In this paper we express special bihyperbolic numbers as paradeterminants and parapermanents of some triangular matrices. Moreover, by applying the connections between these parameters of triangular tables and the determinants and permanents of lower Hessenberg matrices, we obtain another expressions of these numbers, using matrices which are not triangular.

In this article we introduce and analyze the concept of (involutive) bounded BRK-algebras. Additionally, we observe some of the substructures of (involutive) bounded BRK-algebras and find their mutual connections.

A mathematical model is presented that describes the dynamics of a spatially distributed population, incorporating the effects of external migration. The evolution of the population density is governed by a simple integro-differential equation. In the spatially homogeneous case, the model isreduced to the classical logistic equation with an additional constant term and its behavior is fully characterized. In the inhomogeneous case, the dynamics is examined through numerical simulations and typical long-term behavior is illustrated.

In this paper, we introduce bihyperbolic third-order Jacobsthal and third-order Jacobsthal--Lucas numbers. In other words, bihyperbolic numbers whose coefficients are consecutive third-order Jacobsthal and third-order Jacobsthal--Lucas numbers. Furthermore, we study one parameter generalizations of bihyperbolic third-order Jacobsthal and third-order Jacobsthal--Lucas numbers and relations between them.

In this paper we are interested in the existence and uniqueness  of solutions for Dirichlet problem associated with the degenerate nonlinear elliptic equations\begin{align*}&- \mathrm{div}\big[{\mathcal{A}}(x, {\nabla}u)\, {\omega}_2(x)+ {\mathcal{B}}(x, {\nabla}u)\, {\nu}_1(x)\big] + {\mathcal{H}}(x,u,{\nabla}u){\nu}_2 + {\vert u \vert}^{p-2}u\,{\omega}_1\\&= {\rho}_0 - \sum_{j=1}^nD_j{\rho}_j,\\ &  u - {\psi}\, {\in}\, W_0^{1,p}(\Omega , {\omega}_1 , {\omega}_2),\end{align*}in the setting of the weighted Sobolev spaces.

In this note, we shall study a new, short proof of the general theorem of Necas about the solvability of linear partial differential equations in the Banach space setting. The complexity of this proof does not seem to be greater than that of the Lax-Milgram theorem, and since the theorem of Necas is strictly stronger than the Lax-Milgram theorem, the author hopes that his new proof will help the theorem of Necas to gain prominence in PDE and functional analysis lectures.

In the spirit of Hahn 1949, the purpose of this paper is to introduce a new \(q\)-Laplace transform for a Jackson \(q\)-integral \(\int_{0}^{a} f(t,q) \,d_q(t)\), with upper integration boundary \(\frac{1}{s(1-q)}\). For this purpose we redefine this \(q\)-integral with a \(\sigma\)-algebra and a discrete measure supported at the points \(x=aq^{n},\ n\in\mathbb{N}\). Then we prove \(q\)-analogues of many well-known Laplace transform formulas, including the formula for the transform of the delta distribution. The paper concludes with a list of \(q\)-Laplace transforms for (multiple) \(q\)-hypergeometric series, some with function arguments in the first \(q\)-real numbers \(\mathbb{R}_{\oplus_{q}}\). Elsewhere, other \(q\)-real numbers are defined in similar style as function arguments in formal power series.

In this paper, we introduce non-Newtonian balancing type numbers. In non-Newtonian calculus, we examine formulas and identities for classical balancing numbers. We give Binet-type formula for non-Newtonian balancing numbers and the general bilinear index-reduction formula which implies Catalan, Cassini and d’Ocagne identities. Moreover, we give the generating function for balancing numbers in terms of non-Newtonian calculus.

In this paper, we present a new class of elliptic quaternions that incorporate Leonardo, Leonardo–Lucas and modified Leonardo numbers into their components. We explore some fundamental properties associated with these numbers. In particular, we obtain recurrence relations, generating function, Binet formula of these sequences and by using Binet formula we derive Vajda, Cassini, Catalan and d’Ocagne identities. Lastly, we investigate two different matrix representations of these numbers.

A new variant of Mercer’s inequality [A.McD. Mercer, A variant of Jensen’s inequality, J. Inequal. Pure Appl. Math. 4(4) (2003) Article 73] of Jessen’s type is given. Moreover, versions of Chebyshev’s inequality and Hardy–Littlewood– Pólya inequality for some abstract nonnegative linear functionals are obtained.

Ali and Vasudevarao considered the integral operator \(I_{r,s}(z):=\int_{0}^{z}(f'(t))^r(g'(t))^s d t\)  and determined all values of \(r\) and $s$ for which the operator \((f,g)\mapsto I_{r,s}\) maps a specified subclass of Hornich space into another specified subclass of Hornich space. Thus, as it was stated by Kumar and Sahoo, Ali and Vasudevarao studied the range of \(r\) and \(s\) that preserves properties of these specified classes. Based on the Kaplan classes, we introduce the product classes \(K_{a,b}\) for arbitrary finite sequences \(a\) and \(b\) and consider operations similar to Hornich operations.  To this end we improve Sheil-Small's factorization theorem. Moreover, using elaborated techniques, we simplify proofs and solve the generalized problems considered by Causey and Reade, Goodman, Kim and Merkes.

Let \(\mathcal{FM}_{m,n}\) denote the category of fibered manifolds with \(m\)-dimensional bases and \(n\)-dimensional fibres and their fibered local diffeomorphisms. We prove that  if \(m,n\) and \(s\) are positive integers, then any \(\mathcal{FM}_{m,n}\)-natural operator \(C\) transforming tuples \((\lambda_1,\lambda_2)\) of Lagrangians \(\lambda_1,\lambda_2:J^sY\to\bigwedge ^mT^*M\) on \(\mathcal{FM}_{m,n}\)-objects \(Y\to M\) into Legendre maps \(C(\lambda_1,\lambda_2):J^{s}Y\to S^sTM\otimes V^*Y\otimes\bigwedge^m T^*M\) on \(Y\) is of the form \(C(\lambda_1,\lambda_2)=c_1\Lambda(\lambda_1)+c_2\Lambda(\lambda_2)\), \(c_1,c_2\in\mathbf{R}\), where \(\Lambda\) is the Legendre operator. We also prove that if \(m,n,s\) and \(p\) are  positive integers, then any \(\mathcal{FM}_{m,n}\)-natural operator \(C\) transforming tuples \((\lambda,\eta)\) of Lagrangians \(\lambda:J^sY\to\bigwedge ^mT^*M\) and \(p\)-forms \(\eta\in \Omega^p(M)\) into Legendre maps \(C(\lambda,\eta):J^{s}Y\to S^sTM\otimes V^*Y\otimes\bigwedge^m T^*M\) is of the form \(C(\lambda,\eta)=c\Lambda(\lambda)\), \(c\in\mathbf{R}\), where \(\Lambda\) is the Legendre operator.

Let \(Y\) be a fibred manifold with an \(m\)-dimensional basis \(M\).  We describe all Legendre-like operators \(C\), i.e. natural operators transforming tuples \((\lambda,g)\) of Lagrangians \(\lambda:J^sY\to\bigwedge ^mT^*M\) and functions \(g:M\to\mathbf{R}\) (resp. \(g:Y\to\mathbf{R}\)) into Legendre maps \(C(\lambda,g):J^{s}Y\to S^sTM\otimes V^*Y\otimes\bigwedge^m T^*M\). The most important example of such operators is the Legendre operator  (from the variational calculus) being  the one in question depending only on Lagrangians.

The purpose of this paper is to consider five classes of quadratic and cubic hypergeometric transformations in the spirit of Bailey and Whipple. We shall successfully evaluate several hypergeometric functions, of the types \(_{2}\text{F}_{1}(x)\), \(_{3}\text{F}_{2}(x)\), and \(_{4}\text{F}_{3}(x)\), with each function having one or more free parameters, and with the argument $x$ chosen to equal such unusual values as \(x=\pm 1,-8,\frac 14, -\frac 18\), (these four values having been explored initially by Gessel and Stanton). In each case, companion identities and/or inverse transformations are given, which are sometimes proved by a limiting process for a divergent hypergeometric series. Some of the proofs use the Clausen quadratic formula, Euler reflection formula, Legendre duplication, Gauss multiplication formula, Euler transformation, hypergeometric reversion formula and known hypergeometric summation formulas. The proofs in the terminating case are simpler and can lead to mixed summation formulas, which depend on values of a negative integer. Some of the formulas use the Digamma function and a dimension formula is referred to.

A general tool to prove conditional strong laws of larger number is considered. It is shown that a conditional Kolmogorov type inequality implies a conditional Hajek–Renyi type inequality and this implies a strong law of large numbers. Both probability and moment inequalities are considered. Some applications are offered in the last section.

Hybrid numbers are generalization of complex, hyperbolic and dual numbers. In this paper, we define and study hybrid numbers with cobalancing and Lucas-cobalancing coefficients. We derive some fundamental identities for these numbers, among others the Binet formulas and the general bilinear index-reduction formulas which imply the Catalan, Cassini, Vajda, d’Ocagne and Halton identities. Moreover, the generating functions for cobalancing and Lucas-cobalancing hybrid numbers are presented.

Let \(\mathcal{FM}_{m,n}\) denote the category of fibered manifolds with $m$-dimensional bases and \(n\)-dimensional fibres and their fibered diffeomorphisms onto open images. We describe all \(\mathcal{FM}_{m,n}\)-natural  operators \(C\) transforming tuples \((\lambda,g)\) of  Lagrangians \(\lambda:J^sY\to\bigwedge ^mT^*M\) (or formal Lagrangians \(\lambda:J^sY\to V^*J^sY\otimes\bigwedge ^mT^*M\)) on \(\mathcal{FM}_{m,n}\)-objects \(Y\to M\) and functions \(g:M\to\mathbf{R}\) into Euler maps \(C(\lambda,g):J^{2s}Y\to V^*Y\otimes\bigwedge^m T^*M\) on \(Y\). The most important example of such \(C\) is the Euler operator \(E\) (from the variational calculus) (or the formal Euler operator \(\mathbf{E}\)) treated as the operator in question depending only on Lagrangians (or formal Lagrangians).

In this note we provide Holland-Walsh-type characterizations for functions on the \(\mathcal{N}(p,q,s)\) spaces on the unit ball for specific values of \(p\ge 1\). Characterizations for the holomorphic function spaces \(\mathcal{N}(p,q,s)\) were studied extensively by B. Hu  and S. Li.

In this paper, a singular linear \(q\)-Hamiltonian system is considered. The Titchmarsh-Weyl theory for this problem is constructed. Firstly, we provide some necessary fundamental concepts of the \(q\)-calculus. Later, we studied Titchmarsh-Weyl functions \(M\left(  \lambda\right)\) and circles \(\mathcal{C}_{TW}\left(a,\lambda\right)\) for this system. Circles \(\mathcal{C}_{TW}\left(a,\lambda\right)\) are proved to be nested. In the fourth part of the work, the number of square-integrable solutions of this system is studied. In the fifth  part of the work, boundary conditions in the singular case are obtained. Finally, a self-adjoint operator is defined.

In this paper we will investigate reproducing kernel Hilbert spaces of polynomials of degree at most n with three different inner products: given by an integral with a weight, given by the sum of products of values of a polynomial at n + 1 points and given by the sum of products of coefficients of the same power. In the first case we will show that the reproducing kernel depends continuously on deformation of an inner product in a precisely defined sense. In the second and third case we will give a formula for the reproducing kernel.

How many holomorphic connections are there with a prescribed Ricci tensor? How many torsion-free holomorphic connections are there with a prescribed Ricci tensor? These questions are answered by using the holomorphic version of the Cauchy–Kowalevski theorem.

The purpose of this article is to define some intermediate q-Lauricella functions, to find convergence regions in two different forms, and to prove corresponding reduction formulas by using a known lemma from our first book. These convergence regions are given in form of q-additions and q-real numbers. The third q-real number plays a special role in the computations. Generating functions are proved by using the q-binomial theorem. Finally, special cases of q-Lauricella functions as well as confluent forms in the spirit of Chandel Singh and Gupta are given.

The hybrid numbers are generalization of complex, hyperbolic and dual numbers. The hybrinomials are polynomials which generalize hybrid numbers. In this paper, we introduce and study the distance Fibonacci hybrinomials, i.e. hybrinomials with coefficients being distance Fibonacci polynomials.

We prove that every implicative aBE algebra satisfies the transitivity property. This means that every implicative aBE algebra is a Tarski algebra, and thus is also a commutative BCK algebra.

We consider some generalizations of BCK algebras (RML, BE, aBE, BE** and aBE** algebras). We investigate the property of implicativity for these algebras. We prove that for any implicative BE** algebra the commutativity property is equivalent to the property of antisymmetry and show that implicative aBE** algebras are commutative BCK algebras. We also show that the class of implicative BE** algebras is a variety.

Generalized commutative quaternions is a number system which generalizes elliptic, parabolic and hyperbolic quaternions, bicomplex numbers, complex hyperbolic numbers and hyperbolic complex numbers. In this paper we introduce and study generalized commutative quaternion polynomials of the Fibonacci type.

In the present article, an equality is established by using the well-known Riemann-Liouville fractional integrals. With the aid of this equality, some Euler-Maclaurin-type inequalities are given in the case of differentiable convex functions. Moreover, we give an example using graphs in order to show that our main result is correct.

We describe completely all natural affinors on product preserving gauge bundle functors  on double vector bundles. Next, we study torsion of double-linear connections.

We consider relations between the distance of a set \(A\) and the distance of its translated set \(A+x\) from 0, for \(x\in A\), in a normed linear space. If the relation \(d(0,A+x)<d(0,A)+\|x\|\) holds for exactly determined vectors \(x\in A\), where \(A\) is a convex, closed set with positive distance from 0, which we call (TP) property, then this property is equivalent to strict convexity of the space. We show that in uniformly convex spaces the considered property holds.

In this paper, faithful matrix representations of the jet groups \(G^3_n\) are presented, following a detailed description of their components in block form. Such  groups can be used further to study symmetries of differential equations. Elements of these matrix representations are derived.

We completely describe all gauge-natural operators \(C\) which send linear \((p+2)\)-forms \(H\) on vector bundles \(E\) (with sufficiently large dimensional bases) into \(\mathbf{R}\)-bilinear operators \(C_H\) transforming pairs \((X_1\oplus\omega_1,X_2\oplus\omega_2)\) of couples of linear vector fields and linear \(p\)-forms on \(E\) into couples \(C_H(X_1\oplus\omega_1, X_2\oplus\omega_2)\) of linear vector fields and linear \(p\)-forms on \(E\). Further, we extract all \(C\) (as above) such that \(C_0\) is the restriction of the well-known Courant bracket and \(C_H\) satisfies the Jacobi identity in Leibniz form for all closed linear \((p+2)\)-forms \(H\).

We prove that if \(X\) is an \(\ell_{1}\)-predual isomorphic to the space \(c_{0}\) of sequences converging to zero, then for any isomorphism \(T:X\rightarrow c_{0}\) we have \(\Vert T\Vert\, \Vert T^{-1}\Vert\ge1+2r^{*}(X)\), where \(r^{*}(X)\) is the smallest radius of the closed ball of the dual space \(X^{*}\) containing  all the weak\(^{*}\) cluster points of the set of all extreme points of the closed unit ball of  \(X^*\).

We prove under a mild condition that Kurepa's conjecture holds for the set of prime numbers \(p\) such that \((\frac{p-1}{2})! = {2 \overwithdelims () p\;}\) in \(\mathbb{F}_p\).

For a continuous and positive function \(w(\lambda)\), \(\lambda>0\) and a positive measure \(\mu\) on \((0,\infty )\) we consider the following integral transform\[\mathcal{D}( w,\mu ) ( T) :=\int_{0}^{\infty }w(\lambda) (\lambda +T)^{-1} d\mu ( \lambda ) ,\]where the integral is assumed to exist for any positive operator \(T\) on a complex Hilbert space \(H\). In this paper we obtain several upper and lower bounds for the difference \(\mathcal{D}( w,\mu ) ( A) -\mathcal{D}( w,\mu ) ( B)\) under certain assumptions for the operators \(A\) and \(B\). Some natural applications for operator monotone and operator convex functions are also given.

In this article we define a binary linear operator T for holomorphic functions in given open sets \(A\) and \(B\) in the complex plane under certain additional assumptions. It coincides with the classical Hadamard product of holomorphic functions in the case where \(A\) and \(B\) are the unit disk. We show that the operator T exists provided \(A\) and \(B\) are simply connected domains containing the origin. Moreover, T is determined explicitly by means of an integral form. To this aim we prove an alternative representation of the star product \(A*B\) of any sets \(A,B\subset\mathbb{C}\) containing the origin. We also touch the problem of holomorphic extensibility of Hadamard product.

We give complete description of all gauge-natural bilinear operators A transforming pairs of couples of linear vector fields and linear p-forms on a vector bundle E into couples of linear vector fields and linear p-forms on E and satisfying the Jacobi identity in Leibniz form.

In the spirit of our earlier articles on \(q\)-\(\omega\) special functions, the purpose of this article is to present many new \(q\)-number systems, which are based on the \(q\)-addition, which was introduced in our previous articles and books. First, we repeat the concept biring, in order to prepare for the introduction of the \(q\)-integers, which extend the \(q\)-natural numbers from our previous book. We formally introduce a \(q\)-logarithm for the \(q\)-exponential function for later use. In order to find \(q\)-analogues of the corresponding formulas for the generating functions and \(q\)-trigonometric functions, we also introduce \(q\)-rational numbers. Then the so-called \(q\)-real numbers \(\mathbb{R}_{\oplus_{q}}\), with a norm, a \(q\)-deformed real line, and with three inequalities, are defined. The purpose of the more general \(q\)-real numbers \(\mathbb{R}_{q}\) is to allow the other \(q\)-addition too. The closely related JHC \(q\)-real numbers \(\mathbb{R}_{\boxplus_{q}}\) have applications to several \(q\)-Euler integrals. This brings us to a vector version of the \(q\)-binomial theorem from a previous paper, which is associated with a special case of the \(q\)-Lauricella function. New \(q\)-trigonometric function formulas are given to show the application of this umbral calculus. Then, some equalities between \(q\)-trigonometric zeros and extreme values are proved. Finally, formulas and graphs for \(q\)-hyperbolic functions are shown.

Let \(\left( H;\left\langle \cdot ,\cdot \right\rangle \right)\) be a complex Hilbert space and \(f:[0,\infty )\rightarrow \mathbb{R}\) be convex (concave) on \([0,\infty )\). If \(x, y\in H\) with \(Re \left\langle x,y\right\rangle \geq 0\), then\begin{align*}f\left( \frac{\left\Vert x\right\Vert ^{2}+Re \left\langle x,y\right\rangle +\left\Vert y\right\Vert ^{2}}{3}\right) & \leq \left( \geq\right) \int_{0}^{1}f\left( \left\Vert \left( 1-t\right) x+ty\right\Vert^{2}\right) dt \\& \leq \left( \geq \right) \frac{1}{3}\left[ f\left( \left\Vert x\right\Vert^{2}\right) +f\left[ Re \left\langle x,y\right\rangle \right] +f\left(\left\Vert y\right\Vert ^{2}\right) \right] .\end{align*}Some examples for power functions and exponential are also provided.

In this paper, we present new perturbed inequalities of Ostrowski-type, for twice differentiable functions with absolutely continuous first derivative and second-order derivative in some \(L^p\)-space for \(1\leq p\leq \infty\).

In this paper, we study matrix-valued Hahn–Sturm–Liouville equations. We give an existence and uniqueness result. We introduce the corresponding maximal and minimal operators for this system, and some properties of these operators are investigated. Finally, we characterize extensions (maximal dissipative, maximal accumulative and self-adjoint) of the minimal symmetric operator.

If \(m \geq 3\) and \(r \geq 1\), we prove that any natural linear operator \(A\) lifting 2-vector fields \(\Lambda \in \Gamma (\bigwedge^2 TM)\) (i.e., skew-symmetric tensor fields of type (2,0)) on \(m\)-dimensional manifolds \(M\) into 2-vector fields \(A(\Lambda)\) on \(r\)-jet prolongation \(J^rTM\) of the tangent bundle \(TM\) of \(M\) is the zero one.

We show that the Banach–Saks property with respect to a regular positive matrix method of summability is inherited by the real interpolation spaces from a space forming the interpolation family and possessing this property. The proof refers to the Galvin–Prikry theorem on Ramsey sets. The results apply to several matrix methods of summability, such as Cesaro, Nørlund or Holder methods.

By a radial set we understand a non-empty set \(A \subset \mathbb{C} \setminus \{0\}\) such that for every point \(z\in A\) the circle with centre at the origin and passing through \(z\) is included in \(A\). We show in a detailed manner that every continuous and injective function \(F : A \to \mathbb{C} \setminus \{0\}\) can be represented by means of the natural exponential function \(\text{exp}\) and a certain continuous function \(\varPhi : \text{Ei}(A) \to \mathbb{C}\), where \(\text{Ei}(A)\) is the set of all \(z \in \mathbb{C}\) with the property \(\text{exp}(iz) \in A\). The representation is given by \(F(\text{exp}(iz)) = \text{exp}(i\varPhi (z))\) for \(z \in \text{Ei}(A)\). We also touch the problem of the injectivity of \(\varPhi\).

In this paper we establish several natural consequences of some Wirtinger type integral inequalities for p-norms. The corresponding weighted versions and applications related to the weighted trapezoid inequalities, to weighted Gruss’ type inequalities and reverses of Jensen’s inequality are also provided.

In this paper we introduce two-parameter generalization of dualhyperbolic Jacobsthal numbers: dual-hyperbolic (s,p)-Jacobsthal numbers. We present some properties of them, among others the Binet formula, Catalan, Cassini, d’Ocagne identities. Moreover, we give the generating function, matrix generator and summation formula for these numbers.

The gradient-projection algorithm (GPA) plays an important role in solving constrained convex minimization problems. In this paper, we combine the GPA and averaged mapping approach to propose an explicit composite iterative scheme for finding a common solution of a generalized equilibrium problem and a constrained convex minimization problem. Then, we prove a strong convergence theorem which improves and extends some recent results.

In this paper, we establish some interval oscillation criteria for impulsive conformable fractional partial delay differential equations with a forced term. The main results will be obtained by employing Riccati technique. Our results extend and improve some results reported in the literature for the classical differential equations without impulses. An example is provided to illustrate the relevance of the new theorems.

In this paper we present how identical columns affect the Cullis-Radić determinant of an \(m\times n\) matrix, where \(m\leq n\).

We give the complete characterization of members of Kaplan classes of products of power functions with all zeros symmetrically distributed in \(\mathbb{T} := \{z \in\mathbb{C} : |z| = 1\}\) and weakly monotonic sequence of powers. In this way we extend Sheil-Small’s theorem. We apply the obtained result to study univalence of antiderivative of these products of power functions.

In this paper we provide several refinements and reverse operator inequalities for operator monotone functions in Hilbert spaces. We also obtain refinements and a reverse of Lowner-Heinz celebrated inequality that holds in the case of power function.

In real life applications not all signals are obtained by uniform shifts; so there is a natural question regarding analysis and decompositions of these types of signals by a stable mathematical tool. Gabardo and Nashed, and Gabardo and Yu filled this gap by the concept of nonuniform multiresolution analysis and nonuniform wavelets based on the theory of spectral pairs for which the associated translation set \(\Lambda= \{0,r/N\}+2\mathbb{Z}\) is no longer a discrete subgroup of \(\mathbb{R}\) but a spectrum associated with a certain one-dimensional spectral pair and the associated dilation is an even positive integer related to the given spectral pair. In this paper, we introduce a notion of nonuniform periodic wavelet frame on non-Archimedean field. Using the Fourier transform technique and the unitary extension principle, we propose an approach for the construction of nonuniform periodic wavelet frames on non-Archimedean fields.

The hybrid numbers are generalization of complex, hyperbolic and dual numbers. In this paper we consider  a special kind of hybrid numbers, namely the Mersenne hybrid numbers and we give some of their properties.

This paper contains some results for algebraic polynomials in the complex plane involving the polar derivative that are inspired by some classical results of Bernstein. Obtained results yield the polar derivative analogues of some inequalities giving estimates for the growth of derivative of lacunary polynomials.

Let \(F\) be a bundle functor on the category of all fibred manifolds and fibred maps. Let \(\Gamma\) be a general connection in a fibred manifold \(\mathrm{pr}:Y\to M\) and \(\nabla\) be a classical linear connection on \(M\). We prove that the  well-known general connection \(\mathcal{F}(\Gamma,\nabla)\) in \(FY\to M\) is canonical with respect to fibred maps and with respect to natural transformations of bundle functors.

For any infinite dimensional Banach space there exists a lipschitzian retraction of the closed unit ball B onto the unit sphere S. Lipschitz constants for such retractions are, in general, only roughly estimated. The paper is illustrative. It contains remarks, illustrations and estimates concerning optimal retractions onto spherical caps for sequence spaces with the uniform norm.

The purpose of this article is to generalize the ring of \(q\)-Appell polynomials to the complex case. The formulas for \(q\)-Appell polynomials thus appear again, with similar names, in a purely symmetric way. Since these complex \(q\)-Appell polynomials are also \(q\)-complex analytic functions, we are able to give a first example of the \(q\)-Cauchy-Riemann equations. Similarly, in the spirit of Kim and Ryoo, we can define \(q\)-complex Bernoulli and Euler polynomials. Previously, in order to obtain the \(q\)-Appell polynomial, we would make a \(q\)-addition of the corresponding \(q\)-Appell number with \(x\). This is now replaced by a \(q\)-addition of the corresponding \(q\)-Appell number with two infinite function sequences \(C_{\nu,q}(x,y)\) and \(S_{\nu,q}(x,y)\) for the real and imaginary part of a new so-called \(q\)-complex number appearing in the generating function. Finally, we can prove \(q\)-analogues of the Cauchy-Riemann equations.

The allocation scheme of \(n\) indistinguishable particles into \(N\) different cells is studied. Let the random variable \(\mu_0(n,K,N)\) be the number of empty cells among the first \(K\) cells. Let \(p=\frac{n}{n+N}\). It is proved that \(\frac{\mu_0(n,K,N)-K(1-p)}{\sqrt{ K p(1-p)}}\) converges in distribution to the Gaussian distribution with expectation zero and variance one, when \(n,K, N\to\infty\) such that \(\frac{n}{N}\to\infty\) and \(\frac{n}{NK}\to 0\). If \(n,K, N\to\infty\) so that \(\frac{n}{N}\to\infty\) and \(\frac{NK}{n}\to \lambda\), where \(0<\lambda<\infty\), then \(\mu_0(n,K,N)\) converges in distribution to the Poisson distribution with parameter \(\lambda\). Two applications of the results are given to mathematical statistics. First, a method  is offered to test the value of \(n\). Then, an analogue of the run-test is suggested with an application in signal processing.

In this paper we introduce a one-parameter generalization of the split Jacobsthal quaternions, namely the split r-Jacobsthal quaternions. We give a generating function, Binet formula for these numbers. Moreover, we obtain some identities, among others Catalan, Cassini identities and convolution identity for the split r-Jacobsthal quaternions.

In this note we establish an advanced version of the inverse function theorem and study some local geometrical properties like starlikeness and hyperbolic convexity of the inverse function under natural restrictions on the numerical range of the underlying mapping.

This paper is devoted to the study of families of so-called nonlinear resolvents. Namely, we construct polynomial transformations which map the closed unit polydisks onto the coefficient bodies for the resolvent families. As immediate applications of our results we present a covering theorem and a sharp estimate for the Schwarzian derivative at zero on the class of resolvents.

We prove an almost sure random version of a maximum limit theorem, using logarithmic means for \(\max_{1\leq i\leq N_n} X_i\), where \(\{X_n, n \geq 1\}\) is a sequence of identically distributed random variables and \(\{N_n, n \geq 1\}\) is a sequence of positive integer random variables independent of \(\{X_n, n \geq 1\}\). Furthermore, we consider the almost sure random version of a limit theorem for \(k\)th order statistics.

In this paper we study \(2\times 2\) systems of conservation laws with discontinuous fluxes arising in vehicular traffic modeling. The main goal is to introduce an appropriate notion of solution. To this aim we consider physically reasonable microscopic follow-the-leader models. Macroscopic Riemann solvers are then obtained as many particle limits. This approach leads us to develop six models. We propose a unified way to describe such models, which highlights their common property of maximizing the density flow across the interface under appropriate physical restrictions depending on the case at hand.

The paper improves approximation theory based on the Trotter–Kato product formulae. For self-adjoint \(C_0\)-semigroups we develop a lifting of the strongly convergent Chernoff approximation (or product) formula to convergence in the operator-norm topology. This allows to obtain optimal estimate for the rate of operator-norm convergence of Trotter–Kato product formulae for Kato functions from the class \(K_2\).

We introduce three quantities related to orbits of non-elliptic continuous semigroups of holomorphic self-maps of the unit disk, the total speed, the orthogonal speed, and the tangential speed and show how they are related and what can be inferred from those.

In this paper we aim to demonstrate how physical perspective enriches statistical analysis when dealing with a complex system of many interacting agents of non-physical origin. To this end, we discuss analysis of urban public transportation networks viewed as complex systems. In such studies, a multi-disciplinary approach is applied by integrating methods in both data processing and statistical physics to investigate the correlation between public transportation network topological features and their operational stability. These studies incorporate concepts of coarse graining and clusterization, universality and scaling, stability and percolation behavior, diffusion and fractal analysis.

We show stability of preemptive, strictly subcritical EDF networks with Markovian routing. To this end, we prove that the associated fluid limits satisfy the first-in-system, first-out (FISFO) fluid model equations and thus, by an extension of a result of Bramson (2001), the corresponding fluid models are stable. We also demonstrate that in a preemptive multiclass EDF network, after a time large enough to process all the initial customers to completion, the maximal number of partially served customers in the system over a finite time horizon converges to zero in \(L^1\) under fluid scaling.

In this note we explore the concept of the logarithmic norm of a matrix and illustrate its applicability by using it to find conditions under which the convergence of solutions of regularly perturbed systems of ordinary differential equations is uniform globally in time.

In this paper we investigate some applications of the differential subordination and superordination of classes of admissible functions associated with an integral operator. Additionally, differential sandwich-type results are obtained.

In this paper, we define a function \(F : D\times D\times D\to \mathbb{C}\) in terms of \(f\) and show that Re\(F > 0\) for all \(\zeta,z,w \in D\) if and only if \(f\) belongs to the class of convex meromorphic functions.

In this paper we study the class \(\mathcal{U}\) of functions that are analytic in the open unit disk \(D =\{z : |z| < 1\}\), normalized such that\(f(0) = f'(0)-1 = 0\) and satisfy \[\left|\left[\frac{z}{f(z)}\right]^2f'(z) - 1\right|< 1\quad  (z\in D).\]For functions in the class \(\mathcal{U}\) we give sharp estimates of the second and the third Hankel determinant, its relationship with the class of \(\alpha\)-convex functions, as well as certain starlike properties.

This paper deals with the problem of finding some upper bound estimates for the maximum modulus of the derivative and higher order derivatives of a complex polynomial on a disk under the assumption that the polynomial has no zeros in another disk. The estimates obtained strengthen the well-known inequality of Ankeny and Rivlin about polynomials.

In this paper, we establish some inequalities for rational functions with prescribed poles and restricted zeros in the sup-norm on the unit circle in the complex plane. Generalizations and refinements of rational function inequalities of Govil, Li, Mohapatra and Rodriguez are obtained.

In this article we consider the problem of univalence of a function introduced by Breaz and Ularu, improve some of their results and receive not only univalence conditions but also close-to-convex conditions for this function. To this aim, we used our method based on Kaplan classes.

Let \(M(r) := \max_{|z|=r} |f(z)|\), where \(f(z)\) is an entire function. Also let \(\alpha> 0\) and \(\beta>1\). We discuss the behavior of the integrand \(M(r)e^{-\alpha(log r)^\beta}\) as \(r \to \infty\) if \(\int_1^\infty M(r)e^{-\alpha(log r)^\beta}dr\) is convergent.

In this paper we establish some new upper and lower bounds for the difference between the weighted arithmetic and harmonic operator means under various assumptions for the positive invertible operators A, B. Some applications when A, B are bounded above and below by positive constants are given as well.

In 2015, Goebel and Bolibok defined the initial trend coefficient of a mapping and the class of initially nonexpansive mappings. They proved that the fixed point property for nonexpansive mappings implies the fixed point property for initially nonexpansive mappings. We generalize the above concepts and prove an analogous fixed point theorem. We also study the initial trend coefficient more deeply.

In this paper we obtain a condition for analytic square integrable functions \(f,g\) which guarantees the boundedness of products of the Toeplitz operators \(T_fT_{\bar g}\) densely defined on the Bergman space in the polydisk. An analogous condition for the products of the Hankel operators \(H_fH^*_g\) is also given.

For every predual \(X\) of \(\ell_1\) such that the standard basis in \(\ell_1\) is weak\(^*\) convergent, we give explicit models of all Banach spaces \(Y\) for which the Banach-Mazur distance \(d(X,Y)=1\). As a by-product of our considerations, we obtain some new results in metric fixed point theory. First, we show that the space \(\ell_1\), with a predual \(X\) as above, has the stable weak\(^*\) fixed point property if and only if it has almost stable weak\(^*\) fixed point property, i.e. the dual \(Y^*\) of every Banach space \(Y\) has the weak\(^*\) fixed point property (briefly, \(\sigma(Y^*,Y)\)-FPP) whenever \(d(X,Y)=1\). Then, we construct a predual \(X\) of \(\ell_1\) for which \(\ell_1\) lacks the stable \(\sigma(\ell_1,X)\)-FPP but it has almost stable \(\sigma(\ell_1,X)\)-FPP, which in turn is a strictly stronger property than the \(\sigma(\ell_1,X)\)-FPP. Finally, in the general setting of preduals of \(\ell_1\), we give a sufficient condition for almost stable weak\(^*\) fixed point property in \(\ell_1\) and we prove that for a wide class of spaces this condition is also necessary.

We study the so-called inverse problem. Namely, given a prescribed skew-symmetric Ricci tensor we find (locally) a respective linear connection.

If \(m\geq p+1\geq 2\) (or \(m=p\geq 3\)), all  natural bilinear  operators \(A\) transforming pairs of couples of vector fields and \(p\)-forms on \(m\)-manifolds \(M\) into couples of vector fields and \(p\)-forms on \(M\) are described. It is observed that  any natural skew-symmetric bilinear operator \(A\) as above coincides with the generalized Courant bracket up to three (two, respectively) real constants.

In this paper we introduce a two-parameter generalization of the classical Jacobsthal numbers ((s,p)-Jacobsthal numbers). We present some properties of the presented sequence, among others Binet’s formula, Cassini’s identity, the generating function. Moreover, we give a graph interpretation of (s,p)-Jacobsthal numbers, related to independence in graphs.

Let \(\mathcal{T}=(V,\mathcal{E})\) be a  3-uniform linear hypertree. We consider a blow-up hypergraph \(\mathcal{B}[\mathcal{T}]\). We are interested in the following problem. We have to decide whether there exists a blow-up hypergraph \(\mathcal{B}[\mathcal{T}]\) of the hypertree \(\mathcal{T}\), with hyperedge densities satisfying some conditions, such that the hypertree \(\mathcal{T}\) does not appear in a blow-up hypergraph as a transversal. We present an efficient algorithm to decide whether a given set of hyperedge densities ensures the existence of a 3-uniform linear hypertree \(\mathcal{T}\) in a blow-up hypergraph \(\mathcal{B}[\mathcal{T}]\).

Let  \(\{ X_{\bf n}, {\bf n}\in \mathbb{N}^d \}\) be a random field of negatively dependent  random variables.  The complete  convergence results for negatively dependent  random fields  are refined. To obtain the main theorem several lemmas  for convergence of families indexed by \(\mathbb{N}^d\)   have been proved. Auxiliary lemmas have wider application to study  the random walks on the lattice.

We describe all natural operators \(A\) transforming general connections \(\Gamma\) on fibred manifolds \(Y \rightarrow M\) and torsion-free classical linear connections \(\Lambda\) on \(M\) into general connections \(A(\Gamma,\Lambda)\) on the fibred product \(J^{<q>}Y \rightarrow M\) of \(q\) copies of the first jet prolongation \(J^{1}Y \rightarrow M\).

In this paper we introduce generalized Mersenne numbers. We shall present some of their interpretations and matrix generators which are very useful for determining identities.

Some new inequalities of Hermite-Hadamard type for GA-convex functions defined on positive intervals are given. Refinements and weighted version of known inequalities are provided. Some applications for special means are also obtained.

It was asked by Nicholson (Comm. Algebra, 1999) whether or not unit-regular rings are themselves strongly clean. Although they are clean as proved by Camillo-Khurana (Comm. Algebra, 2001), recently Nielsen and Ster showed in Trans. Amer. Math. Soc., 2018 that there exists a unit-regular ring which is not strongly clean. However, we define here a proper subclass of rings of the class of unit-regular rings, called invo-regular rings, and establish that they are strongly clean. Interestingly, without any concrete indications a priori, these rings are manifestly even commutative invo-clean as defined by the author in Commun. Korean Math. Soc., 2017.

The main result establishes that a weak solution of degenerate nonlinear  elliptic equations can be approximated by a sequence of solutions for non-degenerate nonlinear elliptic equations.

The aim of this paper is to investigate the oscillatory and asymptotic behavior of solutions of a third-order delay difference equation. By using comparison theorems, we deduce oscillation of the difference equation from its relation to certain associated first-order delay difference equations or inequalities. Examples are given to illustrate the main results.

Let \(\mathcal{M}f_m\) be the category of \(m\)-dimensional manifolds and local diffeomorphisms and let \(T\) be the tangent functor on \(\mathcal{M}f_m\). Let \(\mathcal{V}\) be the category of real vector spaces and linear maps and let  \(\mathcal{V}_m\) be the category of  \(m\)-dimensional real vector spaces and linear isomorphisms. Let \(w\) be a polynomial in one variable with real coefficients. We describe all regular covariant functors \(F\colon \mathcal{V}_m\to\mathcal{V}\) admitting \(\mathcal{M}f_m\)-natural operators \(\tilde{P}\) transforming classical linear connections \(\nabla\) on \(m\)-dimensional manifolds \(M\) into almost polynomial \(w\)-structures  \(\tilde{P}(\nabla)\) on \(F(T)M=\bigcup_{x\in M}F(T_xM)\).

In this paper, we consider properties of the spectrum of a Sturm-Liouvilleoperator on time scales. We will prove that the regular symmetricSturm-Liouville operator is semi-bounded from below. We will also give someconditions for the self-adjoint operator associated with the singularSturm-Liouville expression to have a discrete spectrum. Finally, we willinvestigate the continuous spectrum of this operator.

Basic properties of branches of pseudo-BCH algebras are described. Next, the concept of a branchwise commutative pseudo-BCH algebra is introduced. Some conditions equivalent to branchwise commutativity are given. It is proved that every branchwise commutative pseudo-BCH algebra is a pseudo-BCI algebra.

We give an example of a Banach lattice with a non-convex modulus of monotonicity, which disproves a claim made in the literature. Results on preservation of the non-strict Opial property and Opial property under passing to general direct sums of Banach spaces are established.

In this article we take over methods for determination of Koebe set based on extremal sets for a given class of functions.

In this paper we prove that for each \(1< p, \tilde{p} < \infty\), the Banach space \((l^{\tilde{p}}, \left\|\cdot\right\|_{\tilde{p}})\) can be equivalently renormed in such a way that  the Banach space \((l^{\tilde{p}},\left\|\cdot\right\|_{L,\alpha,\beta,p,\tilde{p}})\) is LUR and has a diametrically complete set with empty interior. This result extends the Maluta theorem about existence of such a set in \(l^2\) with the Day norm. We also show that the Banach space \((l^{\tilde{p}},\left\|\cdot\right\|_{L,\alpha,\beta,p,\tilde{p}})\) has the weak fixed point property for nonexpansive mappings.

In this paper we introduce a modification of the Day norm in \(c_0(\Gamma)\) and investigate properties  of this norm.

In this paper we show some properties of the eccentric distance sum index which is defined as follows \(\xi^{d}(G)=\sum_{v \in V(G)}D(v) \varepsilon(v)\). This index is widely used by chemists and biologists in their researches. We present a lower bound of this index for a new class of graphs.

In this paper we determine successive extremal trees with respect to the number of all \((A,2B)\)-edge colourings.

In this paper, we obtain existence results of periodic solutions of hamiltonian systems in the Orlicz-Sobolev space \(W^1L^\Phi([0,T])\). We employ the direct method of calculus of variations and we consider  a potential  function \(F\) satisfying the inequality \(|\nabla F(t,x)|\leq b_1(t) \Phi_0'(|x|)+b_2(t)\), with \(b_1, b_2\in L^1\) and  certain \(N\)-functions \(\Phi_0\).