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Arkady Bolotin - One of the best experts on this subject based on the ideXlab platform.
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Quantum state change in light of changes in valuational entropies.
arXiv: Quantum Physics, 2019Co-Authors: Arkady BolotinAbstract:In the statement "The vector is an element of the Closed Linear Subspace of the Hilbert space H", the predicate "... is an element of ..." might be not only determined, that is, either true or false (depending on whether set membership is applicable or inapplicable to the specified vector and Subspace) but also undetermined, that is, neither true nor false. To evaluate the vagueness of set membership among arbitrary vectors and Closed Linear Subspaces of H, the notion of the entropy of the predicate "... is an element of ..." is introduced in the present paper. Since each Closed Linear Subspace in H uniquely represents the atomic proposition P about a quantum system, the entropy of this predicate can also be considered as the valuational entropy that measures the uncertainty about the assignment of truth values to the proposition P. As it is demonstrated in the paper, in the Hilbert space H of the dimension greater than or equal to 2, there always exists a nonempty set S of the Closed Linear Subspaces in H, such that the entropy of the predicate "... is an element of ..." on the given vector of H and all the Subspaces of S cannot be zero. This implies the existence of two different processes of the pure quantum state change: the process which yields no changes in the valuational entropies of the propositions (corresponding to the deterministic and reversible evolution) and the process which brings forth changes in the valuational entropies (corresponding to the gain or loss of information in a quantum measurement).
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Admissibility of truth assignments for quantum propositions in supervaluational logic
arXiv: Quantum Physics, 2018Co-Authors: Arkady BolotinAbstract:The structure of a complete lattice formed by Closed Linear Subspaces of a Hilbert space (i.e., a Hilbert lattice) entails some unreasonable consequences from the physical point of view. Specifically, this structure seems to contradict to the localized variant of the Kochen-Specker theorem according to which the bivaluation of a proposition represented by a Closed Linear Subspace that does not belong to a Boolean algebra shared by the state, in which a quantum-mechanical system is prepared, must be value indefinite. For this reason, the Hilbert lattice structure seems to be too strong and needs to be weakened. The question is, how should it be weakened so that to support the quantum uncertainty principle and the Kochen-Specker theorem? Which logic will a weakened structure identify? The present paper tries to answer these questions.
Bolotin Arkady - One of the best experts on this subject based on the ideXlab platform.
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Quantum state change in light of changes in valuational entropies
2019Co-Authors: Bolotin ArkadyAbstract:In the statement "The vector is an element of the Closed Linear Subspace of the Hilbert space H", the predicate "... is an element of ..." might be not only determined, that is, either true or false (depending on whether set membership is applicable or inapplicable to the specified vector and Subspace) but also undetermined, that is, neither true nor false. To evaluate the vagueness of set membership among arbitrary vectors and Closed Linear Subspaces of H, the notion of the entropy of the predicate "... is an element of ..." is introduced in the present paper. Since each Closed Linear Subspace in H uniquely represents the atomic proposition P about a quantum system, the entropy of this predicate can also be considered as the valuational entropy that measures the uncertainty about the assignment of truth values to the proposition P. As it is demonstrated in the paper, in the Hilbert space H of the dimension greater than or equal to 2, there always exists a nonempty set S of the Closed Linear Subspaces in H, such that the entropy of the predicate "... is an element of ..." on the given vector of H and all the Subspaces of S cannot be zero. This implies the existence of two different processes of the pure quantum state change: the process which yields no changes in the valuational entropies of the propositions (corresponding to the deterministic and reversible evolution) and the process which brings forth changes in the valuational entropies (corresponding to the gain or loss of information in a quantum measurement).Comment: 12 page
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Admissibility of truth assignments for quantum propositions in supervaluational logic
2018Co-Authors: Bolotin ArkadyAbstract:The structure of a complete lattice formed by Closed Linear Subspaces of a Hilbert space (i.e., a Hilbert lattice) entails some unreasonable consequences from the physical point of view. Specifically, this structure seems to contradict to the localized variant of the Kochen-Specker theorem according to which the bivaluation of a proposition represented by a Closed Linear Subspace that does not belong to a Boolean algebra shared by the state, in which a quantum-mechanical system is prepared, must be value indefinite. For this reason, the Hilbert lattice structure seems to be too strong and needs to be weakened. The question is, how should it be weakened so that to support the quantum uncertainty principle and the Kochen-Specker theorem? Which logic will a weakened structure identify? The present paper tries to answer these questions.Comment: 13 pages, 5 figures. Typos correcte
Martin Rmoutil - One of the best experts on this subject based on the ideXlab platform.
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Restricting uniformly open surjections
arXiv: General Topology, 2017Co-Authors: Tomasz Kania, Martin RmoutilAbstract:We employ the theory of elementary submodels to improve a recent result by Aron, Jaramillo and Le Donne (Ann. Acad. Sci. Fenn. Math., to appear) concerning restricting uniformly open, continuous surjections to smaller Subspaces where they remain surjective. To wit, suppose that $X$ and $Y$ are metric spaces and let $f\colon X\to Y$ be a continuous surjection. If $X$ is complete and $f$ is uniformly open, then $X$ contains a~Closed Subspace $Z$ with the same density as $Y$ such that $f$ restricted to $Z$ is still uniformly open and surjective. Moreover, if $X$ is a Banach space, then $Z$ may be taken to be a Closed Linear Subspace. A counterpart of this theorem for uniform spaces is also established.
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Restricting uniformly open surjections
Comptes Rendus Mathematique, 2017Co-Authors: Tomasz Kania, Martin RmoutilAbstract:Abstract We employ the theory of elementary submodels to improve a recent result by Aron, Jaramillo and Le Donne (2017) [1] concerning restricting uniformly open, continuous surjections to smaller Subspaces where they remain surjective. To wit, suppose that X and Y are metric spaces and let f : X → Y be a continuous surjection. If X is complete and f is uniformly open, then X contains a Closed Subspace Z with the same density as Y such that f restricted to Z is still uniformly open and surjective. Moreover, if X is a Banach space, then Z may be taken to be a Closed Linear Subspace. A counterpart of this theorem for uniform spaces is also established.
Sergey S. Platonov - One of the best experts on this subject based on the ideXlab platform.
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Spectral synthesis on zero-dimensional locally compact Abelian groups
Russian Universities Reports. Mathematics, 2019Co-Authors: Sergey S. PlatonovAbstract:Let G be a zero-dimensional locally compact Abelian group whose elements are compact, C(G) the space of continuous complex-valued functions on the group G. A Closed Linear Subspace H⊆ C(G) is called invariant Subspace, if it is invariant with respect to translations τ_y ∶ f(x) ↦ f(x + y), y ∈ G. We prove that any invariant Subspace H admits spectral synthesis, which means that H coincides with the closure of the Linear span of all characters of the group G contained in H.
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On spectral synthesis on element-wise compact Abelian groups
Sbornik: Mathematics, 2015Co-Authors: Sergey S. PlatonovAbstract:Let be an arbitrary locally compact Abelian group and let be the space of all continuous complex-valued functions on . A Closed Linear Subspace is referred to as an invariant Subspace if it is invariant with respect to the shifts , . By definition, an invariant Subspace admits strict spectral synthesis if coincides with the closure in of the Linear span of all characters of belonging to . We say that strict spectral synthesis holds in the space on if every invariant Subspace admits strict spectral synthesis. An element of a topological group is said to be compact if is contained in some compact subgroup of . A group is said to be element-wise compact if all elements of are compact. The main result of the paper is the proof of the fact that strict spectral synthesis holds in for a locally compact Abelian group if and only if is element-wise compact. Bibliography: 14 titles.
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On spectral synthesis on zero-dimensional Abelian groups
Sbornik: Mathematics, 2013Co-Authors: Sergey S. PlatonovAbstract:Let G be a zero-dimensional locally compact Abelian group all of whose elements are compact, and let C(G) be the space of all complex-valued continuous functions on G. A Closed Linear Subspace H⊆C(G) is said to be an invariant Subspace if it is invariant with respect to the translations τ{sub y}:f(x)↦f(x+y), y∈G. In the paper, it is proved that any invariant Subspace H admits spectral synthesis, that is, H coincides with the Closed Linear span of the characters of G belonging to H. Bibliography: 25 titles.
Stanislav Shkarin - One of the best experts on this subject based on the ideXlab platform.
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On the set of hypercyclic vectors for the differentiation operator
arXiv: Functional Analysis, 2012Co-Authors: Stanislav ShkarinAbstract:Let $D$ be the differentiation operator $Df=f'$ acting on the Frechet space $\H$ of all entire functions in one variable with the standard (compact-open) topology. It is known since 1950's that the set $H(D)$ of hypercyclic vectors for the operator $D$ is non-empty. We treat two questions raised by Aron, Conejero, Peris and Seoane-Sepulveda whether the set $H(D)$ contains (up to the zero function) a non-trivial subalgebra of $\H$ or an infinite dimensional Closed Linear Subspace of $\H$. In the present article both questions are answered affirmatively.
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On the set of hypercyclic vectors for the differentiation operator
Israel Journal of Mathematics, 2010Co-Authors: Stanislav ShkarinAbstract:Let D be the differentiation operator Df = f′ acting on the Frechet space H of all entire functions in one variable with the standard (compact-open) topology. It is known since the 1950’s that the set H(D) of hypercyclic vectors for the operator D is non-empty. We treat two questions raised by Aron, Conejero, Peris and Seoane-Sepulveda whether the set H(D) contains (up to the zero function) a non-trivial subalgebra of H or an infinite-dimensional Closed Linear Subspace of H. In the present article both questions are answered affirmatively.
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DECOMPOSITIONS OF SPACES OF MEASURES
Infinite Dimensional Analysis Quantum Probability and Related Topics, 2008Co-Authors: Stanislav ShkarinAbstract:Let 𝔐 be the Banach space of σ-additive complex-valued measures on an abstract measurable space. We prove that any Closed, with respect to absolute continuity norm-Closed, Linear Subspace L of 𝔐 is complemented and describe the unique complement, projection onto L along which has norm 1. Using this fact we prove a decomposition theorem, which includes the Jordan decomposition theorem, the generalized Radon–Nikodým theorem and the decomposition of measures into decaying and non-decaying components as particular cases. We also prove an analog of the Jessen–Wintner purity theorem for our decompositions.
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WHITNEY'S TYPE THEOREMS FOR INFINITE DIMENSIONAL SPACES
Infinite Dimensional Analysis Quantum Probability and Related Topics, 2000Co-Authors: Stanislav ShkarinAbstract:It is proved that for any f ∈ Ck(L,ℝ), where k ∈ ℕ and L is a Closed Linear Subspace of a nuclear Frechet space X, the function f can be extended to a function of class Ck-1 defined on the entire space X. It is also proved that for any f ∈ Ck (L, ℝ), where k ∈ℕ∪{∞} and L is a Closed Linear Subspace of a conjugate X of a nuclear Frechet space, the function f can be extended to a function of class Ck defined on the entire space X. In addition, it is proved that under these conditions, the existence of a Linear extension operator is equivalent to the complementability of the Subspace.
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WHITNEY'S TYPE THEOREMS FOR INFINITE DIMENSIONAL SPACES
Infinite Dimensional Analysis Quantum Probability and Related Topics, 2000Co-Authors: Stanislav ShkarinAbstract:It is proved that for any f ∈ Ck(L,ℝ), where k ∈ ℕ and L is a Closed Linear Subspace of a nuclear Frechét space X, the function f can be extended to a function of class Ck-1 defined on the entire space X. It is also proved that for any f ∈ Ck (L, ℝ), where k ∈ℕ∪{∞} and L is a Closed Linear Subspace of a conjugate X of a nuclear Frechét space, the function f can be extended to a function of class Ck defined on the entire space X. In addition, it is proved that under these conditions, the existence of a Linear extension operator is equivalent to the complementability of the Subspace.