Uniformly Continuous Research Articles

Zero-r Law on the Analyticity and the Uniform Continuity of Fractional Resolvent Families

Zero-r Law on the Analyticity and the Uniform Continuity of Fractional Resolvent Families

Euclidean division by d in base b

Let b⩾2 be an integer. For each positive integer d, let Ed,b be the Euclidean division by d in base b, that is, the function which associates to a word u in {0,…,b−1}⁎, representing an integer n in base b, the unique word of the same length as u representing the quotient of the division of n by d. We describe the pure sequential transducer realizing this function and analyze the algebraic structure of its syntactic monoid. We compute its size, describe its Green's relations and its minimum ideal. As a consequence, we show that it is a group if and only if d and b are coprime numbers, it is a p-group if and only if d is a power of p and b is congruent to 1 modulo p and it is an aperiodic monoid if and only if d divides some power of b. The uniform continuity of Ed,b for the pro-group metric was studied by Reutenauer and Schützenberger in 1995. We launch a similar study for the uniform continuity of Ed,b with respect to the pro-p metric, where p is a prime number.

Bone Regeneration with Dental Pulp Stem Cells in an Experimental Model

Background/Objectives: The therapeutic approach to bone mass loss and bone’s limited self-regeneration is a major focus of research, emphasizing new biomaterials and cell therapy. Tissue bioengineering emerges as a potential alternative to conventional treatments. In this study, an experimental model of a critical bone lesion in rats was used to investigate bone regeneration by treating the defect with biomaterials Evolution® and Gen-Os® (OsteoBiol®, Turín, Italy), with or without mesenchymal stromal cells from dental pulp (DP-MSCs). Methods: Forty-six adult male Wistar rats were subjected to a 5-mm critical bone defect in the right mandible, which does not regenerate without intervention. The rats were randomly assigned to a Simulated Group, Control Group, or two Study Groups (using Evolution®, Gen-Os®, and DP-MSCs). The specimens were euthanized at three or six months, and radiological, histological, and ELISA tests were conducted to assess bone regeneration. Results: The radiological results showed that the DP-MSC group achieved uniform radiopacity and continuity in the bone edge, with near-complete structural defect restitution. Histologically, full bone regeneration was observed, with well-organized, vascularized lamellar bone and no lesion edges. These findings were supported by increases in endoglin, transforming growth factor-beta 1 (TGF-β1), protocollagen, parathormone, and calcitonin, indicating a conducive environment for bone regeneration. Conclusions: The use of DP-MSCs combined with biomaterials with appropriate three-dimensional matrices is a promising therapeutic option for further exploration.

Pointwise Products of Uniformly Continuous Functions

The problem of characterizing the metric spaces on which the pointwise product of any two uniformly continuous real - valued functions is uniformly continuous is investigated. A sufficient condition is given; furthermore, the condition is shown to be necessary for certain types of metric spaces, which include those with no isolated point and all subspaces of Euclidean spaces. It is not known if the condition is always necessary. 2000 Mathematics Subject Classification. Primary: 54C10, 54C30; Secondary: 20M20

Proof-Theoretical Aspects of Nonlinear and Set-Valued Analysis

AbstractThis thesis is concerned with extending the underlying logical approach as well as the breadth of applications of the proof mining program to various (mostly previously untreated) areas of nonlinear analysis and optimization, with a particular focus being placed on topics which involve set-valued operators.For this, we extend the current logical methodology of proof mining by new systems and corresponding so-called logical metatheorems that cover these more involved areas of nonlinear analysis. Most of these systems crucially rely on the use of intensional methods, treating sets with potentially high quantifier complexity in the defining matrix via characteristic functions and axioms that describe only their properties and do not completely characterize the elements of the sets.The applicability of all of these metatheorems is then substantiated by a range of case studies for the respective areas which in particular also highlight the naturalness of the use of intensional methods in the design of the corresponding systems.The first new area covered thereby is the theory of nonlinear semigroups induced by corresponding evolution equations for accretive operators. In that context, we present (besides an initial foray into the area from 2015) essentially the first applications of proof mining to the theory of partial differential equations. Concretely, we provide quantitative versions of four central results on the asymptotic behavior of solutions to such equations.The second new area unlocked in this thesis is that of the continuous dual of a Banach space and its norm (which are also approached via intensional methods). This in particular relies on a proof-theoretically tame treatment of suprema over (certain) bounded sets in this intensional context which is further exploited later on. These systems, which give access to this until now untreated fundamental notion from functional analysis, are then used to provide further substantial extensions to treat various notions from convex analysis like the Fréchet derivative of a convex function, Fenchel conjugates, Bregman distances, and monotone operators on Banach spaces in the sense of Browder.These systems are then utilized to provide applications in the context of Picard- and Halpern-style iterations of so-called Bregman strongly nonexpansive mappings where we provide both new quantitative and qualitative results.Lastly, we discuss the key notion of extensionality of a set-valued operator and its relation to set-theoretic maximality principles in more depth (which was already singled out—to some degree—in previous work). We thereby exhibit an issue arising with treating full extensionality in the context of these intensional approaches to set-valued operators and present useful fragments of the full extensionality statement where these issues are avoided.Corresponding to these fragments, we discuss a range of uniform continuity statements for set-valued operators beyond the usual notion involving the Hausdorff-metric. In particular, in that context, we utilize the previous tame treatment of suprema over bounded sets to also provide the first proof-theoretic treatment of that Hausdorff-metric in the context of systems for proof mining.The applicability of this treatment of the Hausdorff-metric is then in particular substantiated by a last case study where we provide quantitative information for a Mann-type iteration of set-valued mappings which are nonexpansive w.r.t. the Hausdorff-metric.The abstract was taken directly from the thesis.Abstract prepared by Nicholas Pischke, Technische Universität Darmstadt, Darmstadt, GermanyE-mail: pischke@mathematik.tu-darmstadt.de.URL: https://doi.org/10.26083/tuprints-00026584.

On two consequences of CH established by Sierpiński

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.

Scalar BSDEs of iterated-logarithmically sub-linear generators with integrable terminal values

We establish a general existence and uniqueness result of integrable adapted solutions to scalar backward stochastic differential equations with integrable parameters, where the generator g has an iterated-logarithmically uniform continuity in the second unknown variable z. The result improves our previous one in Fan et al. (2023), and can be applied to the (conditional) g-expectation and the star-shaped risk measure.

One dimensional BSDEs with critical integrable terminal values and infinite time horizon

We mainly study the existence and uniqueness of solutions to one-dimensional backward stochastic differential equations with terminal values satisfying a critical integrability and infinite time horizon. The main result is established under the assumption that the generator f exhibits a time-varying monotonicity in y and uniform continuity in z and the terminal value is required to satisfy the critical Lexp(μ02log(1+L))-integrability for a μ0>0, a condition stronger than LlogL-integrability but weaker than Lp(p>1)-integrability.

Uniform Continuity of Multifunctions

Uniform Continuity of Multifunctions

On the properties of one auxiliary function for calculating the global extremum

In this paper, we investigate the properties of one Auxiliary Function (AF) for calculating the global optimum of a function of many variables. The considered AF is constructed by transforming the objective function using the Lebesgue integral and is a function of one variable. Despite the fact that in previous works this AF was used to find the global minimum of smooth functions on convex closed sets using the algorithm of dividing a segment in half, its properties have not been studied. Here, this AF is used to calculate the global extremum of continuous functions defined on bounded closed sets of a multidimensional Euclidean space. The basic properties of the AF for any continuous objective function are established, such as non-negativity, positive uniformity, uniform continuity, differentiability and strict convexity, moreover, derivatives up to a certain order are found. The optimality criterion is formulated. The main criterion of optimality is that the value of a free variable, at which the AF and its derivatives are equal to zero up to some order, coincides with the global minimum of the objective function. It follows from this optimality criterion that to calculate the global minimum of the objective function, it is sufficient to find the zero of the AF or its derivative up to some order.

Reconstructed weighted aggregation operator

Every aggregation operator can embody a certain aggregation style pertaining to some detailed evaluation practice. Actually, with these aggregation styles preserved, additional weight functions can further influence the aggregation processes by reconstructing the innate aggregation structures. The advantage is significant because weight functions are commonly used in information fusion and it sometimes is necessary that the weight functions and the original aggregation styles can be embodied simultaneously. To fulfill this aim, we propose and analyze the reconstructed weighted aggregation operators. The method is based on the principle of factorization-reconstruction. Apart from providing a novel aggregation mechanism to simultaneously consider weights and, in some rough sense, original aggregation styles, the proposed aggregation mechanism has some significant mathematical properties such as uniform continuity with respect to weights.

The uniformly continuous theorem of fractal interpolation surface function and its proof

<abstract> <p>In order to research uniform continuity of fractal interpolation surface function on a closed rectangular area, the accumulation principle was applied to prove uniform continuity of fractal interpolation surface function on a closed rectangular area. First, fractal interpolation surface function was constructed by affine mapping. Second, the continuous concept of fractal interpolation surface function at a planar point in a three-dimensional cartesian coordinate space system and uniform continuity of fractal interpolation surface function on a closed rectangular area were defined in the paper. Finally, the uniformly continuous theorem of fractal interpolation surface function was proven through accumulation principle in the paper. The conclusion showed that fractal interpolation surface was uniformly continuous function on a closed rectangular area.</p> </abstract>

Sharp well-posedness for the cubic NLS and mKdV in

Abstract We prove that the cubic nonlinear Schrödinger equation (both focusing and defocusing) is globally well-posed in $H^s({{\mathbb {R}}})$ for any regularity $s>-\frac 12$ . Well-posedness has long been known for $s\geq 0$ , see [55], but not previously for any $s<0$ . The scaling-critical value $s=-\frac 12$ is necessarily excluded here, since instantaneous norm inflation is known to occur [11, 40, 48]. We also prove (in a parallel fashion) well-posedness of the real- and complex-valued modified Korteweg–de Vries equations in $H^s({{\mathbb {R}}})$ for any $s>-\frac 12$ . The best regularity achieved previously was $s\geq \tfrac 14$ (see [15, 24, 33, 39]). To overcome the failure of uniform continuity of the data-to-solution map, we employ the method of commuting flows introduced in [37]. In stark contrast with our arguments in [37], an essential ingredient in this paper is the demonstration of a local smoothing effect for both equations. Despite the nonperturbative nature of the well-posedness, the gain of derivatives matches that of the underlying linear equation. To compensate for the local nature of the smoothing estimates, we also demonstrate tightness of orbits. The proofs of both local smoothing and tightness rely on our discovery of a new one-parameter family of coercive microscopic conservation laws that remain meaningful at this low regularity.

Sobolev estimates and inverse Hölder estimates on a class of non-divergence variation-inequality problem arising in American option pricing

<p>We studied the Sobolev estimates and inverse Hölder estimates for a class of variational inequality problems involving divergence-type parabolic operator structures. These problems arise from the valuation analysis of American contingent claim problems. First, we analyzed the uniform continuity of the spatially averaged operator with respect to time in a spherical region and the Sobolev estimates for solutions of the variational inequality. Second, by using spatial and temporal truncation, we obtained the Caccioppoli estimate for the variational inequality and consequently derived the inverse Hölder estimate for the solutions.</p>

Fast Reconstruction Model of the Ship Hull NURBS Surface with Uniform Continuity for Calculating the Hydrostatic Elements

The fast reconstruction of the ship hull nonuniform rational B-spline (NURBS) surface with uniform continuity is essential for calculating hydrostatic elements such as waterplane area and molded volume in real time. Thus, this study proposes a fast reconstruction model with uniform continuity to solve the problem of uniform continuity and splicing in the separate model of hull bow and stern surfaces. The proposed model includes the NURBS curve generation (UCG) algorithm with uniform continuity and the hybrid NURBS surface generation (HSG) algorithm. The UCG algorithm initially fits the feature points using the global interpolation algorithm and then precisely constructs straight-line segments in the curve using the improved flattening algorithm. In comparison, the HSG algorithm adaptively selects the surface knot vectors according to the parameters of the section curves. In this study, the profile of discontinuous compartments is uniformly expressed, effectively avoiding various articulation problems in separation modeling. The results of comparative experiments show that the NURBS surface generated using the HSG algorithm can accurately express the characteristics of various parts of the hull with uniform continuity, and the calculation speed of the proposed model can be increased by up to 8.314% compared with the existing best-performing algorithms. Thus, the proposed model is effective and can improve computational efficiency to a certain extent. The NURBS surfaces generated by the proposed model can be further applied to calculating the hydrostatic elements of hulls and compartments.

New stochastic convergence theorems: Overcoming the limitations of LaSalle theorems

LaSalle theorems, sometimes called as invariance-like theorems, have witnessed abundant applications as powerful tools of analysing system convergence. However, these theorems have twofold limitations: On the one hand, the uniformity of convergence with respect to initial state values cannot be offered, which indicates that the possibility of excessively slow convergence could not be ruled out for a given bounded set of initial state values. On the other hand, unbounded time-variations are not allowed in the systems, and meanwhile, the boundedness of all the states are required even if one merely concerns the convergence of partial states, precluding many scenarios with the unboundedness (e.g., induced by stochastic noise) for some states defined on the whole time horizon. Towards the limitations of LaSalle theorems, this paper seeks to further develop convergence theorems in the stochastic framework. First, a Lyapunov-like function based theorem on the convergence owning the uniformity with respect to initial state values is presented for Itô-type stochastic systems, with the infinitesimal of Lyapunov-like functions exploited more delicately than those in stochastic LaSalle theorems. Based on this, a framework of adaptive stabilization with the uniformity of convergence is established for uncertain stochastic nonlinear systems. In particular, the performance specifications involved are impossible to achieve by applying LaSalle theorems as in the related results. Second, an enlarged convergence theorem is established with Lyapunov-like conditions moderately relaxed, which allows the unboundedness for partial states defined on the whole time horizon and has potential applications in the presence of unbounded time-variations. What is notable, the enlarged convergence theorem is nontrivial to verify, which compels us to propose an extended Barbălat lemma with the conventional requirement of uniform continuity relaxed in the stochastic framework.

Coding of real‐valued continuous functions under WKL$\mathsf {WKL}$

AbstractIn the context of constructive reverse mathematics, we show that weak Kőnig's lemma () implies that every pointwise continuous function is induced by a code in the sense of reverse mathematics. This, combined with the fact that implies the Fan theorem, shows that implies the uniform continuity theorem: every pointwise continuous function has a modulus of uniform continuity. Our results are obtained in Heyting arithmetic in all finite types with quantifier‐free axiom of choice.

The Exponential Attractor for Partially Dissipative Lattice System of Infinite Dimension

With the wide application of fractional differential equations, the study of dynamic systems of fractional differential equations has become one of the hotspots in the past decade. The attractor theory is one of the research directions of infinite dimensional lattice dynamical system. In paper, Eden et al. Systematically introduced the exponential attractor of deterministic autonomous dynamical system, which is a compact positive invariant set with finite fractal dimension and attracting orbits at exponential rate. The exponential attractor has a finite fractal dimension. Because of the exponential attractiveness, the exponential attractor is more stable under perturbation than the global attractor. Fitzhugh Nagumo equation is a nonlinear partial equation that describes the periodic oscillation of neuronal action potential under constant current stimulation above the threshold. For Fitzhugh Nagumo system, there are many reports on attractors, but few reports on exponential attractors. In this paper, the infinite dimensional lattice Fitzhugh Nagumo system [4] is studied. The specific contents are as follows: In part 1, the background, development and research status of FitzHugh Nagumo lattice system and the main work of this paper are introduced. In part 2, the basic concepts, inequalities and lemmas required in the research of FitzHugh Nagumo lattice system are introduced. In part 3, the initial value problem of FitzHugh Nagumo lattice system is studied. For any initial value ϕ(0), there exists a unique solution ϕ(t) of system (9) and family of mapping S(t) can generate continuous operator semigroup {S(t)} t≥0 on ,. Through lemma 3.2,3.3,3.4, it can be proved that the continuous operator semigroup {S(t)} t≥0 satisfies uniform continuity and is asymptotically compact, thus proving the existence of the exponential attractor of the system.

From Classical to Quantum: Uniform Continuity Bounds on Entropies in Infinite Dimensions

We prove a variety of improved uniform continuity bounds for entropies of both classical random variables on an infinite state space and of quantum states of infinite-dimensional systems. We obtain the first tight continuity estimate on the Shannon entropy of random variables with a countably infinite alphabet. The proof relies on a new mean-constrained Fano-type inequality. We then employ this classical result to derive a tight energy-constrained continuity bound for the von Neumann entropy. To deal with more general entropies in infinite dimensions, <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">e.g</i> . α-Rényi and α-Tsallis entropies, we develop a novel approximation scheme based on operator Hölder continuity estimates. Finally, we settle an open problem raised by Shirokov [1], [2] regarding the characterisation of states with finite entropy.

About uniform continuous functions which are not Lipschitz

Abstract To approximate a function, some simpler functions are used: simpler as definitory structure (formula) and also as regularity, continuity and/or smoothness. Any Lipschitz function is a uniformly continuous function, but conversely it not always true. We add to the usual examples some other, especially of the class of differentiable functions.