In this paper we study \(\widetilde{J}\)-tangent affine hyperspheres, where \(\widetilde{J}\) is the canonical para-complex structure on \(\mathbb {R}^{2n+2}\). The main purpose of this paper is to give a classification of \(\widetilde{J}\)-tangent affine hyperspheres of an arbitrary dimension with an involutive distribution \(\mathcal {D}\). In particular, we classify all such hyperspheres in the

This paper deals with the Urysohn type integral form of the Schurer operators which is a significant special class of operators that act on some function spaces. Firstly, we construct the Urysohn type Schurer operators and after that we obtain some convergence results for the aforementioned operators. Ultimately, we deal with some of their shape-preserving properties.

We use the Hopf fibration to explicitly compute generators of the second homotopy group of the flag manifolds of a compact Lie group. We show that these 2-spheres have nice geometrical properties such as being totally geodesic surfaces with respect to any invariant metric on the flag manifold, generalizing a result in Burstall and Rawnsley (Springer Lect. Notes Math. 2(84):1424, 1990). This illustrates

Our aim in this paper is to prove the equality between two kinds of joint operator norms in the case of doubly commuting tuples of hyponormal operators on Hilbert spaces. Moreover, a sharp bound for the distance between two doubly commuting tuples of hyponormal operators in terms of the distance between their Harte spectra is proved.

We give characterizations of the hyperbolicity of a nonautonomous one-sided dynamics defined by a sequence of linear operators, in terms of the hyperbolicity of an associated evolution map defined on a Banach space of sequences. We consider a large family of spaces of sequences besides \(L^p\) spaces. Moreover, we discuss the general cases when the dynamics may be invertible only along the unstable

The present paper deals with the non-real eigenvalues of one dimensional p-Laplacian problem with an indefinite weight. The upper bounds on the imaginary parts and the absolute values of non-real eigenvalues for this indefinite p-Laplacian problem with Dirichlet boundary conditions are obtained through a new method, partly inspired by the estimates obtained in Kikonko and Mingarelli (J Differ Equ 261:6221–6232

The Sturm–Liouville operators on the lasso graph are considered. We obtain new regularized trace formulae for this class of differential operators, which are very useful to further study the inverse spectral problems of this kind of operators.

We introduce and study the bilinear fractional maximal operator $$\begin{aligned} {\mathfrak {M}}_\alpha (f,g)(x)=\sup \limits _{r>0}\frac{1}{|B(O,r)|^{1-\frac{\alpha }{n}}}\int _{B(O,r)}|f(x+y)g(x-y)|dy,\ \ \ \ \ x\in {\mathbb {R}}^n, \end{aligned}$$ where \(O=(0,0,\ldots ,0)\in {\mathbb {R}}^n\) and \(0\le \alpha

In this article, we obtain a characterizations and representations of set-valued solutions defined on an Abelian group G with values in a Hausdorff topological vector space of the following generalized bi-quadratic functional equation: $$\begin{aligned}&F(x+y,z+w)+F(x+y,z-w)+F(x-y,z+w)+F(x-y,z-w) \\&\quad =aF(x,z)+bF(x,w)+cF(y,z)+dF(y,w), \end{aligned}$$ for some nonnegative real numbers a, b, c, and

The blow-up of solutions of a class of nonlinear viscoelastic Kirchhoff equation with suitable initial data and Dirichlet boundary conditions is discussed. By constructing a suitable auxiliary function to overcome the difficulty of gradient estimation and making use of differential inequality technique, we establish a finite time blow-up result when the initial data is at arbitrary energy level. Moreover

For a pair of points in a smooth closed convex planar curve \(\gamma \), its mid-line is the line containing its mid-point and the intersection point of the corresponding pair of tangent lines. It is well known that the envelope of the mid-lines (EML) is formed by the union of three affine invariants sets: Affine envelope symmetry sets; mid-parallel tangent locus and affine evolute of \(\gamma \).

The conjugacy problem is one of the important questions in iteration theory. As far as we know, for discontinuous strictly monotone maps there is no complete result. In this paper, we investigate the conjugacy problem of strictly monotone maps with only one jump discontinuity. We give some sufficient and necessary conditions for the conjugacy relationship. And we present some methods to construct all

Varchenko introduced a distance function on chambers of hyperplane arrangements that he called quantum bilinear form. That gave rise to a determinant indexed by chambers whose entry in position (C, D) is the distance between C and D: that is the Varchenko determinant. He showed that that determinant has a nice factorization. Later, Aguiar and Mahajan defined a generalization of the quantum bilinear

The principal objective of this paper is to answer positively the open question whether every invariant submanifold of a paracontact metric \(( \tilde{\kappa },\tilde{\mu })\)-manifold is totally geodesic. Main result is that any invariant submanifold of a paracontact metric \((\tilde{\kappa }, \tilde{\mu })\)-manifold,\(\tilde{\kappa }\ne -1\), is always totally geodesic. Additionally, if \(\tilde{\kappa

We introduce two different notions of disjoint distributional chaos for sequences of continuous linear operators in Fréchet spaces. We focus special attention to the analysis of some specific classes of linear continuous operators having a certain disjoint distributionally chaotic behaviour, providing also a great number of illustrative examples and applications of our abstract theoretical results

In this paper we consider the generalization of the orthogonality equation. Let S be a semigroup, and let H, X be abelian groups. For two given biadditive functions \(A:S^2\rightarrow X\), \(B:H^2\rightarrow X\) and for two unknown mappings \(f,g:S\rightarrow H\) the functional equation $$\begin{aligned} B(f(x),g(y))=A(x,y) \end{aligned}$$ will be solved under quite natural assumptions. This extends

We prove that every continuous conjugate linear mapping from a unital C\(^*\)-algebra A into its dual space, \(A^*\), which is ternary derivable at the unit element of A is a ternary derivation. This is somehow a ternary-counterpart result for (binary) derivations on associative algebras which proves that any linear continuous map from a unital C\(^*\)-algebra A into a Banach A-bimodule which is derivable

We introduce and study the King type operators associated to a couple \( \left( \mathcal {A},\tau \right) \) where \(\mathcal {A}=\left( A_{n}\right) _{n\in \mathbb {N}}\) is a sequence of linear positive operators from \(C\left[ 0,1\right] \) into \(C\left[ 0,1\right] \) and \(\tau :\left[ 0,1\right] \rightarrow \left[ 0,\infty \right) \) a continuous strictly increasing function. Given a sequence

We study the regularity of smooth functions f defined on an open subset of \({\mathbb {R}}^n\) and such that, for certain integers \(p\ge 2\), the powers \(f^p :x\mapsto (f(x))^p\) belong to a Denjoy–Carleman class \({\mathcal {C}}_M\) associated with a suitable weight sequence M. Our main result is a statement analogous to a classic theorem of H. Joris on \({\mathcal {C}}^\infty \) functions: if a

We prove some q-supercongruences modulo the fourth power of a cyclotomic polynomial by making use of the Chinese remainder theorem for coprime polynomials, Watson’s \(_8\phi _7\) transformation, and the ‘creative microscoping’ method introduced by the author and Wadim Zudilin. In particular, we confirm Conjecture 1.5 in (Results Math 74:131, 2019).

Let \(\lambda _1({\tilde{R}}_{2n+1})\) be the first Dirichlet eigenvalues of the Laplacian associated to regular Reuleaux polygons \({\tilde{R}}_{2n+1}\) of width \({\tilde{w}}_{2n+1}=1\), \(n=1,2,3,\ldots \)Let \(T({\tilde{R}}_{2n+1})\) be their torsional rigidities. It is established that \(\lambda _1({\tilde{R}}_{2n+1})\) is a decreasing sequence, and \(T({\tilde{R}}_{2n+1})\) an increasing sequence

A spacetime endowed with a globally defined timelike Killing vector field admits a certain model of warped product, called the standard static spacetime, and, when the volume element is modified by a factor that depends on a smooth function (which is called density function), we say that this ambient is a weighted standard static spacetime. In such spacetimes, we study some aspects of the geometry

Let M be a topological monoid. Our main goal is to introduce the functional equation $$\begin{aligned} g(xy)+\mu (y)g(x\psi (y))=2g(x)g(y),\ \ x,y\in M, \end{aligned}$$ where \(\psi :M\rightarrow M\) is a continuous anti-endomorphism that need not be involutive and \(\mu :M\rightarrow \mathbb {C}\) is a continuous multiplicative function such that \(\mu (x\psi (x))=1\) for all \(x\in M\). We exploit

The Signorini problem for the Laplace operator is considered in a general polygonal domain. It is proved that the coincidence set consists of a finite number of boundary parts plus a finite number of isolated points. The regularity of the solution is described. In particular, we show that the leading singularity is in general \(r_i^{\pi /(2\alpha _i)}\) at transition points of Signorini to Dirichlet

Involving the notion of strongly Schur-convex functions we give a new characterization of inner product spaces among norm spaces. We also present a representation theorem for functions which generate strongly Schur-convex sums.

Recently, Jana and Kalita proved the following supercongruences involving rising factorials \((\frac{1}{d})_k^3\): $$\begin{aligned}&\sum _{k=0}^{N} (-1)^k (2dk+1)\frac{(\frac{1}{d})_k^3}{k!^3}\\&\quad \equiv {\left\{ \begin{array}{ll} p^r&{}\pmod {p^{r+2}},\quad \text {if } r \text { is even};\\ (-1)^{\frac{(p-d+1)r}{d}}(d-1)p^r&{}\pmod {p^{r+2}},\quad \text {if } r \text { is odd}; \end{array}\right

We investigate the group \({\mathcal {H}}_{\mathbb {C}}\) of complexified Heisenberg matrices with entries from an infinite-dimensional complex Hilbert space H. Irreducible representations of the Weyl–Schrödinger type on the space \(L^2_\chi \) of quadratically integrable \({\mathbb {C}}\)-valued functions are described. Integrability is understood with respect to the projective limit \(\chi =\varprojlim

It is well known that if \(f\in H^1\) and \(g\in H^q\), where \(1\le q<\infty \), then the integral means of order q of their Hadamard product \(f*g\) satisfy \(M_q(r,f*g)\le \Vert f\Vert _{H^1}\Vert g\Vert _{H^q}\), uniformly for each \(0

This paper is dedicated to Karl Strambach on the occasion of his 80th birthday. Here we want to describe our work with Prof. Karl Strambach.

Let K be a convex body in \({\mathbb {R}} ^d\), with \(d = 2,3\). We determine sharp sufficient conditions for a set E composed of 1, 2, or 3 points of \(\mathrm{bd}K\), to contain at least one endpoint of a diameter of K. We extend this also to convex surfaces, with their intrinsic metric. Our conditions are upper bounds on the sum of the complete angles at the points in E. We also show that such

We prove that for every \(p\ge 1\) there exists a bounded function in the analytic Besov space \(B^p\) whose derivative is “badly integrable” along every radius. We apply this result to study multipliers and weighted superposition operators acting on the spaces \(B^p\).

This work is an adaptation of the Katok Closing Lemma for non-singular, endomorphisms on a compact Riemannian manifold. In special, it appears the idea of the proof of the hyperbolicity as well as the admissibility of local stable-unstable manifolds of a proper periodic point in adapted Katok Closing Lemma.

We study which fields F can be represented as finite sums of proper subfields. We prove that for any \(n \ge 2\) every field F of infinite transcendence degree over its prime subfield can be represented as an unshortenable sum of n subfields, and every rational function field \(F = K(x_1, \ldots , x_n)\) can be represented as an unshortenable sum of \(n + 1\) subfields. We also show that no subfield

In the paper we will focus on lineability of some subsets of \({\mathbb {R}}^{\left[ 0,1\right] }\) which are called linearly sensitive. A function f is called linearly sensitive with respect to the property (or condition) (P) if f has the property (P) and for any \(a\ne 0\) the function \(f+a\cdot {{\,\mathrm{id}\,}}\) does not have the property (P). We discuss some general method of proving \({\mathfrak

By using quasi-Banach techniques as key ingredient we prove Poincaré- and Sobolev- type inequalities for m-subharmonic functions with finite (p, m)-energy. A consequence of the Sobolev type inequality is a partial confirmation of Błocki’s integrability conjecture for m-subharmonic functions.

We present a solution of Ulam’s stability problem for the functional equation \(f(x\star g(y))=f(x)f(y)\) with vector-valued map f.

The aim of this paper is to introduce two new classes of square matrices, which are called the Drazin-star and star-Drazin matrices, in order to solve some type of matrix equations. Several characterizations of these new matrices are given. Some relations between various well-known generalized inverses and the Drazin-star and star-Drazin matrices are investigated. We present the integral representations

The paper concerns holomorphic functions in complete bounded n-circular domains \({{\mathcal {G}}}\) of the space \({\mathbb {C}}^n\). The object of the present investigation is to solve majorization problem of Temljakov operator. This type of problem has been studied earlier in Liczberski and Żywień (Folia Sci Univ Tech Res 33:37–42, 1986), Liczberski (Bull Technol Sci Univ Łódź 20:29–37, 1988) and

It is well known that boundedness of a subadditive function need not imply its continuity. Here we prove that each subadditive function \(f:X\rightarrow {\mathbb {R}}\) bounded above on a shift–compact (non–Haar–null, non–Haar–meagre) set is locally bounded at each point of the domain. Our results refer to results from Kuczma’s book (An Introduction to the theory of functional equations and inequalities

Let \(f:({\mathbb {C}}^n,0)\rightarrow \left( {\mathbb {C}},0\right) ,\)\(n\le 3,\) be a nondegenerate singularity. In this article we give a combinatorial characterization of the dimension of the critical locus of f in terms of its support. We also show that this dimension can be read off from the Newton diagram of f, which solves one of Arnold’s problems in this case.

We study the evolution equations for a regularized version of Dirac-harmonic maps from closed Riemannian surfaces. We establish the existence of a global weak solution for the regularized problem, which is smooth away from finitely many singularities. Moreover, we discuss the convergence of the evolution equations and address the question if we can remove the regularization in the end.