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Introduction
... The canonical embedding G : G ! G^^ is dened by G(g)() = (g) for every g 2 G and every 2 G^. If G is a topological isomorphism, the topological group G is called reexive. The Pontryagin-Van Kampen theorem states that locally compact abelian groups are reexive. However the class of reexiv ...
... The canonical embedding G : G ! G^^ is dened by G(g)() = (g) for every g 2 G and every 2 G^. If G is a topological isomorphism, the topological group G is called reexive. The Pontryagin-Van Kampen theorem states that locally compact abelian groups are reexive. However the class of reexiv ...
A shorter proof of a theorem on hereditarily orderable spaces
... A topological space is orderable if it is homeomorphic to some linearly ordered topological space (LOTS) (X, <, L(<)) where < is a linear ordering of X and L(<) is the usual open interval topology of <. As the subspace [0, 1] ∪ (2, 3) of the usual space R of real numbers shows, a subspace of a LOTS ...
... A topological space is orderable if it is homeomorphic to some linearly ordered topological space (LOTS) (X, <, L(<)) where < is a linear ordering of X and L(<) is the usual open interval topology of <. As the subspace [0, 1] ∪ (2, 3) of the usual space R of real numbers shows, a subspace of a LOTS ...
Math 304 Answers to Selected Problems 1 Section 5.5
... (a) Show that the vectors 1 and x are orthogonal. (b) Compute k1k and kxk. (c) Find the least squares approximation to x1/3 on [−1, 1] by a linear function `(x) = c1 1 + c2 x. (d) Sketch the graphs x1/3 and `(x) on [−1, 1]. Answer: (a) The vectors 1 and x are orthogonal if h1, xi = 0. ...
... (a) Show that the vectors 1 and x are orthogonal. (b) Compute k1k and kxk. (c) Find the least squares approximation to x1/3 on [−1, 1] by a linear function `(x) = c1 1 + c2 x. (d) Sketch the graphs x1/3 and `(x) on [−1, 1]. Answer: (a) The vectors 1 and x are orthogonal if h1, xi = 0. ...
Schrödinger operators and their spectra
... for standard quantum mechanics) and to be separable (i.e. it has a countable orthonormal basis). The inner product in H will be denoted by (·, ·). A state of the physical system is not described by a point in the phase space, but rather by a unit vector ψ, called wavefunction, in the Hilbert space H ...
... for standard quantum mechanics) and to be separable (i.e. it has a countable orthonormal basis). The inner product in H will be denoted by (·, ·). A state of the physical system is not described by a point in the phase space, but rather by a unit vector ψ, called wavefunction, in the Hilbert space H ...
LECTURES ON ERGODIC THEORY OF GROUP ACTIONS (A VON
... morphism σ : Γ → Aut(X, µ). We’ll often use the notation Γ y (X, µ) to emphasize an action σ, or simply Γ y X if no confusion is possible. We’ll sometimes consider topological groups G other than discrete (typically locally compact or Polish), in which case an action of G on (X, µ) will be a morphis ...
... morphism σ : Γ → Aut(X, µ). We’ll often use the notation Γ y (X, µ) to emphasize an action σ, or simply Γ y X if no confusion is possible. We’ll sometimes consider topological groups G other than discrete (typically locally compact or Polish), in which case an action of G on (X, µ) will be a morphis ...
Recent Developments in the Topology of Ordered Spaces
... As a final part of this chapter, we consider the embedding problem for perfect GOspaces. It is older than the questions of Maurice, Heath, and Nyikos and turns out to be related to them because of work by W-X Shi, as we will explain later. It has been known since the work of Čech that GO-spaces are ...
... As a final part of this chapter, we consider the embedding problem for perfect GOspaces. It is older than the questions of Maurice, Heath, and Nyikos and turns out to be related to them because of work by W-X Shi, as we will explain later. It has been known since the work of Čech that GO-spaces are ...
Notes on Weak Topologies
... (i) Let H = `2 and {en : n ∈ N} be the standard orthonormal basis for H. Recall that en (m) = δn (m), the Dirac√delta function. Then for m, n ∈ N, we have ken − em k2 = 2. Hence the sequence (en ) is not convergent. (1) The unit ball BH = {x ∈ H : kxk2 = 1} is not compact. In general, the unit ball ...
... (i) Let H = `2 and {en : n ∈ N} be the standard orthonormal basis for H. Recall that en (m) = δn (m), the Dirac√delta function. Then for m, n ∈ N, we have ken − em k2 = 2. Hence the sequence (en ) is not convergent. (1) The unit ball BH = {x ∈ H : kxk2 = 1} is not compact. In general, the unit ball ...
Interacting Fock spaces: central limit theorems and quantum
... ii) we are able to construct the approximating sequence of random variables. The main technical tool used to reach such a theorem is given by a special class of interacting Fock spaces (IFS), namely the 1-mode type Free interacting Fock spaces. More precisely, after introducing a new basic operator ...
... ii) we are able to construct the approximating sequence of random variables. The main technical tool used to reach such a theorem is given by a special class of interacting Fock spaces (IFS), namely the 1-mode type Free interacting Fock spaces. More precisely, after introducing a new basic operator ...
New Class of rg*b-Continuous Functions in Topological Spaces
... implies f (V) is a clopen neighbourhood of each of its points. Hence it is a clopen set in X. Therefore f is rg*b-totally continuous. Theorem 3.8. A function f : ( X,τ) → (Y,σ) is rg*b-totally continuous, if its graph function is rg*b-totally continuous. Proof: Let g: X → X×Y be a graph function of ...
... implies f (V) is a clopen neighbourhood of each of its points. Hence it is a clopen set in X. Therefore f is rg*b-totally continuous. Theorem 3.8. A function f : ( X,τ) → (Y,σ) is rg*b-totally continuous, if its graph function is rg*b-totally continuous. Proof: Let g: X → X×Y be a graph function of ...
ON THE PRIME SPECTRUM OF MODULES
... It is well known that if R is a PID and M ax.R/ is not finite, then the intersection every infinite number of maximal ideals of R is zero. Now it is natural to ask the following question: Is the same true when R is a one dimensional integral domain with infinite maximal ideals? In below, we show tha ...
... It is well known that if R is a PID and M ax.R/ is not finite, then the intersection every infinite number of maximal ideals of R is zero. Now it is natural to ask the following question: Is the same true when R is a one dimensional integral domain with infinite maximal ideals? In below, we show tha ...
Topological Pattern Recognition for Point Cloud Data
... Much of mathematics can be characterized as the construction of methods for organizing infinite sets into understandable representations. Euclidean spaces are organized using the notions of vector spaces and affine spaces, which allows one to organize the (infinite) underlying sets into understandable ...
... Much of mathematics can be characterized as the construction of methods for organizing infinite sets into understandable representations. Euclidean spaces are organized using the notions of vector spaces and affine spaces, which allows one to organize the (infinite) underlying sets into understandable ...
On the degree of ill-posedness for linear problems
... characteristic function We are going to use that fact for getting a deeper insight into the non-compact case. The paper is organized as follows: In Section 2 we discuss different concepts for measuring ill-posedness based, only, on smoothing properties of A in the compact as well as in the non-compa ...
... characteristic function We are going to use that fact for getting a deeper insight into the non-compact case. The paper is organized as follows: In Section 2 we discuss different concepts for measuring ill-posedness based, only, on smoothing properties of A in the compact as well as in the non-compa ...
Concepts and Methods of Mathematical Physics - math.uni
... Vector spaces and related structures play an important role in physics because they arise whenever physical systems are linearised, i.e. approximated by linear structures. Linearity is a very strong tool. Non-linear systems such as general relativity or the fluid dynamics governed by Navier Stokes eq ...
... Vector spaces and related structures play an important role in physics because they arise whenever physical systems are linearised, i.e. approximated by linear structures. Linearity is a very strong tool. Non-linear systems such as general relativity or the fluid dynamics governed by Navier Stokes eq ...
The Universal Covering Group of U (n) and Projective Representations
... the unitary or antiunitary operators representing the symmetry transformations are required to preserve transition probabilities, that is, the square of the modulus of the transition amplitudes, but not the transition amplitudes themselves; therefore the operators are determined only up to a phase. ...
... the unitary or antiunitary operators representing the symmetry transformations are required to preserve transition probabilities, that is, the square of the modulus of the transition amplitudes, but not the transition amplitudes themselves; therefore the operators are determined only up to a phase. ...
part I: algebra - Waterloo Computer Graphics Lab
... intersection algorithms are split out in treatment of the various cases: lines and planes, planes and planes, lines and lines, et cetera, need to be treated in separate pieces of code. After all, the outcomes themselves can be points, lines or planes, and those are essentially different in their fur ...
... intersection algorithms are split out in treatment of the various cases: lines and planes, planes and planes, lines and lines, et cetera, need to be treated in separate pieces of code. After all, the outcomes themselves can be points, lines or planes, and those are essentially different in their fur ...
Chapter 4 MANY PARTICLE SYSTEMS
... considered numerous examples drawn from the quantum mechanics of a single particle, the postulates themselves are intended to apply to all quantum systems, including those containing more than one and possibly very many particles. Thus, the only real obstacle to our immediate application of the post ...
... considered numerous examples drawn from the quantum mechanics of a single particle, the postulates themselves are intended to apply to all quantum systems, including those containing more than one and possibly very many particles. Thus, the only real obstacle to our immediate application of the post ...
Completely ultrametrizable spaces and continuous
... It turns out that many of the techniques useful for the countable trees of Polish spaces work just as well for uncountable trees (although not for arbitrarily large trees: as we will see in Section 5, the situation is more complicated for trees of size ℵω and larger). Certain aspects of these proofs ...
... It turns out that many of the techniques useful for the countable trees of Polish spaces work just as well for uncountable trees (although not for arbitrarily large trees: as we will see in Section 5, the situation is more complicated for trees of size ℵω and larger). Certain aspects of these proofs ...
Global exact controllability in infinite time of Schrödinger equation
... In this article, we study the properties of control system on the time half-line R+ instead of a finite interval [0, T ], as in all above cited papers. We study the mapping, which associates to initial condition z0 and control u the ω-limit set of the corresponding trajectory. We consider this mappi ...
... In this article, we study the properties of control system on the time half-line R+ instead of a finite interval [0, T ], as in all above cited papers. We study the mapping, which associates to initial condition z0 and control u the ω-limit set of the corresponding trajectory. We consider this mappi ...
Toward an Understanding of Parochial Observables
... in C. Recall that the GNS theorem allows us to take any C*-algebra of observables A and, having chosen some state ω, represent it via the representation πω as a subalgebra πω (A) ⊆ B(Hω ) for some Hilbert space Hω . Using the weak operator topology on B(Hω ) as a physically relevant notion of approx ...
... in C. Recall that the GNS theorem allows us to take any C*-algebra of observables A and, having chosen some state ω, represent it via the representation πω as a subalgebra πω (A) ⊆ B(Hω ) for some Hilbert space Hω . Using the weak operator topology on B(Hω ) as a physically relevant notion of approx ...
Hilbert space
![](https://commons.wikimedia.org/wiki/Special:FilePath/Standing_waves_on_a_string.gif?width=300)
The mathematical concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space. It extends the methods of vector algebra and calculus from the two-dimensional Euclidean plane and three-dimensional space to spaces with any finite or infinite number of dimensions. A Hilbert space is an abstract vector space possessing the structure of an inner product that allows length and angle to be measured. Furthermore, Hilbert spaces are complete: there are enough limits in the space to allow the techniques of calculus to be used.Hilbert spaces arise naturally and frequently in mathematics and physics, typically as infinite-dimensional function spaces. The earliest Hilbert spaces were studied from this point of view in the first decade of the 20th century by David Hilbert, Erhard Schmidt, and Frigyes Riesz. They are indispensable tools in the theories of partial differential equations, quantum mechanics, Fourier analysis (which includes applications to signal processing and heat transfer)—and ergodic theory, which forms the mathematical underpinning of thermodynamics. John von Neumann coined the term Hilbert space for the abstract concept that underlies many of these diverse applications. The success of Hilbert space methods ushered in a very fruitful era for functional analysis. Apart from the classical Euclidean spaces, examples of Hilbert spaces include spaces of square-integrable functions, spaces of sequences, Sobolev spaces consisting of generalized functions, and Hardy spaces of holomorphic functions.Geometric intuition plays an important role in many aspects of Hilbert space theory. Exact analogs of the Pythagorean theorem and parallelogram law hold in a Hilbert space. At a deeper level, perpendicular projection onto a subspace (the analog of ""dropping the altitude"" of a triangle) plays a significant role in optimization problems and other aspects of the theory. An element of a Hilbert space can be uniquely specified by its coordinates with respect to a set of coordinate axes (an orthonormal basis), in analogy with Cartesian coordinates in the plane. When that set of axes is countably infinite, this means that the Hilbert space can also usefully be thought of in terms of infinite sequences that are square-summable. Linear operators on a Hilbert space are likewise fairly concrete objects: in good cases, they are simply transformations that stretch the space by different factors in mutually perpendicular directions in a sense that is made precise by the study of their spectrum.