In this study, it is shown that the only Lucas numbers which are concatenations of three repdigits are 123, 199, 322, 521, 843, 2207, 5778. The proof depends on lower bounds for linear forms and some tools from Diophantine approximation.

This paper is devoted to the complete algebraic and geometric classification of complex 4-dimensional nilpotent right commutative algebras. The corresponding geometric variety has dimension 15 and decomposes into 5 irreducible components determined by the Zariski closures of four one-parameter families of algebras and a two-parameter family of algebras (see Theorem B). In particular, there are no rigid

We solve the functional equation \(f(xy) + g(\sigma (y)x) = h(x)k(y)\) for complex-valued functions f, g, h, k on groups or monoids generated by their squares, where \(\sigma \) is an involutive automorphism. This contains both classical d’Alembert equations \(g(x + y) + g(x - y) = 2g(x)g(y)\) and \(f(x + y) - f(x - y) = g(x)h(y)\) in the abelian case, but we do not suppose our groups or monoids are

For every nilpotent n-Lie algebra A of dimension d, t(A) is defined by \(t(A)=\left( {\begin{array}{c}d\\ n\end{array}}\right) -\dim {\mathcal {M}}(A)\), where \({\mathcal {M}}(A)\) denotes the Schur multiplier of A. In this paper, we classify all nilpotent n-Lie alegbras A satisfying \(t(A)=9,10\). We also classify all nilpotent n-Lie algebras for \(11\le t(A)\le 16\) when \(n\ge 3\).

In this paper, we consider invariant Einstein–Randers metrics on the Stiefel manifolds \(V_k\mathbb {R}^n\) of all orthonormal k-frames in \(\mathbb {R}^n\), which is diffeomorphic to the homogeneous space \({\mathrm {S}}{\mathrm O}(n)/{\mathrm {S}}{\mathrm O}(n-k)\). We prove that for \(2\le p\le \frac{2}{5}n-1\), there are four different families of \({\mathrm {S}}{\mathrm O}(n)\)-invariant Einstein–Randers

This paper aims to highlight a class of integral linear and positive operators of Landau type which have affine functions as fixed points. We focus to reveal approximation properties both in \(L_p\) spaces and in weighted \(L_p\) spaces \((1\le p<\infty )\). Also, we give an extension of the operators to approximate real-valued vector functions. In this case, the study pursues the approximation of

We establish a unique global strong solution for nonhomogeneous Bénard system with zero density at infinity on the whole two-dimensional (2D) space. In particular, the initial data can be arbitrarily large and the initial density can contain vacuum states and even have compact support. Our method relies heavily on the structure of the system under consideration and spatial dimension.

We consider a nonlinear Dirichlet problem, driven by the p-Laplacian with a reaction involving two parameters \(\lambda \in {\mathbb {R}}, \theta >0\). We view the problem as a perturbation of the classical eigenvalue problem for the Dirichlet problem. The perturbation consists of a parametric singular term and of a superlinear term. We prove a nonexistence and a multiplicity results in terms of the

The concept of moment differentiation is extended to the class of moment summable functions, giving rise to moment differential properties. The main result leans on accurate upper estimates for the integral representation of the moment derivatives of functions under exponential-like growth at infinity, and appropriate deformation of the integration paths. The theory is applied to obtain summability

The concept of penumbra of a convex set with respect to another convex set was introduced by Rockafellar (1970). We study various geometric and topological properties of penumbras, their role in proper and strong separation of convex sets, and their relation to polyhedra and M-decomposable sets.

In this paper, we derive an incompressible Oldroyd-B model with hybrid dissipation and partial damping on stress tensor \(\tau \) via the velocity equations and the generalized constitutive law so that global well-posedness of the model is established in the Sobolev space framework. Precisely speaking, the proof is based on the curl-div free property of \(\tau - {\nabla }\frac{1}{\Delta } \mathbb {P}\mathrm{div}\tau

Using the Euler–Jacobi formula there is a relation between the singular points of a polynomial vector field and their topological indices. Using this formula we obtain the configuration of the singular points together with their topological indices for the polynomial differential systems \(\dot{x}=P(x,y)\), \(\dot{y} =Q(x,y)\) with degree of P equal to 2 and degree of Q equal to m when these systems

Let G be a finite group. An automorphism \(\alpha \) of G is called an \(\text{ IA }\) automorphism if it induces the identity automorphism on the abelianized group \(G/G'\). Let \(\text{ IA }(G)\) be the group of all \(\text{ IA }\) automorphisms of G, and let \(\text{ IA }(G)^*\) be the group of all \(\text{ IA }\) automorphisms of G which fix Z(G) element-wise. Let \(\mathrm {Var}(G)\) be the group

Davison in 2015 used the famous Banach Fixed Point Theorem to prove that a certain class of iterated function systems generated counterparts of the Hutchinson measure in the space of projection-valued measures. In this paper, we generalize this result by considering iterated function systems with infinitely many maps.

In this note we derive some maximum principle for an appropriate function involving u, Du and \(D^2 u\), where \(u=u(x)\) is a convex solution to a Monge–Ampère equation in a plane domain.

In this article we consider some classes of orthogonally additive operators in Köthe–Bochner spaces in the setting of the theory of lattice-normed spaces and dominated operators. The our first main result asserts that the C-compactness of a dominated orthogonally additive operator \(S:E(X)\rightarrow F(Y)\) implies the C-compactness of its exact dominant \(|\!|S|\!|:E\rightarrow F\). Then we show that

Based on the lower and upper solutions method, we propose a new approach to the Hyers–Ulam and Hyers–Ulam–Rassias stability of first-order ordinary differential equations \(u'=f(t,u)\), in the lack of Lipschitz continuity assumption. Apart from extending and improving the literature by dropping some assumptions, our result provides an estimate for the difference between the solutions of the exact and

The sine and cosine addition laws on a (not necessarily commutative) semigroup are \(f(xy) = f(x)g(y) + g(x)f(y)\), respectively \(g(xy) = g(x)g(y) - f(x)f(y)\). Both of these have been solved on groups, and the first one has been solved on semigroups generated by their squares. Quite a few variants and extensions with more unknown functions and/or additional terms have also been studied. Here we extend

In this paper, we study the Bohr phenomenon for functions that are defined on a general simply connected domain of the complex plane. We improve known results of R. Fournier and St. Ruscheweyh for a class of analytic functions. Furthermore, we examine the case where a harmonic mapping is defined in a disk containing \({\mathbb {D}}\) and obtain a Bohr type inequality.

This paper deals with a property which is equivalent to generalised-lushness for separable spaces. It thus may be seemed as a geometrical property of a Banach space which ensures the space to have the Mazur–Ulam property. We prove that a Banach space X enjoys this property if and only if C(K, X) enjoys this property. We also show the same result holds for \(L_\infty (\mu ,X)\) and \(L_1(\mu ,X)\).

We consider kernels of unbounded Toeplitz operators in \(H^p({\mathbb {C}}^{+})\) in terms of a factorization of their symbols. We study the existence of a minimal Toeplitz kernel containing a given function in \(H^p({\mathbb {C}}^{+})\), we describe the kernels of Toeplitz operators whose symbol possesses a certain factorization involving two different Hardy spaces and we establish relations between

In this paper we prove the existence of mild solutions for a problem governed by a semilinear non-autonomous second order differential inclusion where a stabilization of the solution is expected due to the control of the reaction term. In order to obtain our existence theorem, first we study a more general problem with a differential inclusion which involves a perturbation guided by an operator \(N

The class \(\mathbb D\) of generalized continuous functions on \(\mathbb {R}\) known under the common name of Darboux-like functions is usually described as consisting of eight families of maps: Darboux, connectivity, almost continuous, extendable, peripherally continuous, those having perfect road, and having either the Cantor Intermediate Value Property or the Strong Cantor Intermediate Value Property

We prove that the maximal number of pairwise disjoint 4-blocks in a MOL(6) is 3. We recall various proofs for the non-existence of a MOL(6) and show: with the theorem the proofs can be simplified considerably.

Let \(\displaystyle \omega \) be a weight in \(A^*_\infty \) and let \(\displaystyle \mathcal {M}^m_n(\mathbf {f})\) be the multilinear strong maximal function of \(\displaystyle \mathbf {f}=\left( f_1,\ldots ,f_m\right) \), where \(f_1,\ldots ,f_m\) are functions on \(\mathbb R^n\). In this paper, we consider the asymptotic estimates for the distribution functions of \(\displaystyle \mathcal {M}^m_n\)

In recent decades, there has been an increase in the number of publications related to the hypersurfaces of real space forms with two principal curvatures. The works focus mainly on the case when one of the two principal curvatures is simple. The purpose of this paper is to study a slightly more general class of complete minimal hypersurfaces in real space forms of constant curvature c, namely those

In this paper, using Bregman distance, we introduce an iterative algorithm for approximating a common solution of Split Equality Fixed Point Problem and Split Equality Equilibrium Problem in p-uniformly convex and uniformly smooth Banach spaces that are more general than Hilbert spaces. The advantage of the algorithm is that it is done without the prior knowledge of Bregman Lipschitz coefficients and

In this paper, we introduce Riemannian warped product submersions and construct examples and give fundamental geometric properties of such submersions. On the other hand, a necessary and sufficient condition for a Riemannian warped product submersion to be totally geodesic, totally umbilic and minimal is given.

In the first part of the article we study polynomial vector fields of arbitrary degree in \({\mathbb {R}}^3\) having an algebraic surface of revolution invariant by their flows. In the second part, we restrict our attention to an important case where the algebraic surface of revolution is a cubic surface. We characterize all the possible configurations of invariant meridians and parallels that the

A new class of bivariate poly-analytic Hermite polynomials is considered. We show that they are realizable as the Fourier–Wigner transform of the univariate complex Hermite functions and form a nontrivial orthogonal basis of the classical Hilbert space on the two-complex space with respect to the Gaussian measure. Their basic properties are discussed, such as their three term recurrence relations,

We extend an inequality involving the Bernstein basis polynomials and convex functions on [0, 1]. The inequality was originally conjectured by Raşa about thirty years ago, but was proved only recently. Our extension provides an inequality involving q-monotone functions, \(q\in \mathbb N\). In particular, 1-monotone functions are nondecreasing functions, and 2-monotone functions are convex functions

In this paper, we study the problem of finding a common solution to variational inequality and fixed point problems for a countable family of Bregman weak relatively nonexpansive mappings in real reflexive Banach spaces. Two inertial-type algorithms with adaptive step size rules for solving the problem are presented and their strong convergence theorems are established. The usage of the Bregman distances

The Feigenbaum–Kadanoff–Shenker (FKS) equation is a nonlinear iterative functional equation, which characterizes the quasiperiodic route to chaos for circle maps. Instead of FKS equation, we put forward the second type of FKS equation. We use the iterative construction method to obtain a new kind of continuous solutions with platforms for this functional equation. We show some properties of these continuous

We consider the space \(C^1(K)\) of real-valued continuously differentiable functions on a compact set \(K\subseteq \mathbb {R}^d\). We characterize the completeness of this space and prove that the restriction space \(C^1(\mathbb {R}^d|K)=\{f|_K: f\in C^1(\mathbb {R}^d)\}\) is always dense in \(C^1(K)\). The space \(C^1(K)\) is then compared with other spaces of differentiable functions on compact

The purpose of this paper is mainly to prove that if f is a transcendental entire function of hyper-order strictly less than 1 and \(f(z)^{n}+a_{1}f'(z)+\cdots +a_{k}f^{(k)}(z)\) is a periodic function, then f(z) is also a periodic function, where n, k are positive integers, and \(a_{1},\cdots ,a_{k}\) are constants. Meanwhile, we offer a partial answer to Yang’s Conjecture, theses results extend some

In this paper, we introduce the notation of E-weighted core-EP and F-weighted dual core-EP inverse of matrices. We then obtain a few explicit expressions for the weighted core-EP inverse of matrices through other generalized inverses. Further, we discuss the existence of generalized weighted Moore–Penrose inverse and additive properties of the weighted core-EP inverse of matrices. In addition to these

In this paper, we obtain an explicit list of self-similar solutions of inverse mean curvature flow in \(\mathbb {R}^2\).

In this paper, we study the regular geometric behavior of the mean curvature flow (MCF) of submanifolds in the standard Gaussian metric space \(({{\mathbb {R}}}^{m+p},e^{-|x|^2/m}\overline{g})\) where \(({{\mathbb {R}}}^{m+p},\overline{g})\) is the standard Euclidean space and \(x\in {{\mathbb {R}}}^{m+p}\) denotes the position vector. Note that, as a special Riemannian manifold, \(({{\mathbb {R}}}^{m+p}

We consider Lie groupoids of the form \({\mathcal {G}}(M,M_1) := M_0 \times M_0 \sqcup H \times M_1 \times M_1 \rightrightarrows M,\) where \(M_0 = M \setminus M_1\) and the isotropy subgroup H is an exponential Lie group of dimension equal to the codimension of the manifold \(M_1\) in M. The existence of such Lie groupoids follows from integration of almost injective Lie algebroids by Claire Debord

Let \(\mathcal {I}\) be a meager ideal on \(\mathbf {N}\). We show that if x is a sequence with values in a separable metric space then the set of subsequences [resp. permutations] of x which preserve the set of \(\mathcal {I}\)-cluster points of x is topologically large if and only if every ordinary limit point of x is also an \(\mathcal {I}\)-cluster point of x. The analogue statement fails for all

The aim of this work is the construction of a new class of disconnected locally compact near-fields. They are all Dickson near-fields and derived from finite-dimensional division algebras over local fields by means of strong couplings with a finite Abelian Dickson group consisting of inner automorphisms.

In this article starting from some reductions of hyperelliptic integrals of genus 3 into elliptic integrals, due to Michael Roberts (A Tract on the addition of Elliptic and hyperelliptic integrals, Hodger, Foster and Co, 1871) we obtain several identities which, to the best of our knowledge, are all new. The strategy used at this purpose is to evaluate Roberts integrals, in two different ways, on one

In the paper, we study evolution equations of the scalar and Ricci curvatures under the Hamilton’s Ricci flow on a closed manifold and on a complete noncompact manifold. In particular, we study conditions when the Ricci flow is trivial and the Ricci soliton is Ricci flat or Einstein.

We study ruled real hypersurfaces whose shape operators have constant squared norm in nonflat complex space forms. In particular, we prove the nonexistence of such hypersurfaces in the projective case. We also show that biharmonic ruled real hypersurfaces in nonflat complex space forms are minimal, which provides their classification due to a known result of Lohnherr and Reckziegel.

A class of integral transforms, on the planar Gaussian Hilbert space with range in the weighted Bergman space on the bi-disk, is defined as the dual transforms of the 2d fractional Fourier transform associated with the Mehler function for Itô–Hermite polynomials. Some spectral properties of these transforms are investigated. Namely, we study their boundedness and identify their null spaces as well

The paper deals with some selection properties of set-valued functions and different types of set-valued integrals of a Young type. Such integrals are considered for classes of Hölder continuous or with bounded Young p-variation set-valued functions. Two different cases are considered, namely set-valued functions with convex values and without convexity assumptions. The integrals contain as a particular

In this paper, we study compact generalized \(\tau \)-quasi Ricci-harmonic metrics. In the first part, we explore conditions under which generalized \(\tau \)-quasi Ricci-harmonic metrics are harmonic-Einstein and give some characterization results for this case. In the second part, we obtain some rigidity results for compact \((\tau , \rho )\)-quasi Ricci-harmonic metrics which are a special case

Let \(\Gamma _n(\mathcal {\scriptstyle {O}}_{\mathbb {K}})\) denote the Hermitian modular group of degree n over an imaginary quadratic number field \(\mathbb {K}\) and \(\Delta _{n,\mathbb {K}}^*\) its maximal discrete extension in the special unitary group \(SU(n,n;\mathbb {C})\). In this paper we study the action of \(\Delta _{n,\mathbb {K}}^*\) on Hermitian theta series and Maaß spaces. For \(n=2\)

A counterpart of the Ohlin theorem for convex set-valued maps is proved. An application of this result to obtain some inclusions related to convex set-valued maps in an alternative unified way is presented. In particular counterparts of the Jensen integral and discrete inequalities, the converse Jensen inequality and the Hermite–Hadamard inequalities are obtained.

In this paper, we introduce the notion of (uniformly weakly) Banach-compact sets, (uniformly weakly) Banach-compact operators and (uniformly weakly) Banach-nuclear operators which generalize p-compact sets, p-compact operators and p-nuclear operators, respectively. Fundamental properties are investigated. Factorizations and duality theorems are given. Injective and surjective hulls are used to show

In our manuscript, we organize a group of sufficient conditions for the approximate controllability of second-order evolution hemivariational inequalities. By applying a suitable fixed-point theorem for multivalued maps, we prove our results. Lastly, we present an example to illustrate the obtained theory.

In this paper, we mainly prove a congruence conjecture of Z.-W. Sun involving Franel numbers: For any prime \(p>3\), we have $$\begin{aligned} \sum _{k=0}^{p-1}(-1)^kf_k\equiv \left( \frac{p}{3}\right) +\frac{2p^2}{3}B_{p-2}\left( \frac{1}{3}\right) \pmod {p^3}, \end{aligned}$$ where \(B_n(x)\) is the n-th Bernoulli polynomial.

We present an alternative proof of a nonexistence result for displaceable constant sectional curvature Lagrangian submanifolds under certain assumptions on the Lagrangian submanifold and on the ambient symplectically aspherical symplectic manifold. The proof utilizes an index relation relating the Maslov index, the Morse index and the Conley–Zehnder index for a periodic orbit of the flow of a specific

We establish direct estimates of the rate of weighted simultaneous approximation by the Baskakov 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.

Dual truncated Toeplitz operators and other restrictions of the multiplication by the independent variable \(M_z\) on the classical \(L^2\) space on the unit circle are investigated. Commutators are calculated and commutativity is characterized. A necessary and sufficient condition for any operator to be a dual truncated Toeplitz operator is established. A formula for recovering its symbol is stated

We establish a new family of q-supercongruences modulo the fourth power of a cyclotomic polynomial, and give several related results. Our main ingredients are q-microscoping and the Chinese remainder theorem for polynomials.

The isoperimetric problem is one of the fundamental problems in differential geometry. By using the method of the calculus of variations we show that the circle centered at the origin in \({\mathbb {B}}^2(1)\) is a proper maximum of the isoperimetric problem in a 2-dimensional Finsler space of Funk type. We also obtain the formula of area enclosed by a simple closed curve in a spherically symmetric

In this paper, we describe an elementary method for counting the number of non-isomorphic algebras of a fixed, finite dimension over a given finite field. We show how this method works in the case of 2-dimensional algebras over the field \({\mathbb {F}}_{2}\).

So far there has not been paid attention to frames that are balanced, i.e. those frames which sum is zero. In this paper we consider balanced frames, and in particular balanced unit norm tight frames, in finite dimensional Hilbert spaces. Here we discover various advantages of balanced unit norm tight frames in signal processing. They give an exact reconstruction in the presence of systematic errors

In this note, we study the Ulam stability of a functional equation both in Banach and m-Banach spaces. Particular cases of this equation are, among others, equations which characterize multi-additive and multi-Jensen functions. Moreover, it is satisfied by the so-called multi-linear mappings.