For each integer \(n\ge 1\) we consider the unique polynomials \(P, Q\in {\mathbb {Q}}[x]\) of smallest degree n that are solutions of the equation \(P(x)x^{n+1}+Q(x)(x+1)^{n+1}=1\). We derive numerous properties of these polynomials and their derivatives, including explicit expansions, differential equations, recurrence relations, generating functions, resultants, discriminants, and irreducibility

In this paper, we define the localization operator associated with the Riemann–Liouville operator, and show that it is not only bounded, but it is also in the Schatten–von Neumann class. We also give a trace formula when the symbol function is nonnegative.

Let d(n) be the divisor function and denote by [t] the integral part of the real number t. In this short note, we prove that \(\sum _{n\le x} \ d\Big (\Big [\frac{x}{n}\Big ]\Big ) = x\sum _{m\ge 1}\frac{\ d(m)}{m(m+1)} + O_{\varepsilon }\big (x^{11/23+\varepsilon }\big )\) for \(x\rightarrow \infty \).

In this note we confirm two conjectural supercongruences of double sums of binomial coefficients due to El Bachraoui.

The goal of this paper is to give some theorems which relate to the problem of classifying combinatorial (resp. smooth) closed 5-manifolds up to piecewise-linear (PL) homeomorphism. For this, we use the combinatorial approach to the topology of PL manifolds by means of a special kind of edge-colored graphs, called crystallizations. Within this representation theory, Bracho and Montejano introduced

The main aim of this paper is to provide some range-type criteria for the essentially self-adjointness of symmetric linear relations in real or complex Hilbert spaces. The main tool is a matrix whose entries are certain linear relations.

We consider the partitions of n into parts not congruent to 0, \(\pm 3\pmod {12}\) and provide some new results related to the number of these partitions. In this context, we derive a new parity result involving sums of partition numbers p(n) and squares in arithmetic progressions: for \(n\geqslant 0\), $$\begin{aligned} \sum _{8k+1\text { square}} p\big (n-3k\big ) \equiv 1 \pmod {2} \end{aligned}$$

Let K be a field and put \({\mathcal {A}}:=\{(i,j,k,m)\in \mathbb {N}^{4}:\;i\le j\;\text{ and }\;m\le k\}\). For any given \(A\in {\mathcal {A}}\) we consider the sequence of polynomials \((r_{A,n}(x))_{n\in \mathbb {N}}\) defined by the recurrence $$\begin{aligned} r_{A,n}(x)=f_{n}(x)r_{A,n-1}(x)-v_{n}x^{m}r_{A,n-2}(x),\;n\ge 2, \end{aligned}$$ where the initial polynomials \(r_{A,0}, r_{A,1}\in

In this paper we give a short, new proof of a natural generalization of Gerzon’s bound. This bound improves the Delsarte, Goethals and Seidel’s upper bound in a special case. Our proof is a simple application of the linear algebra bound method.

In this paper, we establish a necessary and sufficient criterion for a finite metabelian 2-group G whose abelianized \(G^{ab}\) is of type \((2, 2^m)\), with \(m\ge 2\), to be metacyclic. This criterion is based on the rank of the maximal subgroup of G which contains the three normal subgroups of G of index 4. Then, we apply this result to study the structure of the Galois group of the maximal unramified

We establish direct estimates of the rate of weighted simultaneous approximation by the Szász–Mirakjan operator for smooth functions in the supremum norm on the non-negative semi-axis. We consider Jacobi-type weights. The estimates are stated in terms of appropriate moduli of smoothness or K-functionals.

In this paper, we give some sufficient conditions for the Domínguez–Lorenzo condition in terms of the James type constant \(J_{X,t}(\tau )\), the coefficient of weak orthogonality \(\mu (X)\) and the Domínguez Benavides coefficient R(1, X), which imply the existence of fixed points for multivalued nonexpansive mappings. Our results extend some well known results in the recent literature.

A new generalization of Grassmannians in supergeometry, called \(\nu \)-Grassmannians, are constructed by gluing \(\nu \)-domains. By a \(\nu \)-domain, we mean a superdomain with an odd involution say \(\nu \) on its structure sheaf, as morphism of modules. Then we show that \(\nu \)-Grassmannians are homogeneous superspaces. In addition, in the last section, a supergroup associated to the odd involution

In this note we construct an algorithm generating any discrete distribution with an arbitrary coin (and, as a result, with arbitrary initial distribution). The coin need not be fair and the target distribution can be supported on a countable set.

Let \(\Lambda (n)\) be the von Mangoldt function, and let [t] be the integral part of real number t. In this note we prove that the asymptotic formula $$\begin{aligned} \sum _{n\leqslant x} \Lambda \Big (\Big [\frac{x}{n}\Big ]\Big ) = x\sum _{d\geqslant 1}\frac{\Lambda (d)}{d(d+1)} + O_{\varepsilon }\big (x^{35/71+\varepsilon }\big ) \end{aligned}$$ holds as \(x\rightarrow \infty \) for any \(\varepsilon

It is well known that for each \(n\ge 0\), there is a continuous map \( f :S^{n}\rightarrow \partial \bigtriangleup ^{n+1}\) with the disjoint support property. Since \(S^{n}\) and \(\partial \bigtriangleup ^{n+1}\) are homeomorphic, it is natural to ask whether or not there is a homeomorphism \( h :S^{n}\rightarrow \partial \bigtriangleup ^{n+1}\) with the disjoint support property. In this paper

Suppose P is a symmetric convex polygon in the plane. We give an algorithm, running in polynomial time in the number of sides of the polygon, which decides if P can tile the plane by translations at some level (not necessarily at level one; this is multiple tiling). The main technical contribution is a polynomial time algorithm that selects, if this is possible, for each \(j=1,2,\ldots ,n\) one of

We study the algebraic independence of Laurent series in positive characteristic which can be fast approximated by rational functions. This can be seen as a completion of the results obtained by Chaichana and Laohakosol (Period Math Hung 55(1):35–59, 2007).

We settle J. Wetzel’s 1970’s conjecture and show that a \(30^{\circ }\) circular sector of unit radius can accommodate every planar arc of unit length. Leo Moser asked in 1966 for the (convex) region with the smallest area in the plane that can accommodate each arc of unit length. With area \(\pi /12,\) this sector is the smallest such set presently known. Moser’s question has prompted a multitude

In this paper, we prove that every Diophantine quadruple in \(\mathbb {Z}[i][X]\) is regular. More precisely, we prove that if \(\{a, b, c, d\}\) is a set of four non-zero polynomials from \(\mathbb {Z}[i][X]\), not all constant, such that the product of any two of its distinct elements increased by 1 is a square of a polynomial from \(\mathbb {Z}[i][X]\), then $$\begin{aligned} (a+b-c-d)^2=4(ab+1)(cd+1)

In this paper, a local version of the topological pressure of dynamical systems is presented. It is a function defined on the product space which does not depend on any measure. It is shown that, for any invariant measure, integration of the introduced function with respect to its corresponding diagonal measure results in the metric pressure of the dynamical system.

We prove that the inequality $$\begin{aligned} \Gamma (x+1)\le \frac{x^2+\beta }{x+\beta } \end{aligned}$$ holds for all \(x\in [0,1]\), \(\beta \ge {\beta ^{*}}\), with the best possible constant $$\begin{aligned} \beta ^*=\max _{0.1\le x\le 0.3} f(x)=1.75527\ldots , \end{aligned}$$ where f is given by $$\begin{aligned} f(x)=\frac{x\Gamma (x+1)-x^2}{1-\Gamma (x+1)}. \end{aligned}$$ This refines bounds

Studying eigenvalues of square matrices is a traditional and fundamental direction in linear algebra. Quaternion matrices constitute an important and extensively useful subclass of square matrices. In the paper, the authors (1) introduce the concept of “generalized eigenvalues of quaternion matrix”; (2) give some properties of generalized eigenvalues for a regular quaternion matrix pair; (3) establish

In this paper we introduce a new conformable derivative call it mixed conformable partial derivative, which obeys classical properties, including linearity, product rule, quotient rule and vanishing derivatives for constant functions. As an application, we establish an Opial type inequality for the mixed second order conformable partial derivatives.

A real valued function f defined on a real open interval I is called \(\Phi \)-monotone if, for all \(x,y\in I\) with \(x\le y\) it satisfies$$\begin{aligned} f(x)\le f(y)+\Phi (y-x), \end{aligned}$$where \( \Phi :[0,\ell (I) [ \rightarrow \mathbb {R}_+\) is a given nonnegative error function, where \(\ell (I)\) denotes the length of the interval I. If f and \(-f\) are simultaneously \(\Phi \)-monotone

Let \(\mathbb {N}\) be the set of nonnegative integers. For any set \(A \subset \mathbb {N}\), let \(R_1(A, n)\), \(R_2(A, n)\) and \(R_3(A, n)\) be the number of representations of n as \(n=a+a',a, a'\in A\); \(n=a+a',a, a'\in A\), \(a

In this paper, we consider plane algebraic curves of genus zero defined over number fields having three simple points at infinity such that their minimal fields of definition are totally real, and we determine an upper bound of polynomial type for the size of their integral points.

For a given integer \(k\ge 3\), a sequence A of nonnegative integers is called an \(AP_k\)-covering sequence if there exists an integer \(n_0\) such that, if \(n>n_0\), then there exist \(a_1\in A\), \(\ldots \), \(a_{k-1}\in A\), \(a_1

In this work, we study non-characteristic domains of Heisenberg groups. We prove that bounded domains which are diffeomorphic to the solid torus having the center of the group as rotation axis, are non-characteristic. Then we state the following conjecture: the bounded non-characteristic domains of the Heisenberg group of dimension 1 are diffeomorphic to a solid torus having the center of the group

Let \(\mathbb N\) be the set of positive integers, and denote by $$\begin{aligned} \lambda (A)=\inf \{t>0:\sum _{a\in A} a^{-t}<\infty \} \end{aligned}$$ the convergence exponent of \(A\subset \mathbb N\). For \(0

Let \(P_k\) denote the kth term of the Pell sequence. In this paper we find all solutions of the exponential Diophantine equation \(P_n+P_m = y^s\) in positive integer variables (m, n, y, s) under the assumption \(n \equiv m \pmod 2\).

We explore the existence of irreducible and reducible arc-sections in an irreducible hypersurface singularity germ along finite projections. In particular we provide examples of irreducible isolated hypersurface singularities for which no irreducible arc-sections exist, and show that reducible ones always exist. Moreover, we give an algorithm to check if a given projection allows irreducible arc-sections

The aim of this article is to give a characterization of strongly quasipositive quasi-alternating links and detect new classes of strongly quasipositive Montesinos links and non-strongly quasipositive Montesinos links. In this direction, we show that, if L is an oriented quasi-alternating link with a quasi-alternating crossing c such that \(L_0\) is alternating (where \(L_0\) has the induced orientation)

Let \(\alpha = (1+\sqrt{5})/2\) be the golden ratio, and let \(B(\alpha ) = (\left\lfloor n\alpha \right\rfloor )_{n\ge 1}\) and \(B(\alpha ^2) = \left( \left\lfloor n\alpha ^2\right\rfloor \right) _{n\ge 1}\) be the lower and upper Wythoff sequences, respectively. In this article, we obtain a new estimate concerning the fractional part \(\{n\alpha \}\) and study the sumsets associated with Wythoff

Let \(a>1,b\) be two positive integers where the square-free part of b is 2pq with p, q two distinct odd primes. Recently, Cipu (Proc Am Math Soc 146:983–992, 2018) proved that if one of the following conditions holds: (i) \(2a^2-1\) is not a perfect square, (ii) \(\{p\pmod 8,q\pmod 8\}\ne \{1,3\}\), then the equations $$\begin{aligned} x^2-(a^2-1)y^2=1 \quad \text {and} \quad y^2-bz^2=1 \end{aligned}$$

We survey Ozsváth–Szabó’s bordered approach to knot Floer homology. After a quick introduction to knot Floer homology, we introduce the relevant algebraic concepts (\(\mathcal {A}_\infty \)-modules, type D-structures, box tensor product, etc.), we discuss partial Kauffman states, the construction of the boundary algebra, and sketch Ozsváth and Szabó’s analytic construction of the type D-structure associated

Recently, using a formula of Carlitz, the second author proved a q-congruence conjectured by Tauraso. In this note we utilize Carlitz’s formula to prove two more similar q-congruences. We also propose a conjectural q-congruence that refines one of our q-congruences.

Our paper begins with a revision of the spectral theory for commutative Banach algebras, which enables us to prove the \(L^p_{\omega }\)-conjecture for locally compact abelian groups. We follow an alternative approach to the one known in the literature. In particular, we do not resort to any structural theorems for locally compact groups at this stage. Subsequently, we discuss nilpotent, locally compact

Let (a, b, c) be pairwise relatively prime integers such that \(a^2 + b^2 = c^2\) . In 1956, Jeśmanowicz conjectured that the only solution of \(a^x + b^y = c^z\) in positive integers is \((x,y,z)=(2,2,2)\). In this note we prove a polynomial analogue of this conjecture.

This paper investigates the optimal dividend problem in a jump-diffusion risk model with debit interest. In this model, the insurer could borrow money at a debit interest when the surplus turns negative. However, when the negative surplus attains a certain critical level, the business stops and absolute ruin happens at this moment. A sufficient condition under which the optimal dividend strategy is

We will present proofs for two conjectures stated in Rot (Homotopy classes of proper maps out of vector bundles, 2020. arXiv:1808.08073). The first one is that for an arbitrary manifold W, the homotopy classes of proper maps \(W\times \mathbb {R}^n\rightarrow \mathbb {R}^{k+n}\) stabilise as \(n\rightarrow \infty \), and the second one is that in a stable range there is a Pontryagin–Thom type bijection

In this short note we use methods of Friedl, Livingston and Zentner to show that there are knots that are not algebraically concordant to a connected sum of positive and negative L-space knots.

The generic element of the moduli space of logarithmic connections with parabolic points on holomorphic vector bundle over the Riemann sphere can be represented by a Fuchsian equation with some singularities and some apparent singularities. We analyze the case of rank 3 vector bundle which leads to third order Fuchsian equation. We find coordinates on an open subset of the moduli space and we construct

In this paper we consider a link \(B_{n,\rho }\) between Baskakov type operators \(B_{n,\infty }\) and genuine Baskakov–Durrmeyer type operators \( B_{n,1}\) depending on a positive real parameter \(\rho \). The topic of the present paper is the pointwise limit relation \(\left( B_{n,\rho }f\right) \left( x\right) \rightarrow \left( B_{n,\infty }f\right) \left( x\right) \) as \(\rho \rightarrow \infty

We characterize p-rational real quadratic fields in terms of generalized Fibonacci numbers. We then use this characterization to give numerical evidence to a conjecture of Greenberg asserting the existence of p-rational multi-quadratic fields of arbitrary degree \(2^{t}\), \(t\ge 1\).

By using the Furstenberg–Ellis–Namioka structure theorem, we give a decomposition theorem for the Banach algebra \({\mathcal {M}}(G)\), i.e. the Banach algebra of those complex regular Borel measures on a compact Hausdorff admissible right topological (or simply CHART) group G for which the natural convolution product makes sense, generalizing an existing result due to Lau and Loy. Next, we characterize

In this paper we prove that if \(\{a,b,c\}\) is a Diophantine triple with \(a

Let \(k=k_0(\root 3 \of {d})\) be a cubic Kummer extension of \(k_0=\mathbb {Q}(\zeta _3)\) with \(d>1\) a cube-free integer and \(\zeta _3\) a primitive third root of unity. Denote by \(C_{k,3}^{(\sigma )}\) the 3-group of ambiguous classes of the extension \(k/k_0\) with relative group \(G={\text {Gal}}(k/k_0)=\langle \sigma \rangle \). The aims of this paper are to characterize all extensions \(k/k_0\)

A generalization of the well-known Fibonacci and Lucas sequences are the k-Fibonacci and k-Lucas sequences with some fixed integer \(k\ge 2\). For these sequences the first k terms are \(0,\ldots ,0,1\) and \(0,\ldots ,0,2,1\), respectively, and each term afterwards is the sum of the preceding k terms. Here we find all pairs of k-Fibonacci and k-Lucas numbers multiplicatively dependent.

Any sequence of d-dimensional boxes of edge length smaller than or equal to 1 with total volume not greater than \((3-2\sqrt{2})\cdot 3^{-d}\) can be packed online into the d-dimensional unit cube.

Let \(\Omega \) be an open, simply connected, bounded subset of the complex plane \(\mathbb {C}\) with a rectifiable boundary \({\partial {\Omega }}\). We investigate the relation between the Hardy–Smirnov space \(H^2(\Omega )\) and the closed span of the exponential system \(\{e^{nz}\}_{n=1}^{\infty }\) with respect to the Hardy–Smirnov norm \(\Vert \cdot \Vert _\Omega \). Depending on the “height”

The present paper deals with some particular properties of special series of Fourier coefficients of the class of functions with bounded variation with respect to general orthonormal systems (ONS). The obtained results demonstrate that the properties of the general ONS and of the classical ONSs (trigonometric, Haar, Walsh system) are different in some cases.

In this paper we generalize the result (obtained by the author earlier in the paper titled “Products and coproducts in the category \({{\mathcal {S}}}(B)\) of Segal topological algebras”) about the existence of the coproduct of objects of the category \({\mathcal S}(B)\) from finite collection of objects of \({{\mathcal {S}}}(B)\) to any collection of objects of \({{\mathcal {S}}}(B)\).

Let \((m,\ n)\) be fixed positive integers such that \(m>n,\ \gcd (m,\ n)=1\) and \( mn\equiv 0 \pmod 2\). Then the triple \((m^2-n^2,\ 2mn,\ m^2+n^2)\) is called a primitive Pythagorean triple. In 1956, Jeśmanowicz (Wiadom Math 1(2):196–202, 1955/1956 ) conjectured that the equation \((m^2-n^2)^x+(2mn)^y=(m^2+n^2)^z\) has only the positive integer solution \((x,\ y,\ z)=(2,\ 2,\ 2)\). This problem

In this paper we find all positive integer solutions (x, y, n, a, b) of the equation in the title for non negative integers a and b under the condition that the integers x and y are relatively prime and \( n \ge 3\).

Let N be a positive integer, \({\mathbb {A}}\) be a nonempty subset of \({\mathbb {Q}}\) and \(\alpha =\dfrac{\alpha _{1}}{\alpha _{2}}\in {\mathbb {A}}{\setminus } \{0,N\}\). \(\alpha \) is called an N-Korselt base (equivalently N is said an \(\alpha \)-Korselt number) if \(\alpha _{2}p-\alpha _{1}\) is a divisor of \(\alpha _{2}N-\alpha _{1}\) for every prime p dividing N. The set of all Korselt

We establish an interpolation result for the rearrangement-invariant martingale Hardy spaces. By using this interpolation result, we extend the mapping properties of the martingale transforms and the fractional integrals on martingale function spaces. In particular, we obtain the mapping properties on the martingale Hardy–Orlicz spaces, the grand martingale Hardy spaces and the martingale Hardy–Lorentz–Karamata

In this work, we generalize the von Neumann and Davis theorems on the extension of the convexity of a group-invariant norm (resp. function) from the subspace of diagonal matrices to the whole space of complex (resp. hermitian) matrices. Our generalizations go in two directions: (1) We replace the space of complex (hermitian) matrices and its subspace of diagonal matrices by an Eaton triple and its

First, we prove that the Diophantine system$$\begin{aligned} f(z)=f(x)+f(y)=f(u)-f(v)=f(p)f(q) \end{aligned}$$has infinitely many integer solutions for \(f(X)=X(X+a)\) with nonzero integers \(a\equiv 0,1,4\pmod {5}\). Second, we show that the above Diophantine system has an integer parametric solution for \(f(X)=X(X+a)\) with nonzero integers a, if there are integers m, n, k such that$$\begin{aligned}

In this note, some basic structural conditions that are imposed on a Finsler manifold as a consequence of existence of a conformal gradient field are presented. Then, after reiterating the definition of warped product Finsler manifolds and the constituents of its related variational problem, the correlation between geodesics of this warped product structure and geodesics of its constructing Finsler