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Abstract: Huang and Zhang reviewed cone metric spaces in 2007 [Huang Long-Guang, Zhang Xian, Cone metric spaces and fixed point theorems of contractive mappings, J. Math. Anal. Appl. 332 (2007) 1468–1476]. We shall prove that there are no normal cones with normal constant and for each there are cones with normal constant . Also, by providing non-normal cones and omitting the assumption of normality in some results of [Huang Long-Guang, Zhang Xian, Cone metric spaces and fixed point theorems of contractive mappings, J. Math. Anal. Appl. 332 (2007) 1468–1476], we obtain generalizations of the results. [Copyright &y& Elsevier]

In this paper, we investigate a class of non-autonomous degenerate p -Laplacian equations ∂ t u − div ( a ( x ) | ∇ u | p − 2 ∇ u ) + λ u + f ( u ) = g ( x , t ) in Ω, where a ( x ) is allowed to vanish on a nonempty closed subset with Lebesgue measure zero, g ( x , t ) ∈ L l o c p ′ ( R ; D − 1 , p ′ ( Ω , a ) ) and Ω an unbounded domain in R N . We first establish the well-posedness of these equations by constructing a compact embedding. Then we show the existence of the minimal pullback D μ -attractor, and prove that it indeed attracts the D μ class in L 2 + δ -norm for any δ ∈ [ 0 , ∞ ) . Our results extend some known ones in previously published papers. [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]

This paper considers matrix convex sets invariant under several types of rotations. It is known that matrix convex sets that are free semialgebraic are solution sets of Linear Matrix Inequalities (LMIs); they are called free spectrahedra. We classify all free spectrahedra that are circular, that is, closed under multiplication by e i t : up to unitary equivalence, the coefficients of a minimal LMI defining a circular free spectrahedron have a common block decomposition in which the only nonzero blocks are on the superdiagonal. A matrix convex set is called free circular if it is closed under left multiplication by unitary matrices. As a consequence of a Hahn–Banach separation theorem for free circular matrix convex sets, we show the coefficients of a minimal LMI defining a free circular free spectrahedron have, up to unitary equivalence, a block decomposition as above with only two blocks. This paper also gives a classification of those noncommutative polynomials invariant under conjugating each coordinate by a different unitary matrix. Up to unitary equivalence such a polynomial must be a direct sum of univariate polynomials. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we introduce BBI spaces (“big balls of itself”), which based on the notion of BPI spaces (“big pieces of itself”) used by David and Semmes to study self-similarity. We prove that the “self-similar” construction described by BBI spaces ensures the equivalence of positive Lebesgue measure and nonempty interior. We apply this result to self-conformal sets satisfying the WSC and prove that positive Lebesgue measure implies nonempty interior for such sets. This generalizes Zerner's corresponding result for self-similar sets. [ABSTRACT FROM AUTHOR]

In this paper we study the existence of Lelong numbers of m -subharmonic currents of bidimension ( p , p ) on an open subset of C n , when m + p ≥ n . In the special case of m -subharmonic function φ , we give a relationship between the Lelong numbers of d d c φ and the mean values of φ on spheres and balls. As an application we study the integrability exponent of φ . We express the integrability exponent of φ in terms of volume of sub-level sets of φ and we give a link between this exponent and its Lelong number. [ABSTRACT FROM AUTHOR]

Perfect dyadic operators were first introduced in [1] , where a local T ( b ) theorem was proved for such operators. In [3] it was shown that for every singular integral operator T with locally bounded kernel on R n × R n there exists a perfect dyadic operator T such that T − T is bounded on L p ( d x ) for all 1 < p < ∞ . In this paper we show a decomposition of perfect dyadic operators on real line into four well known operators: two selfadjoint operators, paraproduct and its adjoint. Based on this decomposition we prove a sharp weighted version of the T ( 1 ) theorem for such operators, which implies A 2 conjecture for such operators with constant which only depends on ‖ T ( 1 ) ‖ BMO d , ‖ T ⁎ ( 1 ) ‖ BMO d and the constant in testing conditions for T . Moreover, the constant depends on these parameters at most linearly. In this paper we also obtain sufficient conditions for the two weight boundedness for a perfect dyadic operator and simplify these conditions under additional assumptions that weights are in the Muckenhoupt class A ∞ d . [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]

Motivated by the study of the distribution of zeros of generalized Bessel-type functions, the principal goal of this paper is to identify new research directions in the theory of multiplier sequences. The investigations focus on multiplier sequences interpolated by functions which are not entire and sums, averages and parametrized families of multiplier sequences. The main results include (i) the development of a ‘logarithmic’ multiplier sequence and (ii) several integral representations of a generalized Bessel-type function utilizing some ideas of G.H. Hardy and L.V. Ostrovskii. The explorations and analysis, augmented throughout the paper by a plethora of examples, led to a number of conjectures and intriguing open problems. [ABSTRACT FROM AUTHOR]

In this paper, we improve upon some observations made in recent papers on the subject of increasingly flat interpolation. We shall establish that the corresponding Lagrange functions converge both for a finite set of functions (collocation matrix) and also for kernels (Fredholm matrix). In our analysis, we use a finite Maclaurin expansion of a multivariate function with remainder and some additional matrix theoretic facts. [ABSTRACT FROM AUTHOR]

This paper studies the relations between Pesin–Pitskel topological pressure on an arbitrary subset and measure theoretic pressure of Borel probability measures, which extends Feng and Huang's recent result on entropies [13] for pressures. More precisely, this paper defines the measure theoretic pressure P μ ( T , f ) for any Borel probability measure, and shows that P B ( T , f , K ) = sup ⁡ { P μ ( T , f ) : μ ∈ M ( X ) , μ ( K ) = 1 } , where M ( X ) is the space of all Borel probability measures, K ⊆ X is a non-empty compact subset and P B ( T , f , K ) is the Pesin–Pitskel topological pressure on K . Furthermore, if Z ⊆ X is an analytic subset, then P B ( T , f , Z ) = sup ⁡ { P B ( T , f , K ) : K ⊆ Z is compact } . This paper also shows that Pesin–Pitskel topological pressure can be determined by the measure theoretic pressure. [ABSTRACT FROM AUTHOR]

Mihail and Miculescu introduced the notion of a generalized iterated function system (GIFS in short), and proved that every GIFS generates an attractor. (In our previous paper we gave this notion a more general setting.) In this paper we show that for any m ≥ 2 , there exists a Cantor subset of the plane which is an attractor of some GIFS of order m , but is not an attractor of a GIFS of order m − 1 . In particular, this result shows that there is a subset of the plane which is an attractor of some GIFS, but is not an attractor of an IFS. We also give an example of a Cantor set which is not an attractor of a GIFS. [ABSTRACT FROM AUTHOR]

Abstract In this paper we study the behavior of holomorphic mappings on A -compact sets. Motivated by the recent work of Aron, Çalişkan, García and Maestre (2016), we give several conditions (on the holomorphic mappings and on the λ -Banach operator ideal A) under which A -compact sets are preserved. Appealing to the notion of tensor stability for operator ideals, we first address the question in the polynomial setting. Then, we define a radius of (A ; B) -compactification that permits us to tackle the analytic case. Our approach, for instance, allows us to show that the image of any (p , r) -compact set under any holomorphic function (defined on any open set of a Banach space) is again (p , r) -compact. [ABSTRACT FROM AUTHOR]

Abstract In this paper we prove that the higher order Lipschitz classes behave invariant under the action of a singular integral operator naturally arising in polyanalytic function theory. This result provides a generalization of the well-known theorem by Joseph Plemelj [16] and Ivan Privalov [17]. [ABSTRACT FROM AUTHOR]

Abstract This paper investigates the function theoretic properties of two reproducing kernel functions based on the Mittag-Leffler function that are related through a composition. Both spaces provide one parameter generalizations of the traditional Bargmann–Fock space. In particular, the Mittag-Leffler space of entire functions yields many similar properties to the Bargmann–Fock space, and several results are demonstrated involving zero sets and growth rates. The second generalization, the Mittag-Leffler space of the slitted plane, is a reproducing kernel Hilbert space (RKHS) of functions for which Caputo fractional differentiation and multiplication by z q (for q > 0) are densely defined adjoints of one another. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we derive a Reilly-type inequality for the drifting Laplacian operator L on a self-shrinker of the mean curvature flow. Using this Reilly-type inequality, we obtain some new Poincaré-type inequalities not only on the self-shrinker, but more interestingly, also on its boundary. Moreover, we get some eigenvalue estimates for L operator on the compact self-shrinker and its boundary, and also make some global estimates on generalized mean curvature H ρ on the boundary. [ABSTRACT FROM AUTHOR]

Abstract In this paper we show that each reflexive Banach space (X , ‖ ⋅ ‖) with a Schauder basis has an equivalent norm ‖ ⋅ ‖ 0 such that the Banach space (X , ‖ ⋅ ‖ 0) contains a diametrically complete set with empty interior. [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]

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]

The transcendental equation x e x = z and its solutions, described by the Lambert W function, often occur in physics and mathematics. In the last decades it turned out that the study of similar but more general equations is necessary in molecular physics, in the theory of general relativity and also in the description of Bose–Fermi mixtures as well as in some combinatorial problems. In this paper we offer a full description for the solution set of a one parameter generalization of the above mentioned equation. [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with subquadratic homogeneity elliptic problems with VMO-coefficients in divergence form. We obtain that the weak solution u is locally Hölder continuous besides a singular set by using A -harmonic approximation, where the Hölder exponent α ∈ ( 0 , 1 ) . The Hausdorff dimension of the singular set is less than n − p . [ABSTRACT FROM AUTHOR]

In this article, by introducing a new set and a new concept of φ − ( h , e ) -concave operators, and by using the cone theory and monotone iterative method, we present some new existence and uniqueness theorems of fixed points for increasing φ − ( h , e ) -concave operators without requiring the existence of upper and lower solutions. As an application, we establish the existence and uniqueness of a nontrivial solution for a new form of fractional differential equation with integral boundary conditions. The main results of this paper improve and extend some known results, and present a new method to study nonlinear equation problems. [ABSTRACT FROM AUTHOR]

This paper considers the radii functionals (circumradius, inradius, and diameter) as well as the Minkowski asymmetry for general (possibly non-symmetric) gauge bodies. A generalization of the concentricity inequality (which states that the sum of the inradius and circumradius is not greater than the diameter in general Minkowski spaces) for non-symmetric gauge bodies is derived and a strong connection between this new inequality, extremal sets of the generalized Bohnenblust inequality, and completeness of simplices is revealed. [ABSTRACT FROM AUTHOR]

This paper answers two open problems raised by Mahlon M. Day [4] and James C.S. Wong [17] : Does uniformly topological left amenability imply the existence of left translation continuous measures on a locally compact semitopological semigroup? Let T be a locally compact Borel subsemigroup of a locally compact semitopological semigroup S . Is the existence of a topological T -invariant mean M on S with M ( χ T ) > 0 enough to imply the topological left amenability of T ? We give a negative answer to the first question by providing a counterexample and a positive proof to the second. This example can also be used to show that the property ( α ) provided in Gerard L. Sleijpen [14] can not be removed in one of his results. [ABSTRACT FROM AUTHOR]

In this paper we obtain the pointwise gradient estimates via the nonlinear Wolff potentials for weak solutions of a class of non-homogeneous quasilinear elliptic equations with measure data. [ABSTRACT FROM AUTHOR]

In this paper, the stability of a certain class of time-varying systems evolving over nonuniformly spaced discrete domains is analyzed. Switched systems, used here in the context of dynamic equations over time scale domains, arise naturally in applications when a continuous time system is discretized via a sample-and-hold method with multiple sample rates. The stability of switched systems is typically deduced by appealing to certain interrelated properties of the subsystems (such as pairwise commutativity [24] , simultaneous diagonalization [7] , simultaneous triangularizability [8] , or other Lie algebraic conditions [1] ) which imply the existence of a common quadratic Lyapunov function. A novel approach is used here to determine the existence of quadratic Lyapunov functions which does not rely on how the subsystems interact with each other. This new method instead examines the role they each play in the aggregate system by way of singular value conditions. Results implying switched system stability and instability are developed under the two primary methods of examining such systems: how the system behaves during arbitrary switching and how it behaves under the influence of a particular switching signal. Several examples illuminating the results are provided throughout the text, as well as important insight discussing certain bounds on these results as they apply to stability theory in general. [ABSTRACT FROM AUTHOR]

In this paper, we discuss the continuity property of the α -core with respect to data perturbations in different environments. We show that some collections of abstract economies (or normal games) with the nonempty α -core have the Hadamard well-posedness property. We also show that the α -core points of every game in a (dense) residual subset are essential. [ABSTRACT FROM AUTHOR]

This paper studies the excursion set of a real stationary isotropic Gaussian random field above a fixed level. We show that the standardized Lipschitz–Killing curvatures of the intersection of the excursion set with a window converges in distribution to a normal distribution as the window grows to the d -dimensional Euclidean space. Moreover a lower bound for the asymptotic variance is derived. [ABSTRACT FROM AUTHOR]

This paper concerns with a class of elliptic equations on fractal domains depending on a real parameter. Our approach is based on variational methods. More precisely, the existence of at least two non-trivial weak (strong) solutions for the treated problem is obtained exploiting a local minimum theorem for differentiable functionals defined on reflexive Banach spaces. A special case of the main result improves a classical application of the Mountain Pass Theorem in the fractal setting, given by Falconer and Hu in [14] . [ABSTRACT FROM AUTHOR]

Locally L 0 -convex modules were introduced in Filipovic et al. (2009) [10] as the analytic basis for the study of conditional risk measures. Later, the algebra of conditional sets was introduced in Drapeau et al. (2016) [8] . In this paper we study locally L 0 -convex modules, and find exactly which subclass of locally L 0 -convex modules can be identified with the class of locally convex vector spaces within the context of conditional set theory. Second, we provide a version of the classical James' theorem of characterization of weak compactness for conditional Banach spaces. Finally, we state a conditional version of the Fatou and Lebesgue properties for conditional convex risk measures and, as application of the developed theory, we establish a version of the so-called Jouini–Schachermayer–Touzi theorem for robust representation of conditional convex risk measures defined on a L ∞ -type module. [ABSTRACT FROM AUTHOR]

This paper focuses on the following generalized quasilinear Schrödinger equations − div ( g 2 ( u ) ∇ u ) + g ( u ) g ′ ( u ) | ∇ u | 2 + V ( x ) u = f ( x , u ) , x ∈ R N , where N ≥ 3 , g ( s ) : R → R + is a nondecreasing function with respect to | s | . By using a change of variables and variational methods, we obtain the existence and multiplicity of nontrivial solutions for the above problem when the nonlinearity is superlinear but does not satisfy the Ambrosetti–Rabinowitz type condition. [ABSTRACT FROM AUTHOR]

Abstract: In this paper we find some new conditions to ensure the existence of infinitely many nontrivial solutions for the Dirichlet boundary value problems of the form in a bounded smooth domain. Conditions – in the present paper are somewhat weaker than the famous Ambrosetti–Rabinowitz-type superquadratic condition. Here, we assume that the primitive of the nonlinearity g is either asymptotically quadratic or superquadratic as . [Copyright &y& Elsevier]

Abstract: In the paper Generalized roundness and negative type, Lennard, Tonge, and Weston show that the geometric notion of generalized roundness in a metric space is equivalent to that of negative type. Using this equivalent characterization, along with classical embedding theorems, the authors prove that for , fails to have generalized roundness q for any . It is noted, as a consequence, that the Schatten class , for , has maximal generalized roundness 0. In this paper, we prove that this result remains true for p in the interval . [Copyright &y& Elsevier]

Abstract: We prove an omega estimate related to the general Euler totient function associated to a polynomial Euler product satisfying some natural analytic properties. For convenience, we work with a set of L-functions similar to the Selberg class, but in principle our results can be proved in a still more general setup. In a recent paper the authors treated a special case of Dirichlet L-functions with real characters. Greater generality of the present paper invites new technical difficulties. Effectiveness of the main theorem is illustrated by corollaries concerning Euler totient functions associated to the shifted Riemann zeta function, shifted Dirichlet L-functions and shifted L-functions of modular forms. Results are either of the same quality as the best known estimates or are entirely new. [Copyright &y& Elsevier]

Abstract: We consider a generalized Gause prey–predator model with T-periodic continuous coefficients. In the case where the Poincaré map over time T is well defined, the result of the paper can be explained as follows: we locate a subset U of such that the topological degree equals to +1. The novelty of the paper is that the later is done under only continuity and (some) monotonicity assumptions for the coefficients of the model. A suitable integral operator is used in place of the Poincaré map to cope with possible nonuniqueness of solutions. The paper, therefore, provides a new framework for studying the generalized Gause model with functional differential perturbations and multi-valued ingredients. [Copyright &y& Elsevier]

Abstract: The present paper contains a martingale representation theorem for set-valued martingales defined on a filtered probability space with a filtration generated by a Brownian motion. It is proved that such type martingales can be defined by some generalized set-valued stochastic integrals with respect to a given Brownian motion. The main result of the paper is preceded by short part devoted to the definition and some properties of generalized set-valued stochastic integrals. [Copyright &y& Elsevier]

Abstract: In this paper, we consider the following problem: where and has critical growth. The purpose of this paper is to study the existence of the ground state solution for a class of quasilinear elliptic equations. Some recent results are extended. [Copyright &y& Elsevier]

Recently Nikodem, Rajba and Wąsowicz compared the classes of n -Wright-convex functions and n -Jensen-convex functions by showing that the first one is a proper subclass of the latter one, whenever n is an odd natural number. Till now the case of even n was an open problem. In this paper the complete solution is given: it is shown that the inclusion is proper for any natural n . The classes of strongly n -Wright-convex and strongly n -Jensen-convex functions are also compared (with the same assertion). [ABSTRACT FROM AUTHOR]

It is generally believed that the symmetry of initial value problems (IVP) must leave both the partial differential equations (PDEs) and initial conditions invariant. In this paper, we propose partial symmetry of IVP which needs less restrictive conditions on the PDEs and initial conditions and only leaves the IVP invariant on some non-empty subset of the whole solution set of the governing PDEs. Thus some symmetries which either only leave the PDEs invariant or only leave initial conditions invariant are partial symmetries of IVP. Then considering from two different starting points, we define two types of partial symmetry and give the corresponding algorithms where one starts with the PDEs and the other is from the initial conditions. Applications to Boussinesq equation and a class of nonlinear heat equation are performed. [ABSTRACT FROM AUTHOR]

Consider in a Hilbert space H the Cauchy problem ( P 0 ) : u ′ ( t ) + A u ( t ) + B u ( t ) = f ( t ) , 0 ≤ t ≤ T ; u ( 0 ) = u 0 , where A : D ( A ) ⊂ H → H is the generator of a C 0 -semigroup of contractions and B : H → H is Lipschitzian on bounded sets and monotone. Following the method of artificial viscosity introduced by J.L. Lions, we associate with ( P 0 ) the approximate problem ( P ε ) : − ε u ″ ( t ) + u ′ ( t ) + A u ( t ) + B u ( t ) = f ( t ) , 0 ≤ t ≤ T ; u ( 0 ) = u 0 , u ( T ) = u T , where ε is a positive small parameter. We establish an asymptotic expansion of the solution u ε of ( P ε ) , showing that u ε corrected by a boundary layer function approximates the solution of ( P 0 ) with respect to the sup norm of C ( [ 0 , T ] ; H ) . The same asymptotic expansion still holds if B is not necessarily monotone but is Lipschitzian on H . This paper is a significant extension of a previous one by M. Ahsan and G. Moroşanu [2] so that the framework created here allows the treatment of hyperbolic problems (besides parabolic ones). Specifically, our main result is illustrated with the semilinear telegraph system (thus extending a result by N.C. Apreutesei and B. Djafari Rouhani [3] ) and the semilinear wave equation. [ABSTRACT FROM AUTHOR]

In this paper, we study the dynamics of a non-autonomous SIR model with diffusion. We first prove that the ( L 2 , L 2 ) pullback attractor attracts any bounded subset of L 2 in the topology of L 2 + δ for any δ ⩾ 0 . Then we show that the ( L 2 , L 2 ) pullback attractor is indeed a ( L 2 , L 2 + δ ) pullback attractor. [ABSTRACT FROM AUTHOR]

In the present paper, a new family of integral transforms depending on two parameters and related to the Dunkl transform is introduced. Well-known transforms, such as the fractional Dunkl transform, Dunkl transform, linear canonical transform, canonical Hankel transform, Fresnel transform, etc., can be seen to be special cases of this general transform. Some useful properties of the considered transform such as Riemann–Lebesgue lemma, reversibility property, additivity property, operational formula, Plancherel formula, Bochner type identity and master formula are derived. The intimate connection that exists between this transformation and the quantum harmonic oscillator is developed. [ABSTRACT FROM AUTHOR]

In the present paper we introduce a new characterization of the convexity of a planar domain, based on the convexity constant K ( D ) of a domain D ⊂ C . We show that in the class of simply connected planar domains, K ( D ) = 1 characterizes the convexity of the domain D , and we derive the value of the convexity constant for some classes of doubly connected domains of the form D Ω = D − Ω ‾ , for certain choices of the domains D and Ω. Using the convexity constant of a domain, we derive an extension of the well-known Ozaki–Nunokawa–Krzyz univalence criterion for the case of non-convex domains, and we present some examples, which show that our condition is sharp. [ABSTRACT FROM AUTHOR]

Let β > 1 be a real number. A basic interval of order n is a set of real numbers in ( 0 , 1 ] having the same first n digits in their β -expansion which contains x ∈ ( 0 , 1 ] , denote by I n ( x ) and write the length of I n ( x ) as | I n ( x ) | . In this paper, we prove that the extremely irregular set containing points x ∈ [ 0 , 1 ] whose upper limit of − log β ⁡ | I n ( x ) | n equals to 1 + λ ( β ) is residual for every λ ( β ) > 0 , where λ ( β ) is a constant depending on β . [ABSTRACT FROM AUTHOR]

In the paper we deal with minimal closed subsets invariant with respect to set-valued semiflow. Such sets are known as supports of invariant or even ergodic measures of stochastic processes associated with such semiflows. Our motivation comes from some earlier and recent results connected with bounded noise processes, but we work in the framework of set-valued semiflows with lower semicontinuous members on general metric space rather than mostly studied by many authors continuous and compact-valued ones. Such semiflows appear naturally when nonautonomous/random dynamical systems are considered. [ABSTRACT FROM AUTHOR]

This paper investigates regularity properties of two non-negative sparsity sets: non-negative sparse vectors, and low-rank positive semi-definite matrices. Novel formulae for their Mordukhovich normal cones are given and used to formulate sufficient conditions for non-convex notions of regularity to hold. Our results provide a useful tool for justifying the application of projection methods to certain rank constrained feasibility problems. [ABSTRACT FROM AUTHOR]

For an expanding integer matrix M ∈ M n ( Z ) and two finite subsets D , S ⊂ R n of the same cardinality, the concept of compatible pair ( M − 1 D , S ) (or Hadamard triple ( M , D , S ) ) plays an important role in the spectrality of self-affine measure μ M , D . It is known that ( M − 1 D , S ) is a compatible pair if and only if ( M ⁎ − 1 S , D ) is a compatible pair. An old duality conjecture of Dutkay and Jorgensen states that under the condition of compatible pair ( M − 1 D , S ) , μ M , D is a spectral measure if and only if μ M ⁎ , S is. In this paper, we construct an example of compatible pair ( M − 1 D , S ) to illustrate that the self-affine measure μ M , D is a spectral measure but the self-affine measure μ M ⁎ , S is not. This disproves the above-mentioned conjecture of Dutkay and Jorgensen, and clarifies certain dual relation on the spectrality of self-affine measures. [ABSTRACT FROM AUTHOR]

We will study the linear spectral properties of steady Saffman–Taylor bubbles with surface tension in a Hele-Shaw cell. It was known [21] that the Saffman–Taylor exact solution without surface tension is linearly unstable and the point spectrum is a continuum set. In this paper, we rigorously conclude that the point spectrum of the stability operator of any branch of Saffman–Taylor bubbles is a discrete set when surface tension is not zero. [ABSTRACT FROM AUTHOR]

This paper studies by means of standard analytic tools the small time behavior of the heat content over a bounded Lebesgue measurable set of finite perimeter by working with the set covariance function and by imposing conditions on the heat kernels. Applications concerning the heat kernels of rotational invariant α -stable processes are given. [ABSTRACT FROM AUTHOR]

In this paper we analyze the behavior of a family of steady state solutions of a semilinear reaction–diffusion equation with homogeneous Neumann boundary condition, posed in a two-dimensional thin domain with reaction terms concentrated in a narrow oscillating neighborhood of the boundary. We assume that the domain, and therefore, the oscillating boundary neighborhood, degenerates into an interval as a small parameter ϵ goes to zero. Our main result is that this family of solutions converges to the solution of a one-dimensional limit equation capturing the geometry and oscillatory behavior of the open sets where the problem is established. [ABSTRACT FROM AUTHOR]