Full Paper - World Academic Publishing
... then is a quasi-metric. A metric space is a quasi-metric space (X, d) in such a way that d satisfies the next additional condition (iv) for all x, y X, d(x, y) = d(y, x). In that case, the function d is called a metric. The conjugate of a quasi-metric d on X is the quasi-metric given by (x, y) = d(y ...
... then is a quasi-metric. A metric space is a quasi-metric space (X, d) in such a way that d satisfies the next additional condition (iv) for all x, y X, d(x, y) = d(y, x). In that case, the function d is called a metric. The conjugate of a quasi-metric d on X is the quasi-metric given by (x, y) = d(y ...
products of countably compact spaces
... = card {xntk\} = N0. Now cl {xnrk\} Q C, compact. Thus cl {xntk\} is itself compact. On the other hand, (yn) is a sequence in Y which clusters, since Y is countably compact. So cl (xniky¡ x cl {y„ikyi is countably compact, hence any sequence in it clusters. Thus {(xn,yn)} clusters, and X X Y is coun ...
... = card {xntk\} = N0. Now cl {xnrk\} Q C, compact. Thus cl {xntk\} is itself compact. On the other hand, (yn) is a sequence in Y which clusters, since Y is countably compact. So cl (xniky¡ x cl {y„ikyi is countably compact, hence any sequence in it clusters. Thus {(xn,yn)} clusters, and X X Y is coun ...
Monomial regular sequences - Oklahoma State University
... sequence of monomials. Macaulay [Ma] showed in 1927 that all Hilbert functions over B = k[x1 , · · · , xn ] are attained by lex ideals. Over what quotients of B is this true? Let M be a monomial ideal of B. We say that M is Lex-Macaulay if every Hilbert function over B/M is attained by a lex ideal o ...
... sequence of monomials. Macaulay [Ma] showed in 1927 that all Hilbert functions over B = k[x1 , · · · , xn ] are attained by lex ideals. Over what quotients of B is this true? Let M be a monomial ideal of B. We say that M is Lex-Macaulay if every Hilbert function over B/M is attained by a lex ideal o ...
Local isometries on spaces of continuous functions
... know if local automorphisms of C R (K) are in fact automorphisms when K is compact metric (as it is the case for complex functions). It is worth noting that the proofs of (\) and (]) strongly depend on the Gleason-Kahane-Żelazko theorem (or on some of its generalizations), a result which applies on ...
... know if local automorphisms of C R (K) are in fact automorphisms when K is compact metric (as it is the case for complex functions). It is worth noting that the proofs of (\) and (]) strongly depend on the Gleason-Kahane-Żelazko theorem (or on some of its generalizations), a result which applies on ...
Banach Spaces
... kyM − yN k = k n=N +1 xn k 6 n=N +1 kxn k → 0. Hence yP N is a Cauchy sequence in X, and so converges to y, say. This means that n xn = y. It is also true (see the exercises) that if a normed vector space is such that all its absolutely convergent series converge, then the space is also complete, i. ...
... kyM − yN k = k n=N +1 xn k 6 n=N +1 kxn k → 0. Hence yP N is a Cauchy sequence in X, and so converges to y, say. This means that n xn = y. It is also true (see the exercises) that if a normed vector space is such that all its absolutely convergent series converge, then the space is also complete, i. ...
A Metrics, Norms, Inner Products, and Topology
... typically part of standard beginning mathematics graduate courses are given more detailed attention, while other results are either formulated as exercises (often with hints) or stated without proof. Sources for additional information on the material of this and most of the other appendices include ...
... typically part of standard beginning mathematics graduate courses are given more detailed attention, while other results are either formulated as exercises (often with hints) or stated without proof. Sources for additional information on the material of this and most of the other appendices include ...
QFT on curved spacetimes: axiomatic framework and applications
... A notion of a state can be also defined abstractly, in the following way: Definition 2.4. A state on an involutive algebra A is a linear functional ω : A → C, such that: ω(A∗ A) ≥ 0, and ω(1) = 1 The first condition can be understood as a positivity condition and the second one is the normalization. ...
... A notion of a state can be also defined abstractly, in the following way: Definition 2.4. A state on an involutive algebra A is a linear functional ω : A → C, such that: ω(A∗ A) ≥ 0, and ω(1) = 1 The first condition can be understood as a positivity condition and the second one is the normalization. ...
Abstract Vector Spaces and Subspaces
... To prove (1), suppose that v0 and w0 are both additive identities. Then v0 + w0 = v0 , since w0 is an additive identity. On the other hand, v0 + w0 = w0 , since v0 is an additive identity. We have shown ...
... To prove (1), suppose that v0 and w0 are both additive identities. Then v0 + w0 = v0 , since w0 is an additive identity. On the other hand, v0 + w0 = w0 , since v0 is an additive identity. We have shown ...
2.3 Vector Spaces
... x ≤ π. Which of the following subsets S of C(−π, π) are subspaces? If it is not a subspace say why. If it is, then say why and find a basis. Note: You must show that the basis you choose consists of linearly independent vectors. In what follows a0 , a1 and a2 are arbitrary scalars unless otherwise s ...
... x ≤ π. Which of the following subsets S of C(−π, π) are subspaces? If it is not a subspace say why. If it is, then say why and find a basis. Note: You must show that the basis you choose consists of linearly independent vectors. In what follows a0 , a1 and a2 are arbitrary scalars unless otherwise s ...
Prtop
... exponential in Prtop. Finally, in case the full subconstruct of Prtop is only required to be topological, we present a positive result by constructing a bireflective subconstruct of FrPrtop containing all T1 Fr6chet pretopological spaces and all topological Fr6chet spaces that is Cartesian closed. T ...
... exponential in Prtop. Finally, in case the full subconstruct of Prtop is only required to be topological, we present a positive result by constructing a bireflective subconstruct of FrPrtop containing all T1 Fr6chet pretopological spaces and all topological Fr6chet spaces that is Cartesian closed. T ...
Mixed states and pure states
... i.e. hψ|ψi = 1, in a complex Hilbert space H. Previously, we (and the textbook) just called this a ‘state’, but now we call it a ‘pure’ state to distinguish it from a more general type of quantum states (‘mixed’ states, see step 21). 2. We already know from the textbook that we can define dual vecto ...
... i.e. hψ|ψi = 1, in a complex Hilbert space H. Previously, we (and the textbook) just called this a ‘state’, but now we call it a ‘pure’ state to distinguish it from a more general type of quantum states (‘mixed’ states, see step 21). 2. We already know from the textbook that we can define dual vecto ...
Quantum Field Theory on Curved Backgrounds. I
... Reflection positivity for the measure µ is equivalent to the following inequality for operators on E = L2 (dµ): 0 ≤ Π+ ΘΠ+ where Π+ : E → E+ is the canonical projection. A Gaussian measure with mean zero and covariance C is reflection positive iff C is reflection positive in the operator sense, eqn. ...
... Reflection positivity for the measure µ is equivalent to the following inequality for operators on E = L2 (dµ): 0 ≤ Π+ ΘΠ+ where Π+ : E → E+ is the canonical projection. A Gaussian measure with mean zero and covariance C is reflection positive iff C is reflection positive in the operator sense, eqn. ...
hosted here - Jeffrey C. Morton
... objects and arrows between objects. We emphasize that these are primitive notions in category theory, along with the notion of the source and target object of an arrow, and the composite of arrows. This may not always be possible: arrows f and g can be composed to get a arrow f ◦ g as long as the so ...
... objects and arrows between objects. We emphasize that these are primitive notions in category theory, along with the notion of the source and target object of an arrow, and the composite of arrows. This may not always be possible: arrows f and g can be composed to get a arrow f ◦ g as long as the so ...
1 Summary of PhD Thesis It is well known that the language of the
... most important resource in quantum information, we can mention here a few classical results contained in [11–13]. However in the most of the practical cases we have access to mixed entanglement, which is no longer such useful and universal as pure one. To obtain pure entanglement, usually in the for ...
... most important resource in quantum information, we can mention here a few classical results contained in [11–13]. However in the most of the practical cases we have access to mixed entanglement, which is no longer such useful and universal as pure one. To obtain pure entanglement, usually in the for ...
Linear Algebra Basics A vector space (or, linear space) is an
... (a) If S = {(1, 1)} then argue that span(S) 6= R2 , i.e., S is not a basis for R2 . (Find a vector not in span(S)). (b) Describe all vectors in the span of {(1, 0, 1), (0, 0, 1)}. (Draw xyz-axes and try to visualize span{(1, 0, 1), (0, 0, 1)}.) (c) If {v1 , . . . , vn } is a linearly dependent set o ...
... (a) If S = {(1, 1)} then argue that span(S) 6= R2 , i.e., S is not a basis for R2 . (Find a vector not in span(S)). (b) Describe all vectors in the span of {(1, 0, 1), (0, 0, 1)}. (Draw xyz-axes and try to visualize span{(1, 0, 1), (0, 0, 1)}.) (c) If {v1 , . . . , vn } is a linearly dependent set o ...
we defined the Poisson boundaries for semisimple Lie groups
... a discrete subgroup for which G/Y has finite (left-) invariant measure. To what extent is G determined by a knowledge of Y as an abstract group, and conversely, what is the influence of G on the structure of T? To make this question precise, let us say that G is an envelope of r if an isomorphic cop ...
... a discrete subgroup for which G/Y has finite (left-) invariant measure. To what extent is G determined by a knowledge of Y as an abstract group, and conversely, what is the influence of G on the structure of T? To make this question precise, let us say that G is an envelope of r if an isomorphic cop ...
Postulates of Quantum Mechanics
... systems, such as atoms and photons, whose states admit superpositions. It is a framework onto which other physical theories are built upon. For example, quantum field theories such as quantum electrodynamics and quantum chromodynamics. The central topic of this lecture is a mathematical formulation ...
... systems, such as atoms and photons, whose states admit superpositions. It is a framework onto which other physical theories are built upon. For example, quantum field theories such as quantum electrodynamics and quantum chromodynamics. The central topic of this lecture is a mathematical formulation ...
M05/11
... F (x) = 12 (1 + 3b · x) is not a fuzzy set function. In fact −1 ≤ F (x) ≤ 2. There are other forms of F that will also give T (b), but they are worse in that the corresponding F has a larger range. Consequently, I and 0 are the only projections in {Aη (F ) | 0 ≤ F (x) ≤ 1}. None-the-less, we have an ...
... F (x) = 12 (1 + 3b · x) is not a fuzzy set function. In fact −1 ≤ F (x) ≤ 2. There are other forms of F that will also give T (b), but they are worse in that the corresponding F has a larger range. Consequently, I and 0 are the only projections in {Aη (F ) | 0 ≤ F (x) ≤ 1}. None-the-less, we have an ...
Hilbert space
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.