Demonstratio Mathematica

For a given Banach algebra {\mathcal{ {\mathcal M} }} and a continuous endomorphism \omega on {\mathcal{ {\mathcal M} }} , we define {\mathcal{ {\mathcal M} }} to be {\omega }_{{\mathcal{ {\mathcal L} }}} -biprojective and \overline{\omega } -contractible. We then explore the relationship between them. Additionally, we show that {l}^{1}\left({{\mathbb{N}}}_{\vee }) is a {\omega }_{{\mathcal{ {\mathcal L} }}} -biprojective Banach algebra. Finally, we examine the concept of \omega -pseudo amenability and \omega -approximate biprojectivity in Banach algebras. We demonstrate that for every unital Banach algebra {\mathcal{ {\mathcal M} }} , \omega -approximate biprojectivity and \omega -pseudo contractibility coincide.

This article is dedicated to present various concepts on \alpha -time scale, including power series, Taylor series, binomial series, exponential function, gamma function, and Bessel functions of the first kind. We introduce the \alpha -exponential function as a series, examine its absolute and uniform convergence, and establish its additive identity by employing the \alpha -Gauss binomial formula. Furthermore, we define the \alpha -gamma function and prove \alpha -analogue of the Bohr-Mollerup theorem. Specifically, we demonstrate that the \alpha -gamma function is the unique logarithmically convex solution of f\left(s+1)=\phi \left(s)f\left(s) , f\left(1)=1 , where \phi \left(s) refers to the \alpha -number. In addition, we present Euler’s infinite product form and asymptotic behavior of \alpha -gamma function. As an application, we propose \alpha -analogue of the cylindrical diffusion equation, from which \alpha -Bessel and modified \alpha -Bessel equations are derived. We explore the solutions of the \alpha -cylindrical diffusion equation using the separation of variables technique, revealing analogues of the Bessel and modified Bessel functions of order zero of the first kind. Finally, we illustrate the graphs of the \alpha -analogues of exponential and gamma functions and investigate their reductions to discrete and ordinary counterparts.

In this article, we introduce a new family of \left(\lambda ,\psi ) -Bernstein-Kantorovich operators which depends on a parameter \lambda , derived from the basis functions of Bézier curves and an integrable function \psi . In this approach, all moments and central moments of the new operators can be obtained in terms of two numbers {M}_{1,\psi } and {M}_{2,\psi } , which are the integrals of \psi and {\psi }^{2} , respectively. For operators {L}_{n}\left(f;\hspace{0.33em}x) with {L}_{n}\left(1;\hspace{0.33em}x)=1 , the order of approximation to a function f by {L}_{n}\left(f;\hspace{0.33em}x) is more controlled by the term {L}_{n}\left({\left(t-x)}^{2};\hspace{0.33em}x) . For our operators {K}_{n,\lambda ,\psi }\left(f;\hspace{0.33em}x) , the second central moment {K}_{n,\lambda ,\psi }\left({\left(t-x)}^{2};\hspace{0.33em}x) depend on {M}_{1,\psi } and {M}_{2,\psi } . This means that in our new approach, it is possible to search for a function \psi with different values of {M}_{1,\psi } and {M}_{2,\psi } to make {K}_{n,\lambda ,\psi }\left({\left(t-x)}^{2};\hspace{0.33em}x) smaller. Using this new approach, we show that there exists a function \psi such that the order of approximation to a function f by our new \left(\lambda ,\psi ) -Bernstein-Kantorovich operators is better than the classical \lambda -Bernstein-Kantorovich operators on the interval [0, 1]. Moreover, we obtain some direct and local approximation properties of new operators. We also show that our operators preserve monotonicity properties. Furthermore, we illustrate the approximation results of our operators graphically and numerically.

In this article, we incorporate multivariate Taylor polynomials into the definition of the Bernstein operators to get a faster approximation to multivariate functions by these combined operators. We also give various numerical simulations including graphical illustrations and error estimations. Our results improve not only the linear approximation by classical Bernstein polynomials but also the nonlinear approximation obtained by max-product operations.

Given a square matrix A , we are able to construct numerous equalities involving reasonable mixed operations of A and its conjugate transpose {A}^{\ast } , Moore-Penrose inverse {A}^{\dagger } and group inverse {A}^{\#} . Such kind of equalities can be generally represented in the equation form f\left(A,{A}^{\ast },{A}^{\dagger },{A}^{\#})=0 . In this article, the author constructs a series of simple or complicated matrix equalities composed of A , {A}^{\ast } , {A}^{\dagger } , {A}^{\#} and their algebraic operations, as well as established various explicit formulas for calculating the ranks of these matrix expressions. Many applications of these matrix rank equalities are presented, including a broad range of necessary and sufficient conditions for a square matrix to be range-Hermitian and Hermitian/skew-Hermitian, respectively.

In recent years, research into the multiplicity of solutions to the p -Laplace operator problem has attracted attention, and several important results have been investigated and others still remain open. Problems involving a critical point are indeed interesting and relevant, especially challenging. Motivated by such questions, in this article, we are interested, through a critical point theorem, to investigate the existence of at least three distinct weak solutions for a generalized p -Laplacian problem with parameters under appropriate hypotheses, applicable in physics, for instance, in fluid mechanics, and in Newtonian fluids. In this sense, as a direct consequence of the main result, we finish the work with two other results of weak solutions.

This article is devoted to describing the entire solutions of several systems of the first-order nonlinear partial differential difference equations (PDDEs). Using the Nevanlinna theory and the Hadamard factorization theory of meromorphic functions, we establish some interesting results to reveal the existence and the forms of the finite-order transcendental entire solutions of several systems of the first-order nonlinear PDDEs: \begin{array}{l}\left\{\begin{array}{l}f\left({z}_{1}+{c}_{1},{z}_{2}+{c}_{2})({a}_{1}{g}_{{z}_{1}}+{a}_{2}{g}_{{z}_{2}})={m}_{1},\\ g\left({z}_{1}+{c}_{1},{z}_{2}+{c}_{2})({a}_{3}{f}_{{z}_{1}}+{a}_{4}{f}_{{z}_{2}})={m}_{2},\end{array}\right.\\ \left\{\begin{array}{l}f\left({z}_{1}+{c}_{1},{z}_{2}+{c}_{2})({a}_{1}{f}_{{z}_{1}}+{a}_{2}{g}_{{z}_{1}})={m}_{1},\\ g\left({z}_{1}+{c}_{1},{z}_{2}+{c}_{2})({a}_{3}{f}_{{z}_{2}}+{a}_{4}{g}_{{z}_{2}})={m}_{2},\end{array}\right.\end{array} and \left\{\begin{array}{l}f\left({z}_{1}+{c}_{1},{z}_{2}+{c}_{2})({a}_{1}{f}_{{z}_{2}}+{a}_{2}{g}_{{z}_{1}})={m}_{1},\\ g\left({z}_{1}+{c}_{1},{z}_{2}+{c}_{2})({a}_{3}{f}_{{z}_{1}}+{a}_{4}{g}_{{z}_{2}})={m}_{2},\end{array}\right. where {a}_{1},{a}_{2},{a}_{3},{a}_{4},{c}_{1},{c}_{2}\in {\mathbb{C}} , {m}_{1},{m}_{2}\in {\mathbb{C}}-\left\{0\right\} . Moreover, some examples are given to explain that there are significant differences in the forms of solutions from some previous systems of functional equations.

This article investigates new analytical wave solutions within the beta ( \beta ) fractional framework (F \kappa IIAE and F \kappa IIBE) of the Kuralay II equations, which are significant in the field of nonlinear optics. To achieve this, we employ the improved Kudryashov method, the R method, and the Sardar sub-equation method. The study successfully derives a variety of soliton solutions, including dark, bright, singular, and others, some of which are illustrated through 2D and 3D graphics. For the first time, this research graphically demonstrates the impact of the beta fractional derivative on these solutions. The findings provide valuable insights that may aid in the advancement of future models. The methodologies applied here are not only effective and straightforward to implement but also robust for addressing other fractional, nonlinear partial differential equations.

This work addresses the issue of financial risk analysis by introducing a novel expected shortfall (ES) regression model, which employs expectile regression to define the shortfall threshold in financial risk management. We develop a nonparametric estimator for this model and provide mathematical support by proving both pointwise and uniform complete convergence of the estimator. These asymptotic results are derived under traditional assumptions and include precise convergence rates, emphasizing the impact of the regressor’s dimensionality on the estimation approach. A key feature of our contribution is the straightforward implementation of the estimator, demonstrated through applications on both simulated and real data. Our findings indicate that the new ES-expectile model is more effective than the standard model based on quantile regression, offering improved relevance in financial risk management.

Consider {\mathcal{A}} and {\mathcal{ {\mathcal B} }} to be nonassociative unital algebras. Under the assumption that either one of them has finite dimensions or that both are finite dimensions, a generalized derivation is an additive map {\mathcal{ {\mathcal F} }}:{\mathcal{A}}\to {\mathcal{A}} associated with a derivation {\mathcal{d}} of {\mathcal{A}} if {\mathcal{ {\mathcal F} }}\left(uv)={\mathcal{ {\mathcal F} }}\left(u)v+u{\mathcal{d}}\left(v) for all u,v\in {\mathcal{A}} . The objective of this study is to characterize and elucidate the structure of a generalized derivation on the tensor product of nonassociative algebras. Specifically, we prove that if {\mathcal{ {\mathcal F} }} is a generalized derivation of {\mathcal{A}}\otimes {\mathcal{ {\mathcal B} }} associated with a derivation {\mathcal{d}} of {\mathcal{A}}\otimes {\mathcal{ {\mathcal B} }} , then {\mathcal{ {\mathcal F} }}={{\mathcal{ {\mathcal L} }}}_{u}+{\mathcal{d}} , where {{\mathcal{ {\mathcal L} }}}_{u} is a left multiplication by u and u belongs to the left nucleus of {\mathcal{A}}\otimes {\mathcal{ {\mathcal B} }} (i.e., {{\mathcal{ {\mathcal L} }}}_{u}r=ur for all r\in {\mathcal{A}}\otimes {\mathcal{ {\mathcal B} }} and u\in {N}_{l}\left({\mathcal{A}}\otimes {\mathcal{ {\mathcal B} }}) ). Moreover, every generalized derivation of {\mathcal{A}}\otimes {\mathcal{ {\mathcal B} }} can be represented as the sum of the derivations of the three categories: (i) w+a{\mathcal{d}}u , where u,w\in N\left({\mathcal{A}}\otimes {\mathcal{ {\mathcal B} }}) , (ii) {{\mathcal{ {\mathcal L} }}}_{z}\otimes f , where f is a derivation of {\mathcal{ {\mathcal B} }} and z\in Z\left({\mathcal{A}}) (the center of {\mathcal{A}} ), and (iii) g\otimes {{\mathcal{ {\mathcal L} }}}_{w} , where g is a derivation of {\mathcal{A}} and w\in Z\left({\mathcal{ {\mathcal B} }}) .

In this article, we investigate the following Schrödinger equation: \left\{\begin{array}{ll}-\Delta u=h\left(x)g\left(u)+\lambda u\hspace{1.0em}& \hspace{-0.2em}\text{in}\hspace{0.1em}\hspace{0.33em}{{\mathbb{R}}}^{N},\\ \mathop{\displaystyle \int }\limits_{{{\mathbb{R}}}^{N}}{| u| }^{2}\hspace{0.1em}\text{d}\hspace{0.1em}x=a\hspace{1.0em}& u\in {H}^{1}\left({{\mathbb{R}}}^{N}),\end{array}\right. where N\ge 3 , a\gt 0 , and \lambda \in {\mathbb{R}} arises as a Lagrange multiplier. Assuming that the sublinear nonlinearity f\in C\left({{\mathbb{R}}}^{N}\times {\mathbb{R}},{\mathbb{R}}) is locally defined for | u| small; by using generalized minimax approach, we prove the existence of infinitely many normalized solutions {\{\left({u}_{k},{\lambda }_{k})\}}_{k\in {\mathbb{N}}} for the problem, where {\lambda }_{k}\lt 0 and | {u}_{k}\left(x)| \to 0 as | x| \to \infty . Furthermore, as k\to \infty , we show that {\Vert \nabla {u}_{k}\Vert }_{{L}^{2}\left({{\mathbb{R}}}^{N})}\to 0 , {\Vert {u}_{k}\Vert }_{{L}^{\infty }\left({{\mathbb{R}}}^{N})}\to 0 , and {\lambda }_{k}\to 0 .

This work investigates the existence of singular limit solutions for nonlinear elliptic systems. Our main approach focuses on using the nonlinear domain decomposition method to establish a new Liouville-type result.

The main purpose of this work is to study the ideal topology defined by the minimal and maximal ideals on a topological space. Also, we define and investigate the concepts of ideal quotient and annihilator of any subfamily of {2}^{X} , where {2}^{X} is the power set of X . We obtain some of their fundamental properties. In addition, several relationships among the above notions have been discussed. Moreover, we define a new topology on an ideal topological space, called sharp topology, via the sharp operator defined in this study, which turns out to be finer than the original topology. Furthermore, a decomposition of open sets (in the original topology) has been obtained. Finally, we conclude our work with some interesting applications.

In this article, we present several Geršhgorin-type theorems for {Z}_{1} -eigenvalues of tensors, which improve the results provided by Wang et al. (Some upper bounds on {Z}_{t} -eigenvalues of tensors, Appl. Math. Comput. 329 (2018), 266–277). Bounds for the largest {Z}_{1} -eigenvalue of nonnegative tensors are given. Furthermore, based on the Geršhgorin-type theorems, we introduce some sufficient criteria for the positivity of even order tensors.

Shape-preserving approximation is a significant approximation method that has many application areas, such as computer-based geometric design, image processing, geodesy, chemistry, and robotics. Due to this important fact, the aim of this work is to study the shape-preserving properties of a new class of Kantorovich-type operators, which are constructed via Bézier bases based on shape parameter \lambda \in {[}-1,1] . In pursuance of this goal, we commence our calculations by rewriting these operators as a summation of the classical Kantorovich-type operators and a summation of integral values of first-order divided differences of function f over the interval [0, 1]. Then, adopting this representation, we establish the shape-preserving properties of these operators, such as linearity, positivity, and specifically monotonicity and convexity-preserving properties in connection with function f on the interval [0, 1]. Our results reveal that while operators preserve monotonicity entirely on the interval [0, 1] for all \lambda \in {[}-1,1] , they fail to preserve convexity for some \lambda on the same interval. We endorse this claim by counterexamples and give an altered result on convexity preservation for a specific class of functions when \lambda \in {[}0,1] .

In this article, we are interested in the existence of nontrivial solutions for the following nonhomogeneous Choquard equation involving the p -biharmonic operator: \left\{\begin{array}{l}M\left(\mathop{\displaystyle \int }\limits_{\Omega }{| \Delta u| }^{p}{\rm{d}}x\right){\Delta }_{p}^{2}u-{\Delta }_{p}u=\lambda \left({| x| }^{-\mu }\ast {| u| }^{q}){| u| }^{q-2}u+{| u| }^{{p}^{* }-2}u+f,\hspace{1em}{\rm{in}}\hspace{0.33em}\Omega ,\\ u=\Delta u=0,\hspace{1em}{\rm{on}}\hspace{0.33em}\partial \Omega ,\end{array}\right. where \Omega \subset {{\mathbb{R}}}^{N} , N\ge 3 is a smooth bounded domain, 1\lt p\lt \frac{N}{2} , 0\lt \mu \lt N , p\lt 2q\lt {p}^{* } , {p}^{* }=\frac{Np}{N-2p} denotes the Sobolev conjugate of p . M is a nondecreasing and continuous function, and \lambda \gt 0 is a parameter. {\Delta }_{p}^{2}u:= \Delta \left({| \Delta u| }^{p-2}\Delta u) is the operator of fourth order called the p-biharmonic operator, {\Delta }_{p}u:= \hspace{0.1em}\text{div}\hspace{0.1em}\left({| \nabla u| }^{p-2}\nabla u) is the p -Laplacian operator. f\ge 0 , f\in {L}^{\tfrac{p}{p-1}}\left(\Omega ) , and {| f| }_{\tfrac{p}{p-1}} is sufficiently small. Using the concentration-compactness principle together with the mountain pass theorem, we obtain the existence of nontrivial solutions for the aforementioned problem in both nondegenerate and degenerate cases.

Reverse Pachpatte-type inequalities are concave generalizations of the well-known Bennett-Leindler-type inequalities. We establish reverse nabla Pachpatte-type dynamic inequalities taking account of concavity. It is the first time that converses of Pachpatte-type inequalities are obtained in the nabla time scale calculus as well as for its special cases such as continuous and discrete cases and for the dual results obtained in the delta time scale calculus. Moreover, some of our results extend the related ones when concavity has been removed.

Padé approximations constructed from orthogonal and Faber polynomials on some compact set E serve as tools to detect poles of an approximated function around the set E . The goal of this article is to study the rate of such pole detection using indicators introduced by Gonchar (Poles of rows of the Padé table and meromorphic continuation of functions, Sb. Math. 43 (1981), 527–546). Particularly, we compute the values of these indicators corresponding to our extensions of Padé approximation for the poles of the approximated function within the domain of its meromorphy. Our computations extend the indicator formulas found in the article by Gonchar.

The coexistence of a traveling pulse and a periodic traveling wave was established in a convecting shallow water model when taking a nonlinear buoyancy term u{u}_{x} . In this brief communication, we show that the mechanical balance underlying this coexistence is disrupted by a stronger nonlinear dissipation {\left({u}^{2}{u}_{x})}_{x} , which arises from an enhanced buoyancy term {u}^{2}{u}_{x} . Consequently, the convecting shallow water model exhibits either a unique periodic wave or a unique solitary wave, each within a fixed range of wave speeds. Furthermore, we show that the wave speed is monotonic with respect to the wave amplitude and is smaller than that observed in the model with the buoyancy term u{u}_{x} . A numerical study is performed to verify the theoretical study.

In this work, several bilateral inequalities for K - g -frames in subspaces are established, drawing support from two kinds of operators induced, respectively, by the K - g -frame itself and the K -dual pair, which, compared with previous ones concerning this topic, possess novel structures. It is indicated that new types of inequalities for some other generalized frames can be naturally presented following our approaches.

This study focuses on differential equations incorporating generalized fractional derivatives of the Caputo type. The concept of Ulam-type stability (US) is analyzed in the context of both initial value problems and boundary value problems (BVPs) for the fractional differential equations under investigation. Particular attention is given to addressing certain misconceptions that arise when applying US to BVPs. To mitigate these issues, we propose incorporating a parameter into the boundary conditions as a potential solution. The dependency of the solution on this parameter is established, and a method is outlined for selecting the parameter appropriately. This approach ensures that the solution of the fractional equation is strongly influenced by the arbitrarily chosen solution of the associated inequality. The theoretical findings are further clarified through illustrative examples.

In this study, we investigate a nonlinear heat equation incorporating both a viscoelastic term and a reaction-diffusion term that depends on space-time variables. Initially, we establish the local Hadamard well-posedness results using the standard Faedo-Galerkin method. Subsequently, we demonstrate that the solution exhibits finite-time blowup for initial energy values that are both negative and nonnegative. Finally, we establish the global existence of the solution and provide general decay estimates for the energy functions with small initial energy, utilizing Martinez’s inequality.

In this study, we investigate systems known as nonlinear fractional delay differential (nLFDD) systems, characterized by finite state delays and fractional orders within the range of 0\lt \eta \le 1\lt \zeta \le 2 , situated infinite-dimensional settings. We utilize the controllability Gramian matrix to establish both necessary and sufficient conditions for the controllability of linear fractional delay differential systems that fall within the order range of 0\lt \eta \le 1\lt \zeta \le 2 . Moreover, the Schauder fixed point theorem is employed to delineate the sufficient conditions required for the controllability of nLFDD systems, which are defined by finite state delays and fractional orders in the specified range. To substantiate the theoretical constructs put forth, we provide two illustrative examples.

The aim of this article is to study a semi-functional partial linear regression model (SFPLR) for spatial data with responses missing at random (MAR). The estimators are constructed using the kernel method, and some asymptotic properties, such as the probability convergence rates of the nonparametric component and the asymptotic distribution of the parametric and nonparametric components, are established under certain conditions. Next, the performance and superiority of these estimators are presented and examined through a study on simulated data, comparing our semi-functional partially linear model with the MAR estimator to the semi-functional partially linear model with the full-case estimator, and the functional nonparametric regression model estimator with MAR. The results indicate that the proposed estimators outperform traditional estimators as the amount of randomly missing data increases. Additionally, a study is conducted on real data regarding the modeling of pollution levels using our model, incorporating covariates such as average daily temperature as a functional variable, alongside maximum daily mixing height, total daily precipitation, and daily primary aerosol emission rates as explanatory variables.

The demand for advanced predictive tools has surged in the intricate landscape of global financial markets. Traditional predictive tools based on crisp models offer foundational insights, while the evolving complexities in global financial markets necessitate more nuanced analytical techniques. This research delves deep into Bayesian networks (FBN) as a potential tool for financial risk prediction (FRP). Integrating the probabilistic reasoning of Bayesian Networks with the uncertainty-handling capabilities of fuzzy logic, FBNs present a promising avenue for capturing the multifaceted dynamics of financial data. A comprehensive methodology was employed, encompassing data collection, data preprocessing, and transformation. The FBN model’s construction was rooted in established methodologies, emphasizing feature selection, parameter estimation, and a systematic validation process. The model’s empirical robustness was ensured through rigorous validation and testing mechanisms. The results found that the FBN accuracy achieved a mean absolute error (MAE) of 9.78 and a root mean square error (RMSE) of 11.64, when compared to traditional models such as linear regression, which had MAE and RMSE values of 15.70 and 18.39, respectively. The obtained results illuminate the FBN’s standout performance in FRP. The FBN excels in capturing the underlying intricacies of financial data, offering unparalleled predictive accuracy. Its predictions are closer to actual average value but exhibit fewer large deviations, making it an invaluable tool in the financial analytics arsenal demonstrably outpacing traditional crisp models.