The Legendre polynomials \(P_n(x)\) are defined by $$\begin{aligned} P_n(x)=\sum _{k=0}^n\left( {\begin{array}{c}n+k\\ k\end{array}}\right) \left( {\begin{array}{c}n\\ k\end{array}}\right) \left( \frac{x-1}{2}\right) ^k\quad (n=0,1,2,\ldots ). \end{aligned}$$ In this paper, we prove two congruences concerning Legendre polynomials. For any prime \(p>3\), by using the symbolic summation package Sigma

We consider a planar system \(z'=f(t,z)\) under non-resonance or double resonance conditions and obtain the existence of \(2\uppi \)-periodic solutions by combining a rotation number approach together with Poincaré-Bohl theorem. Firstly, we allow that the angular velocity of solutions of \(z'=f(t,z)\) is controlled by the angular velocity of solutions of two positively homogeneous system \(z'=L_i(t

All Cayley representations of the distant graph \(\Gamma _{\mathbb Z}\) over integers are characterized as Neumann subgroups of the extended modular group. Possible structures of Neumann subgroups are revealed and it is shown that every such structure can be realized.

In this paper, we completely characterize the compactness of the Volterra type integration operators \(J_b\) acting from weighted Bergman spaces \(A^p_{\alpha }\) to Hardy spaces \(H^q\) for all \(0

The paper is devoted to some extremal problems for convex curves and polygons in the Euclidean plane referring to the self Chebyshev radius. In particular, we determine the self Chebyshev radius for the boundary of an arbitrary triangle. Moreover, we derive the maximal possible perimeter for convex curves and boundaries of convex n-gons with a given self Chebyshev radius.

In [2] we initiated the theory of polyanalytic functions of a quaternionic variable in the slice hyperholomorphic setting.

In this paper, we give necessary and sufficient conditions for the \(L^p\)-well-posedness (resp. \(B_{p,q}^s\)-well-posedness) for the third order degenerate differential equation with finite delay: \((Mu)'''(t) + (Nu)''(t)= Au(t) + Bu'(t) + Gu''_t + Fu'_t + Hu_t + f(t)\) on \([0,2\pi ])\) with periodic boundary conditions \((Mu)(0) = (Mu)(2\pi )\), \((Mu)'(0) = (Mu)'(2\pi )\), \((Mu)''(0) = (Mu)''(2\pi

This article studies a type of integrable linear ordinary differential equations considered by Shifner [17] and generalised by Erougin [5], Salakhova and Chebotarev [15]. We will consider a slight generalisation of their results.

In this paper we study block matrices and multipliers with respect to a Schur-type product operation on them. A subclass of the block Toeplitz matrices is explored, obtaining generalizations of Toeplitz’s and Bennett’s classical theorems that show identifications with complex functions and measures on the bitorus. We also investigate the classes of matrices that can be approximated in the operator

C. Kimberling defined the concept of triangle center function in order to describe centers of triangles as points associated to some functions depending on the sidelengths, instead of in terms of geometrical properties. In this article we provide two definitions of n-gon center function for \(n\ge 3\): one of them in terms of the coordinates of the vertices and the other one by means of the lengths

In this paper, we establish quantitative estimates for nonlinear sampling Kantorovich operators in terms of the modulus of smoothness in the setting of Orlicz spaces. This general frame allows us to directly deduce some quantitative estimates of approximation in \(L^{p}\)-spaces, \(1\le p<\infty \), and in other well-known instances of Orlicz spaces, such as the Zygmung and the exponential spaces.

Let C be a plane convex body. The relative distance (or C-distance) of points \(a, b\in C\) is defined by the ratio of the Euclidean distance of a and b to the half of the Euclidean distance of \(a_{1}, b_{1}\in C\), where \(a_{1}b_{1}\) is a longest chord of C parallel to the line-segment ab. Denote by \(\phi _{k}(C)\) the greatest possible number d such that the boundary of C contains k points in

In this work, we study the partial inverse problems for the Schrödinger transmission eigenvalue problem. It is shown that if the potential is partially known a prior, then only partial eigenvalues can uniquely determine the potential. The relationship between the density of the known eigenvalues and the length of the subinterval on the given potential is revealed.

In this paper we show if E is a totally real Galois extension of \({{\mathbb {Q}}}\) of degree s then there is a cubic extension K of E with signature (s, s) such that in the group of units of K there are s multiplicatively independent positive units with the following property: if \(\alpha \) is any of those s units then there are two conjugates of \(\alpha \) over \({{\mathbb {Q}}}\) which are complex

We study the endpoint regularity of the one-dimensional discrete multisublinear fractional maximal operators, both in the centered and uncentered versions. Some new variation inequalities will be proved for the above operators acting on the vector-valued function \(\mathbf {f}=(f_1, \ldots ,f_m)\) with each \(f_j\) belonging to \(\mathrm{BV}({\mathbb {Z}})\) or \(\ell ^1({\mathbb {Z}})\), where \(

In this paper, we establish some necessary and sufficient conditions for the boundedness of generalized multilinear Hausdorff operators on weighted central Morrey, Herz, and Morrey–Herz spaces on the Heisenberg group. Moreover, we also give some sufficient conditions for the boundedness of generalized multilinear Hausdorff operators on such spaces with the Muckenhoupt weights. Our results generalize

We consider inertial iteration methods for Fermat–Weber location problem and primal–dual three-operator splitting in real Hilbert spaces. To do these, we first obtain weak convergence analysis and nonasymptotic O(1/n) convergence rate of the inertial Krasnoselskii–Mann iteration for fixed point of nonexpansive operators in infinite dimensional real Hilbert spaces under some seemingly easy to implement

Let \(\mathrm {AC}(X)\) be the Banach algebra of all absolutely continuous complex-valued functions f on a compact subset \(X\subset \mathbb {R}\) with at least two points under the norm \(\left\| f\right\| _{\Sigma }=\left\| f\right\| _\infty +\mathrm {V}(f)\), where \(\mathrm {V}(f)\) denotes the total variation of f. We prove that every approximate local isometry from \(\mathrm {AC}(X)\) to \(\mathrm

In this paper, we consider the stability problem on perturbation near a physically steady state solution of the 3D generalized incompressible magnetohydrodynamic system in Lei-Lin space. The global stability and analytic estimates for small perturbation are established by the semigroup method in the critical space \(\chi ^{1-2\alpha }(\mathbb {R}^3)\) with \(\frac{1}{2}\le \alpha \le 1\), where linear

We study centro-affine Tchebychev hyperbolic hypersurfaces M in \(\mathbb {R}^{4}\), which satisfy the following conditions: for all \(X,Y,Z \in T_{p}M\): 1. M is flat, it means that the curvature tensor \(\widehat{R}\) associated with the Levi-Civita connection of the centro-affine metric satisfies \(\widehat{R}(X,Y)Z = 0\); 2. The Tchebychev vector field \( T^{\#}\) satisfies \(\widehat{\nabla }_{X}T^{\#}=\alpha

Given a pair of real functions (k, f), we study the conditions they must satisfy for \(k+\lambda f\) to be the curvature in the arc-length of a closed planar curve for all real \(\lambda \). Several equivalent conditions are pointed out, certain periodic behaviours are shown as essential and a family of such pairs is explicitely constructed. The discrete counterpart of the problem is also studied.

In this paper we study a nonlinear system of reaction–diffusion differential equations consisting of an ordinary equation coupled to a fully parabolic chemotaxis system. This system constitutes a mathematical model for the evolution of two populations that are competing for a common resource, one of which is subject to chemotaxis. Under suitable assumptions we prove the global in time existence and

Systems of iterative functional equations with a non-trivial set of contact points are not necessarily solvable, as the resulting intersections may lead to an overdetermination of the system. To obtain existence and uniqueness results additional conditions must be imposed on the system. These are the compatibility conditions, which we define and study in a general setting. An application to the affine

Let X, Y be two Banach spaces, \(f:X\rightarrow Y\) be an \(\varepsilon \)-isometry with \(f(0)=0\) for some \(\varepsilon \ge 0\), and let \(Y_f\equiv \overline{\mathrm{span}}f(X)\). In this paper, we first introduce a notion of \(w^*\)-stability of an \(\varepsilon \)-isometry f. Then we show that stability of f implies its \(w^*\)-stability; the two notions of stability and \(w^*\)-stability coincide

By means of the discrete zero curvature representation, the coupled Bogoyavlensky lattice 1(2) hierarchy related to a \(3\times 3\) matrix problem is derived. Resorting to the characteristic polynomial of Lax matrix for the lattice hierarchy, we introduce a trigonal curve \(\mathcal {K}_{m-1}\) of arithmetic genus \(m-1\) and construct the related Baker–Akhiezer function and meromorphic function. By

Very recently, Masjed-Jamei & Koepf [Some summation theorems for generalized hypergeometric functions, Axioms, 2018, 7, 38, 10.3390/axioms 7020038] established some summation theorems for the generalized hypergeometric functions. The aim of this paper is to establish extensions of some of their summation theorems in the most general form. As an application, several Eulerian-type and Laplace-type integrals

A numerical semigroup S is almost-positioned if for all \(s\in {{\mathbb {N}}}\backslash S\) we have that \(\mathrm{F}(S)+\mathrm{{m}}(S)+1-s\in S\). In this note we give algorithmics for computing the whole set of almost-positioned numerical semigroup with fixed multiplicity and Frobenius number. Moreover, we prove Wilf’s conjecture for this type of numerical semigroups.

The present paper extends Stetkær’s Algebraic Small Dimension Lemma on semigroups (Stetkær in Aequat Math, 2020) by replacing its involution by an anti-homomorphism that need not be involutive. We apply our result, to give an accessible approach to solve the d’Alembert \( \mu \)-functional equation and the Wilson \(\mu \)-functional equation on compact groups with an anti-homomorphism. This generalizes

We show that the Banach–Mazur distance between the parallelogram and the affine-regular hexagon is \(\frac{3}{2}\) and we conclude that the diameter of the family of centrally-symmetric planar convex bodies is just \(\frac{3}{2}\). A proof of this fact does not seem to be published earlier. Asplund announced this without a proof in his paper proving that the Banach–Mazur distance of any planar centrally-symmetric

In this paper, we consider the Cauchy problem of the multi-dimensional generalized MHD system in the whole space and construct global smooth solutions with a class of large initial data by exploring the structure of the nonlinear term. Precisely speaking, our choice of special initial data whose \(L^{\infty }\) norm can be arbitrarily large allows to generate a unique global-in-time solution to the

The aim of this note is to show the generalized Hyers–Ulam stability of a functional equation in four variables. In order to do this, the fixed point method is applied. As corollaries from our main result, some outcomes on the stability of some known equations will be also derived.

Let f be a smooth strictly convex solution of $$\begin{aligned} \det \left( \frac{\partial ^{2}f}{\partial x_{i}\partial x_{j}}\right) =\exp \left\{ \sum _{i=1}^n- a_i\frac{\partial f}{\partial x_{i}} +\sum _{i=1}^n b_ix_i+c\right\} \end{aligned}$$ defined on \({\mathbb {R}}^{n}\), where \(a_i\), \(b_i\) and c are constants, then the graph \(M_{\nabla f}\) of \(\nabla f\) is a space-like translating

Let \(B^\times \) be the biset functor over \(\mathbb {F}_2\) sending a finite group G to the group \(B^\times (G)\) of units of its Burnside ring B(G), and let \(\widehat{B^\times }\) be its dual functor. The main theorem of this paper gives a characterization of the cokernel of the natural injection from \(B^\times \) in the dual Burnside functor \(\widehat{\mathbb {F}_2B}\), or equivalently, an

This article proves the analytical inflexibility of regular developable surfaces and doubly connected surfaces of revolution, which are fixed along a curve on the surface simultaneously with respect to a point and a plane.

Due to Čencov’s theorem, there exists a unique family of invariant symmetric (0, 2)-tensor fields on the space of positive probability measures on a set of n-points indexed by \(n\in \mathbb {N}\) under Markov embeddings. We deform Markov embeddings keeping sufficiency, and prove existence and uniqueness of invariant families under the embeddings.

In this paper, we compare the spectra of bulk and edge Hamiltonians on graphene from the point of view of the theory of quantum graphs. Since the bulk Hamiltonians with even potentials on the graphene has been studied by Kuchment and Post, our main targets here are the edge Hamiltonians on the graphene with boundaries. In this paper, we obtain their spectral properties under the Dirichlet boundary

Let c be a square-free positive integer and p a prime satisfying \(p\not \mid c\). Let \(h(-c)\) denote the class number of the imaginary quadratic field \(\mathbb {Q}(\sqrt{-c})\). In this paper, we consider the Diophantine equation $$\begin{aligned}&cx^2+p^{2m}=4y^n,~~x,y\ge 1, m\ge 0, n\ge 3,\\&\quad \gcd (x,y)=1, \gcd (n,2h(-c))=1, \end{aligned}$$ and we describe all its integer solutions. Our

In this paper we extend the deterministic sublinear FFT algorithm in Plonka et al. (Numer Algorithms 78:133–159, 2018. https://doi.org/10.1007/s11075-017-0370-5) for fast reconstruction of M-sparse vectors \({\mathbf{x}}\) of length \(N= 2^J\), where we assume that all components of the discrete Fourier transform \(\hat{\mathbf{x}}= {\mathbf{F}}_{N} {\mathbf{x}}\) are available. The sparsity of \({\mathbf{x}}\)

We present some applications of the Banach limit in the study of the stability of the linear functional equation in a single variable.

Null scrolls in Lorentz-Minkowski space are ruled surfaces whose rulings are null vectors. They are the only surfaces whose harmonic evolute is not a surface, but a single curve which is spacelike if the mean curvature of a null scroll is not constant. In this paper we do the inverse problem: given a spacelike curve, we find null scrolls having this curve as its harmonic evolute. We also discuss in

In this paper, we prove that the inversions in metric spaces are coarsely bilipschitz continuous with respect to distance ratio metric.

Some errors in the above-named article need to be corrected.

Applying medial limits we describe bounded solutions \(\varphi :S\rightarrow {\mathbb {R}}\) of the functional equation $$\begin{aligned} \varphi (x)=\int _{\Omega }g(\omega )\varphi (f(x,\omega ))d\mu (\omega )+G(x), \end{aligned}$$ where \((\Omega ,{\mathcal {A}},\mu )\) is a measure space, \(S\subset \mathbb R\), \(f:S\times \Omega \rightarrow S\), \(g:\Omega \rightarrow {\mathbb {R}}\) is integrable

We obtain asymptotic formulas with remainder terms for the hyperbolic summations \(\sum _{mn\le x} f((m,n))\) and \(\sum _{mn\le x} f([m,n])\), where f belongs to certain classes of arithmetic functions, (m, n) and [m, n] denoting the gcd and lcm of the integers m, n. In particular, we investigate the functions \(f(n)=\tau (n), \log n, \omega (n)\) and \(\Omega (n)\). We also define a common generalization

In the context of Riordan arrays, the problem of determining the square root of a Bell matrix \(R={\mathcal {R}}(f(t)/t,\ f(t))\) defined by a formal power series \(f(t)=\sum _{k \ge 0}f_kt^k\) with \(f(0)=f_0=0\) is presented. It is proved that if \(f^\prime (0)=1\) and \(f^{\prime \prime }(0)\ne 0\) then there exists another Bell matrix \(H={\mathcal {R}}(h(t)/t,\ h(t))\) such that \(H*H=R;\) in

Let \( \gcd (k,j) \) denote the greatest common divisor of the integers k and j, and let r be any fixed positive integer. Define $$\begin{aligned} M_r(x; f) := \sum _{k\le x}\frac{1}{k^{r+1}}\sum _{j=1}^{k}j^{r}f(\gcd (j,k)) \end{aligned}$$ for any large real number \(x\ge 5\), where f is any arithmetical function. Let \(\phi \), and \(\psi \) denote the Euler totient and the Dedekind function, respectively

In this paper, we study \(\Xi \)-curvature and H-curvature of a special class of Finsler metrics called general \((\alpha ,\beta )\)-metrics. We prove that every general \((\alpha ,\beta )\)-metric of almost vanishing H-curvature is of almost vanishing \( \Xi \)-curvature under certain conditions. Moreover, we study such Finsler metric with vanishing \(\Xi \)-curvature and its interaction to the flag

We discuss the uniform exponential stability of strongly continuous semigroups generated by operators of the form \( A+B \), where B is a bounded perturbation of a generator A. We compare two approaches to the problem: via the Dyson–Phillips formula and via the size of the norm of the commutator of A and B- the method recently developed by M. Gil’. We show that quite often the first approach is more

We consider shift-invariant multiresolution spaces generated by q-elliptic splines in \({\mathbb {R}^{d}},d\ge 2,\) which are tempered distributions characterized by a complex-valued elliptic homogeneous polynomial q of degree \(m>d\). To construct Riesz bases of \(L^{2} ({\mathbb {R}^{d}}),\) a family of non-separable basic smooth functions are obtained by localizing a fundamental solution of the

Let \((S,+)\) be an abelian semigroup, let \((H,+)\) be a uniquely 2-divisible, abelian group and let \(\sigma ,\tau \) be two endomorphisms of S. In this paper, we find the solutions \(f:S\rightarrow H\) of the following Drygas type functional Eq. $$\begin{aligned} f(x+\sigma (y))+f(x+\tau (y))=2f(x)+f(\sigma (y))+f(\tau (y)), \quad x,y\in S, \end{aligned}$$(1) in terms of additive and bi-additive

Some means on the positive matrices can be represented by algebraic midpoint. In this paper, we study such algebraic structures.

In the original publication the author name is incorrectly published.

Let \((a)_{n}=a(a+1)\cdots (a+n-1)\) be the Pochhammer symbol. Recently, Jana and Kalita proved the following two supercongruences: $$\begin{aligned} \sum _{k=0}^{(p-1)/3} (6k+1)\frac{(\frac{1}{3})_k^4 (2k)!}{k!^4 (\frac{2}{3})_{2k}}&\equiv p\pmod {p^3}\quad \text {for primes }\,{ p\equiv 1\pmod {3},} \\ \sum _{k=0}^{(p+1)/3} (6k-1)\frac{(-\frac{1}{3})_k^4 (2k)!}{k!^4 (-\frac{2}{3})_{2k}}&\equiv p\pmod

We illustrate the solvability through weak-renormalized solutions for an invasion system of healthy tissue by a solid tumor under the Dirichlet boundary conditions and the assumptions of no growth conditions and integrable data. This model consists of four parameters: tumor cell density, matrix degradation enzyme concentration, macromolecules concentration, and oxygen concentration.

We prove a version of Whitney’s extension theorem in the ultradifferentiable Beurling setting with controlled loss of regularity. As a by-product we show the existence of continuous linear extension operators on certain spaces of Whitney ultrajets on arbitrary closed sets in \(\mathbb {R}^n\).

In this paper, we consider the generalized Marcum function of the second kind as an analogous function of the so-called generalized Marcum Q-function. We provide the log-convexity (log-concavity) property for its unit complement and improve some of our previous results on it. One of the transformed functions of the generalized Marcum function of the second kind is discussed in details in this paper

In this paper we first discuss weighted mean curvature and volume comparisons on smooth metric measure space \((M, g, e^{-f}dv)\) under the integral Bakry–Émery Ricci tensor bounds. In particular, we add an additional condition on the potential function f to ensure the validity of previous conclusions for some cases proved by the second author. Then, we apply the comparison results to get a new diameter

We obtain a characterization of Ulam stability for the linear difference equation with constant coefficients \(x_{n+p}=a_1x_{n+p-1}+\cdots +a_px_n\) in locally convex spaces. Moreover, for the first order linear difference equation we determine the best Ulam constant.

We study nondegeneracy of ground states of the Hartree equation $$ -\Delta u+u=(I_{2}*u^2)u\quad \text{ in } {\mathbb {R}}^n $$ where \(n=3,4,5\) and \(I_2\) is the Newton potential. As an application of the nondegeneracy result, we use a Lyapunov–Schmidt reduction argument to construct multiple semiclassical solutions to the following Hartree equation with an external potential $$ -\varepsilon ^2\Delta