STOCHASTIC PROPERTIES OF DEGENERATED QUANTUM SYSTEMS

2010 ◽  
Vol 13 (01) ◽  
pp. 65-85 ◽  
Author(s):  
VSEVOLOD Zh. SAKBAEV
Keyword(s):  
Stochastic Process ◽  
Sufficient Conditions ◽  
Quantum Systems ◽  
Additive Measure ◽  
Divergent Sequence ◽  
Quantum Dynamical Semigroup ◽  
Dynamical Semigroup ◽  
The Family ◽  
Necessary And Sufficient ◽  
Stochastic Properties

We study Schrödinger equation with degenerated symmetric but not self-adjoint Hamiltonian. The above properties of the quantum Hamiltonian arise in the description of the asymptotic behavior of the regularizing self-adjoint Hamiltonians sequence. A quantum dynamical semigroup corresponding to degenerated Hamiltonian is defined by means of the passage to the limit for the sequence of the regularizing dynamical semigroups. These semigroups are generated by the regularizing self-adjoint Hamiltonians. The necessary and sufficient conditions are obtained for the convergence of the regularizing semigroups sequence. The description of the divergent sequence of semigroups requires the extension of the stochastic process concept. We extend the stochastic process concept onto the family of measurable functions defined on the space endowed with finite additive measure. The above extension makes it possible to describe the structure of the partial limits set of the regularizing semigroups sequence.

MINIMAL QUANTUM DYNAMICAL SEMIGROUP ON A VON NEUMANN ALGEBRA

1999 ◽  
Vol 02 (02) ◽  
pp. 221-239 ◽  
Author(s):  
DEBASHISH GOSWAMI ◽  
KALYAN B. SINHA
Keyword(s):  
State Space ◽  
Von Neumann Algebra ◽  
Sufficient Conditions ◽  
Markov Semigroup ◽  
Quantum Dynamical Semigroup ◽  
Von Neumann ◽  
Arbitrary State ◽  
Quantum Dynamical ◽  
Dynamical Semigroup ◽  
Necessary And Sufficient

Given a formal unbounded generator, the minimal quantum dynamical semigroup on a von Neumann algebra is constructed. A set of equivalent necessary and sufficient conditions for the conservativity of the minimal semigroup is given and in the case when it is not conservative, a distinguished family of conservative perturbations of the semigroup is studied. Finally, some of these results are applied to the classical Markov semigroup with arbitrary state space.

The Approach to Equilibrium of a Class of Quantum Dynamical Semigroups

1998 ◽  
Vol 01 (04) ◽  
pp. 561-572 ◽  
Author(s):  
Franco Fagnola ◽  
Rolando Rebolledo
Keyword(s):  
Hilbert Space ◽  
Asymptotic Behavior ◽  
Conditional Expectation ◽  
Sufficient Conditions ◽  
Approach To Equilibrium ◽  
Bounded Operators ◽  
Quantum Dynamical Semigroup ◽  
Quantum Dynamical ◽  
Dynamical Semigroup ◽  
Necessary And Sufficient

This paper deals with the asymptotic behavior of a quantum dynamical semigroup [Formula: see text] acting on the algebra of all linear bounded operators on a given Hilbert space. In practice, all these semigroups have a generator which can be written in a well-known form named after Lindblad and Davies. If the semigroup has a faithful normal stationary state ρ, necessary and sufficient conditions are derived for the w*-convergence of [Formula: see text] to [Formula: see text], where [Formula: see text] is the conditional expectation of an element X onto the subalgebra of fixed points. Our main results are expressed in terms of the Lindblad–Davies generator .

Using binary operations to construct a transitive set of block transformations

2020 ◽  
Vol 30 (6) ◽  
pp. 375-389
Author(s):  
Igor V. Cherednik
Keyword(s):  
Binary Operation ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Binary Operations ◽  
The Family ◽  
Transitive Set ◽  
Necessary And Sufficient

AbstractWe study the set of transformations {ΣF : F∈ 𝓑∗(Ω)} implemented by a network Σ with a single binary operation F, where 𝓑∗(Ω) is the set of all binary operations on Ω that are invertible as function of the second variable. We state a criterion of bijectivity of all transformations from the family {ΣF : F∈ 𝓑∗(Ω)} in terms of the structure of the network Σ, identify necessary and sufficient conditions of transitivity of the set of transformations {ΣF : F∈ 𝓑∗(Ω)}, and propose an efficient way of verifying these conditions. We also describe an algorithm for construction of networks Σ with transitive sets of transformations {ΣF : F∈ 𝓑∗(Ω)}.

Separability criteria based on the Bloch representation of density matrices

2007 ◽  
Vol 7 (7) ◽  
pp. 624-638
Author(s):  
J. de Vicente
Keyword(s):  
Sufficient Conditions ◽  
Quantum Systems ◽  
Necessary Condition ◽  
Sufficient Condition ◽  
Pure Product ◽  
Separability Problem ◽  
Arbitrary Dimensions ◽  
Necessary And Sufficient ◽  
Bloch Representation

We study the separability of bipartite quantum systems in arbitrary dimensions using the Bloch representation of their density matrix. This approach enables us to find an alternative characterization of the separability problem, from which we derive a necessary condition and sufficient conditions for separability. For a certain class of states the necessary condition and a sufficient condition turn out to be equivalent, therefore yielding a necessary and sufficient condition. The proofs of the sufficient conditions are constructive, thus providing decompositions in pure product states for the states that satisfy them. We provide examples that show the ability of these conditions to detect entanglement. In particular, the necessary condition is proved to be strong enough to detect bound entangled states.

scholarly journals Gaps in Discrete Random Samples

2009 ◽  
Vol 46 (04) ◽  
pp. 1038-1051 ◽  
Author(s):  
Rudolf Grübel ◽  
Paweł Hitczenko
Keyword(s):  
Analysis Of Algorithms ◽  
Sufficient Conditions ◽  
Borderline Case ◽  
Convergence In Probability ◽  
Enumerative Combinatorics ◽  
Random Samples ◽  
The Family ◽  
Heavy Tailed ◽  
Convergence Almost Surely ◽  
Necessary And Sufficient

Let (X i ) i∈ℕ be a sequence of independent and identically distributed random variables with values in the set ℕ0 of nonnegative integers. Motivated by applications in enumerative combinatorics and analysis of algorithms we investigate the number of gaps and the length of the longest gap in the set {X 1,…,X n } of the first n values. We obtain necessary and sufficient conditions in terms of the tail sequence (q k ) k∈ℕ0 , q k =P(X 1≥ k), for the gaps to vanish asymptotically as n→∞: these are ∑ k=0 ∞ q k+1/q k <∞ and limk→∞ q k+1/q k =0 for convergence almost surely and convergence in probability, respectively. We further show that the length of the longest gap tends to ∞ in probability if q k+1/q k → 1. For the family of geometric distributions, which can be regarded as the borderline case between the light-tailed and the heavy-tailed situations and which is also of particular interest in applications, we study the distribution of the length of the longest gap, using a construction based on the Sukhatme–Rényi representation of exponential order statistics to resolve the asymptotic distributional periodicities.

Finite regular invariant measures for Feller processes

1968 ◽  
Vol 5 (1) ◽  
pp. 203-209 ◽  
Author(s):  
V. E. Beneš
Keyword(s):  
Stochastic Process ◽  
Lyapunov Function ◽  
Invariant Measures ◽  
Sufficient Conditions ◽  
Ergodic Averages ◽  
Feller Process ◽  
The Third ◽  
Sequential Compactness ◽  
Finite Invariant Measure ◽  
Necessary And Sufficient

In the study of dynamical systems perturbed by noise, it is important to know whether the stochastic process of interest has a stationary distribution. Four necessary and sufficient conditions are formulated for the existence of a finite invariant measure for a Feller process on a σ-compact metric (state) space. These conditions link together stability notions from several fields. The first uses a Lyapunov function reminiscent of Lagrange stability in differential equations; the second depends on Prokhorov's condition for sequential compactness of measures; the third is a recurrence condition on the ergodic averages of the transition operator; and the fourth is analogous to a condition of Ulam and Oxtoby for the nonstochastic case.

scholarly journals AUTOMORPHISMS OF THE UHF ALGEBRA THAT DO NOT EXTEND TO THE CUNTZ ALGEBRA

2010 ◽  
Vol 89 (3) ◽  
pp. 309-315 ◽  
Author(s):  
ROBERTO CONTI
Keyword(s):  
Sufficient Conditions ◽  
Extension Property ◽  
Cuntz Algebra ◽  
Maximal Abelian Subalgebra ◽  
Abelian Subalgebra ◽  
Matrix Algebras ◽  
The Family ◽  
The Matrix ◽  
Inner Automorphisms ◽  
Necessary And Sufficient

AbstractThe automorphisms of the canonical core UHF subalgebra ℱn of the Cuntz algebra 𝒪n do not necessarily extend to automorphisms of 𝒪n. Simple examples are discussed within the family of infinite tensor products of (inner) automorphisms of the matrix algebras Mn. In that case, necessary and sufficient conditions for the extension property are presented. Also addressed is the problem of extending to 𝒪n the automorphisms of the diagonal 𝒟n, which is a regular maximal abelian subalgebra with Cantor spectrum. In particular, it is shown that there exist product-type automorphisms of 𝒟n that do not extend to (possibly proper) endomorphisms of 𝒪n.

Planar Linkage Synthesis for Mixed Motion, Path and Function Generation Using Poles and Rotation Angles

2017 ◽  
Author(s):  
Ronald A. Zimmerman
Keyword(s):  
Sufficient Conditions ◽  
Motion Path ◽  
Engineering Practice ◽  
Function Generation ◽  
Problem Types ◽  
The Family ◽  
And Function ◽  
Necessary And Sufficient ◽  
Four Bar Linkage ◽  
Selection Of

The kinematic synthesis of planar linkage mechanisms has traditionally been broken into the categories of motion, path and function generation. Each of these categories of problems has been solved separately. Many problems in engineering practice require some combination of these problem types. For example, a problem requiring coupler points and/or poses in addition to specific input and/or output link angles that correspond to those positions. A limited amount of published work has addressed some specific underconstrained combinations of these problems. This paper presents a general graphical method for the synthesis of a four bar linkage to satisfy any combination of these exact synthesis problems that is not over constrained. The approach is to consider the constraints imposed by the target positions on the linkage through the poles and rotation angles. These pole and rotation angle constraints are necessary and sufficient conditions to meet the target positions. After the constraints are made, free choices which may remain can be explored by simply dragging a fixed pivot, a moving pivot or a pole in the plane. The designer can thus investigate the family of available solutions before making the selection of free choices to satisfy other criteria. The fully constrained combinations for a four bar linkage are given and sample problems are solved for several of them.

scholarly journals The distribution of prime ideals of a Dedekind domain

1974 ◽  
Vol 11 (3) ◽  
pp. 429-441 ◽  
Author(s):  
Anne P. Grams
Keyword(s):  
Class Group ◽  
Sufficient Conditions ◽  
Torsion Group ◽  
Dedekind Domain ◽  
Necessary And Sufficient Conditions ◽  
Prime Ideals ◽  
Class Groups ◽  
Maximal Ideals ◽  
The Family ◽  
Necessary And Sufficient

Let G be an abelian group, and let S be a subset of G. Necessary and sufficient conditions on G and S are given in order that there should exist a Dedekind domain D with class group G with the property that S is the set of classes that contain maximal ideals of D. If G is a torsion group, then S is the set of classes containing the maximal ideals of D if and only if S generates G. These results are used to determine necessary and sufficient conditions on a family {Hλ} of subgroups of G in order that there should exist a Dedekind domain D with class group G such that {G/Hλ} is the family of class groups of the set of overrings of D. Several applications are given.

scholarly journals Defect Effect of Bi-infinite Words in the Two-element Case

2001 ◽  
Vol Vol. 4 no. 2 ◽  
Author(s):  
Ján Maňuch
Keyword(s):  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Finite Alphabet ◽  
Infinite Words ◽  
Infinite Word ◽  
The Family ◽  
International Audience ◽  
Necessary And Sufficient ◽  
Primitive Roots ◽  
Element Set

International audience Let X be a two-element set of words over a finite alphabet. If a bi-infinite word possesses two X-factorizations which are not shiftequivalent, then the primitive roots of the words in X are conjugates. Note, that this is a strict sharpening of a defect theorem for bi-infinite words stated in \emphKMP. Moreover, we prove that there is at most one bi-infinite word possessing two different X-factorizations and give a necessary and sufficient conditions on X for the existence of such a word. Finally, we prove that the family of sets X for which such a word exists is parameterizable.

Sign in / Sign up

Export Citation Format

Share Document