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Extending and completing earlier results on lifting certain continuity properties of aggregation functions by super- and sub-additive transformations (J. Mahani Math. Res. Center 8 (2019) 37–51, and Iranian J. Fuzzy Sets 17 (2020) 2, 165–168), we prove that uniform continuity of multi-dimensional aggregation functions is preserved under super-additive transformations. [ABSTRACT FROM AUTHOR]

The purpose of this study is to introduce an inertial algorithm for approximating a solution of the split equality Hammerstein type equation problem in general reflexive real Banach spaces. Strong convergence results are established under the assumption that the associated mappings are monotone and uniformly continuous. The results in this paper generalize and improve many of the existing results in the literature in the sense that the underlying mappings are relaxed from Lipschitz continuous to uniformly continuous and the spaces under consideration are extended from Hilbert spaces to reflexive real Banach spaces with a more general problem which includes the Hammerstein type equation problems. [ABSTRACT FROM AUTHOR]

It is the purpose of this paper that uniform continuity of fractional order integral of fractal interpolation function (FIF) is researched. First, FIF generated by affine transformation is constructed. Second, the combination between Riemann–Liouville fractional order integral and FIF is defined. Finally, the continuity and the uniform continuity of fractional order integral of FIF are proved by compactness theorem of numbers sequence or finite covering theorem. The result shows that fractional order integral of FIF is uniformly continuous on a closed interval. [ABSTRACT FROM AUTHOR]

We study the relations between two consequences of the Continuum Hypothesis discovered by Wacław Sierpiński, concerning uniform continuity of continuous functions and uniform convergence of sequences of real-valued functions, defined on subsets of the real line of cardinality continuum. [ABSTRACT FROM AUTHOR]

We study divide-and-conquer recurrences of the form f (n) = α f (⌊ n 2 ⌋) + β f (⌈ n 2 ⌉) + g (n) (n ⩾ 2) , with g (n) and f (1) given, where α , β ⩾ 0 with α + β > 0 ; such recurrences appear often in the analysis of computer algorithms, numeration systems, combinatorial sequences, and related areas. We show under an optimum (iff) condition on g (n) that the solution f always satisfies a simple identity f (n) = n log 2 ⁡ (α + β) P (log 2 ⁡ n) − Q (n) , where P is a periodic function and Q (n) is of a smaller order than the dominant term. This form is thus not only an identity but also an asymptotic expansion. Explicit forms for the continuity of the periodic function P are provided, together with a few other smoothness properties. We show how our results can be easily applied to many dozens of concrete examples collected from the literature, and how they can be extended in various directions. Our method of proof is surprisingly simple and elementary, but leads to the strongest types of results for all examples to which our theory applies. [ABSTRACT FROM AUTHOR]

In this paper the properties of space of the G-permutation degree, like: weight, uniform connectedness and index boundedness are studied. It is proved that: (1) If (X, U) is a uniform space, then the mapping s n,G: (Xn, Un) ! (SPn GX, SPn GU) is uniformly continuous and uniformly open, moreover w (U) = w (SPn GU); (2) If the mapping f: (X, U) ! (Y, V) is a uniformly continuous (open), then the mapping SPn Gf: (SPn GX, SPn GU) ! (SPn GY, SPn GV) is also uniformly continuous (open); (3) If the uniform space (X, U) is uniformly connected, then the uniform space (SPn GX, SPn GU) is also uniformly connected. [ABSTRACT FROM AUTHOR]

In 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. [ABSTRACT FROM AUTHOR]

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 UC—meaning that all real valued, continuous functions defined on these spaces are necessarily uniformly continuous. [ABSTRACT FROM AUTHOR]

In this paper we investigate the problem of uniform continuity of nonautonomous superposition operators acting between spaces of functions of bounded Λ-variation. In particular, we give the sufficient conditions for nonautonomous superposition operators to continuously map a space of functions of bounded Λ-variation into itself. The conditions cover the generators being functions of C1-class (in view of two variables), but also allow for less regular functions, including discontinuous generators. [ABSTRACT FROM AUTHOR]

Abstract Let X be a uniform space and U (X) the linear space of real-valued uniformly continuous functions on X. Our main objective is to give a number of properties characterizing the fact that U (X) is stable under pointwise product in case X is a metric space. Some of these characterizations hold in much more general circumstances. [ABSTRACT FROM AUTHOR]

Using concentration-compactness arguments, we prove a variant of the Brézis–Lieb-Lemma under weaker assumptions on the nonlinearity than known before. An intermediate result on the uniform continuity of superposition operators in Sobolev space is of independent interest. [ABSTRACT FROM AUTHOR]

In this article, using the Mizar system [1], [2], we discuss the continuity of bounded linear operators on normed linear spaces. In the first section, it is discussed that bounded linear operators on normed linear spaces are uniformly continuous and Lipschitz continuous. Especially, a bounded linear operator on the dense subset of a complete normed linear space has a unique natural extension over the whole space. In the next section, several basic currying properties are formalized. In the last section, we formalized that continuity of bilinear operator is equivalent to both Lipschitz continuity and local continuity. We referred to [4], [13], and [3] in this formalization. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

We consider a scale invariant Cassinian metric and a Gromov hyperbolic metric. We discuss a distortion property of the scale invariant Cassinian metric under Möbius maps of a punctured ball onto another punctured ball. We obtain a modulus of continuity of the identity map from a domain equipped with the scale invariant Cassinian metric (or the Gromov hyperbolic metric) onto the same domain equipped with the Euclidean metric. Finally, we establish the quasi-invariance properties of both metrics under quasiconformal maps. [ABSTRACT FROM AUTHOR]

A ball of maximal radius inscribed in a convex closed bounded set with a nonempty interior is considered in the class of uniformly convex Banach spaces. It is shown that, under certain conditions, the centers of inscribed balls form a uniformly continuous (as a set function) set-valued mapping in the Hausdorff metric. In a finite-dimensional space of dimension n, the set of centers of balls inscribed in polyhedra with a fixed collection of normals satisfies the Lipschitz condition with respect to sets in the Hausdorff metric. A Lipschitz continuous single-valued selector of the set of centers of balls inscribed in such polyhedra can be found by solving n + 1 linear programming problems. [ABSTRACT FROM AUTHOR]

A d0-algebra, which is a generalization of a D-lattice, is an algebraic structure with one operation, called difference, and a distinguished element denoted by 0. In this paper we investigate those uniformities on a d0-algebra which make the difference uniformly continuous. We prove that such uniformities are univocally determined the neighbourhoods of 0, thus extending the corresponding result known for D-lattices. [ABSTRACT FROM AUTHOR]

Birkhoff's HSP theorem characterizes the classes of models of algebraic theories as those being closed with respect to homomorphic images, subalgebras, and products. In particular, it implies that an algebra B satisfies all equations that hold in an algebra A of the same signature if and only if B is a homomorphic image of a subalgebra of a (possibly infinite) direct power of A. The former statement is equivalent to the existence of a natural map sending term functions of the algebra A to those of B-the natural clone homomorphism. The study of continuity properties of natural clone homomorphisms has been initiated recently by Bodirsky and Pinsker for locally oligomorphic algebras. Revisiting the argument of Bodirsky and Pinsker, we show that for any algebra B in the variety generated by an algebra A, the induced natural clone homomorphism is uniformly continuous if and only if every finitely generated subalgebra of B is a homomorphic image of a subalgebra of a finite power of A. Based on this observation, we study the question as to when Cauchy continuity of natural clone homomorphisms implies uniform continuity. We introduce the class of almost locally finite algebras, which encompasses all locally oligomorphic as well as all locally finite algebras, and show that, in case A is almost locally finite, then the considered natural homomorphism is uniformly continuous if (and only if) it is Cauchy-continuous. In particular, this provides a locally finite counterpart of the result by Bodirsky and Pinsker. Along the way, we also discuss some peculiarities of oligomorphic permutation groups on uncountable sets. [ABSTRACT FROM AUTHOR]

A sequence $$(\alpha _{k})$$ of points in $$\mathbb {R}$$ , the set of real numbers, is called $$\rho $$ -statistically convergent to an element $$\ell $$ of $$\mathbb {R}$$ if for each $$\varepsilon >0$$ , where $$\varvec{\rho }=(\rho _{n})$$ is a non-decreasing sequence of positive real numbers tending to $$\infty $$ such that $$\limsup _{n} \frac{\rho _{n}}{n}<\infty $$ , $$\Delta \rho _{n}=O(1)$$ , and $$\Delta \alpha _{n} =\alpha _{n+1}-\alpha _{n}$$ for each positive integer n. A real-valued function defined on a subset of $$\mathbb {R}$$ is called $$\rho $$ -statistically ward continuous if it preserves $$\rho $$ -statistical quasi-Cauchy sequences where a sequence $$(\alpha _{k})$$ is defined to be $$\rho $$ -statistically quasi-Cauchy if the sequence $$(\Delta \alpha _{k})$$ is $$\rho $$ -statistically convergent to 0. We obtain results related to $$\rho $$ -statistical ward continuity, $$\rho $$ -statistical ward compactness, ward continuity, continuity, and uniform continuity. It turns out that the set of uniformly continuous functions coincides with the set of $$\rho $$ -statistically ward continuous functions not only on a bounded subset of $$\mathbb {R}$$ , but also on an interval. [ABSTRACT FROM AUTHOR]

In this paper, we revisit the discrete lossy Gray-Wyner problem. In particular, we derive its optimal second-order coding rate region, its error exponent (reliability function), and its moderate deviations constant under mild conditions on the source. To obtain the second-order asymptotics, we extend some ideas from Watanabe’s work. In particular, we leverage the properties of an appropriate generalization of the conditional distortion-tilted information density, which was first introduced by Kostina and Verdú. The converse part uses a perturbation argument by Gu and Effros in their strong converse proof of the discrete Gray–Wyner problem. The achievability part uses two novel elements: 1) a generalization of various type covering lemmas and 2) the uniform continuity of the conditional rate-distortion function in both the source (joint) distribution and the distortion level. To obtain the error exponent, for the achievability part, we use the same generalized type covering lemma, and for the converse, we use the strong converse together with a change-of-measure technique. Finally, to obtain the moderate deviations constant, we apply the moderate deviations theorem to probabilities defined in terms of information spectrum quantities. [ABSTRACT FROM PUBLISHER]

We show that a prediction in [8] is inaccurate by constructing quasiconformal mappings onto s-John domains so that the mappings fail to be uniformly continuous between natural distances. These examples also exhibit the sharpeness of the assumptions in [5]. [ABSTRACT FROM AUTHOR]

We prove that functions in the Hajłasz-Sobolev space M1,s on an s-Ahlfors regular metric space are uniformly continuous when s ≤ 1. [ABSTRACT FROM AUTHOR]

It is proved that an endomorphism φ of a hyperbolic group G satisfies a Hölder condition with respect to a visual metric if and only if φ is virtually injective and Gφ is a quasiconvex subgroup of G. If G is virtually free or torsion-free co-hopfian, then φ is uniformly continuous if and only if it satisfies a Hölder condition if and only if it is virtually injective. [ABSTRACT FROM AUTHOR]

We identify yet another category equivalent to that of Kleene–Kreisel continuous functionals. Reasoning constructively and predicatively, all functions from the Cantor space to the natural numbers are uniformly continuous in this category. We do not need to assume Brouwerian continuity axioms to prove this, but, if we do, then we can show that the full type hierarchy is equivalent to our manifestation of the continuous functionals. We construct this manifestation within a category of concrete sheaves, called C-spaces, which form a locally cartesian closed category, and hence can be used to model system T and dependent types. We show that this category has a fan functional and validates the uniform-continuity principle in these theories. Our development is within informal constructive mathematics, along the lines of Bishop mathematics. However, in order to extract concrete computational content from our constructions, we formalized it in intensional Martin-Löf type theory, in Agda notation, and we discuss the main technical aspects of this at the end of the paper. [ABSTRACT FROM AUTHOR]

Several Banach–Stone-like generalizations of Shirota's result for metrizable uniform spaces are proved. Namely, if complete uniform spaces X , Y have isomorphic lattices U ( X ) , U ( Y ) of their real-valued uniformly continuous functions, and both X , Y are either some products of spaces having monotone bases (metrizable or uniformly zero-dimensional), or are locally fine and of non-measurable cardinality, then X and Y are uniformly homeomorphic. [ABSTRACT FROM AUTHOR]

If two uniform spaces have isomorphic lattices of their uniformly continuous real-valued functions then also their sublattices of bounded functions are isomorphic. That result is used to give a different correct proof of Shirota theorem (complete metric spaces are determined by their uniformly continuous real-valued functions) than that in [1] . [ABSTRACT FROM AUTHOR]

Abstract: Investors are naturally interested in the supremum and the infimum of stock prices, also in the maximum gain and the maximum loss over a time period. To shed light on these relatively complicated aspects of sample paths, we consider Brownian motion with and without drift. We provide explicit calculations of the correlation between the supremum and the infimum of Brownian motion with drift. We establish a number of results concerning the distributions of maximum gain and maximum loss. We present simulation studies of maximum gain and of maximum loss of Brownian motion with a range of values for the drift. We conjecture that the correlation between maximum gain and maximum loss has a minimum value of at drift 2. [Copyright &y& Elsevier]

In this paper, we introduced three uniformities V G l , V G r and V G which are induced in a natural way on a strongly topological gyrogroup (G , ⊕ , τ). It is mainly proved that (1) each of the three uniformities is compatible with G ; (2) if H is a subgyrogroup of G , then V G , H l = V H l , V G , H r = V H r and V G , H = V H ; (3) if { (G i , ⊕ i , τ i) : i ∈ I } is a family of strongly topological gyrogroups, then V G l = ∏ i ∈ I V G i l , V G r = ∏ i ∈ I V G i r and V G = ∏ i ∈ I V G i , where G = ∏ i ∈ I G i endowed with the Tychonoff product topology. At the end section, we obtain that every compact strongly topological gyrogroup has property U and every Lindelöf strongly topological gyrogroup has property ω - U. [ABSTRACT FROM AUTHOR]

In this paper, we study local uniform continuity of nonnegative weak solutions to degenerate diffusion-drift equations in the form of u t = Δ u m + ∇ ⋅ B (x , t) u , for m ≥ 1 assuming a vector field B ∈ L t p L x q . Regarding local Hölder continuity, we provide a sharp condition on p and q , which is referred to as the subcritical region. In the critical region, the divergence-free condition is essential to providing uniform continuity which depends on the modulus of continuity of the local norm of B. [ABSTRACT FROM AUTHOR]

In this paper the Lebesgue covering lemma is extended from the setting of metric spaces to the setting of admissible spaces. An admissible space is a topological space endowed with an admissible family of open coverings, and need not be metric. The paper contains applications to uniform continuity and dimension theory. [ABSTRACT FROM AUTHOR]

Cofinal completeness of a metric space, which is a property between completeness and compactness, can be characterized in terms of a measure of local compactness functional [7]. Using this functional, we introduce and then study the variational notion of cofinally complete subset of a metric space. [ABSTRACT FROM AUTHOR]

We produce necessary and sufficient conditions for the pointwise product of two uniformly continuous real-valued functions defined on a metric space to be uniformly continuous. [ABSTRACT FROM AUTHOR]

Abstract: Recently, it has been proved that a real-valued function defined on an interval of R, the set of real numbers, is uniformly continuous on if and only if it is defined on and preserves quasi-Cauchy sequences of points in . In this paper we call a real-valued function statistically ward continuous if it preserves statistical quasi-Cauchy sequences where a sequence is defined to be statistically quasi-Cauchy if the sequence is statistically convergent to 0. It turns out that any statistically ward continuous function on a statistically ward compact subset of R is uniformly continuous on . We prove theorems related to statistical ward compactness, statistical compactness, continuity, statistical continuity, ward continuity, and uniform continuity. [Copyright &y& Elsevier]

Abstract: A subset of a metric space is totally bounded if and only if any sequence of points in has a Cauchy subsequence. We call a sequence statistically quasi-Cauchy if , and lacunary statistically quasi-Cauchy if . We prove that a subset of a metric space is totally bounded if and only if any sequence of points in has a subsequence which is any type of the following: statistically quasi-Cauchy, lacunary statistically quasi-Cauchy, quasi-Cauchy, and slowly oscillating. It turns out that a function defined on a connected subset of a metric space is uniformly continuous if and only if it preserves either quasi-Cauchy sequences or slowly oscillating sequences of points in . [Copyright &y& Elsevier]

A collection of hands-on activities can be used to help students understand the formal definitions for convergent sequences, continuity, and uniform continuity. These activities use feather boas, yarn, and doorknob hangers. [ABSTRACT FROM AUTHOR]

In this work, we give some theorems on (mild) weighted pseudo-almost periodic solutions for some abstract semilinear differential equations with uniform continuity. To facilitate this we give a new composition theorem of weighted pseudo-almost periodic functions. Our composition theorem improves the known one by making use of a uniform continuity condition instead of the Lipschitz condition. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we will study a class of backward stochastic differential equations (BSDEs for short), for which the generator (coefficient) is Lipschitz continuous with respect to and uniformly continuous with respect to . We establish several properties for such BSDEs, including comparison and converse comparison theorems, a representation theorem for and a continuous dependence theorem. Then we introduce a new class of -expectation based on such backward stochastic differential equations, and discuss its properties. [ABSTRACT FROM AUTHOR]

Abstract: If a continuous function on a complete metric space has approximate roots and in a uniform manner at most one root, then it actually has a root. We validate this heuristic principle in Bishop-style constructive mathematics without countable choice, and thus can shed some more light on the role played by the completion when it comes to solving equations. [Copyright &y& Elsevier]

Abstract: We consider the problem of the existence of the global solutions and formation of singularities for a b-family of equations which includes the Camassa–Holm and Degasperis–Procesi equation. We also consider the problem of the uniformly continuity of Degasperis–Procesi equation. [Copyright &y& Elsevier]

We prove a complex analog of Kurzweil’s theorem on the approximation of continuous functions on separable Banach spaces admitting a separating polynomial and obtain a complex analog of new results due to Boiso and Hájek. [ABSTRACT FROM AUTHOR]

Abstract: Let be an ideal of subsets of a metric space . This paper considers a strengthening of the notion of uniform continuity of a function restricted to members of which reduces to ordinary continuity when consists of the finite subsets of X and agrees with uniform continuity on members of when is either the power set of X or the family of compact subsets of X. The paper also presents new function space topologies that are well suited to this strengthening. As a consequence of the general theory, we display necessary and sufficient conditions for continuity of the pointwise limit of a net of continuous functions. [Copyright &y& Elsevier]

In this paper we prove some results of metric spaces including Uniform continuity theorem and Ascoli-Arzela theorem for intuitionistic fuzzy metric spaces. We also prove that every intuitionistic fuzzy metric space has a countably locally finite basis and use this result to conclude that every intuitionistic fuzzy metric space is metrizable. [ABSTRACT FROM AUTHOR]

Abstract: The classical criterion of asymptotic stability for differential equations requires the existence of a Liapunov function with negative definite . Successive efforts have been made to weaken the negative definiteness of to semi-negative definiteness. Recently, it was given an interesting result that, under the boundedness of , the negative definiteness can be weakened to that together with that is negative definite. Unfortunately, its basic lemma is proved to be false by a counter example and cannot support this interesting result. In this paper we re-establish the weak criterion for asymptotic stability with less requirements. [Copyright &y& Elsevier]

We present a teaching approach to uniform continuity on unbounded intervals which, hopefully, may help to meet the following pedagogical objectives: To provide students with efficient and simple criteria to decide whether a continuous function is also uniformly continuous; To provide students with skill to recognize graphically significant classes of both uniformly and nonuniformly continuous functions. Assembling some well-known facts and refining the resulting statement, we establish a useful asymptotic coincidence test for the uniform continuity on unbounded intervals. That test is the core of the present note and yields an easily applicable technique. In particular, one of its immediate consequences is the elementary fact that continuity and existence of horizontal or oblique asymptotes imply uniform continuity. [ABSTRACT FROM AUTHOR]

We construct a metric such that the Cauchy problem for the corresponding nonlinear wave equation admits a nontrivial solution but the solution map is not uniformly continuous. [ABSTRACT FROM AUTHOR]

Abstract: In this paper we suggest a new extension of the method of Piyavskii for global optimization of a Hölder function with exponent . In the one-dimensional case a modification of Piyavskii’s algorithm is introduced and it is based on the construction of sub-estimators which are piecewise linear. The algorithm is then some what easily. Moreover, the results we obtain seem interesting. In the higher dimension, a new variant of the Alienor reducing transformation, which has been devised for exploiting one-dimensional global optimization techniques known for their great efficiency, is used. The method consists in reducing a multidimensional problem to a one-dimensional one using the so called α-dense curves. The convergence of the methods is also studied. [Copyright &y& Elsevier]

In this paper we prove some results of metric spaces including Uniform continuity theorem and Ascoli-Arzela theorem for intuitionistic fuzzy metric spaces. We also prove that every intuitionistic fuzzy metric space has a countably locally finite basis and use this result to conclude that every intuitionistic fuzzy metric space is metrizable. [ABSTRACT FROM AUTHOR]

We find a decomposition of the type of Sobczyk-Hammer for measures with values in l-groups, and also deduce some convergence theorems for such decompositions. Our procedure is based on some theorems of the type of Vitali-Hahn-Saks, and on the so-called Stone extension method. [ABSTRACT FROM AUTHOR]

This paper is essentially a story that can be used to help students understand some of the definitions found in a standard introductory real analysis course. [ABSTRACT FROM AUTHOR]

Abstract: A metric space is called an Atsuji space if every real-valued continuous function on is uniformly continuous. It is well known that an Atsuji space must be complete. A metric space is said to have an Atsuji completion if its completion is an Atsuji space. In this paper, we study twenty-nine equivalent characterisations for a metric space to have an Atsuji completion. [Copyright &y& Elsevier]