Linear canonical transform (LCT), which is of importance in solving differential equations and analyzing optical systems, extends the conventional Fourier transform to the time-frequency domain characterized by a parameter matrix \(\mathbf{A }=(a,b;c,d)\). The main objective of this paper is to study the Balian–Low theorem in the LCT domain. We first state a Balian–Low theorem associated with the LCT

We present a new Kantorovich modification of Szász–Mirakjan operators which reproduce affine functions. We present an upper estimate for the rate of convergence of the new operators in polynomial weighted spaces and characterized all functions for which there is convergence.

We obtain a necessary and sufficient condition for a weighted composition operator to be co-isometric on a general weighted Hardy space of analytic functions in the unit disk whose reproducing kernel has the usual natural form. This turns out to be equivalent to the property of being unitary. The result reveals a dichotomy identifying a specific family of weighted Hardy spaces as the only ones that

In the present paper we continue to study almost geodesic curves and determine in \({\mathbb {R}}^n\) the form of curves \({\mathcal {C}}\) for which every image under an \((n-1)\)-dimensional algebraic torus is also an almost geodesic with respect to an affine connection \(\nabla \) with constant coefficients. We also calculate explicitly the components of \(\nabla \). For the explicit calculation

In this paper, we compute sub-Riemannian limits of Gaussian curvature for a Euclidean \(C^2\)-smooth surface in the BCV spaces and the twisted Heisenberg group away from characteristic points and signed geodesic curvature for Euclidean \(C^2\)-smooth curves on surfaces. We get Gauss–Bonnet theorems in the BCV spaces and the twisted Heisenberg group.

In this paper, new criteria for zero dimensional rings, Gelfand rings, clean rings and mp-rings are given. A new class of rings is introduced and studied, we call them purified rings. Specially, reduced purified rings are characterized. New characterizations for pure ideals of reduced Gelfand rings and mp-rings are provided. It is also proved that if the topology of a scheme is Hausdorff, then the

If a K-contact manifold (M, g) and a D-homothetically deformed K-contact manifold \((M,{\bar{g}})\) are both Ricci almost solitons with the same associated vector field V, then we show (i) that (M, g) and (\(M, {\bar{g}}\)) are both D-homothetically fixed \(\eta \)-Einstein Ricci solitons, and (ii) V preserves \(\phi \). We also show that, if the associated vector field V of a complete K-contact Ricci

A new and elementary proof of the Artin–Zorn theorem that finite alternative division rings are fields is given. The characterisation of finite fields of Glauberman and Heimbeck is also extended to a broader class of fields, the two subjects being connected via geometry.

We use the density function of a harmonic space to obtain estimates for the eigenvalues of the Jacobi operator; when these estimates are sharp, then the harmonic space is a symmetric Osserman space.

The paper focuses on various properties and applications of the homotopy operator, which occurs in the Poincaré lemma. In the first part, an abstract operator calculus is constructed, where the exterior derivative is an abstract derivative and the homotopy operator plays the role of an abstract integral. This operator calculus can be used to formulate abstract differential equations. An example of

In this paper we consider complex form of a modified (perturbed) Bernstein operators attached to an analytic function in a disk of radius \(R>1\) centered at the origin. We obtain quantitative estimates of the convergence in compact disks and exact degree of simultaneous approximation with the help of upper estimates in quantitative form, and of lower estimates obtained from a qualitative Voronovskaja

In this note, using the spinorial description of SU(3) and \(G_2\)-structures obtained recently by other authors, we give necessary and sufficient conditions for harmonicity of the above mentioned structures. We describe the results on appropriate homogeneous spaces. Here, harmonicity means harmonicity of a certain unique section induced by the G-structure in consideration.

In this article, we analyse the behaviour of the new family of Kantorovich type exponential sampling series. We derive the point-wise approximation theorem and Voronovskaya type theorem for the series \((I_{w}^{\chi })_{w>0}.\) Further, we establish a representation formula and an inverse result of approximation for these operators. Finally, we give some examples of kernel functions to which the theory

We consider a superlinear perturbation of the eigenvalue problem for the Robin Laplacian plus an indefinite and unbounded potential. Using variational tools and critical groups, we show that when \(\lambda \) is close to a nonprincipal eigenvalue, then the problem has seven nontrivial solutions. We provide sign information for six of them.

An oriented sphere in \({\mathbb {R}}^n\) is a sphere for which one side (inside or outside) is distinguished as its positive side. Two oriented spheres are said to be properly tangent if they are tangent and the positive side of one sphere is contained in the positive side of the other. A set of 2k spheres \(\{\alpha _1,\dots ,\alpha _k,\beta _1,\dots ,\beta _k\}\) in \({\mathbb {R}}^n\) is called

For an entire function \(f(z) = \sum _{k=0}^\infty a_k z^k, a_k>0\), we show that if f belongs to the Laguerre–Pólya class, and the quotients \(q_k:= \frac{a_{k-1}^2}{a_{k-2}a_k}, k=2, 3, \ldots \) satisfy the condition \(q_2 \le q_3\), then f has at least one zero in the segment \([-\frac{a_1}{a_2},0]\). We also give necessary conditions and sufficient conditions of the existence of such a zero in

Because all the classical asymptotic theorems of Voronovskaya-type for positive and linear operators are in fact based on the Taylor’s formula which is a very particular case of Lagrange–Hermite interpolation formula, in this paper we obtain semi-discrete quantitative Voronovskaya-type theorems based on various other Lagrange–Hermite interpolation formulas. Applications to Bernstein–Kantorovich and

Magnetic curves with respect to the almost cosymplectic structure of the \(\mathrm {Sol}_3\) space are determined and curvature properties of these curves are investigated.

In this paper, we construct a topology on the vector space \(\delta \mathcal F_{m,\chi }(\Omega )\). We also prove that with this topology it is quasi-Banach, non-separable and non-reflexive. We also give a characterization for the class \(\mathcal F_{m, \min \{\chi _1, \chi _2\}}(\Omega )\).

The aim of this paper is to provide alternative characterizations of hyperbolic affine generalized infinite iterated function systems. More precisely, we prove that, for such a system \({\mathcal {F}}=((X,\left\| .\right\| ),(f_{i})_{i\in I})\), among others, the following statements are equivalent: (a) \({\mathcal {F}}\) is hyperbolic. (b) \( {\mathcal {F}}\) has attractor. (c) \({\mathcal {F}}\)

We show that every rationally sampled dilation-and-modulation system is unitarily equivalent to a multi-window Gabor system with special structures. As a consequence, the study of dilation-and-modulation systems in \(L^2({\mathbb {R}}_+)\) reduces to that of multi-window Gabor systems in \(L^2({\mathbb {R}})\) with special structures.

A new non-linear variant of a quantitative extension of the uniform boundedness principle is used to show sharpness of error bounds for univariate approximation by sums of sigmoid and ReLU functions. Single hidden layer feedforward neural networks with one input node perform such operations. Errors of best approximation can be expressed using moduli of smoothness of the function to be approximated

The classical inequality of Bohr asserts that if a power series converges in the unit disk and its sum has modulus less than or equal to 1, then the sum of absolute values of its terms is less than or equal to 1 for the subdisk \(|z|<1/3\) and 1/3 is the best possible constant. Recently, there has been a number of investigations on this topic. In this article, we present related inequalities using

We show that a real normed linear space endowed with the \(\rho \)-orthogonality relation, in general need not be an orthogonality space in the sense of Rätz. However, we prove that \(\rho \)-orthogonally additive mappings defined on some classical Banach spaces have to be additive. Moreover, additivity (and approximate additivity) under the condition of an approximate orthogonality is considered.

In this note, we investigate conformally flat submanifolds of Euclidean space with positive index of relative nullity. Let \(M^n\) be a complete conformally flat manifold and let \(f:M^n\rightarrow \mathord {\mathbb {R}}^m\) be an isometric immersion. We prove the following results: (1) If the index of relative nullity is at least two, then \(M^n\) is flat and f is a cylinder over a flat submanifold

Recently, Michael Gil’ has provided a condition for integrability of norms of semigroups generated by operators of the form \(A+B\) where B is bounded, in terms of the norm of the commutator of A and B. We show that his argument, when slightly refined, leads to a bit more general result than the one presented in his paper. We also argue, by examining an example, that the size of the commutator may

We show that entire functions \(\varphi \), which induce bounded products of Volterra integral operators \(V_g\) (Volterra companion operators \(J_g\)) and composition operators \(C_{\varphi }\) acting between different Fock spaces, must be affine functions, i.e. \(\varphi (z) = az + b\). Then, using this special form of \(\varphi \), we characterize boundedness and compactness of these products in

We give criteria for the closely related concepts of surjectivity, closed range, and Fredholmness of the composition and multiplication operators between possibly different Orlicz spaces over non-atomic measure spaces.

Fractal structures were introduced to characterize non-archimedean quasi-metrization, and there is a strong relationship between these two concepts. In this paper we show how to define a probability measure with the help of a fractal structure, by taking advantage of its recursive nature. One of the keys of this approach is the use of the completion of the fractal structure. Since we want to define

Assume that \((\Omega ,{\mathcal {A}},P)\) is a probability space, \(f:[0,1]\times \Omega \rightarrow [0,1]\) is a function such that \(f(0,\omega )=0\), \(f(1,\omega )=1\) for every \(\omega \in \Omega \), \(g:[0,1]\rightarrow \mathbb R\) is a bounded function such that \(g(0)=g(1)=0\), and \(a,b\in \mathbb R\). Applying medial limits we describe bounded solutions \(\varphi :[0,1]\rightarrow \mathbb

We consider a family of (2, 2)-rational functions given on the set of complex p-adic field \(\mathbb C_p\). Each such function f has two distinct fixed points \(x_1=x_1(f)\), \(x_2=x_2(f)\). We study p-adic dynamical systems generated by the (2, 2)-rational functions. We prove that \(x_1\) is always an indifferent fixed point for f, i.e., \(x_1\) is a center of some Siegel disk \(SI(x_1)\). Depending

A proportionally modular affine semigroup is the set of nonnegative integer solutions of a modular Diophantine inequality \(f_1x_1+\cdots +f_nx_n \bmod b \le g_1x_1+\cdots +g_nx_n\), where \(g_1,\dots ,g_n\), \(f_1,\ldots ,f_n\in \mathbb {Z}\), and \(b\in \mathbb {N}\). In this work, a geometrical characterization of these semigroups is given. On the basis of this geometrical approach, some algorithms

Starting from the inequality $$\begin{aligned} |f(x+y)-f(x)-f(y)+g(x+y)-g(x)g(y)|\leqslant \varepsilon , \quad x,y\in S, \end{aligned}$$ where f is a complex valued function defined on a monoid S, we deal with two problems: the stability problem and the problem of alienation of the approximate additivity condition from the condition of approximate exponentiality.

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