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For a given positive integer n , we consider positive integers a 1 , a 2 … , a t such that a 1 ! a 2 ! ⋯ a t ! = n ! . Luca proved that n − a 1 = 1 if a b c conjecture holds and n is sufficiently large. Erdős, Bhat and Ramachandra gave unconditional upper bounds on n − a 1 which we improve in this paper. Further we solve the equation when P ( n + 1 ) ≤ 79 by using our estimate.

In this paper, we are going to study generalized Berwald manifold with ( α , β ) -metrics. We show that a Finsler manifold with ( α , β ) -Finsler function of sign property is a generalized Berwald manifold if and only if there exists a covariant derivative such that it is compatible with α and β and equivalently if and only if the dual vector field β ♯ is of constant Riemannian length. This generalizes the result previously only known in the case of Randers manifold. Then, we find the necessary and sufficient condition under which an ( α , β ) -metric of sign property is compatible with a semi-symmetric covariant derivative. In the following, we consider the β -change of reversible Finsler manifolds and find some conditions under which these manifolds are generalized Berwaldian. Finally, we study the effect of our theory on left invariant Finsler function on Lie groups.

This paper is concerned with finite volume approximations for a nonlinear parabolic–elliptic system for chemotaxis-growth in R d , d = 2 , 3 . This model describes a process of pattern formation by some chemotactic biological individuals. We present two schemes which make use of a semi-implicit time discretization and an upwind finite volume approximation. For both schemes, we prove existence, uniqueness and nonnegativity of the approximate solutions under some conditions on the time step, and we show (for one of the schemes) that the numerical solution converges to a corresponding weak solution for the studied model. Numerical simulations are performed in two dimensional spaces to demonstrate the efficiency of the schemes to capture the pattern formations and to verify our theoretical results.

In the first part of this paper we give an overview on known general bounds for the most important types of discrepancies of sequences in the unit-interval. It is pointed out that for all these types the van der Corput sequence, or a variant of it, provides an example with lowest possible order of discrepancy. In the second part we slightly improve the until now best known lower bound for the one-dimensional discrepancy constant with respect to extreme discrepancy given by R. Bejian in 1982.

In this paper, we concern ourselves with essential spectra of an operator A which is subjected to structured perturbation of the form A ⟶ A + C D B where B , C are given bounded operators and D is unknown disturbance operator that satisfies ‖ D ‖ e for a given e > 0 . Thereby, we investigate the structured essential pseudospectra of the sum of two operators acting on a Banach space. Furthermore, we establish characterizations of the structured Schechter essential pseudospectrum of a closed, densely defined linear operator.

In this paper, we outline a new approach to the study of structured Schechter and structured Browder essential pseudospectra of closed densely defined linear operators on infinite dimensional Banach spaces via the concept of the measure of noncompactness and the measure of non strict-singularity. Furthermore, we investigate the structured Wolf, structured Schechter and structured Browder essential pseudospectra of the sum of two bounded operators.

We study a higher order difference equation defined on Z with p -Laplacian and containing both advance and retardation. By using the critical point theory, sufficient conditions are obtained for the existence of infinitely many homoclinic solutions of the equation. The proof is based on the fountain theorem in combination with the variational technique. In particular, considerable effort has been made in the paper to construct a variational framework for the problem under study. Some known results in the literature are extended and complemented.

In this paper we present some important generalizations of the Banach contraction principle in which the Lipschitz constant k is replaced by some real-valued control function. For the applications to the fractal, we obtain the fixed point theorem of the some generalized contraction in the fractal space.

The paper examines measurability conditions for a random process for which a general form of the Feynman–Kac formula is valid.

For positive integers k ≤ n let P n , k ( x ) : = ∑ j = 0 k n j x j be the binomial expansion of ( 1 + x ) n truncated at the k th stage. In this paper we show the finiteness of solutions of Diophantine equations of type P n , k ( x ) = P m , l ( y ) in x , y ∈ Z under assumption of irreducibility of truncated binomial polynomials P n − 1 , k − 1 ( x ) and P m − 1 , l − 1 ( x ) . Although the irreducibility of P n , k ( x ) has been studied by several authors, in general, this problem is still open. In addition, we give some results about the possible ways to write P n , k ( x ) as a functional composition of two lower degree polynomials.

In the present paper, we introduce and study the convergence of a sequence of closable linear operators in a Banach space. Moreover, we prove that if T n converges in the generalized sense to T , where T and ( T n ) n ∈ N are closed linear operators, then there exists a non negative integer element n 0 such that, for all n ≥ n 0 , we have the Weyl essential spectrum of T n included in the Weyl essential spectrum of T (see Theorem 3.1). The same study is made for the case of convergence to zero compactly under which weaker results are established (Theorem 3.3 and Corollary 3.1).

In this paper, we generalize the following inequality due to N. Bebiano et al which states that for J -contraction T ∈ M n | [ T x , y ] J | 2 ≥ [ | T | J 2 α x , x ] J [ | T ♯ | J 2 ( 1 − α ) y , y ] J for any x , y ∈ C n with x or y time-like and α ∈ [ 0 , 1 ] . The above inequality gives a reverse weighted mixed Schwarz inequality. Also, we prove some reverse inequalities for indefinite inner product space. Moreover, we establish some refinements for the reverse of Schwarz inequality as well.

Let A be a finite set, and let a symmetric binary relation be given on A . The goal of correlation clustering is to find a partition of A , with minimal conflicts with respect to the relation given. In this paper we investigate correlation clustering of subsets of the positive integers, based upon a relation defined by the help of the greatest common divisor.

The purpose of this paper is to provide upper bounds for the discrepancy of generalized Van-der-Corput–Halton sequences that are built from Halton sequences and the Zeckendorf Van-der-Corput sequence.

In this paper, we employ the Mond--Pe\v{c}ari\'c method to establish some reverses of the operator Bellman inequality under certain conditions. In particular, we show \begin{equation*} \delta I_{\mathscr K}+\sum_{j=1}^n\omega_j\Phi_j\left((I_{\mathscr H}-A_j)^{p}\right)\ge \left(\sum_{j=1}^n\omega_j\Phi_j(I_{\mathscr H}-A_j)\right)^{p} \,, \end{equation*} where $A_j\,\,(1\leq j\leq n)$ are self-adjoint contraction operators with $0\leq mI_{\mathscr H}\le A_j \le MI_{\mathscr H}$, $\Phi_j$ are unital positive linear maps on ${\mathbb B}({\mathscr H})$, $\omega_j\in\mathbb R_+ \,\,(1\leq j\leq n)$, $0 < p < 1$ and $\delta=(1-p)\left(\frac{1}{p}\frac{(1-m)^p-(1-M)^p}{M-m}\right)^{\frac{p}{p-1}}+\frac{(1-M)(1-m)^p-(1-m)(1-M)^p}{M-m}$ . We also present some refinements of the operator Bellman inequality., Comment: 15 pages, to appear in Indag. Math. (N.S.)

We clarify Theorems 5.1 and 5.4 of our paper “Probabilistic properties of the relative tensor degree of finite groups”, published in Indagationes Mathematicae 27 (2016), 147–159.

In this paper we extend a model of Agler for n -hypercontractions on Hilbert space to Banach space. This work also generalizes models of Nagy–Foias, Rovnyak–de Branges for contractions on Hilbert space to Banach space. A version of the similarity model of Rota on Banach space is also given.

This paper investigates the relations between the fixed points of asymptotically contractive operators defined on complete linear metric spaces, in general, and in separable Hilbert spaces in particular and their obtained truncations of the expansions of such operators by using Schauder bases. The truncation used to expand the approximated operators is seen to be specifically relevant for the case when infinite-dimensional Hilbert spaces are truncated by finite-dimensional ones. The discussion of the situation when the fixed points remain unaltered under the mentioned truncations is also highlighted in the formalism. The study is of interest in operator discretization-oriented problems which are approximated by truncated operator representations. Two illustrative worked examples are given.

Optional projections and predictable projections of stochastic processes play important roles in the general theory of stochastic processes, semimartingale theory and stochastic calculus. They share some important properties with ordinary conditional expectations and generalized conditional expectations. While the characterization of ordinary conditional expectations and generalized conditional expectations has been studied by several authors, no similar work has been done for optional projections and predictable projections. This paper aims at undertaking this task by giving Ando–Douglas type characterization theorem for both.

Let M be a differentiable manifold equipped with a Riemannian metric tensor. If we apply translations in the tangent spaces to the Riemannian unit balls such that the translated bodies are keeping the origin in their interiors then they are working as unit balls for a Randers manifold. The unit balls allow us to measure the length of tangent vectors with the help of the induced Minkowski functionals. Analytically these Minkowski functionals are coming from a Riemannian metric tensor by using one-form perturbation in the tangent spaces. Manifolds equipped with a smoothly varying family of Minkowski functionals are called Finsler manifolds. The one-form perturbation of a Riemannian metric in the tangent spaces results in the class of Randers manifolds introduced by G. Randers in 1941. They occur naturally in physical applications related to electron optics, navigation problems or the Lagrangian of relativistic electrons. In this paper we are interested in Randers manifolds with semi-symmetric compatible linear connections. Compatibility means that the parallel transports preserve the length of tangent vectors with respect to the perturbed metric. If the torsion is decomposable in a special way then we speak about a semi-symmetric linear connection. Up to local isometries we characterize Riemannian manifolds admitting a one-form perturbation such that the resulting Randers manifold has a compatible semi-symmetric linear connection. As a paraphrase of our previous work (Vincze, 2006) communicated by Professor J.J. Duistermaat we present an existence theorem of generalized Berwald manifolds with semi-symmetric compatible linear connections. The terminology goes back to M. Matsumoto, M. Hashiguchi and S. Bacso’s work (Bacso et al., 1997).

In this paper we consider a plate equation with viscoelastic damping localized on a part of the boundary. We establish an explicit and general decay rate result that allows a wider class of relaxation functions and generalize previous results existing in the literature.

In this paper we prove Arakelian’s approximation theorem of Runge type in the framework of slice hyperholomorphic functions, namely for slice regular functions of a quaternionic variable and for slice monogenic functions of a paravector variable. Specifically, we show that any slice hyperholomorphic function in a neighborhood of a so-called Arakelian (closed and possible unbounded) set E, can be uniformly approximated on E by sequences of entire slice hyperholomorphic functions.

The paper is dedicated to approximation of plurisubharmonic functions in the Lelong class L ( C n ) by the functions of the form 1 d j log | p j | , where p j are polynomials in C n with degree deg p j ≤ d j . We must say that this result is inspired by Theorem 15.1.6 in Hormander (1983). Some applications to pluripolar sets are also given.

Let P − 1 denote the set of primes minus 1 . A classical theorem of A Sarkőzy says that any set of natural numbers of positive density contains a pair of elements whose difference belongs to P − 1 . An ergodic approach to questions of this type was given by the fourth author, building on work of H. Furstenberg. In this paper we give a proof of the positive characteristic analogue of this result using the same approach.

In this paper, we introduce the notion of Jordan ( θ , ϕ ) -derivations on Hilbert C*-modules and we investigate the generalized Hyers–Ulam–Rassias stability of this notion with an alternative fixed point and direct method. Moreover, we provide Hyers–Ulam, Aoki–Rassias, and Isac–Rassias type stability and superstability of Jordan ( θ , ϕ ) -derivations and Jordan derivations on Hilbert C*-modules.

Our work An et al. (2008) consisted of two main sections. Section 2 concerned non-Archimedean analytic maps to semi-Abelian varieties, and the results of that section are correct, as written. Section 3 concerned non-Archimedean analogs of the work of Noguchi and Winkelmann (2002). Unfortunately, Noguchi and Winkelmann (2002) contains an error, first noticed by Aaron Levin, that we replicated into our work as well. Thus, some of the results in Section 3 of our work are incorrect as stated there, while others require corrected proofs, which we describe in detail here. We also incorporate ideas from work that has appeared since our original paper to obtain a stronger conclusion for one of our results.

In this paper, we show two Marcinkiewicz type interpolation theorems for weak Orlicz martingale spaces by employing the technique of atomic decompositions. As an application, we prove some martingale inequalities with weak Orlicz space norm.

The aim of this paper is to study the dynamics of the real cubic polynomials that have a fixed point of multiplicity two. Such polynomials are conjugate to f a ( x ) = a x 2 ( x − 1 ) + x , a ≠ 0 . We will show that when a > 0 and x ≠ 1 , then | f a n ( x ) | converges to 0 or ∞ and, if a 0 and a belongs to a special subset of the parameter space, then there is a closed invariant subset Λ a of R on which f a is chaotic.

In this paper, a new inclusion range theorem for derivations acting on universally complete f -algebras is given. The principal theorem permits the systematic study of weakly locally one dimensional vector lattices.

For β ∈ C with | β | 1 define the contractions h 0 ( z ) = β z and h 1 ( z ) = β z + 1 and consider the attractor A β of the iterated function system { h 0 , h 1 } . In 1985 Barnsley and Harrington introduced the “Mandelbrot set for pairs of linear maps” which is the set of all β with connected attractor A β . This set has been thoroughly studied by many authors. In the present paper we consider a variant of this Mandelbrot set. In particular, we consider the attractors of the iterated function system { f 0 , f 1 } given by f 0 ( z ) = β z , f 1 ( z ) = 1 + β 2 z and study the associated Mandelbrot set M . Among other things we show that M is connected. The structure of the iterated function system { f 0 , f 1 } is related to the Fibonacci Language which is the subshift of finite type over { 0 , 1 } defined by forbidding the occurrence of two consecutive ones. This language and its difference language play an important role in our proofs.

In this paper, a class of fourth-order nonlinear difference equations are considered. By making use of the critical point method, we establish various sets of sufficient conditions for the nonexistence and existence of solutions for Neumann boundary value problems and give some new results. Our results successfully complement the existing ones.

In this paper, we introduce the notion of b -algebras and we give some related properties. To be more precise, we call a b -algebra any lattice-ordered algebra A the bands of which are closed under multiplication. We obtain that A can be identified with the reals whenever A is an Archimedean b -algebra with unit element e > 0 and such that every positive element has an inverse. This improves a result by Huijsmans who got the same conclusion for f -algebras imposing the extra condition of positivity of inverses. Moreover, we show that the order continuous bidual ( A ∼ ) n ∼ of an Archimedean b -algebra A is a b -algebra with respect to the Arens multiplication. Furthermore, if the b -algebra A has positive squares, then the order bidual A ∼ ∼ is again a b -algebra.

In this paper, we use the Fourier analytic approach to investigate the stability of the L p Shephard problem for volume-normalized L p polar projection bodies. Our results give a complete solution to the stability problem for the L p Shephard problem.

In this paper, we consider third-order linear recurrences { u n } n ≥ 0 satisfying the recurrence relation u n + 3 = u n + 2 + u n + 1 + u n for all n ≥ 0 and investigate the multiplicity of its zeros. We prove that { u n } n ≥ 0 has zero-multiplicity at most 2, except for nonzero multiples of shifts of the Tribonacci sequence which has zero-multiplicity 4 when the indices are extended to all the integers.

In this paper, we establish a change of path formula for function space integrals on the function space C a , b [ 0 , T ] induced by a generalized Brownian motion process. We then apply the change of path formula to obtain the relationship between the generalized analytic Z g -Feynman integral and the ordinary function space integral.

In this paper, we derive the existence and uniqueness of mild solutions for inhomogeneous fractional evolution equations in Banach spaces by means of the method of fractional resolvent. Furthermore, we give the necessary and sufficient conditions for the existence of strong solutions. An example of the fractional diffusion equation is also presented to illustrate our theory.

Ever since Legendre introduced the polynomials that bear his name in 1785, they have played an important role in analysis, physics and number theory, yet their algebraic properties are not well-understood. Stieltjes conjectured in 1890 how they factor over the rational numbers. In this paper, assuming Stieltjes’ conjecture, we formulate a conjecture about the Galois groups of Legendre polynomials, to the effect that they are “as large as possible,” and give theoretical and computational evidence for it.

In this paper we present some new results on the connection between the Siegel norm, the length function and irreducible character values of finite groups. In addition, we provide algorithms to compute the length of a cyclotomic integer and the set of cyclotomic integers with Siegel norm bounded by a given positive real number.

Let R be a commutative ring with identity. We will say that an R -module M satisfies the weak Nakayama property, if I M = M , where I is an ideal of R , implies that for any x ∈ M there exists a ∈ I such that ( a − 1 ) x = 0 . In this paper, we will study modules satisfying the weak Nakayama property. It is proved that if R is a local ring, then R is a Max ring if and only if J ( R ) , the Jacobson radical of R , is T -nilpotent if and only if every R -module satisfies the weak Nakayama property.

The work of the author in measure and integration is based on new inner and outer envelope formations, which replace the traditional Caratheodory outer measure and certain simple suprema and infima. The new formations lead to essential improvements in both the extent and the adequacy of the basic results. However, they did not find an entrance into the recent textbook literature. The present paper seeks to demonstrate their power with the examples of the basic inner and outer extension and representation theorems for set functions and functionals.

Let ( Ω , μ ) be a finite measure space, X a Banach space, and let 1 ≤ p ∞ . The aim of this paper is to give an elementary proof of the Diaz–Mayoral theorem that a subset V ⊆ L p ( μ ; X ) is relatively compact in L p ( μ ; X ) if and only if it is uniformly p -integrable, uniformly tight, and scalarly relatively compact.

This paper is concerned with Riemann–Liouville abstract fractional Cauchy Problems with damping. The notion of Riemann–Liouville fractional ( α , β , c ) resolvent is developed, where 0 β α ≤ 1 . Some of its properties are obtained. By combining such properties with the properties of general Mittag-Leffler functions, existence and uniqueness results of the strong solution of Riemann–Liouville abstract fractional Cauchy Problems with damping are established. As an application, a fractional diffusion equation with damping is presented.

An Archimedean semiprime f -algebra A for which I A ∧ f ∈ A for all f ∈ A is called a Stone f -algebra, where I A is the identity operator on A . Moreover, an operator T between two Stone f -algebras A and B is said to be contractive if f ∈ A and 0 ≤ f ≤ I A imply 0 ≤ T f ≤ I B . The set K ( A , B ) of all positive contractive operators from A into B is a convex set. This paper characterizes extreme points in K ( A , B ) . In this regard, we prove that T ∈ K ( A , B ) is extreme if and only if T is an algebra homomorphism. Furthermore, we show that T ∈ K ( A , B ) is extreme if and only if T is a Stone operator, meaning that, T ( I A ∧ f ) = I B ∧ T f for all f ∈ A .

In this paper we discuss a new metric property that some operator algebras on Hilbert space possess and some resulting consequences concerning transitivity and structure theory of such algebras.

In this paper, we investigate the Hyers–Ulam–Rassias stability of a general equation f ( φ 1 ( x , y ) ) = φ 2 ( f ( x ) , f ( y ) ) in metric spaces. As a consequence, we obtain some stability results in the sense of Hyers–Ulam–Rassias.

The paper establishes the existence of a threshold value kp for which the following holds. In each factorization of an elementary p-group of rank less than or equal to kp into two Z-subsets at least one of the factors does not span the whole group. Further each elementary p-group of rank greater than kp admits factorization into two Z-subsets that both span the whole group. Then it will be shown that kp≤11 for p≥3.

In this paper we introduce the notion of map, which is a notation for the set of occurrences of a symbol in a syntactic expression such as a formula or a λ term. We use binary trees over 0 and 1 as maps, but some well-formedness conditions are required. We develop a representation of lambda terms using maps. The representation is concrete (inductively definable in HOL or Constructive Type Theory) and canonical (one representative per λ term). We define substitution for our map representation, and prove the representation is in substitution preserving isomorphism with both nominal logic λ terms and de Bruijn nameless terms. These proofs are mechanically checked in Isabelle/HOL and Minlog respectively. The map representation has good properties. Substitution does not require adjustment of binding information: neither α conversion of the body being substituted into, nor de Bruijn lifting of the term being implanted. We have a natural proof of the substitution lemma of λ calculus that requires no fresh names, or index manipulation. Using the notion of map we study conventional raw λ syntax. E.g. we give, and prove correct, a decision procedure for α equivalence of raw λ terms that does not require fresh names. We conclude with a definition of β reduction for map terms, some discussion on the limitations of our current work, and suggestions for future work.

In this paper we revisit an unpublished but influential technical report from 1978 by N.G. de Bruijn, written in the framework of the Automath project. This report describes a technique for proving confluence of abstract reduction systems, called the weak diamond property. It paved the way for the powerful technique developed by Van Oostrom to prove confluence of abstract reduction systems, called decreasing diagrams. We first revisit in detail De Bruijn’s old proof, providing a few corrections and hints for understanding. We find that this original criterion and proof technique are still worthwhile. Next, we establish that De Bruijn’s confluence criterion can be used to derive the decreasing diagrams theorem (the reverse was already known). We also provide a short proof of decreasing diagrams in the spirit of De Bruijn. We finally address the issue of completeness of this method.

In memoriam: N.G. de Bruijn. In this article we present a survey of his papers on combinatorics. The section titles show its variety. 1. Common systems of representatives 2. De Bruijn cycles 3. The De Bruijn–Erdos theorem from incidence geometry 4. Bases for integers 5. The BEST theorem 6. The De Bruijn–Erdos theorem from graph theory 7. Factorizations of finite groups 8. Rooted trees in the plane 9. Permutations of a given shape 10. Covering of graphs by dimers 11. Counting (Polya’s fundamental theorem, Color designs) 12. Penrose tilings

In this paper, we construct a risk model with a dependence setting where there exists a specific structure among the time between two claim occurrences, premium sizes and claim sizes. Given that the premium size is exponentially distributed, both the Laplace transforms and defective renewal equations for the expected discounted penalty functions are obtained. Exact representations for the solutions of the defective renewal equations are derived through an associated compound geometric distribution. When the claims are subexponentially distributed, the asymptotic formulae for ruin probabilities are obtained. Finally, when the individual premium sizes have rational Laplace transforms, the Laplace transforms for the expected discounted penalty functions are obtained.