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In an important paper [24] , Reimann showed that the flow mappings of a continuous vector field of Zygmund class Λ ⁎ are quasisymmetric homeomorphisms. In this paper, we will discuss the flow mappings when the vector field belongs to the smooth Zygmund class λ ⁎ or the Sobolev class H 3 2 . [ABSTRACT FROM AUTHOR]

Let G = ( V , E ) be a connected finite graph. In this short paper, we reinvestigate the Kazdan–Warner equation Δ u = c − h e u with c < 0 on G , where h defined on V is a known function. Grigor'yan, Lin and Yang [3] showed that if the Kazdan–Warner equation has a solution, then h ‾ , the average value of h , is negative. Conversely, if h ‾ < 0 , then there exists a number c − ( h ) < 0 , such that the Kazdan–Warner equation is solvable for every 0 > c > c − ( h ) and it is not solvable for c < c − ( h ) . Moreover, if h ≤ 0 and h ≢ 0 , then c − ( h ) = − ∞ . Inspired by Chen and Li's work [1] , we ask naturally: Is the Kazdan–Warner equation solvable for c = c − ( h ) ? In this paper, we answer the question affirmatively. We show that if c − ( h ) = − ∞ , then h ≤ 0 and h ≢ 0 . Moreover, if c − ( h ) > − ∞ , then there exists at least one solution to the Kazdan–Warner equation with c = c − ( h ) . [ABSTRACT FROM AUTHOR]

In this paper we study a general n -dimensional mixed type differential equation with state dependence. Many known works give the existence of solutions with the so-called Return Condition. In this paper, without requiring the Return Condition, we prove a non-local existence of solutions by using a technique of compactification and the dependence of the maximum of functions on the size of their domains. We apply our result to differential equations with iterates. [ABSTRACT FROM AUTHOR]

The main object of study in the paper is the distance from a point to a line in the Riemannian manifold associated with the Heston model. We reduce the problem of computing such a distance to certain minimization problems for functions of one variable over finite intervals. One of the main ideas in this paper is to use a new system of coordinates in the Heston manifold and the level sets associated with this system. In the case of a vertical line, the formulas for the distance to the line are rather simple. For slanted lines, the formulas are more complicated, and a more subtle analysis of the level sets intersecting the given line is needed. We also find simple formulas for the Heston distance from a point to a level set. As a natural application, we use the formulas obtained in the present paper in the study of the small maturity limit of the implied volatility in the Heston model. [ABSTRACT FROM AUTHOR]

Fatou's lemma is a classic fact in real analysis stating that the limit inferior of integrals of functions is greater than or equal to the integral of the inferior limit. This paper introduces a stronger inequality that holds uniformly for integrals on measurable subsets of a measurable space. The necessary and sufficient condition, under which this inequality holds for a sequence of finite measures converging in total variation, is provided. This statement is called the uniform Fatou lemma, and it holds under the minor assumption that all the integrals in the inequality are well-defined. The uniform Fatou lemma improves the classic Fatou lemma in the following directions: the uniform Fatou lemma states a more precise inequality, it provides the necessary and sufficient condition, and it deals with variable measures. Various corollaries of the uniform Fatou lemma are formulated. The examples in this paper demonstrate that: (a) the uniform Fatou lemma may indeed provide a more accurate inequality than the classic Fatou lemma; (b) the uniform Fatou lemma does not hold if convergence of measures in total variation is relaxed to setwise convergence. [ABSTRACT FROM AUTHOR]

This paper discusses the differentiability of a class of functions associated with eigenvalues and eigenvectors of symmetric matrices. Recursive style formulas of partial derivatives for this class of functions are derived and higher order derivatives can be easily obtained from these formulas. Meanwhile, some interesting characteristics of multiple eigenvalues are revealed. Examples involving inverse eigenvalue problems and primary matrix functions are given to illustrate the applications of the results obtained in this paper. [ABSTRACT FROM AUTHOR]

In a previous paper of the authors (Wang et al. (2014) [40] ), the asymptotic estimates of boundary blow-up solutions were established to the infinity Laplace equation Δ ∞ u = b ( x ) f ( u ) in Ω ⊂ R N , with the nonlinearity 0 ≤ f ∈ C [ 0 , ∞ ) regularly varying at ∞, and the weighted function b ∈ C ( Ω ¯ ) positive in Ω and vanishing on the boundary. The present paper gives a further investigation on the asymptotic behavior of boundary blow-up solutions to the same equation by assuming f to be Γ-varying. Note that a Γ-varying function grows faster than any regularly varying function. We first quantitatively determine the boundary blow-up estimates with the first expansion, relying on the decay rate of b near the boundary and the growth rate of f at infinity, and further characterize these results via examples possessing various decay rates for b and growth rates for f . In particular, we pay attention to the second-order estimates of boundary blow-up solutions. It was observed in our previous work that the second expansion of solutions to the infinity Laplace equation is independent of the geometry of the domain, quite different from the classical Laplacian. The second expansion obtained in this paper furthermore shows a substantial difference on the asymptotic behavior of boundary blow-up solutions between the infinity Laplacian and the classical Laplacian. [ABSTRACT FROM AUTHOR]

In a recent paper, J. Lovejoy and the second author conjectured that ranks for four types of unimodal like sequences satisfy certain inequalities. In this paper, we prove these conjectures asymptotically. For this, we use Wright's Circle Method and analyze the asymptotic behavior of certain general partial theta functions. [ABSTRACT FROM AUTHOR]

This paper provides a partial answer to an open problem recently posed by R. Mesiar and J. Kalická regarding the existence of an n -dimensional Bertino copula with a given diagonal section for any n ⩾ 2 . It is known that for any 2-diagonal function, there exists a 2-dimensional Bertino copula that has the given 2-diagonal function as diagonal section. In the present paper, we introduce the notion of a regular n -diagonal function and we characterise for any n ⩾ 3 the sets D n of regular n -diagonal functions for which there exists an n -dimensional Bertino copula whose diagonal section coincides with the given n -diagonal function. We prove that D n + 1 is strictly included in D n , for all n ⩾ 2 , and that D n is the set of all increasing n / ( n − 1 ) -Lipschitz continuous n -diagonal functions. As a by-product, we show that all marginal copulas of an n -dimensional Bertino copula are Bertino copulas themselves. Examples are given to illustrate the construction of an n -dimensional Bertino copula with a given diagonal section and the characterisation of the sets D n . [ABSTRACT FROM AUTHOR]

This article closes a trilogy on the existence of singular functions with non-zero finite derivatives. In two previous papers, the authors had exhibited a continuous strictly increasing singular function from [ 0 , 1 ] into [ 0 , 1 ] with a derivative that takes non-zero finite values at two different zero-measure sets: first, at the points of an uncountable set; then at the points of a dense set in [ 0 , 1 ] . In the present paper, the possibilities are further stretched as the construction is improved to extend it to an uncountable dense set whose intersection with any interval ( a , b ) has Hausdorff dimension one. Another feature of this third article is the construction of the required function using the most paradigmatic of the singular functions: the Cantor–Lebesgue one. [ABSTRACT FROM AUTHOR]

Fractal interpolation function defined with the aid of iterated function system can be employed to show that any continuous real-valued function defined on a compact interval is a special case of a class of fractal functions (self-referential functions). Elements of the iterated function system can be selected appropriately so that the corresponding fractal function enjoys certain properties. In the first part of the paper, we associate a class of self-referential L p -functions with a prescribed L p -function. Further, we apply our construction of fractal functions in L p -spaces in some approximation problems, for instance, to derive fractal versions of the full Müntz theorems in L p -spaces. The second part of the paper is devoted to identify parameters so that the fractal functions affiliated to a given continuous function satisfy certain conditions, which in turn facilitate them to find applications in some one-sided uniform approximation problems. [ABSTRACT FROM AUTHOR]

In this paper, we pursue the investigations started in [18] where the authors provide a construction of the Dunkl intertwining operator for a large subset of the set of regular multiplicity values. More precisely, we make concrete the action of this operator on homogeneous polynomials when the root system is of dihedral type and under a mild assumption on the multiplicity function. In particular, we obtain a formula for the corresponding Dunkl kernel and another representation of the generalized Bessel function already derived in [7] . When the multiplicity function is everywhere constant, our computations give a solution to the problem of counting the number of factorizations of an element from a dihedral group into a fixed number of (non-necessarily simple) reflections. In the remainder of the paper, we supply another method to derive the Dunkl kernel associated with dihedral systems from the corresponding generalized Bessel function. This time, we use the shift principle together with multiple combinations of Dunkl operators in the directions of the vectors of the canonical basis of R 2 . When the dihedral system is of order six and only in this case, a single combination suffices to get the Dunkl kernel and agrees up to an isomorphism with the formula recently obtained by Amri [2, Lemma 1] in the case of a root system of type A 2 . We finally derive an integral representation for the Dunkl kernel associated with the dihedral system of order eight. [ABSTRACT FROM AUTHOR]

In this paper strict monotonicity, lower and upper uniform monotonicities, uniform monotonicity as well as decreasing and increasing uniform monotonicities of Orlicz spaces equipped with the p -Amemiya norm are studied. Criteria for these six properties in the Orlicz spaces L Φ , p are given in the most general case of Orlicz function Φ and for all 1 ≤ p ≤ ∞ . Finally, some applications of the results to the best dominated approximation problems are presented. This paper is a continuation of the studies from [5–7] . [ABSTRACT FROM AUTHOR]

In this paper, the consistency of the nearest neighbor estimator of the density function based on widely orthant dependent (WOD, in short) samples is investigated. The convergence rate of strong consistency, the complete consistency, the uniformly complete consistency and uniformly strong consistency of the nearest neighbor estimator of the density function based on WOD samples are established. Our results established in the paper generalize or improve the corresponding ones for independent samples and some negatively dependent samples. [ABSTRACT FROM AUTHOR]

In this paper, we give the necessary and sufficient conditions for null curves in E13 to be W-slant helix in terms of their curvature functions. Mainly, throughout this paper relationships between the null W-slant helices and their pseudo spherical images are obtained. Furthermore, some illustrative examples of the null W-slant helices and their pseudo spherical indicatrices in E13 are demonstrated. [ABSTRACT FROM AUTHOR]

Abstract: This paper deals with the second-order cone complementarity problem (SOCCP), which is an important class of problems containing various optimization problems. The SOCCP can be reformulated as a system of nonsmooth equations. For solving this system of nonsmooth equations, smoothing Newton methods are widely used. The Jacobian consistency property plays an important role for achieving a rapid convergence of the methods. In this paper, we show the Jacobian consistency of a smoothed Fischer–Burmeister function. Moreover, we estimate the distance between the subgradient of the Fischer–Burmeister function and the gradient of its smoothing function. [Copyright &y& Elsevier]

Abstract: Analytic signals of finite energy in signal analysis are identical with non-tangential boundary limits of functions in the related Hardy spaces. With this identification this paper studies a subclass of the analytic signals that, with the amplitude-phase representation , , satisfy the relation a.e., signals in this subclass are called mono-components, and, in that case, the phase derivative is called the analytic instantaneous frequency of s. This paper proves that when , where is real-valued, band-limited with minimal bandwidth B and is real-valued, as the restriction on the real line of some entire function, then s is an analytic signal if and only if is a linear function, and with there holds . In the case s is a mono-component. This generalizes the corresponding result obtained by Xia and Cohen in 1999 in which is assumed to be a real-valued polynomial. [Copyright &y& Elsevier]

Abstract: The Cheeger problem for a bounded domain , consists in minimizing the quotients among all smooth subdomains and the Cheeger constant is the minimum of these quotients. Let be the p-torsion function, that is, the solution of torsional creep problem in Ω, on ∂Ω, where is the p-Laplacian operator, . The paper emphasizes the connection between these problems. We prove that . Moreover, we deduce the relation where is a constant depending only of N and , explicitely given in the paper. An eigenfunction of the Dirichlet 1-Laplacian is obtained as the strong limit, as , of a subsequence of the family . Almost all t-level sets of u are Cheeger sets and our estimates of u on the Cheeger set yield , where is the unit ball in . For Ω convex we obtain . [Copyright &y& Elsevier]

Abstract In this paper we introduce some local versions of Bishop–Phelps–Bollobás type property for operators. That is, the function η which appears in their definitions depends not only on a given ε > 0 , but also on either a fixed norm-one operator T or a fixed norm-one vector x. We investigate those properties and show differences between local and uniform versions. [ABSTRACT FROM AUTHOR]

Abstract: In this paper we deal with the open problem of convex combinations of continuous triangular norms stated by Alsina, Frank, and Schweizer [C. Alsina, M.J. Frank, B. Schweizer, Problems on associative functions, Aequationes Math. 66 (2003) 128–140, Problems 5 and 6]. They pose a question whether a non-trivial convex combination of triangular norms can ever be a triangular norm. The main result of this paper gives a negative answer to the question for any pair of continuous Archimedean triangular norms with different supports. With the help of this result we show that a non-trivial convex combination of nilpotent t-norms is never a t-norm. The main result also gives an alternative proof to the result presented by Ouyang and Fang [Y. Ouyang, J. Fang, Some observations about the convex combination of continuous triangular norms, Nonlinear Anal., 68 (11) (2008) 3382–3387, Theorem 3.1]. In proof of the main theorem we utilize the Reidmeister condition known from the web geometry. [Copyright &y& Elsevier]

Abstract: In this paper we study iterative roots of PM functions, a special class of non-monotone functions. Problem 2 in [W. Zhang, PM functions, their characteristic intervals and iterative roots, Ann. Polon. Math. LXV (1997) 119–128] is solved partly and Theorem 4 in that paper is generalized. [Copyright &y& Elsevier]

Abstract: The paper introduces a new approach to dynamic modeling, using the variation principle, applied to a functional on trajectories of a controlled random process, and its connection to the process'' information functional. In [V.S. Lerner, Dynamic approximation of a random information functional, J. Math. Anal. Appl. 327 (1) (2007) 494–514, available online 5-24-06], we presented the information path functional with the Lagrangian, determined by the parameters of a controlled stochastic equation. In this paper, the solution to the path functional''s variation problem provides both a dynamic model of a random process and the model''s optimal control, which allows us to build a two-level information model with a random process at the microlevel and a dynamic process at the macrolevel. A wide class of random objects, modeled by the Markov diffusion process and a common structure of the process'' information functional, leads to a universal information structure of the dynamic model, which is specified and identified on a particular object with the applied optimal control functions. The developed mathematical formalism, based on classical methods, is aimed toward the solution of problems identification, combined with an optimal control synthesis, which is practically implemented and also demonstrated in the paper''s example. [Copyright &y& Elsevier]

Abstract: In this paper, by means of a series of counterexamples, we study in a systematic way the relationships among (pseudo, quasi) α-preinvexity, (strict, strong, pseudo, quasi) α-invexity and (strict, strong, pseudo, quasi) αη-monotonicity. Results obtained in this paper can be viewed as a refinement and improvement of the results of Noor and Noor [M.A. Noor, K.I. Noor, Some characterizations of strongly preinvex functions, J. Math. Anal. Appl. 316 (2006) 697–706]. [Copyright &y& Elsevier]

Abstract: Ruelle operator defined by weakly contractive iterated function systems (IFS) satisfying the open set condition was discussed in the paper [K.S. Lau, Y.L. Ye, Ruelle operator with nonexpansive IFS, Studia Math. 148 (2001) 143–169]. There, one of our theorems gave a sufficient condition for the possession of the Perron–Frobenius property. In this paper we consider Ruelle operator defined by nonexpansive IFS on the line instead of by weakly contractive one. And we prove, under the same condition, that the newly defined Ruelle operator has the Perron–Frobenius property. It extends the Ruelle–Perron–Frobenius theorem partially to the nonexpansive IFS. [Copyright &y& Elsevier]

Abstract: The paper is devoted to the maximum principles for functional equations in the space of measurable essentially bounded functions. The necessary and sufficient conditions for validity of corresponding maximum principles are obtained in a form of theorems about functional inequalities similar to the classical theorems about differential inequalities of the Vallee Poussin type. Assertions about the strong maximum principle are proposed. All results are also true for difference equations, which can be considered as a particular case of functional equations. The problems of validity of the maximum principles are reduced to nonoscillation properties and disconjugacy of functional equations. Note that zeros and nonoscillation of a solution in a space of discontinuous functions are defined in this paper. It is demonstrated that nonoscillation properties of functional equations are connected with the spectral radius of a corresponding operator acting in the space of essentially bounded functions. Simple sufficient conditions of nonoscillation, disconjugacy and validity of the maximum principles are proposed. The known nonoscillation results for equation in space of functions of one variable follow as a particular cases of these assertions. It should be noted that corresponding coefficient tests obtained on this basis cannot be improved. Various applications to nonoscillation, disconjugacy and the maximum principles for partial differential equations are proposed. [Copyright &y& Elsevier]

Abstract: In the present paper, a class of critically finite transcendental meromorphic functions having rational Schwarzian derivative is introduced and the dynamics of functions in one parameter family is investigated. It is found that there exist two parameter values and , where and is the real root of , such that the Fatou sets of for and contain parabolic domains. A computationally useful characterization of the Julia set of the function as the complement of the basin of attraction of an attracting real fixed point of is established and applied for the generation of the images of the Julia sets of . Further, it is observed that the Julia set of explodes to whole complex plane for . Finally, our results found in the present paper are compared with the recent results on dynamics of one parameter families , [R.L. Devaney, L. Keen, Dynamics of meromorphic maps: Maps with polynomial Schwarzian derivative, Ann. Sci. École Norm. Sup. 22 (4) (1989) 55–79; L. Keen, J. Kotus, Dynamics of the family , Conform. Geom. Dynam. 1 (1997) 28–57; G.M. Stallard, The Hausdorff dimension of Julia sets of meromorphic functions, J. London Math. Soc. 49 (1994) 281–295] and , [G.P. Kapoor, M. Guru Prem Prasad, Dynamics of : The Julia set and bifurcation, Ergodic Theory Dynam. Systems 18 (1998) 1363–1383]. [Copyright &y& Elsevier]

Abstract: As an important iteration, the Mann and Ishikawa iteration has extensive application in fixed point theory. In 1991, David Borwein and Jonathan Borwein proved the convergence of the Mann iteration on a closed bounded interval in their paper. In this paper, we will extend their result to an arbitrary interval and to the Ishikawa iteration, indicating the necessary and sufficient condition for the convergence of Ishikawa iteration of continuous functions on an arbitrary interval. [Copyright &y& Elsevier]

Abstract: In this paper we investigate the distributional Chébli–Trimèche transforms. We use the so-called kernel method and we are inspired by the papers of Dube and Pandey [L.S. Dube, J.N. Pandey, On the Hankel transform of distributions, Tôhoku Math. J. 27 (1975) 337–354] and Koh and Zemanian [E.L. Koh, A.H. Zemanian, The complex Hankel and I-transformations of generalized functions, SIAM J. Appl. Math. 16 (1968) 945–957] about distributional Hankel transforms. We note that our procedure, supported in a representation of the elements in the corresponding dual spaces, is simpler than the methods described in the above mentioned papers. Some applications of our distributional theory are presented. [Copyright &y& Elsevier]

Abstract: This paper considers Gibbs'' phenomenon for scaling vectors in . We first show that a wide class of multiresolution analyses suffer from Gibbs'' phenomenon. To deal with this problem, in [Contemp. Math. 216 (1998) 63–79], Walter and Shen use an Abel summation technique to construct a positive scaling function , , from an orthonormal scaling function φ that generates . A reproducing kernel can in turn be constructed using . This kernel is also positive, has unit integral, and approximations utilizing it display no Gibbs'' phenomenon. These results were extended to scaling vectors and multiwavelets in [Proceedings of Wavelet Analysis and Multiresolution Methods, 2000, pp. 317–339]. In both cases, orthogonality and compact support were lost in the construction process. In this paper we modify the approach given in [Proceedings of Wavelet Analysis and Multiresolution Methods, 2000, pp. 317–339] to construct compactly supported positive scaling vectors. While the mapping into associated with this new positive scaling vector is not a projection, the scaling vector does produce a Riesz basis for and we conclude the paper by illustrating that expansions of functions via positive scaling vectors exhibit no Gibbs'' phenomenon. [Copyright &y& Elsevier]

Abstract: In this paper we will establish some oscillation criteria for the second-order nonlinear neutral delay dynamic equation on a time scale ; here is a quotient of odd positive integers with and real-valued positive functions defined on . To the best of our knowledge nothing is known regarding the qualitative behavior of these equations on time scales, so this paper initiates the study. [Copyright &y& Elsevier]

The paper presents a formula for the γ-interior of a set under special conditions for γ :expX→expX, more general than those in the previous paper [Acta Math. Hungar. 80 (1998) 89–93]. There are also some applications. [Copyright &y& Elsevier]

Guo (Approx. Theory Appl. 4 (1988) 9–18) introduced the integral modification of Meyer-Konig and Zeller operators Mˆn and studied the rate of convergence for functions of bounded variation. In this paper we introduce the Be´zier variant of these integrated MKZ operators and study the rate of convergence by means of the decomposition technique of functions of bounded variation together with some results of probability theory and the exact bound of MKZ basis functions. Recently, Zeng (J. Math. Anal. Appl. 219 (1998) 364–376) claimed to improve the results of Guo and Gupta (Approx. Theory Appl. 11 (1995) 106–107), but there is a major mistake in the paper of Zeng. For special case our main theorem gives the correct estimate on the rate of convergence, over the result of Zeng. [Copyright &y& Elsevier]

In this paper we consider a viscoelastic laminated beam model. This structure is given by two identical uniform layers on top of each other, taking into account that an adhesive of small thickness is bonding the two surfaces and produces an interfacial slip. We use minimal and general conditions on the relaxation function and establish explicit energy decay formula which gives the best decay rates expected under this level of generality. Our new result generalizes the earlier related results in the literature. [ABSTRACT FROM AUTHOR]

A famous theorem of Coburn says that a nonzero Toeplitz operator on the Hardy space of the unit disk is injective or its adjoint operator is injective. In this paper we study the corresponding problem on the Hardy space of the bidisk. We first show the Coburn type theorem fails generally on the bidisk. But, we show that certain pluriharmonic symbols or product symbols of one variable functions induce Toeplitz operators satisfying the Coburn type theorem. [ABSTRACT FROM AUTHOR]

In this paper we give necessary and sufficient conditions for the convergence of Kurzweil–Stieltjes integrals with respect to regulated functions, using the notion of asymptotical equiintegrability. One thus generalizes several well-known convergence theorems. As applications, we provide existence and closure results for integral problems driven by regulated functions, both in single- and set-valued cases. In the particular setting of bounded variation functions driving the equations, we get features of the solution set of measure integrals problems. [ABSTRACT FROM AUTHOR]

In this paper, we study a type of polynomial Liénard system of degree m ( m ≥ 2 ) with polynomial perturbations of degree n . We prove that the first order Melnikov function of such system has at most n + 1 − [ n + 1 m + 1 ] independent perturbation parameters which can be used to simplify this kind of systems. As an application, we study a type of Lienard systems for m = 4 , n = 19 , 28 and obtain the new lower bounds of the maximal number of limit cycles. [ABSTRACT FROM AUTHOR]

Let N ( a , b , c , d ; n ) and t ( a , b , c , d ; n ) denote the number of representations of n as a x 2 + b y 2 + c z 2 + d w 2 and the number of representations of n as a x ( x + 1 ) 2 + b y ( y + 1 ) 2 + c z ( z + 1 ) 2 + d w ( w + 1 ) 2 , respectively, where a , b , c , d are positive integers, n is a nonnegative integer and x , y , z , w are integers. Sun established many relations between N ( a , b , c , d ; n ) and t ( a , b , c , d ; n ) and posed 23 conjectures. Yao proved five of them by using ( p , k ) -parametrization of theta functions. In this paper, we confirm four conjectures of Sun by employing theta function identities. [ABSTRACT FROM AUTHOR]

The main aim of this paper is to bridge two directions of research generalizing asymptotic density zero sets. This enables to transfer results concerning one direction to the other one. Consider a function g : ω → [ 0 , ∞ ) such that lim n → ∞ ⁡ g ( n ) = ∞ and n g ( n ) does not converge to 0. Then the family Z g = { A ⊆ ω : lim n → ∞ ⁡ card ( A ∩ n ) g ( n ) = 0 } is an ideal called simple density ideal (or ideal associated to upper density of weight g ). We compare this class of ideals with Erdős–Ulam ideals. In particular, we show that there are ⊑-antichains of size c among Erdős–Ulam ideals which are and are not simple density ideals (in [12] it is shown that there is also such an antichain among simple density ideals which are not Erdős–Ulam ideals). We characterize simple density ideals which are Erdős–Ulam as those containing the classical ideal of sets of asymptotic density zero. We also characterize Erdős–Ulam ideals which are simple density ideals. In the latter case we need to introduce two new notions. One of them, called increasing-invariance of an ideal I , asserts that given B ∈ I and C ⊆ ω with card ( C ∩ n ) ≤ card ( B ∩ n ) for all n , we have C ∈ I . This notion is inspired by [3] and is later applied in [12] for a partial solution of [15, Problem 5.8] . Finally, we pose some open problems. [ABSTRACT FROM AUTHOR]

This paper is concerned with the study of harmonic measure on quasicircles and symmetric quasicircles. We investigate how the harmonic measure changes from one side of a Jordan curve to the other side. In particular, we characterize quasicircles and symmetric quasicircles using a type of harmonic symmetry property. We also explore the connection between quasicircles and the harmonic reflection property. [ABSTRACT FROM AUTHOR]

The main purpose of this paper is to explore normality in terms of distances between points and sets. We prove some important consequences on realvalued contractions, i.e. functions not enlarging the distance, showing that as in the classical context of closures and continuous maps, normality in terms of distances based on an appropriate numerical notion of γ-separation of sets , has far reaching consequences on real valued contractive maps, where the real line is endowed with the Euclidean metric. We show that normality is equivalent to (1) separation of γ -separated sets by some Urysohn contractive map , (2) to Katětov–Tong's insertion , stating that for bounded positive realvalued functions, between an upper and a larger lower regular function, there exists a contractive interpolating map and (3) to Tietze's extension theorem stating that certain contractions defined on a subspace can be contractively extended to the whole space. The appropriate setting for these investigations is the category of approach spaces, but the results have (quasi)-metric counterparts in terms of non-expansive maps. Moreover when restricted to topological spaces, classical normality and its equivalence to separation by a Urysohn continuous map, to Katětov–Tong's insertion for semicontinuous maps and to Tietze's extension theorem for continuous maps are recovered. [ABSTRACT FROM AUTHOR]

In this paper, we provide several characterizations on weak recurrence of translation operators on weighted Lebesgue spaces and on continuous function spaces. It is shown that the translation operator admits a nonzero nonwandering point if and only if it has a nonzero recurrent point, if and only if it is hypercyclic; that the translation operator admits a nonzero almost periodic point if and only if it is Devaney chaotic. Moreover, every hypercyclic translation operator has infinite topological entropy, but the converse is not true. Finally, we remark that there exists a linear operator which has the specification property on the whole space. [ABSTRACT FROM AUTHOR]

In this paper, we give the possible orders of growth of solutions to certain linear differential equations near a finite singular point. For that, an asymptotic equality of Wiman–Valiron type is proved for the derivatives of functions which are analytic in the closed complex plane except at a finite singular point. [ABSTRACT FROM AUTHOR]

Perspective functions arise explicitly or implicitly in various forms in applied mathematics and in statistical data analysis. To date, no systematic strategy is available to solve the associated, typically nonsmooth, optimization problems. In this paper, we fill this gap by showing that proximal methods provide an efficient framework to model and solve problems involving perspective functions. We study the construction of the proximity operator of a perspective function under general assumptions and present important instances in which the proximity operator can be computed explicitly or via straightforward numerical operations. These results constitute central building blocks in the design of proximal optimization algorithms. We showcase the versatility of the framework by designing novel proximal algorithms for state-of-the-art regression and variable selection schemes in high-dimensional statistics. [ABSTRACT FROM AUTHOR]

We investigate several possibilities of obtaining a Łojasiewicz inequality for definable multifunctions and give some examples of applications thereof. In particular, we prove that the Hausdorff distance and its extension to closed sets is definable when composed with definable multifunctions. This allows us to obtain Łojasiewicz-type inequalities for definable multifunctions obtained from Clarke's subgradient or the tangent cone. The paper ends with a Łojasiewicz-type subgradient inequality in the spirit of Bolte–Daniilidis–Lewis–Shiota or Phạm. [ABSTRACT FROM AUTHOR]

In this paper we consider the question of under which conditions multi-valued dynamical systems admit invariant measures. We give results on the existence of invariant measures with full support on orbit spaces of multi-valued dynamical systems. We use these measures on the orbit space to induce measures on the original dynamical system. We focus on the question of when a non-atomic invariant measure on the orbit space induces an atomic invariant measure on the multi-valued dynamical system. This phenomenon is an indicator of complicated multi-periodic behaviour. [ABSTRACT FROM AUTHOR]

In this paper we prove that the ball is a maximizer of the Schatten p -norm of some convolution type integral operators with non-increasing kernels among all domains of a given measure in R d . We also show that the equilateral triangle has the largest Schatten p -norm among all triangles of a given area. Some physical motivations for our results are also presented. [ABSTRACT FROM AUTHOR]

Let ψ n = ( − 1 ) n − 1 ψ ( n ) for n ≥ 0 , where ψ ( n ) stands for the psi and polygamma functions. For p , q ∈ R and ρ = min ⁡ ( p , q ) , let D [ x + p , x + q ; ψ n − 1 ] ≡ − ϕ n ( x ) be the divided difference of the functions ψ n − 1 for x > − ρ . In this paper, we establish the necessary and sufficient conditions for the function Φ n ( x , λ ) = ϕ n + 1 ( x ) 2 − λ ϕ n ( x ) ϕ n + 2 ( x ) to be completely monotonic on ( − ρ , ∞ ) . In particular, we find that the function ψ n + 1 2 / ( ψ n ψ n + 2 ) is strictly decreasing from ( 0 , ∞ ) onto ( n / ( n + 1 ) , ( n + 1 ) / ( n + 2 ) ) . These not only generalize and strengthen some known results, but also yield many new and interesting ones. [ABSTRACT FROM AUTHOR]

Let β > 1 be a real number. For any x ∈ [ 0 , 1 ] , the run-length function r n ( x , β ) is defined as the length of the longest run of 0's amongst the first n digits in the β -expansion of x . Let { δ n } n ≥ 1 be a non-decreasing sequence of integers and define E ( { δ n } n ≥ 1 ) = { x ∈ [ 0 , 1 ] : lim sup n → ∞ r n ( x , β ) δ n = 1 } . In this paper, we show that dim H ⁡ E ( { δ n } n ≥ 1 ) = max ⁡ { 0 , 1 − lim inf n → ∞ δ n ⧸ n } . Using the same method, we also study a class of extremely refined subset of the exceptional set in Erdös–Rényi limit theorem. Precisely, we prove that if lim inf n → ∞ δ n n = 0 , then the set E max ( { δ n } n ≥ 1 ) = { x ∈ [ 0 , 1 ] : lim inf n → ∞ r n ( x , β ) δ n = 0 , lim sup n → ∞ r n ( x , β ) δ n = + ∞ } has full Hausdorff dimension. [ABSTRACT FROM AUTHOR]

In this paper, by Karamata regular variation theory and constructing super and subsolutions, we obtain the asymptotic boundary behavior of convex solutions of Monge–Ampère equations with singular righthand sides. [ABSTRACT FROM AUTHOR]

In this paper we give the necessary and sufficient conditions for a function f to be an upper Bohl function of the diagonal discrete linear time-varying systems. [ABSTRACT FROM AUTHOR]