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This paper provides two new Velichko-like theorems for the weak counterpart of the locally convex space C k X of all real-valued functions defined on a Tychonoff space X equipped with the compact-open topology τ k . [ABSTRACT FROM AUTHOR]

In order to ensure reliable and secure image exchange, chaotic systems are often considered for their good performance in information security. In this work, we first propose an extended version of a chaotic tent map (TM)—the multiparametric 1D tent map (MTM). The latter contains six control parameters defined over an unlimited range. These parameters strongly influence the MTM output when they are slightly modified by ∓ 10 − 13 , which makes MTM stronger than the existing TM and other 1D chaotic maps in terms of security key space. Then, this paper proposes a simple, yet powerful method to make uniform the distribution of chaotic sequence values, making the latter suitable for use in cryptosystems. Next, a new image cryptosystem is introduced based on MTM and parallel computing. This computing mode is incorporated to boost the security level of our scheme and to speed up its runtime. Indeed, in only one running round, our encryption scheme generates a security key of space equal to 10 78 × n with n indicating the number of the available CPU cores. Therefore, the suggested scheme achieves a good trade-off between safety and efficiency. The results of the performed comparisons and numerical experiments indicate on the one hand that MTM exhibits good chaotic characteristics in comparison to its original version. On the other hand, the suggested cryptosystem demonstrates good strength against various kinds of attacks (brute force, statistical, classical, noise, differential, etc.). Furthermore, comparison with similar schemes indicate that the proposed scheme is competitive in terms of execution time and superior in terms of security level. [ABSTRACT FROM AUTHOR]

This paper deals with extremally disconnected spaces and extremally disconnected P-spaces. A space X is said to be extremally disconnected if, for every open subset V of X, the closure of V in X is also an open set. P-spaces are spaces in which the intersection of countably many open sets is an open set. The authors present a new characterization of extremally disconnected spaces, and the extremally disconnected P-spaces, by means of selection theory. If X is either an extremally disconnected space or an extremally disconnected P-space, then the usual theorems of extension of real-valued continuous functions for a dense subset S of X can be deduced from our results. A corollary of our outcomes is that every nondiscrete space X of nonmeasurable cardinality has a dense subset S such that S is not C-embedded in X. [ABSTRACT FROM AUTHOR]

In this paper, we provide several Arzelà–Ascoli-type results on the space of all continuous functions from a Tychonoff space X into the fuzzy sets of R n , (F U S C B (R n) , H e n d) , which are upper semi-continuous and have bounded support endowed with the endograph metric. Namely, we obtain positive results when X is considered to be a k r -space and C (X , (F U S C B (R n) , H e n d)) is endowed with the compact open topology, as well as when we assume that X is pseudocompact and C (X , (F U S C B (R n) , H e n d)) is equipped with the uniform topology. [ABSTRACT FROM AUTHOR]

In this paper, we mainly discuss the angle modulus of convexity δ X a (ϵ) and the angle modulus of smoothness ρ X a (ϵ) in a real normed linear space X, which are closely related to the classical modulus of convexity δ X (ϵ) and the modulus of smoothness ρ X (ϵ) . Some geometric properties of the two moduli were investigated. In particular, we obtained a characterization of uniform non-squareness in terms of ρ X a (1) . Meanwhile, we studied the relationships between δ X a (ϵ) , ρ X a (ϵ) and other geometric constants of real normed linear spaces through some equalities and inequalities. Moreover, these two coefficients were computed for some concrete spaces. [ABSTRACT FROM AUTHOR]

Continuing investigations initiated by the first author, we associate relational structures for metric spaces and investigate their model theoretic properties. In this paper, we consider ultrametric spaces. Among others, we show that any elementary substructure of the relational structure associated with a totally bounded ultrametric space X is dense in X. Further, we provide an explicit upper bound for a splitting chain of atomic types in ultrametric spaces of a finite spectrum. For ultrametric spaces, these results improve previous ones of the present authors and may have further practical applications in designing similarity detecting algorithms. [ABSTRACT FROM AUTHOR]

Assume that (X , d , μ) is a metric measure space that satisfies a Q-doubling condition with Q > 1 and supports an L 2 -Poincaré inequality. Let be a nonnegative operator generalized by a Dirichlet form E and V be a Muckenhoupt weight belonging to a reverse Hölder class R H q (X) for some q ≥ (Q + 1) / 2 . In this paper, we consider the Dirichlet problem for the Schrödinger equation − ∂ t 2 u + u + V u = 0 on the upper half-space X × R + , which has f as its the boundary value on X. We show that a solution u of the Schrödinger equation satisfies the Carleson type condition if and only if there exists a square Morrey function f such that u can be expressed by the Poisson integral of f. This extends the results of Song-Tian-Yan [Acta Math. Sin. (Engl. Ser.) 34 (2018), 787-800] from the Euclidean space R Q to the metric measure space X and improves the reverse Hölder index from q ≥ Q to q ≥ (Q + 1) / 2 . [ABSTRACT FROM AUTHOR]

We consider an operator { S (t) } t ≥ 0 on a Banach space X with generator A, characterized by being an α -times-integrated C-regularized semigroup. The adjoint family S * (t) : X * → X * is introduced for analysis. { S * (t) } t ≥ 0 maintains the characteristics of an α -times-integrated C-regularized semigroup, though with strong continuity and Bochner integrals being substituted by weak* continuity and weak* integrals, respectively. Our investigation focuses on the closed subspace X ⊙ , where { S * (t) } t ≥ 0 exhibits strong continuity. Additionally, a comparison between the adjoint A * of A and the generator of the adjoint family is conducted. [ABSTRACT FROM AUTHOR]

We consider the large class G of locally convex spaces that includes, among others, the classes of (D F) -spaces and (L F) -spaces. For a space E in class G we have characterized that a subspace Y of (E , σ (E , E ′)) , endowed with the induced topology, is analytic if and only if Y has a σ (E , E ′) -compact resolution and is contained in a σ (E , E ′) -separable subset of E. This result is applied to reprove a known important result (due to Cascales and Orihuela) about weak metrizability of weakly compact sets in spaces of class G. The mentioned characterization follows from the following analogous result: The space C (X) of continuous real-valued functions on a completely regular Hausdorff space X endowed with a topology ξ stronger or equal than the pointwise topology τ p of C (X) is analytic iff (C (X) , ξ) is separable and is covered by a compact resolution. [ABSTRACT FROM AUTHOR]

Given an injective closed linear operator A defined in a Banach space X , and writing C F D t α the Caputo–Fabrizio fractional derivative of order α ∈ (0 , 1) , we show that the unique solution of the abstract Cauchy problem (∗) C F D t α u (t) = A u (t) + f (t) , t ≥ 0 , where f is continuously differentiable, is given by the unique solution of the first order abstract Cauchy problem u ′ (t) = B α u (t) + F α (t) , t ≥ 0 ; u (0) = − A − 1 f (0) , where the family of bounded linear operators B α constitutes a Yosida approximation of A and F α (t) → f (t) as α → 1. Moreover, if 1 1 − α ∈ ρ (A) and the spectrum of A is contained outside the closed disk of center and radius equal to 1 2 (1 − α) then the solution of (∗) converges to zero as t → ∞ , in the norm of X, provided f and f ′ have exponential decay. Finally, assuming a Lipchitz-type condition on f = f (t , x) (and its time-derivative) that depends on α , we prove the existence and uniqueness of mild solutions for the respective semilinear problem, for all initial conditions in the set S : = { x ∈ D (A) : x = A − 1 f (0 , x) }. [ABSTRACT FROM AUTHOR]

The linear differential operator with constant coefficients D (y) = y (n) + a 1 y (n − 1) + ... + a n y , y ∈ C n (R , X) acting in a Banach space X is Ulam stable if and only if its characteristic equation has no roots on the imaginary axis. We prove that if the characteristic equation of D has distinct roots r k satisfying Re r k > 0 , 1 ≤ k ≤ n , then the best Ulam constant of D is K D = 1 | V | ∫ 0 ∞ | ∑ k = 1 n (− 1) k V k e − r k x | d x , where V = V (r 1 , r 2 , ... , r n) and V k = V (r 1 , ... , r k − 1 , r k + 1 , ... , r n) , 1 ≤ k ≤ n , are Vandermonde determinants. [ABSTRACT FROM AUTHOR]

We investigate the Hyers–Ulam stability of an equation involving a single variable of the form ∥ f (x) − α f (k n (x)) − β f (k n + 1 (x)) ∥ ⩽ u (x) where f is an unknown operator from a nonempty set X into a Banach space Y, and it preserves the addition operation, besides other certain conditions. The theory is employed and stability theorems are proven for various functional equations involving several variables. By comparing this method with the available techniques, it was noticed that this method does not require any restriction on the parity, on the domain, and on the range of the function. Our findings suggest that it is very much easy and more appropriate to apply the proposed method while investigating the stability of functional equations, in particular for several variables. [ABSTRACT FROM AUTHOR]

In this article, we introduce a new class of operators on the Cartesian product of vector lattices. We say that a bilinear operator T : E × F → W defined on the Cartesian product of vector lattices E and F and taking values in a vector lattice W is narrow if the partial operators T x and T y are narrow for all x ∈ E , y ∈ F . We prove that, for order-continuous Köthe–Banach spaces E and F and a Banach space X, the classes of narrow and weakly function narrow bilinear operators from E × F to X are coincident. Then, we prove that every order-to-norm continuous C-compact bilinear regular operator T is narrow. Finally, we show that a regular bilinear operator T from the Cartesian product E × F of vector lattices E and F with the principal projection property to an order continuous Banach lattice G is narrow if and only if | T | is. [ABSTRACT FROM AUTHOR]

Measure-valued Pólya urn processes (MVPP) are Markov chains with an additive structure that serve as an extension of the generalized k-color Pólya urn model towards a continuum of possible colors. We prove that, for any MVPP (μ n) n ≥ 0 on a Polish space X , the normalized sequence (μ n / μ n (X)) n ≥ 0 agrees with the marginal predictive distributions of some random process (X n) n ≥ 1 . Moreover, μ n = μ n − 1 + R X n , n ≥ 1 , where x ↦ R x is a random transition kernel on X ; thus, if μ n − 1 represents the contents of an urn, then X n denotes the color of the ball drawn with distribution μ n − 1 / μ n − 1 (X) and R X n —the subsequent reinforcement. In the case R X n = W n δ X n , for some non-negative random weights W 1 , W 2 , ... , the process (X n) n ≥ 1 is better understood as a randomly reinforced extension of Blackwell and MacQueen's Pólya sequence. We study the asymptotic properties of the predictive distributions and the empirical frequencies of (X n) n ≥ 1 under different assumptions on the weights. We also investigate a generalization of the above models via a randomization of the law of the reinforcement. [ABSTRACT FROM AUTHOR]

Given a complex Banach space X , we investigate the stable character of the property (V E) for a bounded linear operator T : X → X , under commuting perturbations that are Riesz, compact, algebraic and hereditarily polaroid. We also analyze sufficient conditions that allow the transfer of property (V E) from the tensorial factors T and S to its tensor product. [ABSTRACT FROM AUTHOR]