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Given x ∈ ( 0 , 1 ] , let U ( x ) be the set of bases q ∈ ( 1 , 2 ] for which there exists a unique sequence ( d i ) of zeros and ones such that x = ∑ i = 1 ∞ d i ∕ q i . Lü et al. (2014) proved that U ( x ) is a Lebesgue null set of full Hausdorff dimension. In this paper, we show that the algebraic sum U ( x ) + λ U ( x ) and product U ( x ) ⋅ U ( x ) λ contain an interval for all x ∈ ( 0 , 1 ] and λ ≠ 0 . As an application we show that the same phenomenon occurs for the set of non-matching parameters studied by the first author and Kalle (Dajani and Kalle, 2017). [ABSTRACT FROM AUTHOR]

This paper uses a connection between bounded remainder sets in R d and cut-and-project sets in R together with the fact that each one-dimensional Pisot substitution sequence is bounded distance equivalent to some lattice in order to construct several bounded remainder sets with fractal boundary. Moreover it is shown that there are cut-and-project sets being not bounded distance equivalent to each other even if they are locally indistinguishable, more precisely: even if they are contained in the same hull. [ABSTRACT FROM AUTHOR]

This is a companion paper to Adleret al. (in press, 2015). There, we proved the existence of an absorbing invariant tile for the Error Diffusion dynamics on an acute simplex when the input is constant and “ergodic” and we discuss the geometry of this tile. Under the same assumptions we prove here that said invariant tile (a fundamental set of the lattice generated by the vertices of the simplex) which is a finite union of polytopes have the property that any union of the intersections of the tile with the Voronoï regions of the vertices is a tile for a different, explicitly defined lattice. [ABSTRACT FROM AUTHOR]

In this paper we show that a transversely holomorphic foliation in a compact manifold M having an infinite number of invariant hypersurfaces admits a basic transversely meromorphic first integral f : M ⟶ C ¯ . [ABSTRACT FROM AUTHOR]

In this paper, we study the representing and absolutely representing systems in the context of reproducing kernel Hilbert spaces. We prove in particular that, in many classical spaces including weighted Hardy and Dirichlet spaces and de Branges–Rovnyak spaces, there cannot exist absolutely representing systems of reproducing kernels. [ABSTRACT FROM AUTHOR]

A central question in dynamics is whether the topology of a system determines its geometry. This is known as rigidity. Under mild topological conditions rigidity holds for many classical cases, including: Kleinian groups, circle diffeomorphisms, unimodal interval maps, critical circle maps, and circle maps with a break point. More recent developments show that under similar topological conditions, rigidity does not hold for slightly more general systems. In this paper we state a conjecture which describes how topological classes are organized into rigidity classes. [ABSTRACT FROM AUTHOR]

The aim of this paper is to present a new idea to construct the nonlinear fractal interpolation function, in which we exploit the Matkowski and the Rakotch fixed point theorems. Our technique is different from the methods presented in the previous literatures. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the parametric traffic network problems. Afterward, a key hypothesis is introduced by virtue of a parametric gap function to considered problems, and we prove that this hypothesis is not only sufficient but also necessary for the Hausdorff lower semicontinuity and Hausdorff continuity of the solution mapping for parametric traffic network problems. [ABSTRACT FROM AUTHOR]

The aim of this paper is to investigate the existence and uniqueness of solutions for nonlinear fractional q -difference equations with three-point boundary conditions. Our approach relies on a new fixed point theorem of increasing ψ − ( h , r ) − concave operators defined on ordered sets. Further, we can construct a monotone explicit iterative scheme to approximate the unique solution. Finally, the main results are illustrated with the aid of two interesting examples. [ABSTRACT FROM AUTHOR]

Let f : S 1 → S 1 be a C 3 homeomorphism without periodic points having a finite number of critical points of power-law type . In this paper we establish real a-priori bounds , on the geometry of orbits of f , which are beau in the sense of Sullivan, i.e. bounds that are asymptotically universal at small scales. The proof of the beau bounds presented here is an adaptation, to the multicritical setting, of the one given by the second author and de Melo in de Faria and de Melo (1999), for the case of a single critical point. [ABSTRACT FROM AUTHOR]

In this paper, we give a counterexample to Lemma 2.2 proved by Song-il Ri (2016). Further, we improve the result of Song-il Ri by employing a proper setting. [ABSTRACT FROM AUTHOR]

The power graph of a group G is a graph with vertex set G and two distinct vertices are adjacent if and only if one is an integral power of the other. In this paper we find both upper and lower bounds for the spectral radius of power graph of cyclic group C n and characterize the graphs for which these bounds are extremal. Further we compute spectra of power graphs of dihedral group D 2 n and dicyclic group Q 4 n partially and give bounds for the spectral radii of these graphs. [ABSTRACT FROM AUTHOR]

This paper is concerned with the stability of linear relations in Banach and Hilbert spaces. Several important results about the stability of closedness and the stability of indices of operators in Banach spaces are extended to the linear relations in Banach spaces. Moreover, the stability of deficiency indices of dissipative linear relations in Hilbert spaces is studied. As an important application, we discuss the stability of deficiency indices of dissipative linear relations generated by second-order difference expressions. [ABSTRACT FROM AUTHOR]

The power graph Γ G of a finite group G is the graph whose vertex set is G , two distinct elements being adjacent if one is a power of the other. In this paper, we give sharp lower and upper bounds for the independence number of Γ G and characterize the groups achieving the bounds. Moreover, we determine the independence number of Γ G if G is cyclic, dihedral or generalized quaternion. Finally, we classify all finite groups G whose power graphs have independence number 3 or n − 2 , where n is the order of G . [ABSTRACT FROM AUTHOR]

Given Banach algebras A and B , where A be a Banach B -bimodule. In this paper we study the ideal amenability, approximate ideal and cyclic ideal amenability of module Lau Banach algebra A × α B . [ABSTRACT FROM AUTHOR]

In this paper, we consider the class of uniformly locally univalent harmonic mappings in the unit disk and build a relationship between its pre-Schwarzian norm and uniformly hyperbolic radius. Also, we establish eight ways of characterizing uniformly locally univalent sense-preserving harmonic mappings. We also present some sharp distortions and growth estimates and investigate their connections with Hardy spaces. Finally, we study subordination principles of norm estimates. [ABSTRACT FROM AUTHOR]

Let A and B be two factors. For A , B ∈ A , define by A • B = A B + B A ∗ the Jordan ∗ -product of A and B . In this paper, it is proved that a not necessarily linear bijective map Φ : A → B satisfies Φ ( A • B • C ) = Φ ( A ) • Φ ( B ) • Φ ( C ) for all A , B , C ∈ A if and only if Φ is a linear ∗ -isomorphism, or a conjugate linear ∗ -isomorphism, or the negative of a linear ∗ -isomorphism, or the negative of a conjugate linear ∗ -isomorphism. [ABSTRACT FROM AUTHOR]

In the early twentieth century, L.E.J. Brouwer pioneered a new philosophy of mathematics, called intuitionism . Intuitionism was revolutionary in many respects but stands out – mathematically speaking – for its challenge of Hilbert’s formalist philosophy of mathematics and rejection of the law of excluded middle from the ‘classical’ logic used in mainstream mathematics. Out of intuitionism grew intuitionistic logic and the associated Brouwer–Heyting–Kolmogorov interpretation by which ‘there exists x ’ intuitively means ‘an algorithm to compute x is given’. A number of schools of constructive mathematics were developed, inspired by Brouwer’s intuitionism and invariably based on intuitionistic logic, but with varying interpretations of what constitutes an algorithm. This paper deals with the dichotomy between constructive and non-constructive mathematics, or rather the absence of such an ‘excluded middle’. In particular, we challenge the ‘binary’ view that mathematics is either constructive or not. To this end, we identify a part of classical mathematics, namely classical Nonstandard Analysis , and show it inhabits the twilight-zone between the constructive and non-constructive. Intuitively, the predicate ‘ x is standard’ typical of Nonstandard Analysis can be interpreted as ‘ x is computable’, giving rise to computable (and sometimes constructive) mathematics obtained directly from classical Nonstandard Analysis. Our results formalise Osswald’s longstanding conjecture that classical Nonstandard Analysis is locally constructive . Finally, an alternative explanation of our results is provided by Brouwer’s thesis that logic depends upon mathematics . [ABSTRACT FROM AUTHOR]

Allegedly, Brouwer discovered his famous fixed point theorem while stirring a cup of coffee and noticing that there is always at least one point in the liquid that does not move. In this paper, based on a talk in honor of Brouwer at the University of Amsterdam, we will explore how Brouwer’s ideas about this phenomenon spilled over in a lot of different areas of mathematics and how this eventually led to an intriguing geometrical theory we now know as mirror symmetry. [ABSTRACT FROM AUTHOR]

In this paper we analyze the foundations of epistemology from a constructive Brouwerian position. In particular, we consider the famous tripartite account of knowledge as justified true belief, JTB, traditionally attributed to Plato as well as counter-examples by Russell and Gettier. We show that from an intuitionistic perspective, when the constructive character of truth is taken into account, both Russell and Gettier examples no longer refute the principle that JTB yields knowledge . Moreover, we argue that JTB yields knowledge could be accepted given some natural constructivity assumptions. [ABSTRACT FROM AUTHOR]

We discuss dimension theory in the class of all topological groups. For locally compact topological groups there are many classical results in the literature. Dimension theory for non-locally compact topological groups is mysterious. It is for example unknown whether every connected (hence at least 1-dimensional) Polish group contains a homeomorphic copy of [ 0 , 1 ] . And it is unknown whether there is a homogeneous metrizable compact space the homeomorphism group of which is 2-dimensional. Other classical open problems are the following ones. Let G be a topological group with a countable network. Does it follow that dim G = ind G = Ind G ? The same question if X is a compact coset space. We also do not know whether the inequality dim ( G × H ) ≤ dim G + dim H holds for arbitrary topological groups G and H which are subgroups of σ -compact topological groups. The aim of this paper is to discuss such and related problems. But we do not attempt to survey the literature. [ABSTRACT FROM AUTHOR]

A polynomial assignment for a continuous action of a compact torus T on a topological space X assigns to each p ∈ X a polynomial function on the Lie algebra of the isotropy group at p in such a way that a certain compatibility condition is satisfied. The space A T ( X ) of all polynomial assignments has a natural structure of an algebra over the polynomial ring of Lie ( T ) . It is an equivariant homotopy invariant, canonically related to the equivariant cohomology algebra. In this paper we prove various properties of A T ( X ) such as Borel localization, a Chang–Skjelbred lemma, and a Goresky–Kottwitz–MacPherson presentation. In the special case of Hamiltonian torus actions on symplectic manifolds we prove a surjectivity criterion for the assignment equivariant Kirwan map corresponding to a circle in T . We then obtain a Tolman–Weitsman type presentation of the kernel of this map. [ABSTRACT FROM AUTHOR]

In this paper, we study the stability and asymptotic stability of conformable fractional-order nonlinear systems by using Lyapunov function. [ABSTRACT FROM AUTHOR]

Let GL n ( C ) be a complex general linear group and DGL n ( C ) be its subgroup comprising n × n diagonal matrices in which the diagonal elements are non-zero complex numbers. In this paper we present some general results on Jordan triple product maps between GL n ( C ) and GL m ( C ) . The result is applied to determine the structure of Jordan triple product maps between GL n ( C ) and GL 1 ( C ) ≅ C ∗ . As a main result we present the general form of all continuous Jordan triple product maps from the group GL n ( C ) to the group C ∗ . Also, we characterize the general form of all continuous maps between the groups DGL n ( C ) and DGL m ( C ) that preserve the Jordan triple product. These are the continuous maps φ : DGL n ( C ) → DGL m ( C ) which satisfy φ ( V W V ) = φ ( V ) φ ( W ) φ ( V ) , V , W ∈ DGL n ( C ) . [ABSTRACT FROM AUTHOR]

In this paper, we study the existence of even homoclinic solutions for a damped vibration system x ̈ ( t ) + q ( t ) x ̇ ( t ) + V ′ ( t , x ( t ) ) = 0 , where q is a continuously differentiable function and V ∈ C 1 ( R × R N , R ) , V ( t , x ) = − K ( t , x ) + W ( t , x ) . We use a new kind of superquadratic condition instead of the global Ambrosetti–Rabinowitz condition. For the proof we use the Mountain Pass Theorem. [ABSTRACT FROM AUTHOR]

In this paper, we give some new results on the spectral theory for some operator matrices. It focuses on the theory of the measure of non strict singularity which amounts to study the stability of the Schechter essential spectrum of unbounded block 2 × 2 operator matrix in terms of the union of the Schechter essential spectrum of its diagonal operators entries. This result is tested for two-group transport operator model. [ABSTRACT FROM AUTHOR]

In this paper we provide a rigorous derivation of an asymptotic formulae for perturbations in the resonant frequencies of the two-(or three-) dimensional Laplacian operator under geometric variation of the domain. The asymptotic expansion is developed in the presence and with respect to size of the (anisotropic) imperfections of small shapes having constitutive parameters different from the background conductivity. The main feature of the method is to yield a robust procedure making it possible to recover information about the location, shape, and material properties of the anisotropic imperfections. [ABSTRACT FROM AUTHOR]

A space X is called a k R -space, if X is Tychonoff and the necessary and sufficient condition for a real-valued function f on X to be continuous is that the restriction of f to each compact subset is continuous. In this paper, we mainly discuss the k R -property in the free topological groups, and generalize some well-known results of K. Yamada. [ABSTRACT FROM AUTHOR]

This paper studies the limit behaviour of sums of the form T n ( x ) = ∑ 1 ≤ j ≤ n c k j ( x ) , ( n = 1 , 2 , . . . ) where ( c j ( x ) ) j ≥ 1 is the sequence of partial quotients in the regular continued fraction expansion of the real number x and ( k j ) j ≥ 1 is a strictly increasing sequence of natural numbers. Of particular interest is the case where for irrational α , the sequence ( k j α ) j ≥ 1 is uniformly distributed modulo one and ( k j ) j ≥ 1 is good universal. It was observed by the second author, for this class of sequences ( k j ) j ≥ 1 that we have lim n → ∞ T n ( x ) n = + ∞ almost everywhere with respect to Lebesgue measure. The case k j = j ( j = 1 , 2 , … ) is classical and due to A. Ya. Khinchin. Building on work of H. Diamond, Khinchin, W. Philipp, L. Heinrich, J. Vaaler and others, in the special case where k j = j ( j = 1 , 2 , … , ) we examine the asymptotic behaviour of the sequence ( T n ( x ) ) n ≥ 1 in more detail. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the concept of completely positive matrix of linear maps on Hilbert A -modules over locally C ∗ -algebras and prove an analogue of Stinespring theorem for it. We show that any two minimal Stinespring representations for such matrices are unitarily equivalent. Finally, we prove an analogue of the Radon–Nikodym theorem for this type of completely positive n × n matrices. [ABSTRACT FROM AUTHOR]

The main topic of the present paper is a generalization of two theorems about extreme points in the convex set of all positive contractive operators on f -algebras with unit and on Stone f -algebras, respectively, to the more general setting of semiprime f -algebras. Moreover, a lattice characterization of extreme contractive operators is given. [ABSTRACT FROM AUTHOR]

In this paper, we give characterizations for the Zygmund space in terms of higher order radial derivatives of holomorphic function on the unit ball of C N , behaviors of local integral means of those derivatives and the Carleson measure. Also we will solve Gleason’s problem for the Zygmund space. [ABSTRACT FROM AUTHOR]

We investigate the differential equation for the Jacobi-type polynomials which are orthogonal on the interval [ − 1 , 1 ] with respect to the classical Jacobi measure and an additional point mass at one endpoint. This scale of higher-order equations was introduced by J. and R. Koekoek in 1999 essentially by using special function methods. In this paper, a completely elementary representation of the Jacobi-type differential operator of any even order is given. This enables us to trace the orthogonality relation of the Jacobi-type polynomials back to their differential equation. Moreover, we establish a new factorization of the Jacobi-type operator into a product of linear second-order operators, which gives rise to a recurrence relation with respect to the order of the equation. Finally, two interrelations with the differential equation for the symmetric ultraspherical-type polynomials are pointed out. [ABSTRACT FROM AUTHOR]

In this paper, the extent to which the Burkholder Inequalities in classical Stochastic Analysis can be generalized to the new Theory of Stochastic Analysis in Riesz spaces. [ABSTRACT FROM AUTHOR]

We study the topological properties of a class of planar crystallographic replication tiles. Let M ∈ Z 2 × 2 be an expanding matrix with characteristic polynomial x 2 + A x + B ( A , B ∈ Z , B ≥ 2 ) and v ∈ Z 2 such that ( v , M v ) are linearly independent. Then the equation M T + B − 1 2 v = T ∪ ( T + v ) ∪ ( T + 2 v ) ∪ ⋯ ∪ ( T + ( B − 2 ) v ) ∪ ( − T ) defines a unique nonempty compact set T satisfying T o ¯ = T . Moreover, T tiles the plane by the crystallographic group p 2 generated by the π -rotation and the translations by integer vectors. It was proved by Leung and Lau in the context of self-affine lattice tiles with collinear digit set that T ∪ ( − T ) is homeomorphic to a closed disk if and only if 2 | A | < B + 3 . However, this characterization does not hold anymore for T itself. In this paper, we completely characterize the tiles T of this class that are homeomorphic to a closed disk. [ABSTRACT FROM AUTHOR]

In this paper we give answers to questions on multivalued fractals on b -metric spaces proposed in Boriceanu et al. (2010). [ABSTRACT FROM AUTHOR]

Let ( F m ) m ≥ 0 be the Fibonacci sequence given by F 0 = 0 , F 1 = 1 and F m + 2 = F m + 1 + F m , for all m ≥ 0 . In Castillo (2015), it is conjectured that 2 , 5 and 34 are the only Fibonacci numbers of the form n ! + n ( n + 1 ) 2 , for some positive integer n . In this paper, we confirm the above conjecture. [ABSTRACT FROM AUTHOR]

In the paper, the authors present explicit formulas, nonlinear ordinary differential equations, and recurrence relations for Eulerian polynomials, higher order Eulerian polynomials, and their generating functions in terms of the Stirling numbers of the second kind. [ABSTRACT FROM AUTHOR]

In this paper, we make use of the Lagrange interpolation to obtain some algebraic identities, involving one or two infinite sets of variables. [ABSTRACT FROM AUTHOR]

In the current paper, we consider two classes of noncyclic mappings, called quasi-noncyclic relatively nonexpansive and noncyclic relatively u -continuous, and survey the existence of best proximity pairs as well as the structure of best proximity pair sets for these classes of mappings in Busemann convex spaces. We also study the existence of a common best proximity pair for families if noncyclic mappings in Hadamard spaces. In this way, we obtain a generalization of Markov–Kakutani’s fixed point theorem in Hadamard spaces. [ABSTRACT FROM AUTHOR]

In this paper, we use the concept of quasi-compact operators, as a generalization of the class of Riesz operators, to improve the definition of the pseudo-Schechter essential spectrum of a closed densely defined operator acting on Banach space. Moreover, we discuss the incidence of some perturbation results on the behavior of pseudo-essential spectra of the sum of two bounded linear operators. [ABSTRACT FROM AUTHOR]

The aim of this paper is to enlarge some known results from Fredholm and perturbation theory in the Banach algebra of bounded operators on a Banach space to the Fredholm theory in Banach algebra with respect to a subalgebra. The obtained results allow us to characterize the stability of some essential spectra with respect to a subalgebra. [ABSTRACT FROM AUTHOR]

In this paper we define a more general convolution product of functionals on Wiener space and develop the fundamental relationships between the generalized Fourier–Feynman transform and the convolution product. [ABSTRACT FROM AUTHOR]

Erdös–Surányi and Prielipp suggested to study the following problem: For any integers k > 0 and n , are there an integer N and a map ϵ : { 1 , … , N } → { − 1 , 1 } such that (0.1) n = ∑ j = 1 N ϵ ( j ) j k ? Mitek and Bleicher independently solved this problem affirmatively. In this paper we consider the case that for some positive odd integer L the numbers ϵ ( j ) are L -th roots of unity. We show that the answer to the corresponding question is negative if and only if L is a prime power. [ABSTRACT FROM AUTHOR]

Recently, a dual Minkowski inequality and a dual Brunn–Minkowski inequality for volume differences were established. Following this, in this paper we establish a dual Aleksandrov–Fenchel inequality for dual mixed volume differences which generalizes several recent results. [ABSTRACT FROM AUTHOR]

In this paper we introduce the weighted version of fully measurable grand Lebesgue spaces and obtain characterizations for the boundedness of maximal operator, Hilbert transform and the Hardy averaging operator on these spaces. [ABSTRACT FROM AUTHOR]

In the present paper, we study some properties of the Kobayashi metric on unbounded convex domains and on Hartogs domains defined by Ω φ ( C m ) = { ( z , w ) ∈ C m × C : | w | < e − φ ( z ) } , m ∈ N ∗ , where φ is a smooth strictly plurisubharmonic function such that φ ( z ) → + ∞ as | z | → ∞ . [ABSTRACT FROM AUTHOR]

In this paper, we consider equations of Lie triple algebras that are train algebras. We obtain two different types of equations depending on assuming the existence of an idempotent or a pseudo-idempotent. In general Lie triple algebras are not power-associative. However we show that their train equation with an idempotent is similar to train equations of power-associative algebras that are train algebras and we prove that Lie triple algebras that are train algebras of rank ≤ 4 with an idempotent are Jordan algebras. Moreover, the set of non-trivial idempotents has the same expression in Peirce decomposition as that of e -stable power-associative algebras. We also prove that the algebra obtained by 2 -gametization process of a Lie triple algebra is a Lie triple one. [ABSTRACT FROM AUTHOR]

Let b p and h p ( 0 < p < ∞ ) be the harmonic Bergman space and harmonic Hardy space of harmonic functions on the open unit disk U , respectively. Given 1 ≤ p < ∞ , denote by ‖ ⋅ ‖ b p the norm in the space b p and by ‖ ⋅ ‖ h p the norm in the space h p . In this paper we establish some improvements of the constant a p appearing in the inequality ‖ f ‖ b 2 p ≤ a p ‖ f ‖ h p , given on Kalaj and Meštrović (2011) [Theorem 1.1], for p = 4 and f real harmonic, as ‖ f ‖ b 8 ≤ 28 + 35 2 + 24 2 ( 2 + 2 ) + 4 4 ( 2 + 2 ) 64 8 ‖ f ‖ h 4 . [ABSTRACT FROM AUTHOR]