the most serious problem in the
extrusion of thermoplastic materials. This is a flow instability that appears
as an almost regular distorsion of the extruded material. After a panorama
on the known experimental facts, a mathematical model is given in such
a way to emphasize some essential parameters controlling the start of this
phenomenon. In fact has been decovered an adimensional number, Wc ,
(Weissenberg number), characterizing the start of flow instability in all melt
polymers. This allows us to determine the extrudibility characteristics of
polymeric materials.
A. Prástaro, Geometrodynamics of some non-relativistic incompressible fluids, Stochastica 3(2)(1979), 15–31.
Abstract. In some papers [7, 8, 9, 10] we proposed a geometric formulation of continuum mechanics, where a continuum body is seen as a suitable
differentiable fiber bundle C on the Galilean space-time M , beside a differential equation of order k, Ek (C), on C and the assignement of a frame
ψ on M . In the present paper we apply this general theory to some incompressible fluids. The scope is to demonstrate that also for these more
simple materials our theory is a suitable tool in order to understand better
the fundamental principles of continuum mechanics.
A. Prástaro, Spazi derivativi e fisica del continuo in relatività generale, Atti
Accad. Sci. Torino Suppl. 114(1980/81), 289–292.
Abstract. In order to give an axiomatic description of continuum physics,
a derivative space, D k (V, W ), is introduced which allows us to describe
the derivative of order k of a differentiable map f : V → W as section of
the fiber bundle D k (V, W ). Derivative operators and functional differential
operators are seen as useful generalizations of usual differential operators.
With this language we recognize a structural order to all physical entities
which are characteristic in continuum physics.
A. Prástaro, On the general structure of continuum physics.I: Derivative
spaces, Boll. Unione Mat. Ital. (5)17-B(1980), 704–726.
Abstract. In order to give an intrinsic and axiomatic formulation of continuum physics, the derivative space D k (V, W ) is introduced. This allows
us to describe the k-order derivative of a mapping f : V → W , as a section
of the fibre bundle D k (V, W ) → V . This formulation generalizes the concept of jet of a mapping. The corresponding differential calculus is carefully
developed.
A. Prástaro, On the general structure of continuum physics.II: Differential
operators, Boll. Unione Mat. Ital. (5)S.-FM(1981), 69–106.
Abstract. In order to give an intrinsic and axiomatic formulation of continuum physics, the differential operators are studied in the language of
derivative spaces [7, 8]. Useful generalizations of differential operators are
given by introducing derivative operators and functional differential operators. Finally differential equations are considered as suitable subspaces of
derivative spaces.
A. Prástaro, On the general structure of continuum physics.III: The physical
picture, Boll. Unione Mat. Ital. (5)S.-FM(1981), 107–129.
Abstract. An axiomatic and intrinsic formulation of continuum systems is
given by using tools of the modern differential geometry. In this framework
CURRICULUM VITAE AGOSTINO PRÁSTARO – APRIL 2014
(11)
(12)
(13)
(14)
(15)
(16)
29
we are able to give a structural ordering to all the physical entities of
continuum physics.
A. Prástaro, On the intrinsic expression of Euler-Lagrange operator, Boll.
Unione Mat. Ital. (5)18-A(1981), 411–416.
Abstract. An intrinsic representation of the Euler-Lagrange differential
operator is given for the global variational calculus on fibered manifolds.
This expression is of particular interest to be utilized in the Lagrangian
formulation of the field theory, as it gives an explicit derivative dependence
by the field.
A. Prástaro, Dynamic conservation laws and the Korteweg-De Vries equation, Atti convegno su onde e stabilità nei mezzi continui, Catania 1981,
Quaderni CNR-GNFM, Catania,(1982), 272–274.
Abstract. Recently we have introduced a geometric method to describe
conservation laws more general than Noetherian ones. In particular in
[16, 17, 19] we have given a detailed exposition of this method developing
the geometric-differential calculus to build dynamic conserved quantities.
In this communication we shall give an account of some of these results
emphasizing their applications to the Korteweg-de Vries equation.
A. Prástaro, Spinor super bundles of geometric objects on spinG space-time
structures, Boll. Unione Mat. Ital. (6)1-B(1982), 1015–1028.
Abstract. Spinorial fibre bundles are built on space-times manifolds of
type spinG that are fully covariant in the sense of [8]. These fibre bundles
generalize that introduced in [8] and are useful in a unified field theory.
A. Prástaro, Gauge geometrodynamics, Riv. Nuovo Cimento 5(4)(1982),
1–122.
Abstract. In this paper we have a self-contained unitary geometric development of the methods and structures on which the gauge theories are
based. We hope that this geometrical framework shall be useful for a
more clear understanding of continuum physics. Since the framework is
sufficiently generalized, it can be applied to all the situations of physical
interest. Contents: Functors and fibre bundles. Derivative spaces and differential equations. Derivative spaces and variational calculus. Connections
and derivative spaces. Geometrodynamics of gauge continuum systems and
symmetries properties. Classification of gauge continuum systems. Spinor
superbundles of geometric objects and dynamics.
A. Prástaro, Geometry and existence theorems for incompressible fluids, Geometrodynamics Proceedings 1983, A. Prástaro (ed.), Pitagora Ed., Bologna
(1984), 65–90.
Abstract. By utilizing a geometric framework to describe the dynamic
of a continuous medium [7, 8, 9, 10, 11, 12, 13, 14, 15] we give existence
theorems of local and global solutions for an incompressible fluid. The
discussion concerns the Euler equation (E) and the non-isothermic NavierStokes equation (N S).
A. Prástaro, Geometrodynamics of non-relativistic continuous media.I: Spacetime structures, Rend. Sem. Mat. Univ. Politec. Torino 40(2)(1982),
89–117.
Abstract. In order to formulate the non-relativistic continuum mechanics
as a unified field theory on Galilei space-time M , the geometrical structure
30
CURRICULUM VITAE AGOSTINO PRÁSTARO – APRIL 2014
(17)
(18)
(19)
(20)
of M is considered and the space time resolution of bundles of geometric
objects on M are analysed in detail. In particular, the concept of geometric
object gives rigorous meaning to the concept of observed physical quantity.
It clarifies the ambiguity of why “frame dependent” quantities are useful,
even essential, in the kinematic of description of continuum mechanical
bodies. Moreover, it clarifies the paradosical nature of “frame indifferent
statements about frame dpendent quantities”. These turn out to be simply
statements about fields of geometric objects which are not tensor fields.
A. Prástaro, Geometrodynamics of non-relativistic continuous media.II:
Dynamic and constitutive structures, Rend. Sem. Mat. Univ. Politec.
Torino 43(1)(1985), 89–116.
Abstract. An intrinsic formulation of Continuum Mechanics on the affine
Galielan space-time M is given, emphasizing the role of the dynamic equation as a geometric structure. In particular, a continuous body is described
as a geometric structure on M . Thus, the study of symnmetry properties
of this structure allows us to give useful classifications of continuous bodies
and to state generalized forms of Noether’s theorem. These considerations
are applied to incompressible fluids. Existence and uniqueness theorems
for regular solutions are obtained.
A. Prástaro, A geometric point of view for the quantization of non-linear
field theories, Atti VI Convegno Nazionale di Relatività Generale e Fisica
della Gravitazione, Firenze 1984, Pitagora Ed., Bologna (1986), 289–292.
Abstract. The fundamental geometric structure of any field theory is a
fiber bundle π : W → M beside a PDE Ek ⊂ JD k (W ). So we shall recognize in this geometric structure (W, Ek ) suitable properties to interpretate
the meaning of quantization. This communication shortly describe our new
point of view in this field, showing how it is possible to read the meaning
of the quantization in the formal properties of PDEs.
A. Prástaro, Dynamic conservation laws, Geometrodynamics Proceedings
1985, A. Prástaro (ed.), World Scientific Publishing, Singapore (1985), 283–
420.
Abstract. Part A:PDE and Conservation Laws. Basic results on the formal theory of PDE. Pseudogroups and PDE. The Euler-Lagrange operator
for Lagrangians of any order. Cartan form and Noether conservation laws
for Lagrangian of any order. Symmetry properties and dynamic conservation laws. Part B: Quantized PDE and Conservation Laws. Quantum
charges. Quantum situs. Geometric theory of quantized PDE. Quantum
cobordism and Dirac’s approach to quantization. Applications: KleinGordon equation; Maxwell equation; Dirac equation; Einstein equation;
Yang-Mills equation. Appendix. Topological vector spaces, C ∗ -algebras
and spectral theory. Local characterization of some geometric structures
related to PDE.
A. Prástaro & T. Regge, The group structure of supergravity, Ann. Inst.
H. Poincaré Phys. Théor. 44(1)(1986), 39–89.
Abstract. An intrinsic description of the ”group manifold approach” to
supergravity is given. Emphasis is placed on some geometric structures
which allow us to obtain a direct full covariant formulation. In particular,
CURRICULUM VITAE AGOSTINO PRÁSTARO – APRIL 2014
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the geometric theory of partial differential equations allows us to give a dynamic description of space-time. Some applications to physically interesting
situations are discussed in detail.
V. Marino & A. Prástaro, On the geometric generalization of the Noether
theorem, Lecture Notes in Math. 1209, Springer-Verlag, Berlin (1986),
222–234.
Abstract. Task of this paper is to compare some geometric approaches to
obtain conservation laws associated to partial differential equations. More
precisely we intend to consider the methods developed by A. Prástaro and
A. M. Vinogradov. The differential equations are considered from a geometric point of view: namely they are submanifolds of jet-derivative spaces
on fiber bundles. To the symmetries of these submanifolds are associated
conservation laws that are not necessarily of Noetherian type.
A. Prástaro, Quantum gravity and group model gauge theory, Journées Relativistes, Toulouse, France 1986, A. Crumeyrolle (ed.), Univ. Paul Sabatier
Toulouse (1986), 213–222.
[Recent results on the quantum geometrodynamics. Proceedings General
Relativity and Gravitation, vol.1. July, 4-9, 1983, Padova - Italy. (B.
Bertotti, F. de Felice & A. Pascolini eds.) CNR - Roma (1983), 1145.]
Abstract. A new geometric point of view of quantization of PDE’s, founded
on the formal properties of PDE’s, is applied to the quantization of gravity coupled with a Yang-Mills gauge field. The quantization of Einstein
equation in the framework of group model gauge theory is discussed.
V. Marino & A. Prástaro, On the conservation laws of PDE’s, Rep. Math.
Phys. 26(2)(1987/8), 211–225.
Abstract. The general methods of obtaining conservation laws for (nonlinear) partial differential equations (PDE’s) introduced by A. Prástaro in
[12, 19] and by A. M. Vinogradov in Soviet Math. Dokl. (5)18(1977), 1200–
1204; J. Math. Anal. Appl. (1)100(1984), 1–129, are considered and the
general covariance of such methods is studied. In particular, it is shown that
Vinogradov’s method fails to be fully covariant in the non-linear case. The
relations between the number of conservation laws for PDEs and the AtiyahSinger index theorem are studied. A criterion for recognize the wholly
cohomological character of a PDE is given and the link between spectral
sequences and wholly cohomlogical equations is found. Some examples of
interesting PDEs which arise in physics are also considered.
A. Prástaro, On the quantization of Newton equation, Atti IX Congresso
AIMETA, Bari 1988, AIMETA (1988), 13–16.
Abstract. It is proved that the method of formal quantization of PDEs
introduced by the author in some previous papers [18, 25, 28] allows to
obtain in intrinsic way the canonical Dirac quantization for particles of
the lassical mechanics. As examples harmonic oscillator and anharmonic
oscillator are considered. These considerations can be generalized to any
PDE defined on fiber bundles.
A. Prástaro, Wholly cohomological PDE’s, International Conference on Differential Geometry and Applications, Dubrovnick (Yu), 1988, Univ. Beograd
& Univ. Novi Sad (1989), 305–314.
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Abstract. PDE’s are geometric objects to which one can associate conservation laws in relation to their symmetry properties [15, 17, 19, 21, 23].
Then, the wholly-cohomological character of a PDE is its possibility to
represent any (n − 1)-dimensional cohomological class of the n-dimensional
basis manifold by means of a conservation law. In this paper we resume
some recent results in this direction obtained by the author [29] and also
announce some new further results for PDEs defined in the category of
supermanifolds [36].
A. Prástaro, Geometry of quantized PDE’s, Differential Geometry and Applications, J. Janyska & D. Krupka (eds.), World Scientific Publishing,
Singapore (1990), 392–404.
Abstract. In this paper we resume some recent results in the direction
of the formal quantization of PDE’s obtained by the author [24], and also
announce some new further results. The categorial meaning of quantization
of PDE’s is given. Formal quantization results a canonical functor defined
on the category of differential equations. Furthermore, a Dirac-quantization
can be interpreted as a covering in the category of differential equations.
A quantum (pre-)spectral measure is a functor that can be factorized by
means of formal quantization and a (pre-)spectral measure. A relation
between canonical Dirac-quantization and singular solutions of PDE’s is
given. It is proved also that knowledge of Bäcklund correspondeces, as
well conservation laws, can aid the proceeding of canonical quantization of
PDE’s. (See [29].)
A. Prástaro, On the singular solutions of PDE’s, Atti X Congresso Nazionale
AIMETA, Pisa 1990, AIMETA (1990), 17–20.
Abstract. In the geometric formal theory of PDE’s we recognize also
the problem of existence of singular solutions with singularities of ThomBoardman type, i.e., singularities that can be resolved by means of prolongations. Scope of this paper is to give a short account of some fundamental
results in these directions and apply them to some important classic equations of fluid mechanics: Euler equation (E) and Navier-Stokes equation
(N S). Quantum tunneling effects can be described by means of such singular solutions. Furthermore, we show also as singular solutions enter in
the description of canonical quantization of PDE’s. We shall specialize, for
sake of coincision, on equations (E) and (N S). (See [29].)
A. Prástaro, Cobordism of PDE’s, Boll. Unione Mat. Ital. (7)5-B(1991),
977–1001.
Abstract. Cobordism groups of systems of partial differential equations
of any order are considered and their representations by means of suitable
homology groups are given. This approach generalizes previous one by J.
Eliashberg, given for PDE’s of first order, These results are useful also in
order to give an algebraic-topological characterization of the quantum situs
or its classic limit, for PDE’s of any order. (See also [19, 70].)
A. Prástaro, Quantum geometry of PDE’s, Rep. Math. Phys. 30(3)(1991),
273–354.
Abstract. In this paper we present the formal quantization of PDE’s
[18, 19, 22, 24, 26, 27] in categorial language. Formal quantization results
as a canonical functor defined on the category of differential equations.
CURRICULUM VITAE AGOSTINO PRÁSTARO – APRIL 2014
33
Furthermore, a Dirac-quantization can be interpreted as a covering in the
category of differential equations. A quantum (pre-)spectral measure is a
functor that can be factorized by means of formal quantization and a (pre)spectral measure. A relation between canonical Dirac-quantization and
singular solutions of PDE’s is given. It is also proved that the knowledge
of Bäklund correspondences, as well as the conservation laws, can aid the
procedure of canonical quantization of PDE’s. Physically interesting examples are considered. In particular, we give the canonical quantization of
an anharmonic oscillator. A general theory of quantum tunneling effects in
PDE’s is given. In particular, quantum cobordism has been related with
Leray-Serre spectral sequences of PDE’s.
(30) A. Prástaro, Geometry of super PDE’s, Geometry of Partial Differential
Equations, A. Prástaro & Th. M. Rassias (eds.), World Scientific Publishing, River Edge, NJ, (1994), 259–315.
Abstract. Superspaces and supermanifolds are introduced by using the
concept of weak differentiability as usually given for locally convex spaces.
This allows us to consider in algebraic way superdual spaces and superderivative spaces. In this way we obtain a good generalization of just known superstructures general enough to develop a formal theory for super PDE’s that
directly extends previous ones for standard manifolds of finite dimension.
In particular, we give a Goldschmidt-type criterion of formal superintegrability for super PDE’s, and show that a geometric theory of singular supersolutions, with singularities of Thom-Boardman type, can be formulated in
the framework of super PDE’s too. Conservation superlaws associated to
super PDE’s are considered and related with some spectral sequences and
wholly cohomological character of these equations.
(31) V. Lychagin & A. Prástaro, Singularities for Cauchy data, characteristics, cocharacteristics and integral cobordism, Differential Geom. Appl.
4(3)(1994), 283–300.
Abstract. A generalization of the classical concept of characteristic for
partial differential equations (PDE) is given in the framework of the geometric formal theory for PDE’s. In particular, it is given a relation between
singularities of Cauchy data and characteristics in order to obtain integral
manifolds (solutions) generated by means of characteristics. In this direction it is shown that to any PDE we can associate a ”dual” equation
having the same characteristics. These equations can be related by means
of a sort of Bäcklund transformation. Furthermore, a criterion that relates
characteristics and integral cobordism (or quantum cobordism [19, 29]) is
given. Also a relation between quantum cobordism in non-linear PDE’s
and Green’s functions is given.
(32) A. Prástaro & Th. M. Rassias, On a geometric approach to an equation
of J.D’Alembert, Geometry in Partial Differential Equations, A. Prástaro
& Th. M. Rassias (eds.), World Scientific Publishing, Singapore (1994),
316–322.
Abstract. Here we announce some firt results on the J. D’Alembert equa∂2
tion ( ∂x∂y
log f ) = 0. More precisely, by using a geometric framework we
prove that the set of smooth functions of two variables f (x, y), solutions of
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the J. D’Alembert equation, is larger than the set of functions of the form
f (x, y) = h(x).g(y).
A. Prástaro, Th. M. Rassias & J. Šimša, Geometry of the J. D’Alembert
equation, in Finite Sums Decompositions in Mathematical Analysis, Th.
M. Rassias & J. Šimša (eds.), J. Wiley (1995), 133–159.
Abstract. This is the last chapter of a book devoted to a very interesting
and actual problem in Mathematical Analysis. Here the geometric theory of
PDE’s is considered and applied to the d’Alembert equation in its connection with the problem of representation of functions by (partial) separation
of variables.
A. Prástaro & Th. M. Rassias, A geometric approach to an equation of
J.D’Alembert, Proc. Amer. Math. Soc. 123(5)(1995), 1597–1606.
Abstract. By using a geometric framework of PDE’s we prove that the
2
lg f
set of solutions of the D’Alembert equation (∗) ∂∂x∂y
= 0 is larger than the
set of smooth functions of two variables f (x, y) of the form (∗∗) f (x, y) =
h(x).g(y). This agrees with a previous counterexample by Th. M. Rassias
given to a statement by C. M. Stéphanos. More precisely, we have the
following result: The set of 2-dimensional integral manifolds of PDE (∗)
properly contains the ones representable by graphs of 2-jet-derivatives of
functions f (x, y) expressed in the form (∗∗). A generalization of this result
to functions of more than two variables is sketched also by considering the
∂ n log f
equation ∂x
= 0.
1 ...∂xn
A. Prástaro, Geometry of quantized super PDE’s, The Interplay Between
Differential Geometry and Differential Equations, V. Lychagin (ed.), Amer.
Math. Soc. Transl. 2/167(1995), 165–192.
Abstract. In this paper we announce some results on the geometrization of
super PDE’s, i.e., PDE’s defined in the category of supermanifolds. These
results generalize previous ones for PDE’s [29].
A. Prástaro, Quantum geometry of super PDE’s, Rep. Math. Phys. 37(1)(1996),
23–140.
Abstract. In order to extend to super PDEs the theory of quqntization
of PDEs as contained in [29] we first develop a geometric theory for super
PDEs (see also [30, 35]). Superspaces and supermanifolds are introduced
by using the concept of weak differentiability as usually given for locally
convex spaces. This allows us to consider in algebraic way superdual spaces
and superderivative spaces and to develop a formal theory for super PDEs
that directly extends the previous ones for standard manifolds of finite
dimension. In particular, we give a criterion of formal superintegrability
for super PDEs, and show that a geometric theory of singular supersolutions, with singularities of Thom–Boardman type, can be formulated in
the framework of super PDEs too. These results generalize the previous
ones obtained for ordinary manifolds by H.Goldschmidt and by Moscow’s
mathematical school. Conservation superlaws associated to super PDEs
are considered and related with some spectral sequences and wholly cohomological character of these equations. Then, the quantization of super
PDEs is formulated on the ground of quantum cobordism [25, 39]. This is
made in order to give an intrinsic and fully covariant geometric formulation
CURRICULUM VITAE AGOSTINO PRÁSTARO – APRIL 2014
35
of unified quantum field theory. In particular, a theory of quantum supergravity is developed. We explain how canonical quantization and quantum
tunneling effects arise in super PDEs. Furthermore, we explicitly extend
previous results of Witten and Atiyah in topological quantum field theory
to our geometric framework for super PDEs. Obstructions to existence of
quantum cobords in super PDEs are given by means of supercharacteristic
classes. These results can be considered as a generalization of the recent
results obtained by Gibbons and Hawking.
(37) A. Prástaro, (Co)bordism in PDEs and quantum PDEs, Rep. Math. Phys.
38(3)(1996), 443–455.
Abstract. In this paper we announce some recent results on the quantum
and integral (co)bordism in PDEs and quantum PDEs. We shall essentially
prove that the tecnique of (co)bordism, introduced by Pontrjagin and Thom
in algebraic topology, can be generalized in the framework of partial differential equations in order to obtain sufficient criteria that allow to decide
when a p-dimensional compact closed integral manifold contained in a PDE
Ek ⊂ Jnk (W ), is the boundary of a (p + 1)-dimensional integral compact
manifold contained also in Ek (integral bordism) or eventually in the jetspace Jnk (W ) containing Ek (quantum bordism). Furthermore, we shall
prove that such results can be extended to the category of quantum PDEs.
Here, by the term “quantum manifold” (and as a consequence of “quantum
PDEs”) we mean a new structure that extends globally usual concepts of
quantum spaces, and that is very useful for physical applications.
(38) A. Prástaro, Quantum and integral (co)bordisms in partial differential equations, Acta Appl. Math. 51(3)(1998), 243–302.
Abstract. Characterizations of quantum bordisms and integral bordisms
in PDEs by means of subgroups of usual bordism groups are given. More
precisely, it is proved that integral bordism groups can be expressed as extensions of quantum bordism groups and these last are extensions of subgroups of usual bordism groups. Furthermore, a complete cohomological
characterization of integral bordism and quantum bordism is given. Applications to particular important classes of PDEs are considered. Finally, we
give a complete characterization of integral and quantum singular bordisms
by means of some suitable characteristic numbers. Some examples of interesting PDEs which arise in Physics are also considered where existence of
solutions with change of sectional topology (tunnel effect) is proved. As an
application, we relate integral bordism to the spectral term E10,n−1 , that
represents the space of conservation laws for PDEs. This gives, also, a
general method to associate in a natural way a Hopf algebra to any PDE.
(39) A. Prástaro, Quantum and integral bordism groups in the Navier-Stokes
equation, New Developments in Differential Geometry, Budapest 1996, J.
Szenthe (ed.), Kluwer Academic Publishers, Dordrecht (1998), 343-360.
Abstract. In this paper we announce some results concerning theorems
of existence and classification of solutions of the Navier-Stokes equation
(N S). In particular, following our general theory for bordism groups in
PDEs, introduced in [37, 38, 70], the quantum and integral bordism groups
of (N S) are explicitly calculated. (See also [42, 45, 46, 63, 65].)
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(40) A. Prástaro & Th.M.Rassias, On the set of solutions of the generalized
d’Alembert equation, C. R. Acad. Sci. Paris 328(I-5)(1999), 389–394.
Abstract. By using a geometric approach we prove that the set of solutions of the generalized d’Alembert equation ∂ n log f /∂x1 · · · ∂xn = 0,
considered in the domain of the (x1 , · · · , xn )-space Rn , is larger that the
set of the functions that can be represented in the form as f (x1 , · · · , xn ) =
f1 (x2 , · · · , xn ) · · · fn (x1 , · · · , xn−1 ). Here the recent general method introduced by A. Prástaro to calculate integral and quantum (co)bordism groups
in PDE’s [37, 38, 70] is used. This method is very useful in order to prove existence of tunneling effects in PDE’s, i.e., existence of solutions that change
their sectional topology.
(41) A. Prástaro & Th. M. Rassias, A geometric approach of the generalized
d’Alembert equation, J. Comput. Appl. Math. 113(1-2)(2000), 93–122.
Abstract. The following results are obtained: 1) The set Sol(d′ A)n of all
∂ n log f
solutions of the equation ∂x
= 0, (n-d’Alembert equation), (n ≥ 2),
1 ...∂x1
1
considered in domains of the (x , . . . , xn ) ∈ Rn , is larger than the set
of all functions f that can be represented in the form f (x1 , . . . , xn ) =
f1 (x2 , . . . , xn ) . . . fn (x1 , . . . , xn−1 ). 2) In the set of solutions Sol(d′ A)n
of the n-d’Alembert equation, (d′ A)n ⊂ JD n (Rn , R), there are also some
manifolds that have a change of sectional topology (tunneling effect).
(42) A. Prástaro, (Co)bordism groups in PDEs, Acta Appl. Math. 59(2)(1999),
111–201.
Abstract. We introduce a geometric theory of PDEs, by obtaining existence theorems of smooth and singular solutions. In this framework, following our previous results on (co)bordisms in PDEs [37, 38, 70] we give
characterizations of quantum and integral (co)bordism groups and relate
them to the formal integrability of PDEs. An explicitly proof that the
usual Thom-Pontrjagin construction in (co)bordism theory can be generalized also to the case of singular integral (co)bordism in the category of
differential equations is given. In fact, we prove the existence of a spectrum
that characterizes the singular integral (co)bordism groups in PDEs. Moreover, a general method that associates in a natural way Hopf algebras (full
p-Hopf algebras, 0 ≤ p ≤ n − 1), to any PDE Ek ⊂ Jnk (W ), just introduced
in [38, 70], is further studied. Applications to particular important classes
of PDEs are considered. In particular, we carefully consider the NavierStokes equation (N ) and explicitly calculate their quantum and integral
bordism groups. An existence theorem of solutions of (N S) with change of
sectional topology is obtained. Relations between integral bordism groups
and causal integral manifolds, causal tunnel effects, and the full p-Hopf
algebras, 0 ≤ p ≤ 3, for the Navier-Stokes equation are determined.
(43) A. Prástaro, (Co)bordism groups in quantum PDEs, Acta Appl. Math.
64(2)(2000), 111–217.
Abstract. In this paper we formulate a theory of noncommutative manifolds (quantum manifolds) and for such manifolds we develop a geometric
theory of quantum PDEs (QPDEs). In particular, a criterion of formal
integrability is given that extends to QPDEs previous one given by H.
Goldschmidt for PDEs [50] and by us for super PDEs [30, 35, 36]. Quantum manifolds are seen as locally convex manifolds where the model has
CURRICULUM VITAE AGOSTINO PRÁSTARO – APRIL 2014
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ms
1
the structure Am
1 × · · · × As , with A ≡ A1 × · · · × As a noncommutative
algebra that satisfies some particular axioms (quantum algebras). A general
theory of integral (co)bordism for QPDEs is developed, that extends our
previous for PDEs [37, 38, 42, 70]. Then, non-commutative Hopf algebras,
(full quantum p-Hopf algebras, 0 ≤ p ≤ m−1), are canonically associated to
k
any QPDE Êk ⊂ Jˆm
(W ), whose elements represent all the possible invariants that can be recognized for such a structure. Many examples of QPDEs
are considered where we apply our theory. In particular, we carefully study
QPDEs for supergravity. We show that the corresponding regular solutions, observed by means of quantum relativistic frames, give curvature,
torsion, gravitino and electromagnetic fields as A-valued distributions on
space-time, where A is a quantum algebra. For such equations canonical
quantizations are obtained and the quantum and integral bordism groups
and the full quantum p-Hopf algebras, 0 ≤ p ≤ 3, are explicitly calculated.
Then, the existence of (quantum) tunnel effects for quantum superstrings
in supergravity is proved.
(44) A. Prástaro, Theorems of existence of local and global solutions of PDEs
in the category of noncommutative quaternionic manifolds, Quaternionic
Structures in Mathematics and Physics, Rome 1999, S. Marchiafava, P.
Piccinni & M. Pontecorvo (eds.), World Scientific Publishing, Singapore
(2001), 329–337.
Abstract. In this paper we apply our recent geometric theory of noncommuttive (quantum) manifolds and noncommutative (quantum) PDEs
[37, 38, 42, 70, 43] to the category of quantum quaternionic manifolds.
These are manifolds modelled on spaces built starting from quaternionic
algebras. For PDEs considered in such category we determine theorems
of existence of local and global quaternionic solutions. We show also that
such a category of quantum quaternionic manifolds properly contains that
of manifolds with (almost) quaternionic structure. So our theorems of existence of quantum quaternionic manifolds for PDEs produce a cascade of
new solutions with nontrivial topology.
(45) A. Prástaro, Local and global solutions of the Navier-Stokes equation, Steps
in Differential Geometry, Proceedings of the Colloquium on Differential
Geometry, 25–30 July, 2000, Debrecen, Hungary, L. Kozma, P. T. Nagy &
L. Tomassy (eds.), Univ. Debrecen (2001), 263–271.
Abstract. A brief report is given on our recent results [42, 46] proving
existence of (smooth) global solutions of the 3D nonisothermal NavierStokes equation (N S), and (non) uniqueness of such solutions for (smooth)
boundary value problems.
(46) A. Prástaro, Navier-Stokes equation. Global existence and uniqueness, (A
geometric way to solve the “(N S)-problem”.), published as: Addendum I:
Bordism Groups and the (NS)-Problem, in Quantized. Partial Differential
Equations, World Scientific Publishing, Singapore, (2004), 377–434.
Abstract. Here we report on our recent results on the integral bordism
groups of the 3D nonisothermal Navier-Stokes equation (N S) [39, 42, 45]
that proved the existence of global (smooth) solutions. In particular, we
go in some further results emphasizing surgery techniques that allow us
to better understand this geometric proof of existence of (smooth) global
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solutions for any (smooth) boundary condition. Furthermore, a theorem
of nonuniqueness of such solutions for general boundary conditions is given
on the ground of the symmetry properties of (N S) and just by using our
results on the integral bordism groups of (N S). A comparison with the
isothermal case, in the zero viscosity limit condition, (Euler equation), is
considered. (See also [63].)
A. Prástaro & Th. M. Rassias, A geometric approach to a noncommutative
generalized d’Alembert equation, C. R. Acad. Sc. Paris 330(I-7)(2000),
545–550.
Abstract. In this paper the authors provide an account of some of their
results concerning the J. D’Alembert equation especially in a suitable category of noncommutative manifolds, proving that the geometric theory of
PDE’s introduced by A. Prástaro is an handable framework where problems
in the theory of partial differential equations find their natural solutions.
In fact, the J. d’Alembert equation is one such applications.
A. Prástaro & Th. M. Rassias, Results on the J.d’Alembert equation, Ann.
Acad. Paed. Cracoviensis. Studia. Math. 1(2001), 117–128.
Abstract. A new m-d’Alembert equation, m ≥ 2, is introduced in the
category of quantum manifolds [37, 43, 70], that extends the commutative
generalized d’Alembert equation just considered in [47]. For such a new
equation we give theorems of existence of local and global solutions.
A. Prástaro, Quantum manifolds and integral (co)bordism groups in quantum partial differential equations, Nonlinear Anal. Theory Methods Appl.
47/4(2001), 2609–2620.
Abstract. In this paper it is given a short account of some recent theorems
by A. Prástaro on the existence of local and global solutions of QPDEs, i.e.
partial differential equations built in the category of quantum manifolds
[43]. This theory is then applied to the quantum Navier-Stokes equation,
obtaining a generalization of the previous results by Prástaro on the NavierStokes equation [37, 42, 45, 46].
A. Prástaro, Dirac quantization, Encyclopaedia Math. Suppl.III., M. Hazwinkel
(ed.), Kluwer Academic Publishers, Dordrecht (2002), 127–129.
Abstract. A panorama on the modern developments of quantizations of
PDEs is given. In particular it is emphasized that on the framework of a
geometric theory of PDEs, A. Prástaro has given a formulation of canonical
quantization of partial differential equations, without assuming that these
should be of variational type and/or linear. Furthermore, the generalization
of this geometric approach to PDEs in the category of quantum manifolds,
given more recently by A. Prástaro, has been considered. Relations with
other recent works in noncommutative geometry are given.
A. Prástaro, Integral bordisms and Green kernels in PDEs, Cubo Matematica Educacional 4(2)(2002), 316–370.
Abstract. Integral bordisms of (nonlinear) PDEs are characterized by
means of geometric Green kernels and prove that these are invariant for the
classic limit of statistical sets of formally integrable PDEs. Such geometric characterization of Green kernels is related to the geometric approach
of canonical quantization of (nonlinear) PDEs, previously introduced by us
[29]. Some applications are given where particle fields on curved space-times
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having physical or unphysical masses, (i.e., bradions, luxons and massive
neutrinos), are canonically quantized respecting microscopic causality.
A. Prástaro & Th. M. Rassias, On the Ulam stability in geometry of PDE’s,
Functional Equations Inequality and Applications, Th. M. Rassias (ed.),
Kluwer Academic Publishers, Dordrecht (2003), 139–147.
Abstract. The article is concerned with the problem of unstability of flows
corresponding to solutions of the Navier-Stokes equation in relation with
the stability of a new functional equation that is stable as well as superstable
in an extended Ulam sense. In such a framework a natural characterization
of global stable laminar flow is given also.
A. Prástaro & Th. M. Rassias, Ulam stability in geometry of PDE’s, Nonlinear Funct. Anal. Appl. 8(2)(2003), 259–278.
Abstract. The unstability of characteristic flows of solutions of PDE’s is
related to the Ulam stability of functional equations. In particular, we
consider, as master equation, the Navier-Stokes equation. The integral
(co)bordism groups, that have recently been introduced by A. Prástaro to
solve the problem of existence of global solutions of the Navier-Stokes equation [37, 38, 39, 42, 70], lead to a new application of the Ulam stability for
functional equations. This allowed us here to prove that the characteristic
flows associated to perturbed solutions of global laminar solutions of the
Navier-Stokes equation, can be characterized by means of a stable (as well
superstable) functional equation (functional Navier-stokes equation). In
such a framework a natural criterion to recognize stable laminar solutions
is given also.
A. Prástaro, Quantum super Yang-Mills equations: Global existence and
mass-gap, Dynamic Syst. Appl. 4(2004), 227–232. (Eds. G. S. Ladde,
N. G. Madhin and M. Sambandham), Dynamic Publishers, Inc., Atlanta,
USA. ISBN:1-890888-00-1.
Abstract. Quantum super Yang-Mills equations are considered in the
framework of some noncommutative manifolds (quantum supermanifolds)
and for such equations existence theorems of local and global solutions
are obtained by using some geometric methods recently introduced by
A.Prástaro [37, 43, 44, 70]. In particular, global properties of solutions
are characterized by means of integral bordism groups. A criterion to recognize global solutions with mass gap is given. (See also the book quoted
in [74].)
A. Prástaro, Geometry of PDE’s.I: Integral Bordism Groups in PDE’s, J.
Math. Anal. Appl. 319(2)(2006), 547–566.
Abstract. We improve some our previous theorems on the calculation of
integral bordism groups of formally integrable and completely integrable
PDE’s, emphasizing the role played by singular solutions and weak solutions. Some applications to interesting PDE’s, defined on finite dimensional
manifolds, are also considered.
A. Prástaro, Geometry of PDE’s.II: Variational PDE’s and integral bordism
groups, J. Math. Anal. Appl. 321(2)(2006), 930–948.
Abstract. In the framework of the geometry of PDE’s, we classify variational equations of any order with respect to their formal properties. Following our previous results [55], we relate constrained variational PDEs to
40
CURRICULUM VITAE AGOSTINO PRÁSTARO – APRIL 2014
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(58)
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their integral bordism groups. In this way we are able to characterize global
solutions of constrained variational PDEs and to relate them to the structure of global solutions for the corresponding constraint equations. Some
applications are also considered.
A. Prástaro, Conservation laws in quantum super PDE’s, Proceedings of
the Conference on Differential & Difference Equations and Applications,
(eds. R. P. Agarwal & K. Perera), Hindawi Publishing Corporation, New
York (2006), 943–952.
Abstract. Conservation laws are considered for PDE’s built in the category QS of quantum supermanifolds. These are functions defined on the
integral bordism groups of such equations and belonging to suitable Hopf
algebras (full quantum Hopf algebras). In particular, we specialize our calculations on the quantum super Yang-Mills equations and quantum black
holes.
A. Prástaro, (Co)bordism groups in quantum super PDE’s.I: Quantum supermanifolds, Nonlinear Anal. Real World Appl. 8(2)(2007), 505–538.
Abstract. Following our previous works on noncommutative manifolds
and noncommutative PDE’s [37, 43, 44, 54, 57, 70, 74, 76], we consider
in these series of three papers, some further results on quantum supermanifolds and quantum super PDE’s. In particular, in this first part we
focus our attention on quantum supermanifolds. These structures globalize
the notion of quantum superalgebras, obtaining noncommutative manifolds
that are useful to give a fully covariant description of noncommutative
geometric structures, hence of quantum physics. We study the geometry of quantum supermanifolds and characterize their (co)homological and
(co)bordism properties. Covariant quantizations of super PDE’s are given,
obtaining examples of quantum supermanifolds justifying their definitions.
A. Prástaro, (Co)bordism groups in quantum super PDE’s.II: Quantum
super PDE’s, Nonlinear Anal. Real World Appl. 8(2)(2007), 480–504.
Abstract. Following our previous works on the integral (co)bordism groups
of quantum PDE’s [37, 43, 44, 59, 70, 74], we specialize, now, on quantum
super partial differential equations, i.e., partial differential equations built
in the category of quantum supermanifolds. These are manifolds modeled
on locally convex topological vector spaces built starting from quantum
algebras endowed also with a Z2 -gradiation, and a Z2 -graded Lie algebra
structure, (quantum superalgebra). Then, we extend to these manifolds,
with such richer structure, our previous results, and build a geometric theory of quantum super PDEs, that allows us to obtain theorems of existence
of (smooth) local and global solutions in the category of quantum supermanifolds. Some quantum (super) PDE’s, arising from the Dirac quantization of some classical (super) PDE’s, are considered in some details.
A. Prástaro, (Co)bordism groups in quantum super PDE’s.III: Quantum
super Yang-Mills equations, Nonlinear Anal. Real World Appl. 8(2)(2007),
447–479.
Abstract. In this third part of a series of three papers devoted to the study
of geometry of quantum super PDE’s [58, 59], we apply our theory, developed in the first two parts, to quantum super Yang-Mills equations and
quantum supergravity equations. For such equations we determine their
CURRICULUM VITAE AGOSTINO PRÁSTARO – APRIL 2014
41
integral bordism groups, and by using some surgery techniques, we obtain
theorems of existence of global solutions, also with nontrivial topology, for
Cauchy problems and boundary value problems. Quantum tunnelling effects are described in this context. Furthermore, for quantum supergravity
equations we prove existence of solutions of the type quantum black holes
evaporation processes just by using an extension to quantum super PDEs
of our theory of integral (co)bordism groups. Our proof is constructive,
i.e., we give geometric methods to build such solutions. In particular a criterion to recognize quantum global (smooth) solutions with mass-gap, for
the quantum super Yang-Mills equation, is given. Finally it is proved that
quantum super PDE’s contain also solutions that come from Dirac quantization of their superclassical counterparts. This proves that quantum
super PDE’s are (nonlinear) generalizations of Dirac quantized superclassical PDE’s. Applications of this result to free quantum super Yang-Mills
equations are given.
(61) R. Agarwal & A. Prástaro, Geometry of PDE’s.III(I): Webs on PDE’s
and integral bordism groups. The general theory, Adv. Math. Sci. Appl.
17(1)(2007), 239–266.
Abstract. Web structures are recognized on any partial differential equation (PDE) that bring new insights in the geometric theory of PDE’s. Relations with integrability properties of PDE’s, and their integral bordism
groups, are obtained also, emphasizing the role played by singular and
weak solutions. Applications to some important PDE’s of the Mathematical Physics are given too.
(62) R. Agarwal & A. Prástaro, Geometry of PDE’s.III(II): Webs on PDE’s
and integral bordism groups. Applications to Riemannian geometry PDE’s,
Adv. Math. Sci. Appl. 17(1)(2007), 267–285.
Abstract. By using previous results by A.Prástaro on integral bordism
groups of PDE’s, and some issues of the companion paper in [61], we characterize in a geometric way local and global solutions of (generalized) Yamabe equations and Ricci-flow equations. We prove that such results help to
find natural linear and parallel webs on a large category of PDE’s, that are
important in order to find regular and singular solutions on such PDE’s. In
particular, by applying algebraic topologic methods on the Ricci-flow equation we definitively prove that the Poincaré conjecture on the 3-dimensional
manifolds is true.
(63) A. Prástaro, Geometry of PDE’s.IV: Navier-Stokes equation and integral
bordism groups, J. Math. Anal. Appl. 338(2)(2008), 1140–1151.
Abstract. Following our previous results on this subject [39, 42, 45, 46],
integral bordism groups of the Navier-Stokes equation are calculated for
smooth, singular and weak solutions respectively. Then a characterization
of global solutions is made on this ground. Enough conditions to assure
existence of global smooth solutions are given and related to nullity of integral charecteristic numbers of the boundaries. Stability of global solutions
are related to some characteristic numbers of the space-like Cauchy data.
Global solutions of variational problems constrained by (N S) are classified
by means of suitable integral bordism groups too.
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CURRICULUM VITAE AGOSTINO PRÁSTARO – APRIL 2014
(64) A. Prástaro, (Un)stability and bordism groups in PDE’s, Banach J. Math.
Anal. 1(1)(2007), 139–147.
Abstract. In this paper, by using the theory of integral bordism groups
in PDE’s, previously introduced by Prástaro, we give a new interpretation
of the concept of (un)stability in the framework of the geometric theory of
PDE’s. A geometric criterium to identify stable PDE’s and stable solutions
of PDE’s is given.
(65) A. Prástaro, Extended crystal PDE’s stability.I: The general theory, Math.
Comput. Modelling. (2008).
Abstract. This work, divided in two parts, follows some our previous
works devoted to the algebraic topological characterization of PDE’s. In
this first part, the stability of PDE’s is studied in details in the framework
of the geometric theory of PDE’s, and bordism groups theory of PDE’s.
In particular we identify criteria to recognize PDE’s that are stable (in
extended Ulam sense) and in their regular smooth solutions do not occur
finite time unstabilities, (stable extended crystal PDE’s). Applications to
some important PDE’s are considered in some details. (In the second part
a stable extended crystal PDE encoding anisotropic incompressible magnetohydrodynamics is obtained.)
(66) A. Prástaro, Extended crystal PDE’s stability.II: The extended crystal MHDPDE’s, Math. Comput. Modelling. (2008).
Abstract. This paper is the second part of a work devoted to the algebraic topological characterization of PDE’s stability and its relation with
an important class of PDE’s called extended crystals PDE’s. In fact, their
integral bordism groups can be considered extensions of subgroups of crystallographic groups. This allows us to identify a characteristic class that
measures the obstruction to the existence of global solutions. In part I we
identified criteria to recognize PDE’s that are stable (in extended Ulam
sense) and in their regular smooth solutions do not occur finite time unstabilities, (stable extended crystal PDE’s). Here we study in some details a
new PDE encoding anisotropic incompressible magnetohydrodynamics. A
stable extended crystal MHD-PDE’s is obtained where in its smooth solutions do not occurr unstabilities in finite times. These results are considered
first for systems without body energy source and after by introducing also
a contribution by energy source in order to take into account of nuclear energy production. A condition in order solutions satisfy the second principle
of thermodynamics is given.
(67) A. Prástaro, On the extended crystal PDE’s stability.I: The n-d’Alembert
extended crystal PDE’s, Appl. Math. Comput. 204(1)(2008), 63–69.
Abstract. Our recent results on extended crystal PDE’s and geometric
theory on PDE’s stability, are applied to the generalized n-d’Alembert
PDE’s, (d′ A)n , n ≥ 2. We prove that these are extended crystal PDE’s
for any n ≥ 2. For suitable n, (d′ A)n becomes an extended 0-crystal PDE
and also a 0-crystal PDE. An equation, having all the same smooth solutions of (d′ A)n , but without unstabilities at “finite time” is obtained for
each n ≥ 2.
(68) A. Prástaro, On the extended crystal PDE’s stability.II: Entropy-regularsolutions in MHD-PDE’s, Appl. Math. Comput. 204(1)(2008), 82–89.
CURRICULUM VITAE AGOSTINO PRÁSTARO – APRIL 2014
43
Abstract. Local and global existence theorems of entropy-regular-solutions
in the geometric framework of MHD-PDE’s, recently introduced by A.Prástaro,
are given. Stability properties of such solutions are studied. In particular
it is shown how to stabilize the smooth entropy-regular-solutions, in order
to avoid finite-times unstabilities.
(69) A. Prástaro, On quantum black-hole solutions of quantum super Yang-Mills
equations, Proceedings Dynamic Systems Appls. 5(2008), 407–414. (Eds.
G. S. Ladde, N. G. Madhin C. Peng & M. Sambandham), Dynamic Publishers, Inc., Atlanta, USA. ISBN: 1-890888-01-6.
Abstract. The category Q, (resp. QS ), of quantum manifolds, (resp.
quantum supermanifolds), introduced by A.Prástaro, gives a natural framework where implement a geometric theory of quantum (super) partial differential equations (PDE’s). The interest for quantum supermanifolds is
motivated by the fact that these structures allow us to describe the unification of all the four fundamental forces (gravity, electromagnetic, weak
nuclear, strong nuclear), at the quantum level. In this note we present
some recent developments in this direction. In particular we will consider
quantum black holes as solutions of quantum super Yang-Mills equations.
These interpret very high energy level production of particles, where the
effects of strong-quantum-gravity become dominant.
(70) A. Prástaro, Surgery and bordism groups in quantum partial differential
equations.I: The quantum Poincaré conjecture, Nonlinear Anal. Theory
Methods Appl. 71(12)(2009), 502–525.
Abstract. In this work, in two parts, we continue to develop the geometric theory of quantum PDE’s, introduced by us starting from 1996. (The
second part is quoted in ref.[71].) This theory has the purpose to build a
rigorous mathematical theory of PDE’s in the category DS of noncommutative manifolds (quantum (super)manifolds), necessary to encode physical
phenomena at microscopic level (i.e., quantum level). Aim of the present
paper is to report on some new issues in this direction, emphasizing an
interplaying between surgery, integral bordism groups and conservations
laws. In particular, a proof of the Poincaré conjecture, generalized to the
category DS , is given by using our geometric theory of PDE’s just in such
a category.
(71) A. Prástaro, Surgery and bordism groups in quantum partial differential
equations.II: Variational quantum PDE’s, Nonlinear Anal. Theory Methods Appl. 71(12)(2009), 526–549.
Abstract. This is the second part of a work devoted to the interplay between surgery, integral bordism groups and conservation laws, in order to
characterize the geometry of PDE’s in the category QS of quantum (super)manifolds. (Part I is quoted in ref.[70].) In this paper we will consider
variational problems, in the category QS , constrained by partial differential
equations. We get theorems of existence for local and global solutions. The
characterization of global solutions is made by means of integral bordism
groups. Applications to some important examples of the Mathematical
Physics, as quantum super-black-hole solutions of quantum super YangMills equations, are discussed in some details. Quantum supermanifolds
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CURRICULUM VITAE AGOSTINO PRÁSTARO – APRIL 2014
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allow us to unify, at the quantum level, the four fundamental forces, (gravitational, electromagnetic, weak-nuclear, strong-nuclear), in an unique geometric structure. The geometric theory of PDE’s, built in the category QS
of quantum supermanifolds, gives us the right mathematic tool to describe
quantum phenomena also at very high energy levels, where quantum-gravity
becomes dominant.
R. P. Agarwal & A. Prástaro, Singular PDE’s geometry and boundary value
problems, J. Nonlinear Conv. Anal. 9(3)(2008), 417–460.
Abstract. Local and global existence theorems for boundary value problems in singular PDE’s are considered. In particular, surgery techniques and
integral bordism groups are utilized, following previous works by A.Prástaro
on PDE’s, in order to build global solutions crossing also singular points
and to study their stability properties.
R. P. Agarwal & A. Prástaro, On singular PDE’s geometry and boundary
value problems, Appl. Anal. 88(8)(2009), 1115-1131.
Abstract. A geometric formulation of singular PDE’s is considered. Surgery
techniques and integral bordism groups are utilized, following previous
works by A.Prástaro on PDE’s, in order to build global solutions crossing also singular points and to study their stability properties. A detailed
proof on the integral characterization of singular ODE’s is given.
A. Prástaro, Extended crystal PDE’s. Mathematics Without Boundaries:
Surveys in Pure Mathematics. P. M. Pardalos and Th. M. Rassias (Eds.)
Springer-Heidelberg New York Dordrecht London, (to appear).
arXiv:0811.3693[math.AT].
Abstract. In this paper we show that between PDE’s and crystallographic
groups there is an unforeseen relation. In fact we prove that integral bordism groups of PDE’s can be considered extensions of crystallographic subgroups. In this respect we can consider PDE’s as extended crystals. Then an
algebraic-topological obstruction (crystal obstruction), characterizing existence of global smooth solutions for smooth boundary value problems, is
obtained. Applications of this new theory to the Ricci-flow equation and
Navier-Stokes equation are given that solve some well-known fundamental
problems. These results, are also extended to singular PDE’s, introducing
(extended crystal singular PDE’s). An application to singular MHD-PDE’s,
is given following some our previous results on such equations, and showing
existence of (finite times stable smooth) global solutions crossing critical
nuclear energy production zone.
A. Prástaro, Quantum extended crystal PDE’s, Nonlinear Studies 18(3)(2011),
447–485. arXiv:1105.0166[math.AT].
Abstract. Our recent results on extended crystal PDE’s are generalized to
PDE’s in the category QS of quantum supermanifolds. Then obstructions
to the existence of global quantum smooth solutions for such equations are
obtained, by using algebraic topologic techniques. Applications are considered in details to the quantum super Yang-Mills equations. Furthermore,
our geometric theory of stability of PDE’s and their solutions, is also generalized to quantum extended crystal PDE’s. In this way we are able to
identify quantum equations where their global solutions are stable at finite
CURRICULUM VITAE AGOSTINO PRÁSTARO – APRIL 2014
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45
times. These results, are also extended to quantum singular (super)PDE’s,
introducing (quantum extended crystal singular (super) PDE’s).
A. Prástaro, Quantum extended crystal super PDE’s. Nonlinear Anal. Real
World Appl. 13(6)(2012), 2491–2529. arXiv:0906.1363[math.AT].
Abstract. We generalize our geometric theory on extended crystal PDE’s
and their stability, to the category QS of quantum supermanifolds. By
using algebraic topologic techniques, obstructions to the existence of global
quantum smooth solutions for such equations are obtained. Applications
are given to encode quantum dynamics of nuclear nuclides, identified with
graviton-quark-gluon plasmas, and study their stability. We prove that
such quantum dynamical systems are encoded by suitable quantum extended crystal Yang-Mills super PDE’s. In this way stable nuclear-charged
plasmas and nuclides are characterized as suitable stable quantum solutions
of such quantum Yang-Mills super PDE’s. An existence theorem of local
and global solutions with mass-gap, is given for quantum super Yang-Mills
\
\ ⊂ (Y
\
PDE’s, (Y
M ), by identifying a suitable constraint, (Higgs)
M ), Higgs
quantum super PDE, bounded by a quantum super partial differential rela\
\
tion (Goldstone)
⊂ (Y
M ), quantum Goldstone-boundary. A global solution
\
V ⊂ (Y
M ), crossing the quantum Goldstone-boundary acquires (or loses)
mass. Stability properties of such solutions are characterized.
A. Prástaro, Exotic heat PDE’s, Commun. Math. Anal. 10(1)(2011),
64–81. arXiv:1006.4483[math.GT].
Abstract. Exotic heat equations that allow to prove the Poincaré conjecture, some related problems and suitable generalizations too are considered.
The methodology used is the PDE’s algebraic topology, introduced by A.
Prástaro in the geometry of PDE’s, in order to characterize global solutions.
A. Prástaro, Exotic heat PDE’s.II. Essays in Mathematics and its Applications. In Honor of Stephen Smale’s 80th Birthday. P. M. Pardalos and
Th. M. Rassias (Eds.) Springer-Heidelberg New York Dordrecht London
(2012), 369–419. ISBN 978-3-642-28820-3 (Print) 978-3-28821-0 (Online).
DOI: 10.1007/978-3-642-28821-0. arXiv: 1009.1176[math.AT].
Abstract. Exotic heat equations that allow to prove the Poincaré conjecture and its generalizations to any dimension are considered. The methodology used is the PDE’s algebraic topology, introduced by A. Prástaro in
the geometry of PDE’s, in order to characterize global solutions. In particular it is shown that this theory allows us to identify n-dimensional exotic
spheres, i.e., homotopy spheres that are homeomorphic, but not diffeomorphic to the standard S n .
A. Prástaro, Exotic n-d’Alembert PDE’s and stability. Nonlinear Analysis.
Stability, Approximation and Inequalities. Series: Springer Optimization
and its Applications Vol 68. P. M. Pardalos, P. G. Georgiev and H. M.
Srivastava (Eds.). Springer Optimization and its Applications Volume 68
(2012), 571–586. ISBN 978-1-4614-3498-6. arXiv:1011.0081[math.AT].
Abstract. In the framework of the PDE’s algebraic topology, previously
introduced by A. Prástaro, exotic n-d’Alembert PDE’s are considered. These
are n-d’Alembert PDE’s, (d′ A)n , admitting Cauchy manifolds N ⊂ (d′ A)n
identifiable with exotic spheres, or such that ∂N , can be exotic spheres. For
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CURRICULUM VITAE AGOSTINO PRÁSTARO – APRIL 2014
such equations local and global existence theorems and stability theorems
are obtained.
(80) A. Prástaro, Exotic PDE’s. Mathematics Without Boundaries: Surveys
in Interdisciplinary Research. P. M. Pardalos and Th. M. Rassias (Eds.)
Springer-Heidelberg New York Dordrecht London, (to appear).
arXiv:1101.0283[math.AT].
Abstract. In the framework of the PDE’s algebraic topology, previously
introduced by A. Prástaro, are considered exotic differential equations, i.e.,
differential equations admitting Cauchy manifolds N identifiable with exotic spheres, or such that their boundaries ∂N are exotic spheres. For such
equations are obtained local and global existence theorems and stability
theorems. In particular the smooth (4-dimensional) Poincaré conjecture is
proved. This allows to complete the previous Theorem 4.59 in [22] also for
the case n = 4.
(81) A. Prástaro, Quantum exotic PDE’s. Nonlinear Anal. Real World Appl.
14(2)(2013), 893–928. arXiv:1106.0862[math.AT].
Abstract. Following the previous works on the A. Prástaro’s formulation of algebraic topology of quantum (super) PDE’s, it is proved that a
canonical Heyting algebra (integral Heyting algebra) can be associated to
any quantum PDE. This is directly related to the structure of its global
solutions. This allows us to recognize a new inside in the concept of quantum logic for microworlds. Furthermore, the Prástaro’s geometric theory
of quantum PDE’s is applied to the new category of quantum hypercomplex manifolds, related to the well-known Cayley-Dickson construction for
algebras. Theorems of existence for local and global solutions are obtained
for (singular) PDE’s in this new category of noncommutative manifolds.
Finally the extension of the concept of exotic PDE’s, recently introduced
by A.Prástaro, has been extended to quantum PDE’s. Then a smooth
quantum version of the quantum (generalized) Poincaré lemma is given
too. These results extend ones for quantum (generalized) Poincaré lemma,
previously given by A. Prástaro.
(82) A. Prástaro, Strong reactions in quantum super PDE’s. I: Quantum hypercomplex exotic super PDE’s.
arXiv:1205.2984[math.AT]. (Part I and Part II are unified in arXiv.)
Abstract. In order to encode strong reactions of the high energy physics,
by means of quantum nonlinear propagators in the Prástaro’s geometric
theory of quantum super PDE’s, some related geometric structures are further developed and characterized. In particular super-bundles of geometric
objects in the category QS of quantum supermanifolds are considered and
quantum Lie derivative of sections of super bundle of geometric objects are
calculated. Quantum supermanifolds with classic limit are classified with
respect to the holonomy groups of these last commutative manifolds. A
theorem characterizing quantum super manifolds with structured classic
limit as super bundles of geometric objects is obtained. A theorem on the
characterization of chi-flow on suitable quantum manifolds is proved. This
solves a previous conjecture too. Quantum instantons and quantum solitons are defined are useful generalizations of the previous ones, well-known
in the literature. Quantum conservation laws for quantum super PDEs are
CURRICULUM VITAE AGOSTINO PRÁSTARO – APRIL 2014
47
characterized. Quantum conservation laws are proved work for evaporating
quantum black holes too. Characterization of observed quantum nonlinear
propagators, in the observed quantum super Yang-Mills PDE, by means of
conservation laws and observed energy is obtained. Some previous results
by A. Prástaro about generalized Poincaré conjecture and quantum exotic
spheres, are generalized to the category Qhyper,S of hypercomplex quantum
supermanifolds. (This is the first part of a work divided in two parts. For
part II see [83].)
(83) A. Prástaro, Strong reactions in quantum super PDE’s. II: Nonlinear quantum propagators.
arXiv:1205.2984[math.AT]. (Part I and Part II are unified in arXiv.)
Abstract. In this second part, of a work devoted to encode strong reactions of the high energy physics, in the algebraic topologic theory of
quantum super PDE’s, (previously formulated by A. Prástaro), decomposition theorems of integral bordisms in quantum super PDEs are obtained.
(For part I see [82].) In particular such theorems allow us to obtain representations of quantum nonlinear propagators in quantum super PDE’s,
by means of elementary ones (quantum handle decompositions of quantum
nonlinear propagators). These are useful to encode nuclear and subnuclear
reactions in quantum physics. Prástaro’s geometric theory of quantum
PDE’s allows us to obtain constructive and dynamically justified answers
to some important open problems in high energy physics. In fact a Reggetype relation between reduced quantum mass and quantum phenomenological spin is obtained. A dynamical quantum Gell-Mann-Nishijima formula is given. An existence theorem of observed local and global solutions with electric-charge-gap, is obtained for quantum super Yang-Mills
\
\
\
PDE’s, (Y
M )[i], by identifying a suitable constraint, (Y
M )[i]w ⊂ (Y
M )[i],
quantum electromagnetic-Higgs PDE, bounded by a quantum super partial
\
\
differential relation (Goldstone)[i]
w ⊂ (Y M )[i], quantum electromagnetic
Goldstone-boundary. An electric neutral, connected, simply connected ob\
served quantum particle, identified with a Cauchy data of (Y
M )[i], it is
\
proved do not belong to (Y M )[i]w . Existence of Q-exotic quantum non\
linear propagators of (Y
M )[i], i.e., quantum nonlinear propagators that do
not respect the quantum electric-charge conservation is obtained. By using integral bordism groups of quantum super PDE’s, a quantum crossing
symmetry theorem is proved. As a by-product existence of massive photons
and massive neutrinos are obtained. A dynamical proof that quarks can be
broken-down is given too. A quantum time, related to the observation of any
quantum nonlinear propagator, is calculated. Then an apparent quantum
time estimate for any reaction is recognized. A criterion to identify solutions
of the quantum super Yang-Mills PDE encoding (de)confined quantum systems is given. Supersymmetric particles and supersymmetric reactions are
classified on the ground of integral bordism groups of the quantum super
\
Yang-Mills PDE (Y
M ). Finally, existence of the quantum Majorana neutrino is proved. As a by-product, the existence of a new quasi-particle, that
we call quantum Majorana neutralino, is recognized made by means of two
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CURRICULUM VITAE AGOSTINO PRÁSTARO – APRIL 2014
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quantum Majorana neutrinos, a couple (νee , νēe ), supersymmetric partner of
(νe , ν̄e ), and two Higgsinos.
A. Prástaro, Strong reactions in quantum super PDE’s. III: Exotic quantum
supergravity. arXiv:1206.4956[math.AT].
Abstract. of quantum super PDE’s, quantum nonlinear propagators in
\
the observed quantum super Yang-Mills PDE, (Y
M )[i], are further characterized. In particular, quantum nonlinear propagators with non-zero
lost quantum electric-charge, are interpreted as exotic-quantum supergravity effects. As an application, the recently discovered bound-state called
Zc(3900), is obtained as a neutral quasi-particle, generated in a Q-quantum
exotic supergravity process. Quantum entanglement is justified by means
of the algebraic topologic structure of quantum nonlinear propagators. Ex\
istence theorem for solutions of (Y
M )[i] admitting negative local temperatures (quantum thermodynamic-exotic solutions) is obtained too and related
to quantum entanglement.
A. Prástaro, The Landau’s problems. I: The Goldbach’s conjecture proved.
arXiv:1208.2473[math.GM].
Abstract. We give a direct proof of the Goldbach’s conjecture, (GC), in
number theory, in the Euler’s form. The proof is also constructive, since it
gives a criterion to find two prime numbers ≥ 1, such that their sum gives a
fixed even number ≥ 2. The proof is obtained by recasting the problem in
the framework of the Commutative Algebra and Algebraic Topology. Even
if in this paper we consider 1 as a prime number, our proof of the GC works
also for the restricted Goldbach conjecture, (RGC), i.e., by excluding 1 from
the set of prime numbers.
A. Prástaro, The Landau’s problems. II: Landau’s problems solved.
arXiv:1208.2473[math.GM].
Abstract. Three of the well known four Landau’s problems are solved in
this paper. (In [85] the proof of the Goldbach’s conjecture has been already
given.)
A. Prástaro, The Riemann hypothesis proved.
arXiv:1305.6845[math.GM].
Abstract. The Riemann hypothesis is proved by extending the zeta Riemann function to a quantum mapping between quantum 1-spheres with
quantum algebra A = C, in the sense of A. Prástaro [70, 81]. Algebraic
topologic properties of quantum-complex manifolds and suitable bordism
groups of morphisms in the category QC of quantum-complex manifolds are
utilized.
A. Prástaro, Quantum Geometrodynamic Cosmology.
(Submitted for publication.)
Abstract. By utilizing Prástaro’s quantum supergravity, it is proved that
the Universe’s expansion at the Planck epoch is justified by the fact that
it is encoded by a quantum nonlinear propagator having thermodynamic
quantum exotic components in its boundary. This effect produces also an
increasing of energy in the Universe at the Einstein epoch: Planck-epochlegacy on the boundary of our Universe. This is the main source of the
Universe’s expansion and solves the problem of the non-apparent energymatter (dark-energy-matter) in the actual Universe.
CURRICULUM VITAE AGOSTINO PRÁSTARO – APRIL 2014
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ABSTRACTS OF MONOGRAPHS AND TEXTS
(89) A. Prástaro, Geometry of PDEs and Mechanics, World Scientific Publishing, River Edge, NJ, 1996, 760 pp.
Abstract. 1. Algebraic Geometry: Algebraic Complements, Affine spaces,
Differential Manifolds, Grassmann Manifolds, Spectral Sequences. 2. Differential Equations (PDES): Geometry of Differential Equations, Ordinary
Differential Equations (ODEs), Characteristics of PDEs, Affine PDEs and
Green Functions, Spectral Sequences in PDEs, Tunnel Effects in PDEs,
Cobordism Groups in PDEs. 3. Mechanics: Structure of Galilean SpaceTime, One-Body Dynamics, Important Formulas, Fundamental Theorems
of Dynamics, Lagrangian Mechanics for Perfect Holonomic Systems, RigidBody Dynamics. 4. Continuum Mechanics: Flow, Stress Tensor and Moment of Stress Tensor, Local Dynamic Equations, Thermodynamics of Continuum Media, Rheological Classification of Materials, Rheoptics, Multicomponent Continuum Systems, Variational Field Theory. 5. Quantum
Field Theory: Locally Convex Manifolds and Derivative Spaces, Differential Geometry of Quantum Situs, Mathematical Logic and Quantization
of PDEs, Formal and Dirac Quantizations of PDEs, Canonical Quantization of PDEs. 6. Geometry of Quantum PDEs: Differential Geometry of
Quantum Manifolds, Cohomology of Quantum Manifolds, Formal Theory of
Quantum PDEs, Cartan Spectral Sequences of Quantum PDEs, Quantum
Distribution Solutions and Singular Solutions of Quantum PDEs, Tunnel
Effects in Quantum PDEs, Quantum PDEs Non-holonomic Connections,
Gauge Quantum PDEs, Supergravity Quantum PDEs.
(90) A. Prástaro, Elementi di Meccanica Razionale, VII edizione, Aracne Editrice, Roma, 2010,446 pp.
Abstract. 1. This monography is addressed to Italian university students
in Mathematics, Physics and Engineering. It develops with a modern geometric language the methods of classical mechanics and geometry of (partial) differential equations. The presentation, even if elementary, gives the
actual mathematics situation in classical mechanics.
Indice. Algebra: matrici ed applicazioni lineari; prodotto tensoriale
fra spazi vettoriali; tensori; spazi vettoriali Euclidei; componenti covarianti e controvarianti; tensori simmetrici ed antisimmetrici; orientazione di
spazi vettoriali; gruppi GL(V ), O(V ), SO(V ); equazione agli autovalori;
spazio affine; gruppo affine; soluzioni di equazione affine. 2. Equazioni
differenziali: varietà differenziale; spazio tangente; campi tensoriali e loro
immagine reciproca; forma volume ed orientazione di varietà differenziale;
integrazione su varietà differenziale; varietà Riemanniana; gruppi di Lie e
loro caratterizzazione tramite costanti di struttura; elementi di teoria geometrica delle equazioni differenziali. 3. Connessioni differenziali: derivata
covariante; connessione di Levi-Civita; curvatura di una connessione; complesso di de Rham e gruppi di comologia; importanti operatori differenziali su varietà Riemanniane; componenti fisiche di oggetti geometrici. 4.
Spazio-tempo Galileiano: struttura dello spazio-tempo Galileiano; moto;
velocità ed accelerazione del moto; osservatore e moto osservato; gruppo
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di Galileo; formule di Frenet; espressione della velocità ed accelerazione
tramite prametro di linea; teorema di Coriolis; osservatori inerziali ed osservatori rigidi; moti rigidi e teorema fondamentale della cimematica dei
corpi rigidi; angoli di Euler; precessioni; traiettorie polari (base e ruletta).
5. Dinamica di un elemento materiale: vincoli di ordine 0 ≤ n ≤ 2; forze ed
equazione di Newton; forze conservative; equazione di Lagrange; equazione
di Hamilton e campo vettoriale di Euler; struttura simplettica della meccanica; simmetrie dinamiche; teorema di Noether generalizzato; sistemi dinamici con un numero finito N di particelle; vincoli con attrito. 6. Teoremi
fondamentali della dinamica: teorema dell’impulso; teorema della conservazione dell’impulso; momento totale e sue proprietà; momento assiale;
coppia; lavoro; potenza; energia cinetica; stabilità dell’equilibrio; equazioni
cardinali della dinamica; teorema di conservazione; principio dei lavori virtuali; teorema di Köenig. 7. Meccanica Lagrangiana: Equazione di Lagrange e sue forme particolari. Potenziali generalizzati e forza di Lorentz;
stabilità ed equazioni di Lagrange linearizzate. Leggi di conservazione e
coordinate ignorabili; equazione di Lagrange e calcolo variazionale; configurazioni di stato stazionario e loro stabilità. 8. Meccanica dei sistemi
rigidi: baricentri di sistemi continui; tensore momento d’inerzia; equazioni
cardinali per corpo rigido; moto alla Poinsot e sue leggi di conservazione. 9.
Esercizi: (ventuno esercizi completamente risolti). 10. Meccanica dei continui: flusso; osservatore proprio di un sistema continuo; oggetti geometrici
associati ad un flusso; tensore degli sforzi e suo momento; equazioni di
Euler; equazione dinamica dei sistemi continui; termodinamica covariante
dei sistemi continui; classificazione reologica dei sistemi continui; esempi di
flussi e deformazioni. 11. Reottica. Bibliografia. 12. Equazioni di Maxwell
e relatività ristretta. 13. Esercizi complementari (completamente svolti):
Esercizi di Geometria; Esercizi di Meccanica dei sistemi rigidi; Esercizi di
Meccanica dei sistemi continui. Indice analitico. Indice dei simboli.
(91) A. Prástaro, Quantized Partial Differential Equations, World Scientific Publishing, River Edge, NJ, 2004, 500 pp.
Abstract. This book contains three chapters and two addenda. Quantized PDE’s.I. In this first part we consider quantum (super) manifolds as
topological spaces locally identified with open sets of some locally convex
topological vector spaces built starting from suitable topological algebras
A, quantum (super)algebras. The noncommutative character of such quantum (super)manifolds is given by the underlying noncommutative algebras
A. In fact, here A plays the role of “fundamental algebra of numbers”, like
K = R, C does for usual commutative manifolds. Therefore, quantum (super)manifolds are the natural generalizations of manifolds , when one substitutes commutative numbers with noncommutative ones. Commutative
manifolds are contained into quantum (super)manifolds, as quantum (super)algebras A are required to contain K. This aspect is also reflected by the
fact that the class of differentiability Qkw for pseudogroup structures defining quantum (super)manifolds, contains the usual C k differentiability for
manifolds. In fact, the class of differentiability of such topological manifolds
is defined by requiring weak differentiability and Z-linearity of the derivatives, where Z is the center of the underlying quantum (super)algebras.
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51
We give (co)homological characterizations of quantum (super)algebras and
quantum (super)manifolds, by applaying to these noncommutative topological manifolds standard methods of algebraic topology. In particular, we
calculate also (co)bordism groups in quantum (super)manifolds. Quantized
PDE’s.II. Here we give a geometric theory of PDE’s in the category of quantum (super)manifolds. This theory is the natural extension of the geometric
theory of PDE’s in the category of commutative (super)manifolds. Emphasis is put on some new algebraic topological techniques that allow us to calculate the integral (co)bordism groups of quantum (super)PDE’s, hence to
characterize global properties of solutions of quantum (super)PDE’s. Many
applicatio ns to important equations of quantum field theory are considered
also. Quantized PDE’s.III. Here we consider a process that allows us to associate to a (super)PDE, defined in the category of (super)commutative
manifolds, a quantum (super)PDE. This process is the covariant quantization. We describe it in some steps. In fact, we first define quantizations
of PDE’s in the framework of the mathematical logic, by means of evaluations of the logic of a PDE Ek , that is the Boolean algebra of subsets of the
classic limit Ω(Ek )c of the quantum situs Ω(Ek ) of Ek , into quantum logics
A ⊂ L(H), that are algebras of (self-adjoint) operators on a locally convex
topological vector (Hilbert) space H, in such a way to define (pre-)spectral
measures on Ω(Ek )c : Ω(Ek )c ◦→L(H). We show that these quantizations
can be obtained by means of a geometric process called covariant quantization, (or canonical quantization), of PDE’s, that is, roughly speaking
the covariant quantization observed by a physical frame. In fact, in a
purely geometric context, we prove that any physical observable deforms
the classical PDE, Ek ⊂ JD k (W ), around its solutions. In this way we can
associate to the Lie filtered (super)algebra of the (super)classical observables, B, of Ek , a filtered quantum (super)algebra B̂, defined by means
of distributive kernels, G̃q , propagators, canonically associated to Ek . We
characterize also the propagators of PDE’s by means oftheir integral bordism groups. The final step is the relation between the formal properties
E∞ · · · → Ek+1 → Ek → · · · of the classical equation Ek , with quantum
ones Ê∞ · · · → Êk+1 → Êk → · · · . These are obtained in the category
of QPDE’s, where the quantum (super)algebra B̂, so obtained as covariant quantization of Ek , identifies a quantum (super)PDE. Addendum I. In
refs.[38, 41] are calculated, for the first time, the integral bordism groups
of the 3D nonisothermal Navier-Stokes equation (N S). A direct consequence of these results is the proof of existence of global (smooth) solutions
for (N S). Here we go in some further results emphasizing surgery techniques that allow us to better understand this geometric proof of existence
of (smooth) global solutions for any (smooth) boundary condition. Addendum II. In the framework of the geometry of PDE’s, we classify variational
equations of any order with respect to their formal properties. A variational sequence is introduced for constrained variational PDE’s that extends previous ones for variational calculus on fiber bundles. Such extended
variational sequence allows us to locally and globally solve variational problems, constrained by PDE’s of any order, Ek ⊂ Jnk (W ), by means of some
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CURRICULUM VITAE AGOSTINO PRÁSTARO – APRIL 2014
cohomological properties of Ek . Moreover, we relate constrained variational PDE’s to the integral (co)bordism groups for PDE’s. In this way
we are able to characterize the structure properties of global solutions of
constrained variational PDE’s and to relate them to the structure of global
solutions for the corresponding constraint equations. Contents: Quantized
PDE’s.I. Noncommutative Manifolds: Algebraic topology; Quantum algebras; Quantum manifolds; Quantum supermanifolds. Quantized PDE’s.II.
Noncommutative PDE’s: Quantum PDE’s; The quantum Navier-Stokes
equation; Quantum super PDE’s; The quantum super Yang-Mills equations. Quantized PDE’s.III. Quantizations of commutative PDE’s: Integral (co)bordism groups in PDE’s; Algebraic geometry of PDE’s; Spectral
measures of PDE’s; Quantizations of PDE’s; Covariant and canonical quantizations of PDE’s. Addendum I: Bordism groups and the (N S)-problem.
Addendum II: Bordism groups and variational PDE’s. References. Index.
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ABSTRACTS OF BOOKS (EDITOR AND COAUTHOR)
(92) A. Prástaro, Geometrodynamics Proceedings 1983, Pitagora Ed., Bologna
1984.
Abstract. The Hamilton-Jacobi Equation for a Hamiltonian Action: S.
Benenti. Twisteurs Sans Twisteurs: A. Crumeyrolle. General Covariance
and Minimal Gravitational Coupling in Newtonian Space-Time: H. P. Kunzle. Canonical Cartan Equations for Higher Order Variational Problems:
J. M. Masqué. Bäcklund Problem and Group Theory: J. F. Pommaret.
Group Structure of Non-linear Field Theories: J. F. Pommaret. Geometry
and Existence Theorems for Incompressible Fluids: A. Prástaro. Relativistic Hydrodynamics as a Symplectic Field Theory: W. M. Tulczyjew.
(93) A. Prástaro, Geometrodynamics Proceedings 1985, World Scientific Publishing, Singapore 1985.
Abstract. A Geometrical Interpretation of the 1-cocycles of a Lie Group:
S. Benenti, W. M. Tulzyjew. Supermanifolds and Supergravity: Y. ChoquetBruhat. Self-dual Yang-Mills Fields and the Penrose Trasnform: A. Crumeyrolle. On Smooth and Analytic Functions in Gauge Field Theory: J. Czyz.
An Application of Topological Methods to the Study of Periodic Solutions of
Hamiltonian Systems: G. F. Dell’Antonio. Introducing Spinors, Isospinors,
etc. in Globally Nontrivial Space-times: L. Dabrowski. The Radon Transform on Compact Symmetric Spaces: H. Goldschmidt. Unconstrained Degrees of Freedom of Gravitational Field and the Positivity of Gravitational
Energy: J. Kijowski. Polynomial Identities Satisfied by Realizations of
Lie Algebras: M. Iosifescu, H. Scutaru. Free Motions in Multidimensional
Universes: G. Marmo. Harmonically Immersed Lorentz Surfaces: T. Milnor. On the Symmetry Properties of Constrained Hamiltonian System:
M. Mintchev. On a Property of Higher Order Poincaré-Cartan Forms in
the Constructive Approach: J. M. Masqué. Covariant Canonical Formalism for Gravity Theories: J. E. Nelson, T. Regge. Invariant Differential
Techniques: A. Nijenhuis. A Utiyama Type Theorem in the C-K-S- Gauge
Approach to Gravity: A. Pérez-Rendòn. Dynamic Conservation Laws: A.
Prástaro. The Doulbeault-Kostant Complex and Geometric Quantization:
M. Puta. Symplectic Origin of Some Properties of Generally Covariant
Field Theories: A. Smolski. Symplectic Scattering Theory: S. Sternberg.
(94) A. Prástaro & Th. M. Rassias, Geometry in Partial Differential Equations,
World Scientific Publishing, River Edge, NJ, 1994.
Abstract. Some Applications of the Corea Formula to Partial Differential
Equations: F. Bethuel, J.-M. Chidaglia. Large Solutions for the Equation
of Surfaces of Prescribed Mean Curvature: F. Bethuel, O. Rey. Optical
Hamiltonian Functions: M. Bialy, L. Polterovich. On the Geometry of
the Hodge-de Rham Laplace Operator: M. Craioveanu, M. Puta, Th. M.
Rassias. Minimal Surfaces in Economic Theory: J. Donato. The Morimoto
Problem: B. Doubrov, A. Hushner. Asymptotic Expansions in Spectral Geometry: P. B. Giley. Deformations and Recursion Operators for Evolution
Equations: I. S. Krasil’shchik, P. H. M. Kersten. Geometric Hamiltonian
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Forms for the Kadomtsev-Petviashvili and Zabolotskaya-Khokhlov Equations, B. A. Kupershmidt. Classification of Mixed Type Momnge-Ampere
Equations: A. Kushner. Non-Holonomic Filtration: Algebraic and Geometric Aspects of Non-Integrability: V. Lychagin, V. Rubtsov. Spencer
Cohomologies: V. Lychagin, L. Zilbergleit. Hawking’s Relation via Fourier
Integral Operators: P. E. Parker. Geometry of Super PDE’s: A. Prástaro.
On a Geometric Approach to an Equation of J.d’Alembert: A. Prástaro,
Th. M. Rassias. Geometric Prequantization of the Einstein’s Vacuum
Field: M. Puta. On Differential Equations and Cartan’s Projective Connections: Y. R. Romanovsky. Smooth Marginal Analysis of Bifurcation
of Extremals: Yu I. Sapronov. On the Schrödinger Equation for an N Electron Atom: C. S. Sharma. Higher Symmetries and Conservation Laws
of Euler-Darboux Equations: V. E. Shemarulin. Strings and Menbranes:
K. S. Stelle. Methods for Solving Two-Dimensional Nonstationary MHD
Equations at Small Alfven-Mach Numbers: V. S. Titov.
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ABSTRACTS OF PATENTS
(95) A. Prástaro et al., 28/5/1975 - No 23.797 A/75. Process of production of
fibrous structures with high degree of birefringence.
Abstract. The present invention refers to a process to produce fibrous
structures having an high degreee of monoaxial orientation, by means of
controlled extrusion of solutions, emulsions, suspensions of fibrogenous thermoplastic polymers. In particular, one fixes the optimum conditions of the
extrusor design in order to obtain a birefringence higher than 0.1 10−4 .
(96) A. Prástaro et al., 11/7/1975 - No 25.334 A/75. Process of production of
synthetic polymers by means of flash-spinning of polymers solutions.
Abstract. The present invention refers to a process to produce fibrous
structures of synthetic polymers, in the form of plexus-filament of single
little fibers, by means of the technique of the “flash-spinning” of polymers
solutions. In particular, one fixes the optimum thermodynamical conditions to obtain plexus-filaments that extend ranges previously found by
USA-patents. This has been possible by using a new mathematical model
of flash-spinning, purposely formulated, and put on informatic support. Of
particular importance has been the discovery of metastable states in the
extruded solution, similar to ones used in the bubble-chambers for dedecting sub-atomic collisions. These thermodynamical conditions, beside some
suitable dynamical conditions, allow us to get optimum control conditions
in such a process.
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FIG. 8. Prástaro’s mathematicians poster 2005-2006.
CURRICULUM VITAE AGOSTINO PRÁSTARO – APRIL 2014
Fig. 9. Prástaro’s mathematicians poster 2007-2008.
57
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Fig. 10. Prástaro’s mathematicians poster 2009-2012.
CURRICULUM VITAE AGOSTINO PRÁSTARO – APRIL 2014
59
5. INDEX
General Informations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
Summary of Principal results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
Publications. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
Papers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
Monographs and Texts. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
Books (Editor and Coauthor). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
Patents. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
Abstracts of Publications. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
Abstracts of Papers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
Abstracts of Monographs and Texts. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
Abstracts of Books (Editor and Coauthor). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
Abstracts of Patents. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
Photos and Posters.
Title and A.Prástaro’s picture. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
University Address. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
Home Address. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
Acknowledgments (Prástaro’s mathematicians poster 2009-2013). . . . . . . . .3
School of Engineering, University of Roma La Sapienza. . . . . . . . . . . . . . . . . . 6
Florida Institute of Technology, Melbourne, FL - USA. . . . . . . . . . . . . . . . . . 20
ICM 2006 SC: Advances in PDE’s Geometry, Madrid - poster. . . . . . . . . . 26
Prástaro’s mathematicians poster 2005-2006. . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
Prástaro’s mathematicians poster 2007-2008. . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
Prástaro’s mathematicians poster 2009-2012. . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
Prástaro’s mathematicians poster 2013-2014. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
Roma - Spagna Square. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
Fig. 11. Roma - Spagna Square.
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6. CONTENTS
1
2
3
4
5
-
General Informations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
Summary of Principal Results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
Publications. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .21
Abstracts of Publications. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .27
Index. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
Curriculum Vitae Agostino Prástaro - Edition April 2014