Band 21, Heft 1

In this preface of the Special Issue on Future Directions of Further Developments in Mathematics, we discuss about mathematics, how it should be, what it was and how we hope it will develop.

Camassa-Holm type equations arise as models for the unidirectional propagation of shallow water waves over a flat bottom. They also describe finite length, small amplitude radial deformation waves in cylindrical compressible hyperelastic rods. Under appropriate assumption on the initial data, on the time T T , and on the coefficients of such equation, we prove the well-posedness of the classical solutions for the Cauchy problem.

The Cauchy problem in R n {{\mathbb{R}}}^{n} , n ≥ 2 n\ge 2 , for u t = Δ u − ∇ ⋅ ( u S ⋅ ∇ v ) , 0 = Δ v + u , ( ⋆ ) \begin{array}{r}\left\{\phantom{\rule[-1.25em]{}{0ex}}\begin{array}{l}{u}_{t}=\Delta u-\nabla \cdot \left(uS\cdot \nabla v),\\ 0=\Delta v+u,\end{array}\right.\hspace{2.0em}\hspace{2.0em}\hspace{2.0em}\left(\star )\end{array} is considered for general matrices S ∈ R n × n S\in {{\mathbb{R}}}^{n\times n} . A theory of local-in-time classical existence and extensibility is developed in a framework that differs from those considered in large parts of the literature by involving bounded classical solutions. Specifically, it is shown that for all non-negative initial data belonging to BUC ( R n ) ∩ L p ( R n ) {\rm{BUC}}\left({{\mathbb{R}}}^{n})\cap {L}^{p}\left({{\mathbb{R}}}^{n}) with some p ∈ [ 1 , n ) p\in \left[1,n) , there exist T max ∈ ( 0 , ∞ ] {T}_{\max }\in \left(0,\infty ] and a uniquely determined u ∈ C 0 ( [ 0 , T max ) ; BUC ( R n ) ) ∩ C 0 ( [ 0 , T max ) ; L p ( R n ) ) ∩ C ∞ ( R n × ( 0 , T max ) ) u\in {C}^{0}\left(\left[0,{T}_{\max });\hspace{0.33em}{\rm{BUC}}\left({{\mathbb{R}}}^{n}))\cap {C}^{0}\left(\left[0,{T}_{\max });\hspace{0.33em}{L}^{p}\left({{\mathbb{R}}}^{n}))\cap {C}^{\infty }\left({{\mathbb{R}}}^{n}\times \left(0,{T}_{\max })) such that with v ≔ Γ ⋆ u v:= \Gamma \star u , and with Γ \Gamma denoting the Newtonian kernel on R n {{\mathbb{R}}}^{n} , the pair ( u , v ) \left(u,v) forms a classical solution of ( ⋆ \star ) in R n × ( 0 , T max ) {{\mathbb{R}}}^{n}\times \left(0,{T}_{\max }) , which has the property that if T max < ∞ , then both limsup t ↗ T max ‖ u ( ⋅ , t ) ‖ L ∞ ( R n ) = ∞ and limsup t ↗ T max ‖ ∇ v ( ⋅ , t ) ‖ L ∞ ( R n ) = ∞ . \hspace{0.1em}\text{if}\hspace{0.1em}\hspace{0.33em}{T}_{\max }\lt \infty ,\hspace{1.0em}\hspace{0.1em}\text{then both}\hspace{0.1em}\hspace{0.33em}\mathop{\mathrm{limsup}}\limits_{t\nearrow {T}_{\max }}\Vert u\left(\cdot ,t){\Vert }_{{L}^{\infty }\left({{\mathbb{R}}}^{n})}=\infty \hspace{1.0em}\hspace{0.1em}\text{and}\hspace{0.1em}\hspace{1.0em}\mathop{\mathrm{limsup}}\limits_{t\nearrow {T}_{\max }}\Vert \nabla v\left(\cdot ,t){\Vert }_{{L}^{\infty }\left({{\mathbb{R}}}^{n})}=\infty . An exemplary application of this provides a result on global classical solvability in cases when ∣ S + 1 ∣ | S+{\bf{1}}| is sufficiently small, where 1 = diag ( 1 , … , 1 ) {\bf{1}}={\rm{diag}}\hspace{0.33em}\left(1,\ldots ,1) .

The purpose of this review article is to present some recent results on the modeling and control of large systems of agents. We focus on particular applications where the agents are capable of independent actions instead of simply reacting to external forces. In the literature, such agents were referred to as autonomous, intelligent, self-propelled, greedy, and others. The main applications we have in mind are social systems (as opinion dynamics), pedestrians’ movements (also called crowd dynamics), animal groups, and vehicular traffic. We note that the last three examples include physical constraints; however, the agents are able to inject energy into the system, thus preventing the typical conservation of momentum and energy. In addition, the control problems posed by such systems are new and require innovative methods. We illustrate some ideas developed recently, including the use of sparse controls, limiting the total variation of controls, and defining new control problems for measures. After reviewing various approaches, we discuss some future research directions of potential interest. The latter encompasses both new types of equations and new types of limiting procedures to connect several scales at which a system can be represented. We conclude by illustrating a recent real-life experiment using autonomous vehicles on an open highway to smooth traffic waves. This opens the door to a new era of interventions to control real-time multi-agent systems and to increase the societal impact of such interventions guided by control research.

We illustrate how a variety of logical methods and techniques provide useful, though currently underappreciated, tools in the foundations and applications of reasoning under uncertainty. The field is vast spanning logic, artificial intelligence, statistics, and decision theory. Rather than (hopelessly) attempting a comprehensive survey, we focus on a handful of telling examples. While most of our attention will be devoted to frameworks in which uncertainty is quantified probabilistically, we will also touch upon generalisations of probability measures of uncertainty, which have attracted a significant interest in the past few decades.

This article deals with the existence, multiplicity, minimal complexity, and global structure of the subharmonic solutions to a class of planar Hamiltonian systems with periodic coefficients, being the classical predator-prey model of V. Volterra its most paradigmatic example. By means of a topological approach based on techniques from global bifurcation theory, the first part of the paper ascertains their nature, multiplicity and minimal complexity, as well as their global minimal structure, in terms of the configuration of the function coefficients in the setting of the model. The second part of the paper introduces a dynamical system approach based on the theory of topological horseshoes that permits to detect, besides subharmonic solutions, “chaotic-type” solutions. As a byproduct of our analysis, the simplest predator-prey prototype models in periodic environments can provoke chaotic dynamics. This cannot occur in cooperative and quasi-cooperative dynamics, as a consequence of the ordering imposed by the maximum principle.

We classify the non-negative critical points in W 0 1 , p ( Ω ) {W}_{0}^{1,p}\left(\Omega ) of J ( v ) = ∫ Ω H ( D v ) − F ( x , v ) d x , J\left(v)=\mathop{\int }\limits_{\Omega }\hspace{0.15em}H\left(Dv)-F\left(x,v){\rm{d}}x, where H H is convex and positively p p -homogeneous, while t ↦ ∂ t F ( x , t ) / t p − 1 t\hspace{0.33em}\mapsto \hspace{0.33em}{\partial }_{t}F\left(x,t)\hspace{0.1em}\text{/}\hspace{0.1em}{t}^{p-1} is non-increasing. Since H H may not be differentiable and F F has a one-sided growth condition, J J is only lower semi-continuous on W 0 1 , p ( Ω ) {W}_{0}^{1,p}\left(\Omega ) . We use a weak notion of critical point for non-smooth functionals, derive sufficient regularity of the latter without an Euler-Lagrange equation available, and focus on the uniqueness part of the results in the study of Brezis and Oswald, using a non-smooth Picone inequality.

An optimal, complete, continuous theory of the Luenberger dynamic compensator (or state estimator or state observer) is obtained for the recently studied class of heat-structure interaction partial differential equation (PDE) models, with structure subject to high Kelvin-Voigt damping, and feedback control exercised either at the interface between the two media or else at the external boundary of the physical domain in three different settings. It is a first, full investigation that opens the door to numerous and far reaching subsequent work. They will include physically relevant fluid -structure models, with wave- or plate-structures, possibly without Kelvin-Voigt damping, as explicitly noted in the text, all the way to achieving the ultimate discrete numerical theory, so critical in applications. While the general setting is functional analytic, delicate PDE-energy estimates dictate how to define the interface/boundary feedback control in each of the three cases.

In this article, we consider a nonlocal problem with a strong singular term and a general weight function. By using Ekeland’s variational principle, we prove a necessary and sufficient condition for the existence of a positive solution. Moreover, a method of algebraic analysis is used to deal with the multiplicity of solutions. Compared with the existing literature, our problems and results are novel.

We mainly consider the existence and stability results of the positive solutions for the fractional differential equation with Hadamard-type by applying fixed point theorems, if the nonlinearity may be continuous or singular. We also construct some examples to show the applicability of the results.

In this article, we introduce and study a boundary value problem for ( k , χ ¯ * ) \left(k,{\bar{\chi }}_{* }) -Hilfer generalized proportional fractional differential equation of order in an interval (1, 2], equipped with integro-multipoint nonlocal boundary conditions. In the scalar case setting, the existence results are proved via Leray-Schauder nonlinear alternative and Krasnosel’skiĭ’s fixed point theorem, while the existence of a unique solution is established by applying Banach’s contraction mapping principle. In Banach’s space setting, an existence result is proved via Mönch’s fixed point theorem and the measure of noncompactness. Finally, the obtained theoretical results are well illustrated by constructed examples.

This article presents a comprehensive literature survey on fractional-order optimal control problems. Fractional-order differential equation is extensively used nowadays to model real-world systems accurately, which exhibit fractal dimensions, memory effects, as well as chaotic behaviour. These versatile features attract engineers to concentrate more on this, and it is widely used in the broad domain of science and technology. The mentioned numerical tools take the necessary optimal conditions into account, which makes it a two-point boundary value problem of non-integer order. In this review article, some numerical approaches for the approximation have been stated for obtaining the solution to fractional optimal control problems (FOCPs). Here, few numerical approaches including Grunwald-Letnikov approximation, Adams type predictor-corrector method, generalized Euler’s method, Caputo-Fabrizio method Bernoulli and Legendre polynomials method, Legendre operational method, and Ritz’s and Jacobi’s method are treated as an advanced method to obtain the solution of FOCP. Fractional delayed optimal control is selected for our investigation. It refers to a type of control problem where the control action is delayed by a fractional amount of time. In other words, the control input at a given time depends not only on the current state of the system but also on its past state at fractional times. The fractional delayed optimal control problem is formulated as an optimization problem that seeks to minimize a cost function subject to a set of constraints that represent the dynamics of the system and the fractional delay in the control input. The solution to this problem typically involves the use of fractional polynomials types, i.e. Chebyshev and Bassel polynomials.

In this article, we develop a unified framework for studying some derivatives defined as limits. This framework, the θ \theta -derivative, is used to investigate the relationships between these derivatives and their relation to the ordinary derivative. It is shown that the existence of any of these derivatives is equivalent to the existence of the ordinary derivative. By using these results, we show that two derivatives that appear in the literature under different names are actually identical, and an infinite family of derivatives actually consists of only one member. We also give a unified form for the integral corresponding to these derivatives, generalize the standard analysis theorems to this setting, and relate our results to those of other researchers. Finally, we address the question of whether these derivatives should be considered fractional derivatives.

In this article, we survey and unify a large class or L L -functionals of the conditional distribution of the response variable in regression models. This includes robust measures of location, scale, skewness, and heavytailedness of the response, conditionally on covariates. We generalize the concepts of L L -moments (G. Sillito, Derivation of approximants to the inverse distribution function of a continuous univariate population from the order statistics of a sample , Biometrika 56 (1969), no. 3, 641–650.), L L -skewness, and L L -kurtosis (J. R. M. Hosking, L-moments: analysis and estimation of distributions using linear combinations or order statistics , J. R. Stat. Soc. Ser. B Stat. Methodol. 52 (1990), no. 1, 105–124.) and introduce order numbers for a large class of L L -functionals through orthogonal series expansions of quantile functions. In particular, we motivate why location, scale, skewness, and heavytailedness have order numbers 1, 2, (3,2), and (4,2), respectively, and describe how a family of L L -functionals, with different order numbers, is constructed from Legendre, Hermite, Laguerre, or other types of polynomials. Our framework is applied to models where the relationship between quantiles of the response and the covariates follows a transformed linear model, with a link function that determines the appropriate class of L L -functionals. In this setting, the distribution of the response is treated parametrically or nonparametrically, and the response variable is either censored/truncated or not. We also provide a framework for asymptotic theory of estimates of L L -functionals and illustrate our approach by analyzing the arrival time distribution of migrating birds. In this context, a novel version of the coefficient of determination is introduced, which makes use of the abovementioned orthogonal series expansion.

In this article, we are concerned with minimal-time optimal problems for the class of controllable linear control system on low-dimensional nonnilpotent solvable Lie groups and their homogeneous spaces.

In this article, we study a class of new p ( x )-Kirchhoff problem without satisfying the Ambrosetti-Rabinowitz type growth condition. Under some suitable superliner conditions, we introduce new methods to show the boundedness of Cerami sequences. By using the mountain pass lemma and the symmetric mountain pass lemma, we prove that the p ( x )-Kirchhoff problem has a nontrivial weak solution and infinitely many solutions.

In this article, we obtain an extension of the classical Hermite-Hadamard inequality for convex functions (concave functions) extending it to the power functions [ f ( x ) ] n {[f\left(x)]}^{n} . Some related inequalities are also introduced. By applying those results in analysis, we obtain new upper and lower bounds for the error function.

The main purpose is to establish the variational structure of a fourth-order ordinary differential system with both instantaneous and non-instantaneous impulses. Then by using the mountain pass lemma and symmetric mountain pass lemma, the existence and multiplicity results of non-trivial weak solutions are obtained.

The relay fusion frames have been recently introduced in Hilbert spaces to model sensor relay networks and distributed sensor relay systems, which are deeply connected with compressed sensing. In this article, we introduce the notions of relay fusion frames and Banach relay fusion frames in Banach spaces and study certain attractive properties of relay fusion frames in this more general setting. In a particular sense, Schauder frames can be shown to be a special case of relay fusion frames. Moreover, the stability issue of relay fusion frames and Banach relay fusion frames will be addressed.

Ratio monotonicity, a property stronger than both log-concavity and the spiral property, describes the behavior of the coefficients of many classical polynomials. It is known that the coordinator polynomials of the root lattice of type B n {B}_{n} possess log-concavity. In this paper, we show that the coordinator polynomials of type B n {B}_{n} have a refined version of ratio monotonicity except for the first and the last terms. To our knowledge, this refined version is novel.

In this article, the general Liénard system d x d t = ϕ ( y ) − F ( x ) , d y d t = − g ( x ) \left\{\begin{array}{l}\frac{{\rm{d}}x}{{\rm{d}}t}=\phi (y)-F\left(x),\\ \frac{{\rm{d}}y}{{\rm{d}}t}=-g\left(x)\end{array}\right. is studied. By using the Filippov transformation, combined with the careful estimation of divergence along the closed orbit, we prove the sufficient conditions for the uniqueness of limit cycles in this system. Our results extend almost all the related existing studies on the Liénard system.

Let μ \mu be a positive Borel measure on the interval [ 0 , 1 ) \left[0,1) . The Hankel matrix H μ = ( μ n , k ) n , k ≥ 0 {{\mathcal{ {\mathcal H} }}}_{\mu }={\left({\mu }_{n,k})}_{n,k\ge 0} with entries μ n , k = μ n + k {\mu }_{n,k}={\mu }_{n+k} , where μ n = ∫ [ 0 , 1 ) t n d μ ( t ) {\mu }_{n}={\int }_{\left[0,1)}{t}^{n}{\rm{d}}\mu \left(t) , induces formally the operator as follows: DH μ ( f ) ( z ) = ∑ n = 0 ∞ ∑ k = 0 ∞ μ n , k a k ( n + 1 ) z n , z ∈ D , {{\mathcal{D {\mathcal H} }}}_{\mu }(f)\left(z)=\mathop{\sum }\limits_{n=0}^{\infty }\left(\mathop{\sum }\limits_{k=0}^{\infty }{\mu }_{n,k}{a}_{k}\right)\left(n+1){z}^{n},\hspace{1em}z\in {\mathbb{D}}, where f ( z ) = ∑ n = 0 ∞ a n z n f\left(z)={\sum }_{n=0}^{\infty }{a}_{n}{z}^{n} is an analytic function in D {\mathbb{D}} . In this article, we characterize those positive Borel measures on [ 0 , 1 ) \left[0,1) for which DH μ {{\mathcal{D {\mathcal H} }}}_{\mu } is bounded (resp. compact) from Dirichlet spaces D α ( 0 < α ≤ 2 ) {{\mathcal{D}}}_{\alpha }\hspace{0.33em}\left(0\lt \alpha \le 2) into D β ( 2 ≤ β < 4 ) {{\mathcal{D}}}_{\beta }\hspace{0.33em}\left(2\le \beta \lt 4) .

We consider the problem of scheduling jobs with equal lengths and arbitrary sizes on uniform parallel batch machines with different capacities. Each machine can only process the jobs whose sizes are not larger than its capacity. Several jobs can be processed as a batch simultaneously on a machine, as long as their total size does not exceed the machine’s capacity. The objective is to minimize makespan. Under a divisibility constraint, we obtain two efficient exact algorithms. For the general problem, we obtain an efficient 2-approximation algorithm. Previous work has shown that the problem cannot be approximated to within an approximation ratio better than 2, unless P = NP, even when all machines have identical speeds and capacities.

In this article, we study the following quasilinear Schrödinger equation: − Δ u + V ( ∣ x ∣ ) u − κ u Δ ( u 2 ) + q h 2 ( ∣ x ∣ ) ∣ x ∣ 2 ( 1 + κ u 2 ) u + q ∫ ∣ x ∣ + ∞ h ( s ) s ( 2 + κ u 2 ( s ) ) u 2 ( s ) d s u = ( I α ∗ ∣ u ∣ p ) ∣ u ∣ p − 2 u , x ∈ R 2 , -\Delta u+V\left(| x| )u-\kappa u\Delta \left({u}^{2})+q\frac{{h}^{2}\left(| x| )}{| x\hspace{-0.25em}{| }^{2}}\left(1+\kappa {u}^{2})u+q\left(\underset{| x| }{\overset{+\infty }{\int }}\frac{h\left(s)}{s}\left(2+\kappa {u}^{2}\left(s)){u}^{2}\left(s)\hspace{0.1em}\text{d}\hspace{0.1em}s\right)u=\left({I}_{\alpha }\ast | u\hspace{-0.25em}{| }^{p})| u\hspace{-0.25em}{| }^{p-2}u,\hspace{1em}x\in {{\mathbb{R}}}^{2}, where κ \kappa , q > 0 q\gt 0 , p > 8 p\gt 8 , I α {I}_{\alpha } is a Riesz potential, α ∈ ( 0 , 2 ) \alpha \in \left(0,2) and V ∈ C ( R 2 , R ) V\in {\mathcal{C}}\left({{\mathbb{R}}}^{2},{\mathbb{R}}) . By using Jeanjean’s monotone trick, it can be explored that the aforementioned equation has a ground state solution under appropriate assumptions.

In this article, we give another method to calculate the values of multiple Hurwitz zeta function at non-positive integers by means of Raabe ’s formula and the Bernoulli numbers and we simplify these values by symbolic computation techniques.

In this article, we study some results on the value distribution of differential polynomials and mainly prove the following theorem: suppose that P P is a polynomial with deg P ≥ 3 {\rm{\deg }}\hspace{0.33em}P\ge 3 and f f is a transcendental meromorphic function. Let α \alpha be a small function of f f . If α \alpha is a constant, we also require that there exists a constant A ≠ α A\ne \alpha such that P ( z ) − A P\left(z)-A has a zero of multiplicity at least 3. Then, for any 0 < ε < 1 0\lt \varepsilon \lt 1 , we have T ( r , f ) ≤ k N ¯ r , 1 P ( f ) − α + S ( r , f ) , T\left(r,f)\le k\overline{N}\left(r,\frac{1}{P(f)-\alpha }\right)+S\left(r,f), where if P ′ ( z ) P^{\prime} \left(z) has only one zero, then k = 1 deg P − 2 k=\frac{1}{{\rm{\deg }}\hspace{0.33em}P-2} ; if P ′ ( z ) P^{\prime} \left(z) has two distinct zeros a a and b b with P ( a ) ≠ P ( b ) P\left(a)\ne P\left(b) and α \alpha is nonconstant, then k = 1 1 − ε k=\frac{1}{1-\varepsilon } ; otherwise k = 1 k=1 .

We prove the lower bound for the number of Lucas non-Wieferich primes in arithmetic progressions. More precisely, for any given integer k ≥ 2 k\ge 2 , there are ≫ log x \gg \hspace{0.25em}\log x Lucas non-Wieferich primes p ≤ x p\le x such that p ≡ ± 1 ( mod k ) p\equiv \pm 1\hspace{0.25em}\left({\rm{mod}}\hspace{0.33em}k) , assuming the a b c abc conjecture for number fields.

In this article, we focus on the scattering analysis of Sturm-Liouville type singular operator including an impulsive condition and turning point. In the classical literature, there are plenty of papers considering the positive values of the weight function in both continuous and discontinuous cases. However, this article differs from the others in terms of the impulsive condition appearing at the turning point. We generate the scattering function, resolvent operator, and discrete spectrum of the operator using the hyperbolic representations of the fundamental solutions. Finally, we create an example to show the article’s primary conclusions.

The author of this article concerns with the existence, uniqueness, and stability of the weak solution to the variation-inequality problem. The Kirchhoff operator is a non-divergence form with space variable parameter. The existence of generalized solution is proved by the Leray-Schauder principle and parabolic regularization. The uniqueness and stability of the solution are also discussed by contradiction.

Let ( C , E , s ) \left({\mathcal{C}},{\mathbb{E}},{\mathfrak{s}}) be an extriangulated category. Given a composition of two commutative squares in C {\mathcal{C}} , if two commutative squares are homotopy cartesian, then their composition is also a homotopy cartesian square. This covers the result by Mac Lane [ Categories for the Working Mathematician , Second edition, Graduate Texts in Mathematics, Vol. 5, Springer-Verlag, New York, 1998] for abelian categories and by Christensen and Frankland [ On good morphisms of exact triangles , J. Pure Appl. Algebra 226 (2022), no. 3, 106846] for triangulated categories.

Using notions from linear algebraic graph theory, this article provides a unified perspective on some autocorrelation measures in different fields. They are as follows: (a) Orcutt’s first serial correlation coefficient, (b) Anderson’s first circular serial correlation coefficient, (c) Moran’s r 11 {r}_{11} , and (d) Moran’s I I . The first two are autocorrelation measures for one-dimensional data equally spaced, such as time series data, and the last two are for spatial data. We prove that (a)–(c) are a kind of (d). For example, we show that (d) such that its spatial weight matrix equals the adjacency matrix of a path graph is the same as (a). The perspective is beneficial because studying the properties of (d) leads to studying the properties of (a)–(c) at the same time. For example, the bounds of (a)–(c) can be found from the bounds of (d).

Let D = ( V , A ) D=\left(V,A) be a simple digraph with vertex set V V , arc set A A , and no isolated vertex. A total Roman dominating function (TRDF) of D D is a function h : V → { 0 , 1 , 2 } h:V\to \left\{0,1,2\right\} , which satisfies that each vertex x ∈ V x\in V with h ( x ) = 0 h\left(x)=0 has an in-neighbour y ∈ V y\in V with h ( y ) = 2 h(y)=2 , and that the subdigraph of D D induced by the set { x ∈ V : h ( x ) ≥ 1 } \left\{x\in V:h\left(x)\ge 1\right\} has no isolated vertex. The weight of a TRDF h h is ω ( h ) = ∑ x ∈ V h ( x ) \omega \left(h)={\sum }_{x\in V}h\left(x) . The total Roman domination number γ t R ( D ) {\gamma }_{tR}\left(D) of D D is the minimum weight of all TRDFs of D D . The concept of TRDF on a graph G G was introduced by Liu and Chang [ Roman domination on strongly chordal graphs , J. Comb. Optim. 26 (2013), no. 3, 608–619]. In 2019, Hao et al. [ Total Roman domination in digraphs , Quaest. Math. 44 (2021), no. 3, 351–368] generalized the concept to digraph and characterized the digraphs of order n ≥ 2 n\ge 2 with γ t R ( D ) = 2 {\gamma }_{tR}\left(D)=2 and the digraphs of order n ≥ 3 n\ge 3 with γ t R ( D ) = 3 {\gamma }_{tR}\left(D)=3 . In this article, we completely characterize the digraphs of order n ≥ k n\ge k with γ t R ( D ) = k {\gamma }_{tR}\left(D)=k for all integers k ≥ 4 k\ge 4 , which generalizes the results mentioned above.

In this article, well-posedness for two types of bilevel vector equilibrium problems with variable domination structures are introduced and studied. With the help of cosmically upper continuity or Hausdorff upper semi-continuity for variable domination structures, sufficient and necessary conditions are given for such problems to be Levitin-Polyak (LP) well-posed and LP well-posedness in the generalized sense. As variable domination structure is a valid generalization of fixed one, the main results obtained in this article extend and develop some recent works in the literature.

Legendre was the first to evaluate two well-known integrals involving sines and exponentials. One of these integrals can be used to prove Binet’s second formula for the logarithm of the gamma function. Here, we show that the other integral leads to a specific case of Hermite’s generalization of Binet’s formula. From the analogs of Legendre’s integrals, with sines replaced by cosines, we obtain two integration identities involving logarithms and trigonometric functions. Using these identities, we then subsequently derive generalizations of Binet’s and Hermite’s formulas involving the integral of a complex logarithm.

In this article, without requiring solidness of the underlying cone, a kind of new convergence for sequences in cone b b -metric spaces over Banach algebras and a new kind of completeness for such spaces, namely, wrtn-completeness, are introduced. Under the condition that the cone b b -metric spaces are wrtn-complete and the underlying cones are normal, we establish a common fixed point theorem of contractive conditions with vector-valued coefficients in the non-solid cone b b -metric spaces over Banach algebras, where the coefficients s ≥ 1 s\ge 1 . As consequences, we obtain a number of fixed point theorems of contractions with vector-valued coefficients, especially the versions of Banach contraction principle, Kannan’s and Chatterjea’s fixed point theorems in non-solid cone b b -metric spaces over Banach algebras. Moreover, some valid examples are presented to support our main results.

The main purpose of this article is to prove a result of convergence in variation for a family of multidimensional sampling-Kantorovich operators in the case of averaged-type kernels. The setting in which we work is that one of BV -spaces in the sense of Tonelli.

In this article, we study a class of pseudomonotone split variational inequality problems (VIPs) with non-Lipschitz operator. We propose a new inertial extragradient method with self-adaptive step sizes for finding the solution to the aforementioned problem in the framework of Hilbert spaces. Moreover, we prove a strong convergence result for the proposed algorithm without prior knowledge of the operator norm and under mild conditions on the control parameters. The main advantages of our algorithm are: the strong convergence result obtained without prior knowledge of the operator norm and without the Lipschitz continuity condition often assumed by authors; the minimized number of projections per iteration compared to related results in the literature; the inertial technique employed, which speeds up the rate of convergence; and unlike several of the existing results in the literature on VIPs with non-Lipschitz operators, our method does not require any linesearch technique for its implementation. Finally, we present several numerical examples to illustrate the usefulness and applicability of our algorithm.

In this work, the authors study some convergence results including weak law of large numbers, strong law of large numbers, complete convergence, and complete moment convergence for weighted sums of coordinatewise asymptotically negatively associated random vectors in Hilbert spaces. These results improve or extend some corresponding ones in the literature.

A set D ⊆ V ( G ) D\subseteq V\left(G) is a super dominating set of a graph G G if for every vertex u ∈ V ( G ) \ D u\in V\left(G)\setminus D , there exists a vertex v ∈ D v\in D such that N ( v ) \ D = { u } N\left(v)\setminus D=\left\{u\right\} . The super domination number of G G , denoted by γ s p ( G ) {\gamma }_{sp}\left(G) , is the minimum cardinality among all super dominating sets of G G . In this article, we show that if G G is a cactus graph with k ( G ) k\left(G) cycles, then γ s p ( G ) ≤ γ 2 ( G ) + k ( G ) {\gamma }_{sp}\left(G)\le {\gamma }_{2}\left(G)+k\left(G) , where γ 2 ( G ) {\gamma }_{2}\left(G) is the 2-domination number of G G . In addition, and as a consequence of the previous relationship, we show that if T T is a tree of order at least three, then γ s p ( T ) ≤ α ( T ) + s ( T ) − 1 {\gamma }_{sp}\left(T)\le \alpha \left(T)+s\left(T)-1 and characterize the trees attaining this bound, where α ( T ) \alpha \left(T) and s ( T ) s\left(T) are the independence number and the number of support vertices of T T , respectively.

The method of brackets is a symbolic approach to the computation of integrals over R n {{\mathbb{R}}}^{n} based on a deep result by Ramanujan. Its usefulness to obtain new and difficult integrals has been demonstrated many times in the last few years. This article shows that this method is consistent with most classical rules for the computation of integrals, such as the fundamental theorem of calculus, the Laplace transform, the reduction formula for the integration of functions with spherical symmetry, the Cauchy-Schlömilch transformation, and explicit evaluations for multivariate integrals of product of Bessel functions as obtained by Exton and Srivastava. This work is part of a program dedicated to the derivation of solid theoretical grounds for the use of this attractive integration method.

The Collatz-Sinogowitz irregularity index is the oldest known numerical measure of graph irregularity. For a simple and connected graph G G of order n n and size m m , it is defined as CS ( G ) = λ 1 − 2 m / n , \hspace{0.1em}\text{CS}\hspace{0.1em}\left(G)={\lambda }_{1}-2m\hspace{0.1em}\text{/}\hspace{0.1em}n, where λ 1 {\lambda }_{1} is the largest eigenvalue of the adjacency matrix of G G , and 2 m / n 2m\hspace{0.1em}\text{/}\hspace{0.1em}n is the average vertex degree of G G . Here, the Collatz-Sinogowitz inverse irregularity problem is studied. For every integer i ≥ 0 i\ge 0 , it is shown that there exists a graph G G such that CS ( G ) = i \hspace{0.1em}\text{CS}\hspace{0.1em}\left(G)=i . Also, for every interval I i = ( i , i + 1 ) {I}_{i}=\left(i,i+1) , it is shown that there are infinitely many graphs whose Collatz-Sinogowitz irregularity lies in I i {I}_{i} .

This article discusses the existence of positive solutions for the system of second-order ordinary differential equation boundary value problems − u ″ ( t ) = f ( t , u ( t ) , v ( t ) , u ′ ( t ) ) , t ∈ [ 0 , 1 ] , − v ″ ( t ) = g ( t , u ( t ) , v ( t ) , v ′ ( t ) ) , t ∈ [ 0 , 1 ] , u ( 0 ) = u ( 1 ) = 0 , v ( 0 ) = v ( 1 ) = 0 , \left\{\begin{array}{l}-{u}^{^{\prime\prime} }\left(t)=f\left(t,u\left(t),v\left(t),u^{\prime} \left(t)),\hspace{1em}t\in \left[0,1],\\ -{v}^{^{\prime\prime} }\left(t)=g\left(t,u\left(t),v\left(t),v^{\prime} \left(t)),\hspace{1em}t\in \left[0,1],\\ u\left(0)=u\left(1)=0,\hspace{1em}v\left(0)=v\left(1)=0,\end{array}\right. where f , g : [ 0 , 1 ] × R + × R + × R → R + f,g:\left[0,1]\times {{\mathbb{R}}}^{+}\times {{\mathbb{R}}}^{+}\times {\mathbb{R}}\to {{\mathbb{R}}}^{+} are continuous. Under the related conditions that the nonlinear terms f ( t , x , y , p ) f\left(t,x,y,p) and g ( t , x , y , q ) g\left(t,x,y,q) may be super-linear growth or sub-linear growth on x , y , p x,y,p , and q q , we obtain the existence results of positive solutions. For the super-linear growth case, the Nagumo condition ( F 3 ) \left(F3) is presented to restrict the growth of f ( t , x , y , p ) f\left(t,x,y,p) and g ( t , x , y , q ) g\left(t,x,y,q) on p p and q q . The super-linear growth or sub-linear growth of the nonlinear terms f f and g g is described by related inequality conditions instead of the usual independent inequality conditions about f f and g g . The discussion is based on the fixed point index theory in cones.

In this study, a vector-borne epidemic model with multi-edge infection on complex networks is built. Using the method of next-generation matrix, the basic reproduction number R 0 {R}_{0} is calculated, and if R 0 < 1 {R}_{0}\lt 1 , the disease-free equilibrium E 0 {E}_{0} is globally asymptotically stable; if R 0 > 1 {R}_{0}\gt 1 , there exists a unique endemic equilibrium i ∗ = ( i 1 ∗ , i 2 ∗ , … , i n ∗ ) {i}^{\ast }=\left({i}_{1}^{\ast },{i}_{2}^{\ast },\ldots ,{i}_{n}^{\ast }) that is globally attractive. Moreover, three control strategies are proposed to control the spread of infectious diseases. Finally, some numerical simulations are given to illustrate our theoretical results.

In this article, we investigate the Cauchy problem for Klein-Gordon equations with combined power-type nonlinearities. Coefficients in the nonlinearities depend on the space variable. They are sign preserving functions except one of the coefficients, which may change its sign. We study completely the structure of the Nehari manifold. By using the potential well method, we give necessary and sufficient conditions for nonexistence of global solution for subcritical initial energy by means of the sign of the Nehari functional. When the energy is positive, we propose new sufficient conditions for finite time blow up of the weak solutions. One of these conditions is independent of the sign of the scalar product of the initial data. We also prove uniqueness of the weak solutions under slightly more restrictive assumptions for the powers of the nonlinearities.

In this article, we are interested in uniformly p r pr -ideals with order ≤ 2 \le 2 (which we call 2 r 2r -ideals) introduced by Rabia Üregen in [ On uniformly pr-ideals in commutative rings , Turkish J. Math. 43 (2019), no. 4, 18781886]. Several characterizations and properties of these ideals are given. Moreover, the comparison between the (nonzero) 2 r 2r -ideals and certain classes of classical ideals gives rise to characterizations of certain rings based only on the properties of the ideals consisting only of zero-divisors. Namely, among other things, we compare the class of (nonzero) 2 r 2r -ideals with the class of (minimal) prime ideals, the class of minimal prime ideals and their squares, and the class of primary ideals. The study of 2 r 2r -ideal in polynomial rings allows us to give a new characterization of the rings satisfying the famous A A -property.

Confidence intervals are valuable tools in statistical practice for estimating binomial proportions, with the most well-known being the Wald and Clopper-Pearson intervals. However, it is known that these intervals perform poorly in terms of coverage probability and expected mean length, leading to the proposal of alternative intervals in the literature, although these may also have deficiencies. In this work, we investigate the performance of several of these confidence intervals using the parametric family p ^ c = X + c n + 2 c {\widehat{p}}_{c}=\frac{X+c}{n+2c} with c ≥ 0 c\ge 0 to estimate the parameter p p . Rather than using the confidence intervals approach, this analysis is done from the hypothesis tests approach. Our primary goal with this work is to identify values of c c that result in better-performing tests and to establish an optimal procedure.

Let G G be a π \pi -separable group for some set π \pi of primes, let χ ∈ B π ( G ) \chi \in {B}_{\pi }\left(G) and let N ◃ G N\hspace{0.33em}<mml:mpadded xmlns:ali="http://www.niso.org/schemas/ali/1.0/" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance">\triangleleft </mml:mpadded>\hspace{0.33em}G . In this article, we explore how to construct a nucleus for an irreducible constituent of χ N {\chi }_{N} via the given nucleus ( W , γ ) \left(W,\gamma ) for χ \chi .

In this article, we study a class of variational inequality problems with non-Newtonian polytropic parabolic operators. We introduce a mapping with an adjustable parameter to control the polytropic term, which exactly meets the conditions of Leray-Schauder fixed point theory. At the same time, we construct a penalty function to transform the variational inequality into a regular parabolic initial boundary value problem. Thus, the existence is treated with a Leray-Schauder fixed point theory as well as a suitable version of Aubin-Lions lemma. Then, the uniqueness and stability of the solution are analyzed.

This research focuses on the stochastic functional differential equations driven by G-Brownian motion (G-SFDEs) with infinite delay. It is proved that the trivial solution of a G-SFDE with infinite delay is exponentially stable in mean square. An example is also presented to illustrate the effectiveness of the obtained theory.

In this article, we obtain some sufficient conditions for the boundedness of commutators of p p -adic Hardy-Littlewood operators with symbols in central bounded mean oscillation space and Lipschitz space on the p p -adic function spaces with variable exponents such as the p p -adic local central Morrey, p p -adic Morrey-Herz, and p p -adic local block spaces with variable exponents.

In this article, the coupled matrix nonlinear Schrödinger (NLS) type equations are gauge equivalent to the equation of Schrödinger flow from R 1 {{\mathbb{R}}}^{1} to complex Grassmannian manifold G ˜ n , k = GL ( n , C ) ∕ GL ( k , C ) × GL ( n − k , C ) , {\widetilde{G}}_{n,k}={\rm{GL}}\left(n,{\mathbb{C}})/{\rm{GL}}\left(k,{\mathbb{C}})\times {\rm{GL}}\left(n-k,{\mathbb{C}}), which generalizes the correspondence between Schrödinger flow to the complex 2-sphere C S 2 ( 1 ) ↪ C 3 {\mathbb{C}}{{\mathbb{S}}}^{2}\left(1)\hspace{0.33em}\hookrightarrow \hspace{0.33em}{{\mathbb{C}}}^{3} and the coupled Landau-Lifshitz (CLL) equation. This gives a geometric interpretation of the matrix generalization of the coupled NLS equation (i.e., CLL equation) via Schrödinger flow to the complex Grassmannian manifold G ˜ n , k {\widetilde{G}}_{n,k} . Finally, we explicit soliton solutions of the Schrödinger flow to the complex Grassmannian manifold G ˜ 2 , 1 {\widetilde{G}}_{2,1} .

In this article, we introduce the dual index and dual core generalized inverse (DCGI). By applying rank equation, generalized inverse, and matrix decomposition, we give several characterizations of the dual index when it is equal to 1. We realize that if DCGI exists, then it is unique. We derive a compact formula for DCGI and a series of equivalent characterizations of the existence of the inverse. It is worth noting that the dual index of A ^ \widehat{A} is equal to 1 if and only if its DCGI exists. When the dual index of A ^ \widehat{A} is equal to 1, we study dual Moore-Penrose generalized inverse (DMPGI) and dual group generalized inverse (DGGI) and consider the relationships among DCGI, DMPGI, DGGI, Moore-Penrose dual generalized inverse, and other dual generalized inverses. In addition, we consider symmetric dual matrix and its dual generalized inverses. Finally, two examples are given to illustrate the application of DCGI in linear dual equations.

We focus on the problem of generalized orthogonality of matrix operators in operator spaces. Especially, on ℬ ( l 1 n , l p n ) ( 1 ≤ p ≤ ∞ ) {\mathcal{ {\mathcal B} }}\left({l}_{1}^{n},{l}_{p}^{n})\left(1\le p\le \infty ) , we characterize Birkhoff orthogonal elements of a certain class of matrix operators and point out the conditions for matrix operators which satisfy the Bhatia-Šemrl property. Furthermore, we give some conclusions which are related to the Bhatia-Šemrl property. In a certain class of matrix operator space, such as ℬ ( l ∞ n ) {\mathcal{ {\mathcal B} }}\left({l}_{\infty }^{n}) , the properties of the left and right symmetry are discussed. Moreover, the equivalence condition for the left symmetry of Birkhoff orthogonality of matrix operators on ℬ ( l p n ) ( 1 < p < ∞ ) {\mathcal{ {\mathcal B} }}\left({l}_{p}^{n})\left(1\lt p\lt \infty ) is obtained.

Let f f be a transcendental entire function of finite order with a Picard exceptional value β ∈ C \beta \in {\mathbb{C}} , q ∈ C \ { 0 , 1 } q\in {\mathbb{C}}\setminus \left\{0,1\right\} and c c are complex constants. The authors prove that D q , c f ( z ) − a f ( z ) − a = a a − β , \frac{{{\mathfrak{D}}}_{q,c}f\left(z)-a}{f\left(z)-a}=\frac{a}{a-\beta }, if D q , c f ( z ) {{\mathfrak{D}}}_{q,c}f\left(z) and f ( z ) f\left(z) share value a ( ≠ β ) a\left(\ne \beta ) CM, where D q , c f ( z ) = f ( q z + c ) − f ( z ) ( q − 1 ) z + c {{\mathfrak{D}}}_{q,c}f\left(z)=\frac{f\left(qz+c)-f\left(z)}{\left(q-1)z+c} is the Hahn difference operator. This result generalizes the related results of Zongxuan Chen [ On the difference counterpart of Brück’s conjecture , Acta Math. Sci. 34B (2014), 653–659].

We establish certain delicate L p {L}^{p} bounds for a class of generalized Marcinkiewicz integral operators along submanifolds with rough kernels. These bounds allow us to use Yano’s extrapolation method to prove the L p {L}^{p} boundedness of the aforementioned integral operators under very weak assumptions on the kernels. Our results in this article improve and generalize many previously known results.

For a non-empty set X X , denote the full transformation semigroup on X X by T ( X ) T\left(X) and suppose that E E is an equivalence relation on X X . Evidently, T E ∗ ( X ) = { α ∈ T ( X ) ∣ ( x , y ) ∈ E if and only if ( x α , y α ) ∈ E for all x , y ∈ X } {T}_{{E}^{\ast }}\left(X)=\left\{\alpha \in T\left(X)| \left(x,y)\in E\hspace{0.33em}\hspace{0.1em}\text{if and only if}\hspace{0.1em}\hspace{0.33em}\left(x\alpha ,y\alpha )\in E\hspace{0.33em}\hspace{0.1em}\text{for all}\hspace{0.1em}\hspace{0.33em}x,y\in X\right\} is a subsemigroup of T ( X ) T\left(X) . In this article, we investigate Green relations, Green ∗ \ast -relations and Green ∼ \sim -relations, various kinds of regularities, ℱ {\mathcal{ {\mathcal F} }} -abundant and G {\mathcal{G}} -abundant elements and left and right magnifying elements in T E ∗ ( X ) {T}_{{E}^{\ast }}\left(X) . More specifically, we first obtain the necessary and sufficient conditions under which ℒ {\mathcal{ {\mathcal L} }} (respectively, ℒ ∗ {{\mathcal{ {\mathcal L} }}}^{\ast } , ℒ ˜ \widetilde{{\mathcal{ {\mathcal L} }}} , ℛ {\mathcal{ {\mathcal R} }} , ℛ ∗ {{\mathcal{ {\mathcal R} }}}^{\ast } , and ℛ ˜ \widetilde{{\mathcal{ {\mathcal R} }}} ) is (left, right) compatible, ℛ = ℛ ∗ {\mathcal{ {\mathcal R} }}={{\mathcal{ {\mathcal R} }}}^{\ast } or ℒ = ℒ ˜ {\mathcal{ {\mathcal L} }}=\widetilde{{\mathcal{ {\mathcal L} }}} . Then, we give the sufficient and necessary conditions such that T E ∗ ( X ) {T}_{{E}^{\ast }}\left(X) is left regular (respectively, right regular, completely regular, intra-regular, and completely simple). Finally, we characterize the ℱ {\mathcal{ {\mathcal F} }} -abundant (respectively, G {\mathcal{G}} -abundant) and left (respectively, right) magnifying elements in T E ∗ ( X ) {T}_{{E}^{\ast }}\left(X) .

We study the one-parameter discrete Lane-Emden systems with Minkowski curvature operator Δ Δ u ( k − 1 ) 1 − ( Δ u ( k − 1 ) ) 2 + λ μ ( k ) ( p + 1 ) u p ( k ) v q + 1 ( k ) = 0 , k ∈ [ 2 , n − 1 ] Z , Δ Δ v ( k − 1 ) 1 − ( Δ v ( k − 1 ) ) 2 + λ μ ( k ) ( q + 1 ) u p + 1 ( k ) v q ( k ) = 0 , k ∈ [ 2 , n − 1 ] Z , Δ u ( 1 ) = u ( n ) = 0 = Δ v ( 1 ) = v ( n ) , \left\{\begin{array}{ll}\Delta \left[\frac{\Delta u\left(k-1)}{\sqrt{1-{\left(\Delta u\left(k-1))}^{2}}}\right]+\lambda \mu \left(k)\left(p+1){u}^{p}\left(k){v}^{q+1}\left(k)=0,& k\in {\left[2,n-1]}_{{\mathbb{Z}}},\\ \Delta \left[\frac{\Delta v\left(k-1)}{\sqrt{1-{\left(\Delta v\left(k-1))}^{2}}}\right]+\lambda \mu \left(k)\left(q+1){u}^{p+1}\left(k){v}^{q}\left(k)=0,& k\in {\left[2,n-1]}_{{\mathbb{Z}}},\\ \Delta u\left(1)=u\left(n)=0=\Delta v\left(1)=v\left(n),& \\ \end{array}\right. where n ∈ N n\in {\mathbb{N}} with n > 4 n\gt 4 , max { p , q } > 1 \max \left\{p,q\right\}\gt 1 , λ > 0 \lambda \gt 0 , Δ u ( k − 1 ) = u ( k ) − u ( k − 1 ) \Delta u\left(k-1)=u\left(k)-u\left(k-1) , and μ ( k ) > 0 \mu \left(k)\gt 0 for all k ∈ [ 2 , n − 1 ] Z k\in {\left[2,n-1]}_{{\mathbb{Z}}} . The existence of zero at least one or two positive solutions for the system are obtained according to the different intervals of λ \lambda . Our main tools are based on topological methods, critical point theory, and lower and upper solutions.

Numerical methods for computing Steklov eigenvalues have attracted the attention of academia for their important physical background and wide applications. In this article we discuss the multigrid discretization scheme based on the shifted inverse iteration for the Steklov eigenvalue problem in inverse scattering, and give the error estimation of the proposed scheme. In addition, on the basis of the a posteriori error indicator, we design an adaptive multigrid algorithm. Finally, we present numerical examples to show the efficiency of the proposed scheme.

In this article, the existence and nonexistence of solutions for the quasilinear elliptic equations involving multiple critical terms under Dirichlet boundary conditions on bounded smooth domains Ω ⊂ R N ( N ≥ 3 ) \Omega \subset {R}^{N}(N\ge 3) are proved by using the variational method and Pohozaev identity, respectively.

Let m ∈ N m\in {\mathbb{N}} and be a generalized Orlicz function. We obtained some interpolation inequalities for derivatives in generalized Orlicz-Sobolev spaces W m , φ ( R n ) {W}^{m,\varphi }\left({{\mathbb{R}}}^{n}) . As applications, we established a compact Sobolev embedding on domain and a Landau-Kolmogorov-type inequality in generalized Orlicz spaces. And we introduced the Sobolev φ \varphi -capacity and studied some of its properties.

For a real number α \alpha , the general Randić index of a graph G G , denoted by R α ( G ) {R}_{\alpha }\left(G) , is defined as the sum of ( d ( u ) d ( v ) ) α {\left(d\left(u)d\left(v))}^{\alpha } for all edges u v uv of G G , where d ( u ) d\left(u) denotes the degree of a vertex u u in G G . In particular, R − 1 2 ( G ) {R}_{-\tfrac{1}{2}}\left(G) is the ordinary Randić index, and is simply denoted by R ( G ) R\left(G) . Let α \alpha be a real number. In this article, we show that (1) if α ≥ 0 \alpha \ge 0 , R α ( L ( G ) ) ≥ 2 R α ( G ) {R}_{\alpha }\left(L\left(G))\ge 2{R}_{\alpha }\left(G) for any graph G G with δ ( G ) ≥ 3 \delta \left(G)\ge 3 ; (2) if α ≥ 0 \alpha \ge 0 , R α ( L ( G ) ) ≥ R α ( G ) {R}_{\alpha }\left(L\left(G))\ge {R}_{\alpha }\left(G) for any connected graph G G which is not isomorphic to P n {P}_{n} ; (3) if α < 0 \alpha \lt 0 , R α ( L ( G ) ) ≥ R α ( G ) {R}_{\alpha }\left(L\left(G))\ge {R}_{\alpha }\left(G) for any k k -regular graph G G with k ≥ 2 − 2 α + 1 k\ge {2}^{-2\alpha }+1 ; (4) R ( L ( S ( G ) ) ) ≥ R ( S ( G ) ) R\left(L\left(S\left(G)))\ge R\left(S\left(G)) for any graph G G with δ ( G ) ≥ 3 \delta \left(G)\ge 3 , where S ( G ) S\left(G) is the graph obtained from G G by inserting exactly one vertex into each edge.

We show that even if a Hilbert space does not admit a reproducing kernel, there could still be a kernel function that realizes the Riesz representation map. Constructions in spaces that are the Fourier transform of weighted L 2 {L}^{2} spaces are given. With a mild assumption on the weight function, we are able to reproduce Riesz representatives of all functionals through a limit procedure from computable integrals over compact sets, despite that the kernel is not necessarily in the underlying Hilbert space. Distributional kernels are also discussed.

Let N N be a sufficiently large integer. In this article, it is proved that, with at most O ( N 1 12 + ε ) O\left({N}^{\tfrac{1}{12}+\varepsilon }) exceptions, all even positive integers up to N N can be represented in the form p 1 2 + p 2 2 + p 3 3 + p 4 3 + p 5 3 + p 6 3 {p}_{1}^{2}+{p}_{2}^{2}+{p}_{3}^{3}+{p}_{4}^{3}+{p}_{5}^{3}+{p}_{6}^{3} , where p 1 , p 2 , p 3 , p 4 , p 5 {p}_{1},{p}_{2},{p}_{3},{p}_{4},{p}_{5} , and p 6 {p}_{6} are prime variables. This result constitutes a large improvement upon the previous result of Liu [ On a Waring-Goldbach problem involving squares and cubes , Math. Slovaca. 69 (2019), no. 6, 1249–1262].

When singular moduli of Hauptmodules generate ring class fields (resp. ray class fields) of imaginary quadratic fields, using the theory of Shimura reciprocity law, we determine a necessary and sufficient condition for singular moduli of r r th roots of the Hauptmodules to generate the same ring class fields (resp. ray class fields) as do the singular moduli of the Hauptmodules.

In this article, we investigate the classification of self-adjoint boundary conditions of odd-order differential operators. We obtain that for odd-order self-adjoint boundary conditions under some assumptions, there are exactly two basic types of self-adjoint boundary conditions: coupled and mixed. Moreover we determine the number of possible conditions for each type, which is different from the even-order cases. Our construction will play an important role in the canonical forms and in the spectral analysis of these operators.

This article analyzes the JK iteration process with the class of mappings that are essentially endowed with a condition called condition (E). The convergence of the iteration toward a fixed point of a specific mapping satisfying the condition (E) is obtained under some possible mild assumptions. It is worth mentioning that the iteration process JK converges better toward a fixed point compared to some prominent iteration processes in the literature. This fact is confirmed by a numerical example. Furthermore, it has been shown that the iterative scheme JK is stable in the setting of generalized contraction. The data dependence result is also established. Our results are new in the iteration theory and extend some recently announced results of the literature.

In this article, we introduce anisotropic mixed-norm Herz spaces K ˙ q → , a → α , p ( R n ) {\dot{K}}_{\overrightarrow{q},\overrightarrow{a}}^{\alpha ,p}\left({{\mathbb{R}}}^{n}) and K q → , a → α , p ( R n ) {K}_{\overrightarrow{q},\overrightarrow{a}}^{\alpha ,p}\left({{\mathbb{R}}}^{n}) and investigate some basic properties of those spaces. Furthermore, we establish the Rubio de Francia extrapolation theory, which resolves the boundedness problems of Calderón-Zygmund operators and fractional integral operator and their commutators, on spaces K ˙ q → , a → α , p ( R n ) {\dot{K}}_{\overrightarrow{q},\overrightarrow{a}}^{\alpha ,p}\left({{\mathbb{R}}}^{n}) and K q → , a → α , p ( R n ) {K}_{\overrightarrow{q},\overrightarrow{a}}^{\alpha ,p}\left({{\mathbb{R}}}^{n}) . Especially, the Littlewood-Paley characterizations of anisotropic mixed-norm Herz spaces are also gained. As the generalization of anisotropic mixed-norm Herz spaces, we introduce anisotropic mixed-norm Herz-Hardy spaces H K ˙ q → , a → α , p ( R n ) H{\dot{K}}_{\overrightarrow{q},\overrightarrow{a}}^{\alpha ,p}\left({{\mathbb{R}}}^{n}) and H K q → , a → α , p ( R n ) H{K}_{\overrightarrow{q},\overrightarrow{a}}^{\alpha ,p}\left({{\mathbb{R}}}^{n}) , on which atomic decomposition and molecular decomposition are obtained. Moreover, we gain the boundedness of classical Calderón-Zygmund operators.

In this article, we present some necessary or sufficient conditions for the hyponormality of Toeplitz operator T φ {T}_{\varphi } on the Bergman space A 2 ( D ) {A}^{2}\left({\mathbb{D}}) . In particular, we consider the Toeplitz operator T φ {T}_{\varphi } with nonharmonic symbols φ \varphi under certain assumptions.

Let G G be a finite nonabelian group. For any integer m ≥ 2 m\ge 2 , let A 1 , … , A m {A}_{1},\ldots ,{A}_{m} be nonempty subsets of G G . If A 1 , … , A m {A}_{1},\ldots ,{A}_{m} are mutually disjoint and if the subset product A 1 … A m = { α 1 … α m ∣ α v ∈ A v , v = 1 , 2 , … , m } {A}_{1}\ldots {A}_{m}=\left\{{\alpha }_{1}\ldots {\alpha }_{m}| {\alpha }_{v}\in {A}_{v},v=1,2,\ldots ,m\right\} coincides with G G , then ( A 1 , … , A m ) \left({A}_{1},\ldots ,{A}_{m}) is called a complete decomposition of G G of order m m . In this article, we let G G be the generalized quaternion groups Q 2 n {Q}_{{2}^{n}} , which is a finite nonabelian group of order 2 n {2}^{n} with group presentation given by ⟨ x , y ∣ x 2 n − 1 = 1 , y 2 = x 2 n − 2 , y x = x 2 n − 1 − 1 y ⟩ \langle x,y| {x}^{{2}^{n-1}}=1,{y}^{2}={x}^{{2}^{n-2}},yx={x}^{{2}^{n-1}-1}y\rangle for positive integer n ≥ 3 n\ge 3 . We determine the existence of complete decompositions of Q 2 n {Q}_{{2}^{n}} of order k k , for k ∈ { 2 , 3 , … , 2 n − 1 } k\in \left\{2,3,\ldots ,{2}^{n-1}\right\} , and show that Q 2 n {Q}_{{2}^{n}} can be written in the product of 2 n − 1 {2}^{n-1} subsets, i.e., Q 2 n = A 1 A 2 ⋯ A 2 n − 1 {Q}_{{2}^{n}}={A}_{1}{A}_{2}\cdots {A}_{{2}^{n-1}} , where ∣ A j ∣ = 2 | {A}_{j}| =2 , for j ∈ { 1 , 2 , … , 2 n − 1 } j\in \left\{1,2,\ldots ,{2}^{n-1}\right\} . In addition, we construct the non-complete decomposition of Q 2 n {Q}_{{2}^{n}} of order k k , for k ∈ { 2 , 3 , … , 2 n − 1 } k\in \left\{2,3,\ldots ,{2}^{n-1}\right\} , using the non-exhaustive subsets of Q 2 n {Q}_{2n} .

Let M M be a right R R -module with S = End ( M R ) S={\rm{End}}\left({M}_{R}) . Given two cardinal numbers α \alpha and β \beta and a row-finite matrix A ∈ RFM β × α ( S ) A\in {{\rm{RFM}}}_{\beta \times \alpha }\left(S) , S M {}_{S}M is called injective relative to A A if every left S S -homomorphism from S ( β ) A {S}^{\left(\beta )}A to M M extends to one from S ( α ) {S}^{\left(\alpha )} to M M . It is shown that S M {}_{S}M is injective relative to A A if and only if the right R R -module M β ∕ A M α {M}^{\beta }/A{M}^{\alpha } is cogenerated by M M . S S is called left coherent relative to A ∈ S β × α A\in {S}^{\beta \times \alpha } if Ker ( S ( β ) S → S ( β ) S A ) \left({}_{S}S^{\left(\beta )}\to {}_{S}S^{\left(\beta )}A) is finitely generated. It is shown that S S is left coherent relative to A A if and only if M n ∕ A M α {M}^{n}/A{M}^{\alpha } has an add ( M ) {\rm{add}}\left(M) -preenvelope. As applications, we obtain the necessary and sufficient conditions under which M n ∕ A M α {M}^{n}/A{M}^{\alpha } has an add ( M ) {\rm{add}}\left(M) -preenvelope, which is monic (resp., epic, having the unique mapping property). New characterizations of left n n -semihereditary rings and von Neumann regular rings are given.

A kind of fourth-order boundary value problem with eigenparameter-dependent boundary and transmission conditions is investigated. By constructing the characteristic function, we prove that the problems consist of a finite number of eigenvalues. We obtain that the number of eigenvalues of the problems not only depend on the order of the equation but also depend on the partition of the domain interval, the boundary conditions, and the eigenparameter-dependent transmission conditions.

There are strong relations between the theory of continued fractions and groups of linear fractional transformations. We consider the group G 3 , 3 {G}_{3,3} generated by the linear fractional transformations a = 1 − 1 ∕ z a=1-1/z and b = z + 2 b=z+2 . This group is the unique subgroup of the modular group PSL ( 2 , Z ) {\rm{PSL}}(2,{\mathbb{Z}}) with index 2. We calculate the cusp point of an element given as a word in generators. Conversely, we use the continued fraction expansion of a given rational number p ∕ q p/q , to obtain an element in G 3 , 3 {G}_{3,3} with cusp point p ∕ q p/q . As a result, we say that the action of G 3 , 3 {G}_{3,3} on rational numbers is transitive.

We deal with multiplicity of solutions to the following Schrödinger-Poisson-type system in this article: Δ H u − μ 1 ϕ 1 u = ∣ u ∣ 2 u + F u ( ξ , u , v ) , in Ω , − Δ H v + μ 2 ϕ 2 v = ∣ v ∣ 2 v + F v ( ξ , u , v ) , in Ω , − Δ H ϕ 1 = u 2 , − Δ H ϕ 2 = v 2 , in Ω , ϕ 1 = ϕ 2 = u = v = 0 , on ∂ Ω , \left\{\begin{array}{ll}{\Delta }_{H}u-{\mu }_{1}{\phi }_{1}u={| u| }^{2}u+{F}_{u}\left(\xi ,u,v),\hspace{1.0em}& \hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}\Omega ,\\ -{\Delta }_{H}v+{\mu }_{2}{\phi }_{2}v={| v| }^{2}v+{F}_{v}\left(\xi ,u,v),\hspace{1.0em}& \hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}\Omega ,\\ -{\Delta }_{H}{\phi }_{1}={u}^{2},\hspace{1.0em}-{\Delta }_{H}{\phi }_{2}={v}^{2},\hspace{1.0em}& \hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}\Omega ,\\ {\phi }_{1}={\phi }_{2}=u=v=0,\hspace{1.0em}& \hspace{0.1em}\text{on}\hspace{0.1em}\hspace{0.33em}\partial \Omega ,\end{array}\right. where Δ H {\Delta }_{H} is the Kohn-Laplacian and Ω \Omega is a smooth bounded region on the first Heisenberg group H 1 {{\mathbb{H}}}^{1} , μ 1 {\mu }_{1} , and μ 2 {\mu }_{2} are some real parameters, and F = F ( x , u , v ) , F u = ∂ F ∂ u F=F\left(x,u,v),{F}_{u}=\frac{\partial F}{\partial u} , F v = ∂ F ∂ u {F}_{v}=\frac{\partial F}{\partial u} satisfying natural growth conditions. By the limit index theory and the concentration compactness principles, we prove that the aforementioned system has multiplicity of solutions for μ 1 , μ 2 < ∣ Ω ∣ − 1 2 S {\mu }_{1},{\mu }_{2}\lt {| \Omega | }^{-\tfrac{1}{2}}S , where S S is the best Sobolev constant. The novelties of this article are the presence of critical nonlinear term, and the system is set on the Heisenberg group.

In this article, we study the existence of mild solutions and the approximate controllability for a class of stochastic elastic systems with structural damping and infinite delay in Hilbert spaces. The estimation of the control function is discussed, where the expression of the control function is constructed by the defined resolvent operator. Under this estimate, the existence of mild solutions for this system is obtained by the Schauder fixed point theorem and the stochastic analysis theory, and sufficient conditions for the approximate controllability are formulated and proved by using the so-called resolvent operator type condition. Finally, an example is given to illustrate the applicability of our conclusion.

Let G G be a simple graph with degree sequence D ( G ) = ( d 1 , d 2 , … , d n ) D\left(G)=\left({d}_{1},{d}_{2},\ldots ,{d}_{n}) . The first degree-based entropy of G G is defined as I 1 ( G ) = ln ∑ i = 1 n d i − 1 ∑ i = 1 n d i ∑ i = 1 n ( d i ln d i ) {I}_{1}\left(G)=\mathrm{ln}{\sum }_{i=1}^{n}{d}_{i}-\frac{1}{{\sum }_{i=1}^{n}{d}_{i}}{\sum }_{i=1}^{n}\left({d}_{i}\mathrm{ln}{d}_{i}) . In this article, we give sharp upper and lower bounds for the first degree-based entropy of graphs in C ( n , k ) {\mathcal{C}}\left(n,k) and characterize the corresponding extremal graphs when each bound is attained, where C ( n , k ) {\mathcal{C}}\left(n,k) is the set of all cacti with n n vertices and k k cycles.

Our objective is the compressed enumeration (based on wildcards) of all minimal hitting sets of general hypergraphs. To the author’s best knowledge, the only previous attempt towards compression, due to Toda, is based on binary decision diagrams and much different from our techniques. Traditional one-by-one enumeration schemes cannot compete when the number of minimal hitting sets is large and the degree of compression is high. Our method works particularly well in these two cases: either compressing all minimum cardinality hitting sets, or compressing all exact hitting sets.

This article shows the existence and multiplicity of solutions for the following p p -Kirchhoff-type equation: a + b ∫ R N ( ∣ ∇ u ∣ p + V ( x ) ∣ u ∣ p ) d x ( − △ p u + V ( x ) ∣ u ∣ p − 2 u ) = λ g ( x ) ∣ u ∣ r − 2 u − h ( x ) ∣ u ∣ q − 2 u , in R N . \left(a+b\mathop{\int }\limits_{{{\mathbb{R}}}^{N}}\left({| \nabla u| }^{p}+V\left(x){| u| }^{p}){\rm{d}}x\right)\left(-{\bigtriangleup }_{p}u+V\left(x){| u| }^{p-2}u)=\lambda g\left(x){| u| }^{r-2}u-h\left(x){| u| }^{q-2}u,\hspace{1.0em}\hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}{{\mathbb{R}}}^{N}. where λ \lambda is a real parameter and 1 < p < q < ∞ 1\lt p\lt q\lt \infty , a a and b b are positive constants. Depending on the relationship of p , q p,q , and r r , we obtain the existence, multiplicity, and nonexistence of solutions to the abovementioned equation using variational methods.

The prime objective of the approach is to give geometric classifications of k k -almost Ricci solitons associated with paracontact manifolds. Let M 2 n + 1 ( φ , ξ , η , g ) {M}^{2n+1}\left(\varphi ,\xi ,\eta ,g) be a paracontact metric manifold, and if a K K -paracontact metric g g represents a k k -almost Ricci soliton ( g , V , k , λ ) \left(g,V,k,\lambda ) and the potential vector field V V is Jacobi field along the Reeb vector field ξ \xi , then either k = λ − 2 n k=\lambda -2n , or g g is a k k -Ricci soliton. Next, we consider K K -paracontact manifold as a k k -almost Ricci soliton with the potential vector field V V is infinitesimal paracontact transformation or collinear with ξ \xi . We have proved that if a paracontact metric as a k k -almost Ricci soliton associated with the non-zero potential vector field V V is collinear with ξ \xi and the Ricci operator Q Q commutes with paracontact structure φ \varphi , then it is Einstein of constant scalar curvature equals to − 2 n ( 2 n + 1 ) -2n\left(2n+1) . Finally, we have deduced that a para-Sasakian manifold admitting a gradient k k -almost Ricci soliton is Einstein of constant scalar curvature equals to − 2 n ( 2 n + 1 ) -2n\left(2n+1) .

The present article is devoted to the positive periodic solution for impulsive discrete hematopoiesis model. This model is described by a first-order nonlinear difference equation with multiple delays and impulses. The existence result is obtained by applying the Krasnosel’skii fixed point theorem. A sufficient condition that guarantees that there is at least one positive periodic solution is established. Moreover, illustrative example and numerical simulation analysis are provided to show the existence of positive periodic solutions and the effect of impulses on positive periodic solutions.

We establish necessary conditions for the existence of solutions to various systems of partial differential inequalities in the plane. The obtained conditions provide new Hermite-Hadamard-type inequalities for differentiable functions in the plane. In particular, we obtain a refinement of an inequality due to Dragomir [ On the Hadamard’s inequality for convex functions on the co-ordinates in a rectangle from the plane , Taiwan. J. Math. 5 (2001), no. 4, 775–788] for convex functions on the coordinates.

In this work, we investigate the existence and uniqueness of solutions for retarded equations with non-instantaneous impulses, non-local conditions, and infinite delay. To achieve this goal, we select an appropriate phase space that satisfies the axiomatic theory developed by Hale and Kato for studying differential equations with infinite delay. Then, we reformulate the problem of existence of solutions as the problem of finding fixed points of an operator. To this end, we apply the Karakostas fixed point theorem, which is an extension of the well-known Krasnoselskii fixed point theorem. Under certain conditions, we establish the uniqueness of solutions for our problem. Finally, we analyze the prolongation of solutions and demonstrate that, given certain conditions, these solutions are globally defined. Our research has important implications for the study of delayed dynamical systems, including models of population dynamics, physiological processes, and engineering systems.

The main purpose of this article is to use elementary methods and properties of classical Gauss sums to determine identities for the number of residue systems of a a mod p p such that a a , a + a ¯ a+\bar{a} , and a − a ¯ a-\bar{a} are all quadratic residues, improving on results of Wang and Lv.

Let Q Q be an integral positive definite quadratic form of level N N in 2 k 2k variables. Further, we assume that ( − 1 ) k N {\left(-1)}^{k}N is a fundamental discriminant. We express the average theta function of Q Q as an explicit sum of Eisenstein series, which are Hecke eigenforms by applying Atkin-Lehner involution.

In this article, we prove the existence of eigenvalues for the problem ( ϕ p ( u ′ ( t ) ) ) ′ + λ h ( t ) ϕ p ( u ( t ) ) = 0 , t ∈ ( 0 , 1 ) , A u ( 0 ) − A ′ u ′ ( 0 ) = 0 , B u ( 1 ) + B ′ u ′ ( 1 ) = 0 \left\{\begin{array}{l}\left({\phi }_{p}\left(u^{\prime} \left(t)))^{\prime} +\lambda h\left(t){\phi }_{p}\left(u\left(t))=0,\hspace{1em}t\in \left(0,1),\\ Au\left(0)-A^{\prime} u^{\prime} \left(0)=0,\hspace{1em}Bu\left(1)+B^{\prime} u^{\prime} \left(1)=0\end{array}\right. under hypotheses that ϕ p ( s ) = ∣ s ∣ p − 2 s , p > 1 {\phi }_{p}\left(s)={| s| }^{p-2}s,p\gt 1 , and h h is a nonnegative measurable function on ( 0 , 1 ) \left(0,1) , which may be singular at 0 and/or 1. For the result, we establish the existence of connected components of positive solutions for the following problem: ( ϕ p ( u ′ ( t ) ) ) ′ + λ h ( t ) f ( u ( t ) ) = 0 , t ∈ ( 0 , 1 ) , u ( 0 ) = 0 , a u ′ ( 1 ) + c ( λ , u ( 1 ) ) = 0 , \left\{\begin{array}{l}\left({\phi }_{p}\left(u^{\prime} \left(t)))^{\prime} +\lambda h\left(t)f\left(u\left(t))=0,\hspace{1em}t\in \left(0,1),\\ u\left(0)=0,\hspace{1em}au^{\prime} \left(1)+c\left(\lambda ,u\left(1))=0,\end{array}\right. where λ \lambda is a real parameter, a ≥ 0 a\ge 0 , f ∈ C ( ( 0 , ∞ ) , ( 0 , ∞ ) ) f\in C\left(\left(0,\infty ),\left(0,\infty )) satisfies inf s ∈ ( 0 , ∞ ) f ( s ) > 0 {\inf }_{s\in \left(0,\infty )}f\left(s)\gt 0 and limsup s → 0 s α f ( s ) < ∞ {\mathrm{limsup}}_{s\to 0}{s}^{\alpha }f\left(s)\lt \infty for some α > 0 \alpha \gt 0 .

Hyperharmonic numbers were introduced by Conway and Guy ( The Book of Numbers , Copernicus, New York, 1996), whereas harmonic numbers have been studied since antiquity. Recently, the degenerate hyperharmonic and degenerate harmonic numbers were introduced as their respective degenerate versions. The aim of this article is to further investigate some properties and identities involving the degenerate hyperharmonic numbers and degenerate harmonic numbers. Especially, we derive some identities by making use of the transfer formula for associated sequences from umbral calculus.

In this article, we prove ergodic convergence for a sequence of nonexpansive mappings in Hadamard (complete CAT ( 0 ) {\rm{CAT}}\left(0) ) spaces. Our result extends the nonlinear ergodic theorem by Baillon for nonexpansive mappings from Hilbert spaces to Hadamard spaces. We further establish the standard nonlinear ergodic theorem and apply our results to the problem of finding a common fixed point of a countable family of nonexpansive mappings. Finally, we propose some applications of our results to solve convex optimization problems in Hadamard spaces.

Our main purpose is to introduce and investigate the concepts of some forms of spaces via topological ideals. Some characterizations of some forms of spaces via topological ideals are established. Moreover, several properties of ⋆ \star -monotonically normal ideal topological spaces are considered.

Let X X be a complex Banach space with dimension at least two and B ( X ) B\left(X) the algebra of all bounded linear operators on X X . We show that a bijective linear map Φ \Phi preserves asymptotic equivalence if and only if it preserves equivalence, and in turn, if and only if there exist invertible bounded linear operators T T and S S such that either Φ ( A ) = T A S \Phi \left(A)=TAS or Φ ( A ) = T A * S \Phi \left(A)=T{A}^{* }S for all A ∈ B ( X ) A\in B\left(X) .

In this article, the effect of Coleman-Gurtin’s and Gurtin-Pipkin’s thermal laws on the displacement of a Timoshenko beam system with suspenders is studied. Using the existing semi-group theory and energy method, the existence and uniqueness of weak global solution, as well as a stability result without imposing any conditions on the coefficient parameters, are established.

A new class of stochastic differential equations (SDEs) is introduced in this article, which is driven by the generalized stochastic mixed variational inequality (GS-MVI). First, the property of the solution sets of the GS-MVI is proved by Fan-Knaster-Kuratowski-Mazurkiewicz (FKKM) theorem and Aumann’s measurable selection theorem. Next, we obtain the Carathéodory property of the solution set, with which the discussed SDEs can be transformed to stochastic differential inclusions (SDIs). The solution set of the proposed SDEs is proved to be nonempty through the existence of the solutions of the corresponding SDIs by the tools of fixed point theorem.

In this article, we investigate the precise expression forms of entire solutions for two certain Fermat-type ordinary differential equations: ( a 0 f + a 1 f ′ ) 2 + ( a 0 f + a 2 f ′ ) 2 = p {\left({a}_{0}f+{a}_{1}{f}^{^{\prime} })}^{2}+{\left({a}_{0}f+{a}_{2}{f}^{^{\prime} })}^{2}=p and ( a 0 f + a 1 f ′ ) 2 + ( a 0 f + a 2 f ′ ) 2 = e g \hspace{0.39em}{\left({a}_{0}f+{a}_{1}{f}^{^{\prime} })}^{2}+{\left({a}_{0}f+{a}_{2}{f}^{^{\prime} })}^{2}={e}^{g} in C {\mathbb{C}} by making use of the Nevanlinna theory for meromorphic functions, where a 0 {a}_{0} , a 1 {a}_{1} , and a 2 {a}_{2} are the complex numbers with a 0 ≠ 0 {a}_{0}\ne 0 and ∣ a 1 ∣ + ∣ a 2 ∣ ≠ 0 | {a}_{1}| +| {a}_{2}| \ne 0 , while p p and g g are the polynomials in C {\mathbb{C}} . Moreover, some examples are given to illustrate the existence of entire solutions for the aforementioned two certain equations.

In this study, we consider generalized Lie n n -derivations of an arbitrary triangular algebra T T through the constructed triangular algebra T 0 {T}_{0} , where T 0 {T}_{0} is constructed using the notion of maximal left (right) ring of quotients.

This article presents an approximation of discrete Markov decision processes with small noise on Borel spaces with an infinite horizon and an expected total discounted cost by the corresponding deterministic Markov process. In both cases, the dynamics evolve through a system consisting of two coupled difference equations. It is assumed that the difference equations of the system are perturbed by a small noise. Under our assumptions, a bound for the stability index is given, and the optimal cost convergence rate is estimated using a small perturbation parameter. Moreover, the convergence of the optimal policy on compact subsets is verified. Finally, two examples are presented to illustrate the developed theory.

This article focuses on the boundedness of the commutators generated by pseudodifferential operators with Lipschitz functions and obtains a sufficient condition such that these operators are bounded from L p ( R n ) {L}^{p}\left({{\bf{R}}}^{n}) into the homogeneous Triebel-Lizorkin space F ˙ p γ , ∞ ( R n ) {\dot{F}}_{p}^{\gamma ,\infty }\left({{\bf{R}}}^{n}) and from L p ( R n ) {L}^{p}\left({{\bf{R}}}^{n}) into L q ( R n ) {L}^{q}\left({{\bf{R}}}^{n}) .

We investigate when there is a partition of a positive integer n n , n = f ( λ 1 ) + f ( λ 2 ) + ⋯ + f ( λ ℓ ) , n=f\left({\lambda }_{1})+f\left({\lambda }_{2})+\cdots +f\left({\lambda }_{\ell }), satisfying that 1 = χ p ( λ 1 ) λ 1 + χ p ( λ 2 ) λ 2 + ⋯ + χ p ( λ ℓ ) λ ℓ , 1=\frac{{\chi }_{p}\left({\lambda }_{1})}{{\lambda }_{1}}+\frac{{\chi }_{p}\left({\lambda }_{2})}{{\lambda }_{2}}+\cdots +\frac{{\chi }_{p}\left({\lambda }_{\ell })}{{\lambda }_{\ell }}, where χ p {\chi }_{p} is the Legendre symbol modulo prime p p and f ( k ) = k f\left(k)=k or the k k th m m -gonal number with m = 3 m=3 , 4, or 5.

We construct a family of processes, from a renewal process, that have realizations that converge almost surely to the Brownian motion, uniformly on the unit time interval. Finally, we compute the rate of convergence in a particular case.

A scheme for the stochastization of systems of ordinary differential equations (ODEs) based on Itô calculus is presented in this article. Using the presented techniques, a system of stochastic differential equations (SDEs) can be constructed in such a way that eliminating the stochastic component yields the original system of ODEs. One of the main benefits of this scheme is the ability to construct analytical solutions to SDEs with the use of special vector-valued functions, which significantly differs from the randomization approach, which can only be applied via numerical integration. Moreover, using the presented techniques, a system of ODEs and SDEs can be constructed from a given diffusion function, which governs the uncertainty of a particular process.

The two-distance vertex-distinguishing index χ d 2 ′ ( G ) {\chi }_{d2}^{^{\prime} }\left(G) of graph G G is defined as the smallest integer k k , for which the edges of G G can be properly colored using k k colors. In this way, any pair of vertices at distance of two have distinct sets of colors. The two-distance vertex-distinguishing edge coloring of graphs can be used to solve some network problems. In this article, we used the method of discharging to prove that if G G is a graph with mad ( G ) < 8 3 \left(G)\lt \frac{8}{3} , then χ d 2 ′ ( G ) ≤ max { 7 , Δ + 2 } {\chi }_{d2}^{^{\prime} }\left(G)\le \max \left\{7,\Delta +2\right\} , which improves the result that a graph G G of Δ ( G ) ≥ 4 \Delta \left(G)\ge 4 has χ d 2 ′ ( G ) ≤ Δ ( G ) + 2 {\chi }_{d2}^{^{\prime} }\left(G)\le \Delta \left(G)+2 if mad ( G ) < 5 2 \left(G)\lt \frac{5}{2} , and χ d 2 ′ ( G ) ≤ Δ ( G ) + 3 {\chi }_{d2}^{^{\prime} }\left(G)\le \Delta \left(G)+3 if mad ( G ) < 8 3 \left(G)\lt \frac{8}{3} .

Let P ( X ) P\left(X) be a partial transformation semigroup on a non-empty set X X . For a fixed non-empty subset Y Y of X X , let P T ¯ ( X , Y ) = { α ∈ P ( X ) ∣ ( dom α ∩ Y ) α ⊆ Y } . \overline{PT}\left(X,Y)=\left\{\alpha \in P\left(X)| \left({\rm{dom}}\hspace{0.33em}\alpha \cap Y)\alpha \subseteq Y\right\}. Then, P T ¯ ( X , Y ) \overline{PT}\left(X,Y) consists of all the mapping in P ( X ) P\left(X) that leave Y ⊆ X Y\subseteq X as an invariant. It is a generalization of P ( X ) P\left(X) since P T ¯ ( X , X ) = P ( X ) \overline{PT}\left(X,X)=P\left(X) . In this article, we present the necessary and sufficient conditions for elements of P T ¯ ( X , Y ) \overline{PT}\left(X,Y) to be regular, left regular, and right regular. The results are used to describe the relationships between these elements and determine their number when X X is a finite set. Moreover, we show that P T ¯ ( X , Y ) \overline{PT}\left(X,Y) is always abundant.

In this article, the Liouville theorems for the Kirchhoff-type parabolic equations on the Heisenberg group were established. The main technique for proving the result relies on the method of test functions.

Let X X be a compact Riemann surface of genus g ≥ 2 g\ge 2 , G G be a semisimple complex Lie group and ρ : G → GL ( V ) \rho :G\to {\rm{GL}}\left(V) be a complex representation of G G . Given a principal G G -bundle E E over X X , a vector bundle E ( V ) E\left(V) whose typical fiber is a copy of V V is induced. A ( G , ρ ) \left(G,\rho ) -Higgs pair is a pair ( E , φ ) \left(E,\varphi ) , where E E is a principal G G -bundle over X X and φ \varphi is a holomorphic global section of E ( V ) ⊗ L E\left(V)\otimes L , L L being a fixed line bundle over X X . In this work, Higgs pairs of this type are considered for G = Spin ( 8 , C ) G={\rm{Spin}}\left(8,{\mathbb{C}}) and the three irreducible eight-dimensional complex representations which Spin ( 8 , C ) {\rm{Spin}}\left(8,{\mathbb{C}}) admits. In particular, the reduced notions of stability, semistability, and polystability for these specific Higgs pairs are given, and it is proved that the corresponding moduli spaces are isomorphic, and a precise expression for the stable and not simple Higgs pairs associated with one of the three announced representations of Spin ( 8 , C ) {\rm{Spin}}\left(8,{\mathbb{C}}) is described.

This study investigates some topological properties of locally semi-compact Ir-topological groups and establishes the relationship between Ir-topological groups and semi-compact spaces. The proved theorems generalize the corresponding results of Ir-topological group. Finally, we define a quotient topology on the Ir-topological group and study some topological properties of the space.

Our purpose in this article is to describe the solutions of several product-type nonlinear partial differential equations (PDEs) ( a 1 u + b 1 u z 1 + c 1 u z 2 ) ( a 2 u + b 2 u z 1 + c 2 u z 2 ) = 1 , \left({a}_{1}u+{b}_{1}{u}_{{z}_{1}}+{c}_{1}{u}_{{z}_{2}})\left({a}_{2}u+{b}_{2}{u}_{{z}_{1}}+{c}_{2}{u}_{{z}_{2}})=1, and ( a 1 u + b 1 u z 1 + c 1 u z 2 ) ( a 2 u + b 2 u z 1 + c 2 u z 2 ) = e g , \left({a}_{1}u+{b}_{1}{u}_{{z}_{1}}+{c}_{1}{u}_{{z}_{2}})\left({a}_{2}u+{b}_{2}{u}_{{z}_{1}}+{c}_{2}{u}_{{z}_{2}})={e}^{g}, where g ( z ) g\left(z) is a nonconstant polynomial and a j , b j {a}_{j},{b}_{j} , and c j ( j = 1 , 2 ) {c}_{j}\left(j=1,2) are constants in C {\mathbb{C}} . The finite-order transcendental entire solution u u of the first equation is of the following forms: u ( z 1 , z 2 ) = ± 1 a 1 a 2 + η 0 e 1 D [ ( a 2 c 1 − a 1 c 2 ) z 1 + ( a 1 b 2 − a 2 b 1 ) z 2 ] , u\left({z}_{1},{z}_{2})=\pm \frac{1}{\sqrt{{a}_{1}{a}_{2}}}+{\eta }_{0}{e}^{\tfrac{1}{D}{[}\left({a}_{2}{c}_{1}-{a}_{1}{c}_{2}){z}_{1}+\left({a}_{1}{b}_{2}-{a}_{2}{b}_{1}){z}_{2}]}, or u ( z 1 , z 2 ) = 1 2 a 1 e Q ( z 1 , z 2 ) + 1 2 a 2 e − Q ( z 1 , z 2 ) + η 0 e 1 D [ ( a 2 c 1 − a 1 c 2 ) z 1 + ( a 1 b 2 − a 2 b 1 ) z 2 ] , u\left({z}_{1},{z}_{2})=\frac{1}{2{a}_{1}}{e}^{Q\left({z}_{1},{z}_{2})}+\frac{1}{2{a}_{2}}{e}^{-Q\left({z}_{1},{z}_{2})}+{\eta }_{0}{e}^{\tfrac{1}{D}{[}\left({a}_{2}{c}_{1}-{a}_{1}{c}_{2}){z}_{1}+\left({a}_{1}{b}_{2}-{a}_{2}{b}_{1}){z}_{2}]}, where D = b 1 c 2 − b 2 c 1 D={b}_{1}{c}_{2}-{b}_{2}{c}_{1} , η 0 ∈ C − { 0 } {\eta }_{0}\in {\mathbb{C}}-\left\{0\right\} , and Q ( z 1 , z 2 ) = − 1 D [ ( a 1 c 2 + a 2 c 1 ) z 1 − ( a 1 b 2 + a 2 b 1 ) z 2 ] + η 1 , η 1 ∈ C . Q\left({z}_{1},{z}_{2})=-\frac{1}{D}\left[\left({a}_{1}{c}_{2}+{a}_{2}{c}_{1}){z}_{1}-\left({a}_{1}{b}_{2}+{a}_{2}{b}_{1}){z}_{2}]+{\eta }_{1},\hspace{1em}{\eta }_{1}\in {\mathbb{C}}. The description of the forms of the solutions for these PDEs demonstrates that our results are some improvements of the previous results given by Liu, Cao, and Xu [L. Xu and T. B. Cao, Solutions of complex Fermat-type partial difference and differential-difference equations, Mediterr. J. Math. 15 (2018), 227], and [K. Liu and T. B. Cao, Entire solutions of Fermat type difference differential equations, Electron. J. Diff. Equ. 2013 (2013), No. 59, 1–10.]. Meantime, we list some examples to explain that the forms of solutions of our theorems are precise to some extent.

This study is concerned with the stability of Nash equilibria for a class of n n -person noncooperative differential games. More precisely, due to a preorder induced by a convex cone on a real linear normed space, we define a new concept called ordering stability of equilibria against the perturbation of the right-hand side functions of state equations for the differential game. Moreover, using the set-valued analysis theory, we present the sufficient conditions of the ordering stability for such differential games.

In this article, a new reverse half-discrete Hilbert-type inequality with one partial sum involving one derivative function of higher order is obtained, by using the weight functions, the mid-value theorem, and the techniques of real analysis. A few equivalent statements of the best possible constant factor related to several parameters are considered. As applications, the equivalent forms and some particular inequalities are provided.

In this study, we comment about a wrong proof, at least incomplete, of the closed Newton Cotes error formulas for integration in a closed interval [ a , b ] . \left[a,b]. These error formulas appear as an intuitive generalization of the simple proof for the error formula of the trapezoidal rule, and their proofs present one controversial step, which converts the proofs in mischievous, or at least, this step needs a clear clarification that it is not easy to derive. The correct proof of such formulas comes from a technique based on the Peano kernel.

Let M 3 {M}^{3} be a strictly almost cosymplectic three-manifold whose Ricci operator is weakly ϕ \phi -invariant. In this article, it is proved that Ricci curvatures of M 3 {M}^{3} are invariant along the Reeb flow if and only if M 3 {M}^{3} is locally isometric to the Lie group E ( 1 , 1 ) E\left(1,1) of rigid motions of the Minkowski 2-space equipped with a left-invariant almost cosymplectic structure.

In this article, we investigate the concepts of monotonicity, Schur-geometric convexity, Schur-harmonic convexity, and Schur-power convexity for the lower and upper limits of the integral mean, focusing on convex functions on coordinate axes. Furthermore, we introduce novel and fascinating inequalities for binary means as a practical application.

In this article, the notions of ∼ \sim admissible congruences and ∼ \sim normal congruences on a weakly type B semigroup are characterized and the relationship between ∼ \sim admissible congruences and ∼ \sim normal congruences is investigated. In particular, some properties of such congruences on a weakly type B semigroup are given using an approach of kernel-trace. Finally, we extend the congruence pair on an inverse semigroup to the case of a weakly type B semigroup and obtain some results.

The focus of this article is to explore the Bohr inequality for a specific subset of harmonic starlike mappings introduced by Ghosh and Vasudevarao ( Some basic properties of certain subclass of harmonic univalent functions , Complex Var. Elliptic Equ. 63 (2018), no. 12, 1687–1703.). This set is denoted as ℬ H 0 ( M ) ≔ { f = h + g ¯ ∈ ℋ 0 : ∣ z h ″ ( z ) ∣ ≤ M − ∣ z g ″ ( z ) ∣ } {{\mathcal{ {\mathcal B} }}}_{H}^{0}\left(M):= \{f=h+\overline{g}\in {{\mathcal{ {\mathcal H} }}}_{0}:| z{h}^{^{\prime\prime} }\left(z)| \le M-| z{g}^{^{\prime\prime} }\left(z)| \} for z ∈ D z\in {\mathbb{D}} , where 0 < M ≤ 1 0\lt M\le 1 . It is worth mentioning that the functions belonging to the class ℬ H 0 ( M ) {{\mathcal{ {\mathcal B} }}}_{H}^{0}\left(M) are recognized for their stability as starlike harmonic mappings. With this in mind, this research has a twofold goal: first, to determine the optimal Bohr radius for this specific subclass of harmonic mappings, and second, to extend the Bohr-Rogosinski phenomenon to the same subclass.

For the first-order differential-difference equations of the form A ( z ) f ( z + 1 ) + B ( z ) f ′ ( z ) + C ( z ) f ( z ) = F ( z ) , A\left(z)f\left(z+1)+B\left(z)f^{\prime} \left(z)+C\left(z)f\left(z)=F\left(z), where A ( z ) , B ( z ) , C ( z ) A\left(z),B\left(z),C\left(z) , and F ( z ) F\left(z) are polynomials, the existence, growth, zeros, poles, and fixed points of their nonconstant meromorphic solutions are investigated. It is shown that all nonconstant meromorphic solutions are transcendental when deg B ( z ) < deg { A ( z ) + C ( z ) } + 1 {\rm{\deg }}B\left(z)\lt {\rm{\deg }}\left\{A\left(z)+C\left(z)\right\}+1 and all transcendental solutions are of order at least 1. For the finite-order transcendental solution f ( z ) f\left(z) , the relationship between ρ ( f ) \rho (f) and max { λ ( f ) , λ ( 1 ∕ f ) } \max \left\{\lambda (f),\lambda \left(1/f)\right\} is discussed. Some examples for sharpness of our results are provided.

In the present article, we consider a double-phase eigenvalue problem with large exponents. Let λ ( p n , q n ) 1 {\lambda }_{\left({p}_{n},{q}_{n})}^{1} be the first eigenvalues and u n {u}_{n} be the first eigenfunctions, normalized by ‖ u n ‖ ℋ n = 1 \Vert {u}_{n}{\Vert }_{{{\mathcal{ {\mathcal H} }}}_{n}}=1 . Under some assumptions on the exponents p n {p}_{n} and q n {q}_{n} , we show that λ ( p n , q n ) 1 {\lambda }_{\left({p}_{n},{q}_{n})}^{1} converges to Λ ∞ {\Lambda }_{\infty } and u n {u}_{n} converges to u ∞ {u}_{\infty } uniformly in the space C α ( Ω ) {C}^{\alpha }\left(\Omega ) , and u ∞ {u}_{\infty } is a nontrivial viscosity solution to a Dirichlet ∞ \infty -Laplacian problem.

Let G G be a graph. A perfect matching of G G is a regular spanning subgraph of degree one. Enumeration of perfect matchings of a (molecule) graph is interest in chemistry, physics, and mathematics. But the enumeration problem of perfect matchings for general graphs (even in bipartite graphs) is non-deterministic polynomial (NP)-hard. Xiao et al. [C. Xiao, H. Chen, L. Liu, Perfect matchings in random pentagonal chains , J. Math. Chem. 55 (2017), 1878–1886] have studied the problem of perfect matchings for random odd-polygonal chain (i.e., with odd polygons). In this article, we further present simple counting formulae for the expected value of the number of perfect matchings in random even-polygonal chains (i.e., with even polygons). Based on these formulae, we obtain the average values of the number for perfect matchings with respect to the set of all even-polygonal chains with n n polygons.

In this article, we define evolutoids and pedaloids of frontals on timelike surfaces in Minkowski 3-space. The evolutoids of frontals on timelike surfaces are not only the generalization of evolutoids of curves in the Minkowski plane but also the generalization of caustics in Minkowski 3-space. As an application of the singularity theory, we classify the singularities of evolutoids and reveal the relationships between the singularities and geometric invariants of frontals. Furthermore, we find that there exists a close connection between the pedaloids of frontals and the pedal surfaces of evolutoids. Finally, we give some examples to demonstrate the results.

In this study, by virtue of a derivative formula for the ratio of two differentiable functions and with aid of a monotonicity rule, the authors expand a logarithmic expression involving the cosine function into the Maclaurin power series in terms of specific determinants and prove a decreasing property of the ratio of two logarithmic expressions containing the cosine function.

In this article, we study the m {\mathfrak{m}} -WG ∘ {}^{\circ } inverse which presents a generalization of the m {\mathfrak{m}} -WG inverse in the Minkowski space. We first show the existence and the uniqueness of the generalized inverse. Then, we discuss several properties and characterizations of the m {\mathfrak{m}} -WG ∘ {}^{\circ } inverse by using the core-EP decomposition. Applying the generalized inverse, we obtain the solutions of some matrix equations in Minkowski space.

The Lord Shulman swelling porous thermo-elastic soil system with the presence of a distributed delay term is studied in this work. We will establish the well-posedness of the system and the exponential stability of the system is derived.

In this article, without assuming the compactness of semigroup, we deal with the existence and uniqueness of a mild solution for semilinear impulsive evolution equation with nonlocal condition in a reflexive Banach space by applying the approximate solvability method and Yosida approximations of the infinitesimal generator of C 0 -semigroup.

In this article, we consider the problem of extending Hamilton’s principle to the class of natural mechanical systems with random perturbing forces of white noise type. By the method of moment functions, we construct the functionals taking a stationary value on the solutions of a given stochastic equation of Lagrangian structure.

We have established the existence and uniqueness of the local solution for (0.1) ∂ t u + ∂ x 5 u − u ∂ x u = 0 , 0 < x < 1 , t > 0 , u ( x , 0 ) = φ ( x ) , 0 < x < 1 , u ( 0 , t ) = h 1 ( t ) , u ( 1 , t ) = h 2 ( t ) , ∂ x u ( 1 , t ) = h 3 ( t ) , ∂ x u ( 0 , t ) = h 4 ( t ) , ∂ x 2 u ( 1 , t ) = h 5 ( t ) , t > 0 , \left\{\begin{array}{ll}{\partial }_{t}u+{\partial }_{x}^{5}u-u{\partial }_{x}u=0,& 0\lt x\lt 1,\hspace{1.0em}t\gt 0,\\ u\left(x,0)=\varphi \left(x),& 0\lt x\lt 1,\\ u\left(0,t)={h}_{1}\left(t),u\left(1,t)={h}_{2}\left(t),\hspace{0.33em}{\partial }_{x}u\left(1,t)={h}_{3}\left(t),& \\ {\partial }_{x}u\left(0,t)={h}_{4}\left(t),\hspace{0.33em}{\partial }_{x}^{2}u\left(1,t)={h}_{5}\left(t),& t\gt 0,\end{array}\right. in the study of Zhao and Zhang [ Non-homogeneous boundary value problem of the fifth-order KdV equations posed on a bounded interval , J. Math. Anal. Appl. 470 (2019), 251–278]. A question arises naturally: Can the local solution be extended to a global one? This article will address this question. First, through a series of logical deductions, a global a priori estimate is established, and then the local solution is naturally extended to a global solution.

The main objective of this article is to study the ordered partial transformations PO ( X ) {\mathcal{PO}}\left(X) of a poset X X . The findings show that the set of all partial transformations of a poset with a pointwise order is not necessarily a pomonoid. Some conditions are implemented to guarantee that PO ( X ) {\mathcal{PO}}\left(X) is a pomonoid and this pomonoid is denoted by PO ↑ ( X ) {{\mathcal{PO}}}^{\uparrow }\left(X) . Moreover, we determine the necessary conditions in order that the partial order-embedding transformations define the ordered version of the symmetric inverse monoid. The findings show that this set is an inverse pomonoid and we will denote it by ℐPO ↑ ( X ) {{\mathcal{ {\mathcal I} PO}}}^{\uparrow }\left(X) . In case the order on the poset X X is total, we explore some properties of PO ↑ ( X ) {{\mathcal{PO}}}^{\uparrow }\left(X) and ℐPO ↑ ( X ) {{\mathcal{ {\mathcal I} PO}}}^{\uparrow }\left(X) , including regressive, unitary, and reversible.

In this article, we present new fractional integral inequalities via Euler’s beta function in terms of s s -convex mappings. We develop some new generalizations of fractional trapezoid- and midpoint-type inequalities using the class of differentiable s s -convexity. The results obtained in this study extended other related results reported in the literature.

In this article, we develop an efficient Legendre-Galerkin approximation based on a reduced-dimension scheme for the fourth-order equation with singular potential and simply supported plate (SSP) boundary conditions in a circular domain. First, we deduce the equivalent reduced-dimension scheme and essential pole condition associated with the original problem, based on which a class of weighted Sobolev spaces are defined and a weak formulation and its discrete scheme are also established for each reduced one-dimensional problem. Second, the existence and uniqueness of the weak solution and the approximation solutions are given using the Lax-Milgram theorem. Then, we construct a class of projection operators, give their approximation properties, and then prove the error estimates of the approximation solutions. In addition, we construct a set of effective basis functions in approximate space using orthogonal property of Legendre polynomials and derive the equivalent matrix form of the discrete scheme. Finally, a large number of numerical examples are performed, and the numerical results illustrate the validity and high accuracy of our algorithm.

Gaussian solitons are important examples in the theory of Riemannian measure space. In the first part, we explicitly characterize the first eigenfunctions of the drift Laplacian in a Gaussian shrinking soliton, which shows that apart from each coordinate function, other first eigenfunctions must involve exponential functions and the so-called error functions. Moreover, the second eigenfunctions are also described. In the second part, we discuss the corresponding issues in Finsler Gaussian shrinking solitons, which is a natural generalization of Gaussian shrinking solitons. For the first eigenfunction, we complement an example to show that if a coordinate function is a first eigenfunction, then the Finsler Gaussian shrinking soliton must be a Euclidean measure space. For the second eigenfunction, we give some necessary and sufficient conditions for these spaces to be Euclidean measure spaces.

In this article, we researched the existence of the solution to the fractional Navier-Stokes equations with the Coriolis force under initial data, which belong to the Lei-Lin-Gevrey spaces. Moreover, we showed a blow-up criterion, i.e., when the maximal time of existence T * {T}^{* } is finite, we proved that the norm of this same solution, in a specific Lei-Lin-Gevrey space, goes to infinity, as time tends to the maximal time of its existence.

In this article, the main aim is to consider the boundedness of the nonlinear commutator of p p -adic sharp maximal operator ℳ p ♯ {{\mathcal{ {\mathcal M} }}}_{p}^{\sharp } with symbols belonging to the p p -adic Lipschitz spaces in the context of the p p -adic version of (variable) Lebesgue spaces, by which some new characterizations of the Lipschitz spaces are obtained in the p p -adic field context.

Recently, regime-switching option pricing based on fractional diffusion models has been used, which explains many significant empirical facts about financial markets better. There are many methods to solve the problem, but to the best of our knowledge, effective preconditioners for the second-order schemes have not been proposed. Thus, in this article, an implicit numerical scheme is developed for a regime-switching European option pricing problem under a multi-state tempered fractional model. The scheme is proven to be unconditionally stable and converges quadratically in space and linearly in time. Besides, the resulting linear system is solved using an iterative method, and a preconditioner is proposed to accelerate the rate of convergence. The preconditioner is constructed through circulant approximations to the Toeplitz blocks due to the coefficient matrix, which is is a block matrix with Toeplitz blocks. The spectral analysis of the preconditioned matrix is given, which demonstrates that the spectrum of the preconditioned matrix is clustered around 1. Numerical examples show the efficiency of the proposed method, and an empirical study is also provided.

After introducing a graph connectedness induced by a given set of paths of the same length, we focus on the 2-adjacency graph on the digital line Z {\mathbb{Z}} with a certain set of paths of length n n for every positive integer n n . The connectedness in the strong product of three copies of the graph is used to define digital Jordan surfaces. These are obtained as polyhedral surfaces bounding the polyhedra that can be face-to-face tiled with digital tetrahedra.

In this article, we investigate a spherically symmetric backward heat conduction problem, starting from the final temperature. This problem is severely ill posed: the solution (if it exists) does not depend continuously on the final data. A conditional stability result of its solution is given. Further, we propose a quasi-boundary value regularization method to solve this ill-posed problem. Two Hölder type error estimates between the approximate solution and its exact solution are obtained under an a priori and an a posteriori regularization parameter choice rule, respectively.

One of the symbolic parameters to measure the fault tolerance of a network is its connectivity. The H H -structure connectivity and H H -substructure connectivity extend the classical connectivity and are more practical. For a graph G G and its connected subgraph H H , the H H -structure connectivity κ ( G ; H ) \kappa \left(G;\hspace{0.33em}H) (resp. H H -substructure connectivity κ s ( G ; H ) {\kappa }^{s}\left(G;\hspace{0.33em}H) ) of G G is the cardinality of a minimum subgraph set such that every element of the set is isomorphic to H H (resp. every element of the set is isomorphic to a connected subgraph of H H ) in G G , whose vertices removal disconnects G G . In this article, we investigate the H H -structure connectivity and H H -substructure connectivity of the n n -dimensional burnt pancake network BP n {{\rm{BP}}}_{n} for each H ∈ { K 1 , K 1 , 1 , … , K 1 , n − 1 , P 4 , … , P 7 , C 8 } H\in \left\{{K}_{1},{K}_{1,1},\ldots ,{K}_{1,n-1},{P}_{4},\ldots ,{P}_{7},{C}_{8}\right\} .

We compute an asymptotic formula for the divisor class numbers of real cubic function fields K m = k ( m 3 ) {K}_{m}=k\left(\sqrt[3]{m}) , where F q {{\mathbb{F}}}_{q} is a finite field with q q elements, q ≡ 1 ( mod 3 ) q\equiv 1\hspace{0.3em}\left(\mathrm{mod}\hspace{0.3em}3) , k ≔ F q ( T ) k:= {{\mathbb{F}}}_{q}\left(T) is the rational function field, and m ∈ F q [ T ] m\in {{\mathbb{F}}}_{q}\left[T] is a cube-free polynomial; in this case, the degree of m m is divisible by 3. For computation of our asymptotic formula, we find the average value of ∣ L ( s , χ ) ∣ 2 {| L\left(s,\chi )| }^{2} evaluated at s = 1 s=1 when χ \chi goes through the primitive cubic even Dirichlet characters of F q [ T ] {{\mathbb{F}}}_{q}\left[T] , where L ( s , χ ) L\left(s,\chi ) is the associated Dirichlet L L -function.

In this article, we study the uniqueness of exponential polynomials and mainly prove: Let n n be a positive integer, let p i ( z ) ( i = 1 , 2 , … , n ) {p}_{i}\left(z)\hspace{0.33em}\left(i=1,2,\ldots ,n) be nonzero polynomials, and let c i ≠ 0 ( i = 1 , 2 , … , n ) {c}_{i}\ne 0\hspace{0.33em}\left(i=1,2,\ldots ,n) be distinct finite complex numbers. Suppose that f ( z ) f\left(z) is an entire function, g ( z ) = p 1 ( z ) e c 1 z + p 2 ( z ) e c 2 z + ⋯ + p n ( z ) e c n z g\left(z)={p}_{1}\left(z){e}^{{c}_{1}z}+{p}_{2}\left(z){e}^{{c}_{2}z}+\cdots +{p}_{n}\left(z){e}^{{c}_{n}z} . If f ( z ) f\left(z) and g ( z ) g\left(z) share a a and b b CM (counting multiplicities), where a a and b b are two distinct finite complex numbers, then one of the following cases must occur: (i) n = 1 n=1 . If a ≠ 0 a\ne 0 , b = 0 b=0 , then either f ( z ) ≡ g ( z ) f\left(z)\equiv g\left(z) or f ( z ) g ( z ) ≡ a 2 f\left(z)g\left(z)\equiv {a}^{2} ; If a = 0 a=0 , b ≠ 0 b\ne 0 , then either f ( z ) ≡ g ( z ) f\left(z)\equiv g\left(z) or f ( z ) g ( z ) ≡ b 2 f\left(z)g\left(z)\equiv {b}^{2} ; If a ≠ 0 a\ne 0 , b ≠ 0 b\ne 0 , then either f ( z ) ≡ g ( z ) f\left(z)\equiv g\left(z) or f ( z ) g ( z ) ≡ ( a + b ) g ( z ) − a b f\left(z)g\left(z)\equiv \left(a+b)g\left(z)-ab . (ii) n ≥ 2 n\ge 2 , f ( z ) ≡ g ( z ) f\left(z)\equiv g\left(z) . This is an extension of the result obtained in an earlier study on meromorphic functions in 1974.

This study systematically develops error estimates tailored to a specific set of general quadrature rules that exclusively incorporate first derivatives. Moreover, it introduces refined versions of select generalized Ostrowski’s type inequalities, enhancing their applicability. Through the skillful application of Hayashi’s celebrated inequality to specific functions, the provided proofs underpin these advancements. Notably, this approach extends its utility to approximate integrals of real functions with bounded first derivatives. Remarkably, it employs Newton-Cotes and Gauss-Legendre quadrature rules, bypassing the need for stringent requirements on higher-order derivatives.