Uniformly Continuous Research Articles

On uniformly continuous functions between pseudometric spaces and the Axiom of Countable Choice

In this note we show that the Axiom of Countable Choice is equivalent to two statements from the theory of pseudometric spaces: the first of them is a well-known characterization of uniform continuity for functions between (pseudo)metric spaces, and the second declares that sequentially compact pseudometric spaces are $$\mathbf {UC}$$ —meaning that all real valued, continuous functions defined on these spaces are necessarily uniformly continuous.

АППРОКСИМАЦИОННЫЕ ПОЛИНОМЫ ДИРИХЛЕ И НЕКОТОРЫЕ СВОЙСТВА L-ФУНКЦИЙ ДИРИХЛЕ

In this paper we study the analytic properties of Dirichlet L -functions in the critical strip, characteristic for almost periodic functions. The research is based on Approximation approach, consisting in the construction of Dirichlet polynomials, which are almost periodic functions, "rapidly convergent"in the critical strip to Dirichlet L -functions. On this path, for any rectangle lying in the critical strip, the existence of ε -almost period for the Dirichlet L-function, we obtain the estimate constants of uniform continuity. Issues related to studying other properties of Dirichlet L -functions are discussed.

\u03bc-pseudo almost automorph mild solutions to the fractional integro-differential equation with uniform continuity

Our aim in the article is to study the existence of μ-pseudo almost automorph mild solutions to the following fractional integro-differential equation: \t\t\tDαu(t)=Au(t)+∫−∞ta(t−s)Au(s)ds+f(t,u(t)),t∈R,\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$ D^{\\alpha}u(t)=Au(t)+ \\int_{-\\infty}^{t}a(t-s)Au(s)\\,ds+f\\bigl(t,u(t)\\bigr), \\quad t\\in\\mathbb{R}, $$\\end{document} where for alpha>0, the fractional derivative D^{alpha} is understood in the sense of Weyl, and A is a closed linear operator defined on Banach space mathbb{X}, ain L_{mathrm{loc}}^{1}(mathbb {R_{+}}) is a scalar-valued kernel. The novelty of this work is a study of this equation with a μ-S^{p}-pseudo almost automorph nonlinear term satisfying the condition of “uniform continuity” instead of some “Lipschitz” type conditions supposed in the literature. We utilize Schauder’s fixed point theorem. An example is provided to explain our abstract results.

Lp (1 < p ⩽ 2) solutions of one-dimensional BSDEs whose generator is weakly monotonic in y and non-Lipschitz in z

ABSTRACTIn this paper, we start with establishing the existence of a minimal (maximal) Lp (1 < p ⩽ 2) solution to a one-dimensional backward stochastic differential equation (BSDE), where the generator g satisfies a p-order weak monotonicity condition together with a general growth condition in y and a linear growth condition in z. Then, we propose and prove a comparison theorem of Lp (1 < p ⩽ 2) solutions to one-dimensional BSDEs with q-order (1 ⩽ q < p) weak monotonicity and uniform continuity generators. As a consequence, an existence and uniqueness result of Lp (1 < p ⩽ 2) solutions is also given for BSDEs whose generator g is q-order (1 ⩽ q < p) weakly monotonic with a general growth in y and uniformly continuous in z.

Adaptation of the Alicki—Fannes—Winter method for the set of states with bounded energy and its use

We describe a modification of the Alicki-Fannes-Winter method which allows to prove uniform continuity on the set of quantum states with bounded energy of any locally almost affine function having limited growth with increasing energy. Some applications in quantum information theory are considered. The asymptotic continuity of the relative entropy of entanglement and of its regularization under the energy constraint on one subsystem is proved. Channel-independent continuity bounds for the quantum mutual information at the output of a local channel and for the output Holevo quantity under the input energy constraint are obtained.

A fixed point theorem and some properties of v-generalized metric spaces

We give a characterization of complete v-generalized metric spaces in terms of the fixed point property. Moreover, for such spaces, we prove Matkowski’s fixed point theorem. In the case of two-generalized metric spaces, we obtain a characterization of uniform continuity in terms of the distance function between two sets.

Some Basic Properties of Uniformly Symmetrically Continuous (Real Valued) Functions on Metric Spaces

We give the definition of uniform symmetric continuity for real valued functions defined on a metric space. Then we investigate the basic properties of uniformly symmetrically continuous functions and compare them with those of symmetrically continuous functions and uniformly continuous functions. Several examples are also given.

Optimal Control Existence for Degenerate Infinite Dimensional Systems of Fractional Order

Abstract Optimal control problems solvability are researched for infinite dimensional control systems, described by semilinear evolution equations in Banach spaces with degenerate linear operator at the Caputo fractional derivative. The pair of linear operators in the equation is relatively bounded and the nonlinear operator satisfies some smoothness conditions, in particular the condition of uniform Lipschitz continuity, and one of two types additional conditions: independence of degeneracy subspace elements or non-belonging of the operator image to the degeneracy subspace. The control system is endowed by the generalized Showalter — Sidorov initial conditions, which are natural for degenerate evolution equations. Optimal control have to belong to a convex closed set of admissible controls and to minimize a convex, bounded from below, lower semicontinuous and coercive cost functional. Solvability conditions are found for the optimal control problem of this class. If the existence of the initial problem solution with an admissible control is obvious, it is shown that the local Lipschitz continuity in phase variables that uniform with respect to time is sufficient for the optimal control existence. Abstract results are illustrated by optimal control problem for the equations system of the fractional viscoelastic Kelvin — Voigt fluid dynamics.

Finite dimensionality of global attractor for the solutions to 3D viscous primitive equations of large-scale moist atmosphere

Under general boundary conditions we consider the finiteness of the Hausdorff and fractal dimensions of the global attractor for the strong solution of the 3D moist primitive equations with viscosity. Firstly, we obtain time-uniform estimates of the first-order time derivative of the strong solutions in \begin{document}$L^2(\mho)$\end{document} . Then, to prove the finiteness of the Hausdorff and fractal dimensions of the global attractor, the common method is to obtain the uniform boundedness of the strong solution in \begin{document}$H^2(\mho)$\end{document} to establish the squeezing property of the solution operator. But it is difficult to achieve due to the boundary conditions and complicated structure of the 3D moist primitive equations. To overcome the difficulties, we try to use the uniform boundedness of the derivative of the strong solutions with respect to time \begin{document}$t$\end{document} in \begin{document}$L^2(\mho)$\end{document} to prove the uniform continuity of the global attractor. Finally, using the uniform continuity of the global attractor we establish the squeezing property of the solution operator which implies the finiteness of the Hausdorff and fractal dimensions of the global attractor.

On the Energy-Constrained Diamond Norm and Its Application in Quantum Information Theory

We consider the family of energy-constrained diamond norms on the set of Hermitian-preserving linear maps (superoperators) between Banach spaces of trace class operators. We prove that any norm from this family generates the strong (pointwise) convergence on the set of all quantum channels (which is more adequate for describing variations of infinite-dimensional channels than the diamond norm topology). We obtain continuity bounds for information characteristics (in particular, classical capacities) of energy-constrained quantum channels (as functions of a channel) with respect to the energy-constrained diamond norms which imply uniform continuity of these characteristics with respect to the strong convergence topology.

Amortized entanglement of a quantum channel and approximately teleportation-simulable channels

This paper defines the amortized entanglement of a quantum channel as the largest difference in entanglement between the output and the input of the channel, where entanglement is quantified by an arbitrary entanglement measure. We prove that the amortized entanglement of a channel obeys several desirable properties, and we also consider special cases such as the amortized relative entropy of entanglement and the amortized Rains relative entropy. These latter quantities are shown to be single-letter upper bounds on the secret-key-agreement and PPT-assisted quantum capacities of a quantum channel, respectively. Of especial interest is a uniform continuity bound for these latter two special cases of amortized entanglement, in which the deviation between the amortized entanglement of two channels is bounded from above by a simple function of the diamond norm of their difference and the output dimension of the channels. We then define approximately teleportation- and positive-partial-transpose-simulable (PPT-simulable) channels as those that are close in diamond norm to a channel which is either exactly teleportation- or PPT-simulable, respectively. These results then lead to single-letter upper bounds on the secret-key-agreement and PPT-assisted quantum capacities of channels that are approximately teleportation- or PPT-simulable, respectively. Finally, we generalize many of the concepts in the paper to the setting of general resource theories, defining the amortized resourcefulness of a channel and the notion of ν-freely-simulable channels, connecting these concepts in an operational way as well.

Simulations and bisimulations for analysis of stability with respect to inputs of hybrid systems

Simulation and bisimulation relations define pre-orders on processes which serve as the basis for approximation based verification techniques, and have been extended towards the design of continuous and hybrid systems with complex logic specifications. We study pre-orders between hybrid systems which preserve stability properties with respect to input. We show that these properties are not bisimulation invariant, and hence propose stronger notions which strengthen simulation and bisimulation relations with uniform continuity constraints. We show that uniform continuity is necessary on the relations corresponding to both the state-space and the input-space, and continuity itself does not suffice. Finally, we demonstrate the satisfiability of our definitions by casting the well-known Lyapunov function based techniques for stability analysis as constructing a simple one-dimensional system which is stable and uniformly continuously simulates the original system.

English

English

SOS specifications for uniformly continuous operators

Behavioral metric semantics provide formal notions to compare probabilistic systems, giving a notion of behavioral distance characterizing how far the behavior of two systems is apart. Compositional reasoning over probabilistic systems with respect to behavioral metric semantics requires the language operators to be uniformly continuous, which ensures that a limited change in the behavior of a subsystem implies a smooth and limited change in the behavior of the whole system. We consider a hierarchy of uniform continuity properties for process algebra operators and, for each of these properties, we propose a Structural Operational Semantics specification format (namely a set of syntactical constraints on the form of the SOS rules) ensuring that the property is satisfied by construction by all operators captured by the format.

Tight uniform continuity bounds for the quantum conditional mutual information, for the Holevo quantity, and for capacities of quantum channels

We start with Fannes’ type and Winter’s type tight (uniform) continuity bounds for the quantum conditional mutual information and their specifications for states of special types. Then we analyse continuity of the Holevo quantity with respect to nonequivalent metrics on the set of discrete ensembles of quantum states. We show that the Holevo quantity is continuous on the set of all ensembles of m states with respect to all the metrics if either m or the dimension of underlying Hilbert space is finite and obtain Fannes’ type tight continuity bounds for the Holevo quantity in this case. In the general case, conditions for local continuity of the Holevo quantity for discrete and continuous ensembles are found. Winter’s type tight continuity bound for the Holevo quantity under constraint on the average energy of ensembles is obtained and applied to the system of quantum oscillators. The above results are used to obtain tight and close-to-tight continuity bounds for basic capacities of finite-dimensional channels (significantly refining the Leung-Smith continuity bounds).

Lacunary Statistically Slowly Oscillating Continuity in Abstract Spaces

The notion of lacunary statistically slowly oscillating sequences was introduced, and lacunary statistically slowly oscillating continuous functions in topological vector space valued cone metric space was studied in this paper. The concept of ideally ward continuity was also introduced and obtain results were discussed for ward continuity, uniform continuity, ordinary continuity in topological vector space valued cone metric space.

A Zero-Sum Stochastic Differential Game with Impulses, Precommitment, and Unrestricted Cost Functions

We study a zero-sum stochastic differential game (SDG) in which one controller plays an impulse control while their opponent plays a stochastic control. We consider an asymmetric setting in which the impulse player commits to, at the start of the game, performing less than q impulses (q can be chosen arbitrarily large). In order to obtain the uniform continuity of the value functions, previous works involving SDGs with impulses assume the cost of an impulse to be decreasing in time. Our work avoids such restrictions by requiring impulses to occur at rational times. We establish that the resulting game admits a value, and in turn, the existence and uniqueness of viscosity solutions to an associated Hamilton-Jacobi-Bellman-Isaacs quasi-variational inequality.

Uniform continuity and quantization on bounded symmetric domains

We consider Toeplitz operators $T_f^{\lambda}$ with symbol $f$ acting on the standard weighted Bergman spaces over a bounded symmetric domain $\Omega\subset \mathbb{C}^n$. Here $\lambda > genus-1$ is the weight parameter. The classical asymptotic semi-commutator relation $\lim_{\lambda \rightarrow \infty} \big{\|}T_f^{\lambda} T_g^{\lambda} -T_{fg}^{\lambda} \big{\|}=0$ with $f,g \in C(\overline{\mathbb{B}^n})$, where $\Omega=\mathbb{B}^n$ denotes the complex unit ball, is extended to larger classes of bounded and unbounded operator symbol-functions and to more general domains. We deal with operator symbols that generically are neither continuous inside $\Omega$ (Section 4) nor admit a continuous extension to the boundary (Section 3 and 4). Let $\beta$ denote the Bergman metric distance function on $\Omega$. We prove that the semi-commutator relation remains true for $f$ and $g$ in the space ${\rm UC}(\Omega)$ of all $\beta$-uniformly continuous functions on $\Omega$. Note that this space contains also unbounded functions. In case of the complex unit ball $\Omega=\mathbb{B}^n \subset \mathbb{C}^n$ we show that the semi-commutator relation holds true for bounded symbols in ${\rm VMO}(\mathbb{B}^n)$, where the vanishing oscillation inside $\mathbb{B}^n$ is measured with respect to $\beta$. At the same time the semi-commutator relation fails for generic bounded measurable symbols. We construct a corresponding counterexample using oscillating symbols that are continuous outside of a single point in $\Omega$.

On a Finite Range Decomposition of the Resolvent of a Fractional Power of the Laplacian II. The Torus

In previous papers, [M1, M2], [M3], we proved the existence as well as regularity of a finite range decomposition for the resolvent $G_{\alpha} (x-y,m^2) = ((-\Delta)^{\alpha\over 2} + m^{2})^{-1} (x-y) $, for $0<\alpha <2$ and all real $m$, in the lattice ${\bf Z}^{d}$ for dimension $d\ge 2$. In this paper, which is a continuation of the previous one, we extend those results by proving the existence as well as regularity of a finite range decomposition for the same resolvent but now on the lattice torus ${\bf Z}^{d}/L^{N+1}{\bf Z}^{d} $ for $d\ge 2$ provided $m\neq 0$ and $0<\alpha <2$. We also prove differentiability and uniform continuity properties with respect to the resolvent parameter $m^{2}$. Here $L$ is any odd positive integer and $N\ge 2$ is any positive integer.

The crystal morphology effect of Iridium tris-acetylacetonate on MOCVD iridium coatings

Iridium tris-acetylacetonate is the most commonly used precursor for the metal organic chemical vapour deposition (MOCVD) of iridium coating. In this paper, the crystal morphology effect of iridium tris-acetylacetonate on iridium coatings prepared by MOCVD was studied. Two kinds of Ir(acac)3 crystalline powder were prepared. A precursor sublimation experiment in a fixed bed reactor and an iridium deposition experiment in a cold-wall atmospheric CVD reactor were designed. It is found that the volatility of the hexagonal columnar crystals is better than that of the tetragonal flake crystals under the experimental conditions. It’s due to the hexagonal columnar crystals exposed more crystal faces than the tetragonal flake crystals, increasing its contact area with the transport gas. An adequate supply of iridium tris-acetylacetonate during the pre-deposition period contributed to obtain an iridium coating with a smooth and uniform continuity surface.