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The present paper deals with various recurrence relations, generating functions and series expansion formulas for two families of orthogonal polynomials in two variables, given Laguerre–Laguerre Koornwinder polynomials and Laguerre–Jacobi Koornwinder polynomials in the limit cases. Several families of bilinear and bilateral generating functions are derived. Furthermore, some special cases of the results presented in this study are indicated.

In this paper, we consider the boundary value problem for the one-dimensional Dirac system with spectral parameter in the boundary condition. We completely study the oscillatory properties of components of the eigenvector functions of this problem.

This paper considers properties of the spectrum of q-Sturm–Liouville operator derived from the q-Sturm–Liouville expression $$\begin{aligned} Ly:=-\frac{1}{q}D_{q^{-1}}\left( p(x)D_{q}y(x)\right) +r\left( x\right) y(x),\quad 0

This paper focuses on the multidimensional spherically symmetric full compressible Euler equations and constructs rigorously a family of global self-similar bounded weak solutions for all positive time to its initial value problem with constant initial data. The main approach is to reduce the full compressible Euler equations to an autonomous system of ordinary differential equations under the spherically symmetric and self-similar assumptions. We establish the detailed structures of solutions as well as their existence by analyzing carefully the properties of the integral curves of the autonomous ODE system.

In this paper, we find all Pell and Pell–Lucas numbers expressible as sums of two base 10 repdigits.

A doubleRoman dominating function of a digraph D is a function $$f:V\longrightarrow \{0,1,2,3\}$$ such that every vertex u for which $$f(u)=0$$ has an in-neighbor v for which $$f(v)=3$$ or at least two in-neighbors assigned 2 under f, while if $$f(u)=1$$, then the vertex u must have at least one in-neighbor assigned 2 or 3. The weight of a double Roman dominating function is the value $$f(V)=\sum _{u\in V}f(u)$$. The minimum weight of a double Roman dominating function of a digraph D is called the double Roman domination number of D, denoted by $$\gamma _{dR}\left( D\right) $$. This paper gives a descriptive characterization for some classes of digraphs satisfying $$\gamma _{dR}(D)=2\left( n-\Delta ^{+}(D)\right) +1$$. Also, a descriptive characterization for digraphs D of order $$n\ge 4$$ for which $$\gamma _{dR}(D)+\gamma _{dR}({\overline{D}})=2n+3$$ holds where $${\overline{D}}$$ is the complement of D.

Let $$G=(V,E)$$ be a graph with no isolated vertex. A subset $$S\subseteq V(G)$$ is a total dominating set of graph G if every vertex in V(G) is adjacent to at least one vertex in S. A total dominating set S of graph G is a locating-total dominating set if for every pair of distinct vertices $$u_1$$ and $$u_2$$ in $$V(G)-S$$, $$N(u_1)\cap S\ne N(u_2)\cap S$$. The locating-total domination number of graph G, denoted by $$\gamma _t^L(G)$$, is the minimum cardinality of a locating-total dominating set of G. In this paper, we investigate the bounds of locating-total domination number of grid graphs.

In this paper, we characterize the graded post-Lie algebra structures and a class of shifting post-Lie algebra structures on the Witt algebra. We obtain some new Lie algebras and give a class of their modules. As an application, the homogeneous Rota-Baxter operators and a class of non-homogeneous Rota-Baxter operators of weight $1$ on the Witt algebra are studied., Comment: 24 pages

Let G be a finite group. Let $$\psi ^{\prime }(G)= \prod _{g \in G} o(g), $$ where o(g) denotes the order of $$g \in G$$. In this paper, we prove that $$\mathrm{PSL}(2, 7)$$ and $$\mathrm{PSL}(2, 11)$$ are uniquely determined by their product of element orders. Furthermore, we prove that $$\mathrm{PSL}(2, 5)$$ and $$\mathrm{PSL}(2, 13)$$ are uniquely determined by their orders and the product of element orders.

In this paper, we study the following first-order Hamiltonian system $$\begin{aligned} \dot{z}=\mathscr {J}H_{z}(t,z). \end{aligned}$$Under some suitable conditions on the nonlinearity, we establish the existence of ground state homoclinic orbits by using variational methods. Moreover, we also explore some properties of these homoclinic orbits, such as compactness of set of and exponential decay of ground state homoclinic orbits.

In the present paper, we establish some fixed-point theorems for a $$2\times 2$$ block operator matrix involving multivalued maps acting on Banach spaces. These results are formulated in terms of weak sequential continuity and the technique of measures of weak noncompactness. The results obtained are then applied to a coupled system of nonlinear equations.

An entropy-stable residual distribution (RD) method is developed for the systems of two-dimensional shallow water equations (SWE). The construction of entropy stability for the residual distribution method is derived from finite volume method principles, albeit using a multidimensional approach. The paper delves into in-depth discussions on how finite volume methods achieve entropy stability in a “one-dimensional” sense for two-dimensional systems of SWE unlike the residual distribution methods. Results herein demonstrate the superiority of the entropy-stable RD methods relative to their finite volume counterparts, especially on highly irregular triangular grids. The comparative results with other established RD methods are also included, depicting similar performances with the Lax–Wendroff method for unsteady smooth flows but more accurate than the multidimensional upwind approaches (N, LDA) on both smooth and discontinuous test cases.

A strong edge-coloring of a graph G is a proper edge-coloring such that every path of length 3 uses three different colors. The strong chromatic index of a graph G, denoted by $$\chi _{s}^\prime (G)$$, is the minimum number of colors needed for a strong edge-coloring of G. In this paper, we show that $$\chi _{s}^\prime (G)\le 3\Delta -2$$ for any pseudo-Halin graph G with $$\Delta \ge 4$$.

In the present paper, we introduce a new kind of Riemannian submersion such that the fibers of such submersion are generic submanifolds in the sense of Ronsse that we call generic submersion. Some examples are given for generic submersion. Necessary and sufficient conditions are found for the integrability and totally geodesicness of the distributions which are mentioned in the definition. The geometry of the fibers is investigated. New results are obtained by considering the parallelism condition of canonical structures.

In this paper, the Cauchy problem associated with the biharmonic equation is investigated. We prove that in principle, the problem is severely ill-posed in the sense of Hadamard. Therefore, we propose a quasi-boundary value-type regularization method for stabilizing the ill-posed problem. Very sharp convergence estimates are established based on some a priori information on the exact solution. Finally, several numerical examples with random data are provided to show the effectiveness of the proposed method.

Let $$F_q$$ be a finite field of q elements, $$\mathbb {V}$$ an n-dimensional vector space over $$F_q$$, and $$\mathbb {V}^*$$ the dual space of $$\mathbb {V}$$, i.e., the vector space of all linear function over $$\mathbb {V}$$. The graph $$\hbox {DG}(\mathbb {V})$$, called the dual graph of $$\mathbb {V}$$, is defined to be a bipartite graph, whose vertex set is partitioned into two coloring sets, respectively, consisting of all one-dimensional subspaces of $$\mathbb {V}$$ and all one-dimensional subspaces of $$\mathbb {V}^*$$, and there is an undirected edge between an one-dimensional subspace [v] of $$\mathbb {V}$$ and an one-dimensional subspace [f] of $$\mathbb {V}^*$$ if and only if $$f(v) = 0$$. In this paper, the domination number, independence number, diameter and girth of $$\hbox {DG}(\mathbb {V})$$ are, respectively, determined; some automorphisms of $$\hbox {DG}(\mathbb {V})$$ are introduced, and such a graph is proved to be distance transitive.

In this paper, a class of nonlinear viscoelastic Kirchhoff equation in a bounded domain with a time-varying delay in the weakly nonlinear internal feedback is considered, where the global existence of solutions in suitable Sobolev spaces by means of the energy method combined with Faedo–Galerkin procedure is proved with respect to the condition of the weight of the delay term in the feedback and the weight of the term without delay and the speed of delay. Furthermore, a general stability estimate using some properties of convex functions is given.

The revised Szeged index of a connected graph G is defined as $$\begin{aligned} Sz^*(G)=\sum _{e=uv\in E(G)}\left( n_u(e|G)+{\frac{n_0(e|G)}{2}}\right) \left( n_v(e|G)+{\frac{n_0(e|G)}{2}}\right) , \end{aligned}$$where E(G) is the edge set of G, and for any $$e = uv\in E(G)$$, $$n_u(e|G)$$ is the number of vertices of G lying closer to vertex u than to v, $$n_v(e|G)$$ is the number of vertices of G lying closer to vertex v than to u, and $$n_0(e|G)$$ is the number of vertices with equal distances from both end vertices of the edge e. In this paper, we characterize the graph with the minimum revised Szeged index among all the unicyclic graphs with given order and diameter.

In this paper, we consider the regularity criteria, where a condition is added only on the velocity field to guarantee the smoothness of whole unknowns, and the global existence of the solutions with small initial data for the 3D tropical climate model with diffusion.

In this paper, the boundedness problem of factorizable four-dimensional matrices on the space of double sequences is investigated. As an application of our results, the lower bounds and the operator norms of four-dimensional Cesaro matrix and four-dimensional Copson matrix are obtained, which provide an extension of Hardy’s discrete inequality and Copson’s discrete inequality to double series, respectively. Finally, we present complementary results for the operator norm and lower bound of the four-dimensional Hausdorff matrices.

In this paper, nonlinear Fredholm integral equation of the second kind is solved by using parameter continuation method. Then, we propose parameter continuation method to solve perturbed nonlinear Fredholm integral equation of the second kind, which appear as an extension of the method of contractive mapping and parameter continuation method for solving nonlinear Fredholm integral equation of the second kind. Illustrative examples are presented to show the effectiveness and convenience of parameter continuation method.

In this paper, a mathematical model is formulated to investigate the dynamics of HIV infection. The model incorporates two routes of infection (namely cell-to-cell transmission and virus-to-cell infection), two types of adaptive immune responses (i.e., cellular and antibody immune response) and two intracellular delays (viz., the eclipse phase and virus production period). By constructing Lyapunov functionals, we show that the global dynamics of the model can be explicitly determined by five reproduction numbers. Using the uncertainty and sensitivity analyses, we obtain the mean values and standard deviation of the five reproduction numbers and find that the reproduction numbers are most sensitive to the death rate of infected cells, viral clearance rate and immune parameters. We further compare four related HIV models and show that cell-to-cell transmission, immune responses and time delays can substantially affect the dynamical behavior of the system. Specifically, cell-to-cell transmission increases the concentration of infected cells. Inclusion of cytotoxic T lymphocyte and virus production delay can significantly reduce the viral load and infected cell concentration, and generate a higher level of uninfected CD4+ T cells. The analytical and numerical results may help to improve the understanding of HIV dynamics.

Prime partitions are partitions of integers into prime parts. In this paper, we first consider prime partitions with distinct parts. By using generating functions, we obtain some inductive formulas to calculate the number of prime partitions with distinct parts. Our formulas give two generalizations of the Euler’s formula for the integer partition case. Then, we consider general prime partitions with not necessarily distinct parts. By keeping track of the recurrence of primes in a partition and finding bijections between different prime partitions, we get some inductive formulas to calculate the number of general prime partitions. Finally, by numerical experimentation we find an approximation of some analytical formulas for the number of general prime partitions.

In this paper, we study sets of positive integers with the property that the product of any two elements in the set increased by the unity is a square. It is shown that if the two smallest elements have the form $${ KA}^2$$, $$4{ KA}^4 \pm 4 A$$ for some positive integers A and K, and the third one is chosen canonically, then any such set consisting of three elements can be contained in a unique such set with four elements.

In this paper, we consider some regular boundary value problems generated by a third-order differential equation and some boundary conditions. In particular, we construct maximal self-adjoint, maximal dissipative and maximal accumulative extensions of the minimal operator. Further using Lax–Phillips scattering theory and Sz.-Nagy–Foias characteristic function theory we prove a completeness theorem.

In 2014, J. Pipan and M. Schechter discussed periodic solutions for second-order Hamiltonian systems with resonance. In this paper, we will generalize their results. To this end, we will first establish an index theory for second-order linear Hamiltonian systems with coefficient matrix in $$L^1$$. Then we propose generalized assumptions. We also investigate multiple solutions for symmetric second-order Hamiltonian systems.

A single-cone tree (unicyclic graph) is the join of a complete graph $$K_1$$ and a tree (unicyclic graph). Suppose $$\pi =(d_1, d_2, \ldots , d_n)$$ and $$\pi ^{\,\prime }=(d_1^{\,\prime }, d_2^{\,\prime }, \ldots , d_n^{\,\prime })$$ are two non-increasing degree sequences. We say $$\pi $$ is majorizated by $$\pi ^{\,\prime }$$, denoted by $$\pi \lhd \pi ^{\,\prime }$$, if and only if $$\pi \ne \pi ^{\,\prime }$$, $$\sum \nolimits _{i=1}^{n} d_i=\sum \nolimits _{i=1}^{n} d_i^{\,^{\,\prime }}$$, and $$\sum \nolimits _{i=1}^j d_i\le \sum \nolimits _{i=1}^j d_i^{\,^{\,\prime }}$$ for all $$j=1, 2, \ldots , n-1$$. We use $$J_{\pi }$$ to denote the class of single-cone trees (unicyclic graphs) with degree sequence $$\pi $$. Suppose that $$\pi $$ and $$\pi ^{\,\prime }$$ are two different non-increasing degree sequences of single-cone trees (unicyclic graphs). Let $$\rho $$ and $$\rho ^{\,\prime }$$ be the largest spectral radius of the graphs in $$J_{\pi }$$ and $$J_{\pi ^{\,\prime }}$$, respectively, $$\mu $$ and $$\mu ^{\,\prime }$$ be the largest signless Laplacian spectral radius of the graphs in $$J_{\pi }$$ and $$J_{\pi ^{\,\prime }}$$, respectively. In this paper, we prove that if $$\pi \lhd \pi ^{\,\prime }$$, then $$\rho

In this paper, we found the sharp bound of the fifth coefficient of strongly starlike functions of order $$\alpha $$ with real coefficients for all $$\alpha \in (0,1]$$ .

In the present paper, the estimate of the third Hankel determinant $$\begin{aligned} \begin{aligned} H_{3,1}(f)&= \begin{vmatrix} a_{1}&a_{2}&a_{3} \\ a_{2}&a_{3}&a_{4} \\ a_{3}&a_{4}&a_{5} \end{vmatrix} \end{aligned} \end{aligned}$$ for the class of starlike functions, i.e., for the class of analytic functions f standardly normalized such that $${{\mathrm{Re}}}(zf'(z)/f(z)) > 0,\ z\in {{\mathbb {D}}}:=\{z \in {\mathbb {C}} : |z

In this paper, we attempted to characterize the exponential q-distribution through the q-memorylessness property using the q-addition operator and Jackson integral. Moreover, an extended version of k-gamma q-distribution is introduced and the q-moments of this family is computed. Finally, we suggested a new q-inversion method to simulate data from a q-distribution.

In this paper, for the IBVP of a fourth-order nonlinear parabolic equation, which is related to image analysis, we studied the existence and uniqueness of weak solutions. Moreover, we also considered the asymptotic behavior and the regularity of solutions of such problem.

In this paper, we show the existence of nontrivial ground state solutions of fractional p-Kirchhoff problem $$\begin{aligned} \left\{ \begin{array}{ll} m\left( \int _{{\mathbb {R}}^{2N}}\frac{|u(x)-u(y)|^{p}}{|x-y|^{N+sp}}\mathrm{d}x\mathrm{d}y\right) (-\Delta )_p^s u=f(x,u) ~&{}\text {in}~\Omega , \\ u=0 ~&{}\text {in}~{\mathbb {R}}^N{\setminus } \Omega , \end{array}\right. \end{aligned}$$where $$(-\Delta )_p^s$$ is the fractional p-Laplacian operator with $$0

Let G be a graph and $$v\in V(G)$$. The neighbors of v, denoted by N(v), are the set of vertices adjacent to v. Let $$N[v]=N(v)\cup \{v\}$$ and $$J(u,v)=\{w\in N(u)\cap N(v):N(w)\subseteq N[u]\cup N[v]\}$$. A graph G is called quasi-claw-free if $$J(u,v)\ne \emptyset $$ for any $$u,v\in V(G)$$ with $$d(u,v)=2$$. In this paper, we show that if G is a connected quasi-claw-free graph with $$\sigma _{k+1}(G)\ge |G|-k,$$ then G contains a spanning k-ended tree, which generalizes some known results.

On a real hypersurface M in a non-flat complex space form, two types of connections can be defined: the Levi-Civita connection, which is torsion free and, for any nonnull constant k, the kth generalized Tanaka-Webster connection, which is an affine connection with torsion. So the associated covariant derivatives can be considered. Moreover, the Lie derivative and a derivative of Lie type associated with the kth generalized Tanaka-Webster connection can be defined. In this paper, real hypersurfaces for which both covariant derivatives or Lie derivative and the derivative of Lie type associated with the kth generalized Tanaka-Webster connection coincide when they act on the operator $$\phi A-A\phi $$ , where $$\phi $$ denotes the structure operator and A the shape operator of M, either in the direction of the structure vector field $$\xi $$ or in any direction of the maximal holomorphic distribution of M are classified.

In this paper, we consider the Cauchy problem to the 2D generalized tropical climate model. We establish a new blow-up criterion only in terms of the first baroclinic mode of the vector velocity in Besov spaces with negative indices.

In this paper, we establish a new blow-up criterion for the two-component Dullin–Gottwald–Holm system. We show that strong solutions must blowup in finite time if the integral concerning initial momentum changes sign at some point $$x_0\in {\mathbb {R}}$$.

In this paper, we consider two kinds of nonlinear matrix equations $$X+ \sum _{i=1}^{m}B_{i}^*X^{t_{i}}B_{i}=I\;(01,\;s\ge 1)$$ . By means of the integral representation of matrix functions, properties of Kronecker product and the monotonic p-concave operator fixed point theorem, we derive necessary conditions and sufficient conditions for the existence and uniqueness of the Hermitian positive definite solution for the matrix equations. We also obtain some properties of the Hermitian positive definite solutions, the bounds of the determinant’s sum for $$A_{i}^{*}A_{i}$$ and the spectral radius of $$A_{i}$$ .

In this paper, we study some basic properties of Garding’s cones and k-convex cones. Inclusion relations of these cones are established in lower-dimensional cases ($$n=2, 3, 4$$) and higher-dimensional cases ($$n\ge 5$$), respectively. Admissibility and ellipticity of several differential operators defined on such cones are discussed.

The present paper is concerned with the existence of solutions, in infinite- dimensional Banach spaces, for impulsive differential inclusions and equations of order $$q\in (1,2)$$, with anti-periodic conditions and involving the Caputo derivative whether in the generalized sense (via the Riemann–Liouville fractional derivative,) or in the normal sense and whether the lower limit on each impulsive subinterval $$ (t_{i} ,t_{i+1}],\, i=0,1,\ldots ,m$$ is keep at zero or is set at $$t_{i}$$. Using the technique of fixed point and the properties of the measure of noncompactness, existence results are obtained. Moreover, we derive in reflexive Banach spaces, by using a new version weakly convergent criteria in the space of piecewise continuous functions, an existence result of solutions without assuming any condition on the multivalued function in terms of measure of noncompactness. Some examples will be given to illustrate the obtained results.

In this paper, we present a new fractional Tikhonov regularization method for solving an inverse problem for a time-fractional diffusion equation which is highly ill-posed in the two-dimensional setting. Fractional Tikhonov regularization method not only retains the advantage of classical Tikhonov method, but also overcomes the effect of over-smoothing of classical Tikhonov method. We give the selection of regularization parameters of the new method and the corresponding error estimation. Furthermore, numerical results show that the fractional Tikhonov method outperforms the classical one.

In this paper, we present a construction to extend number system by using the vector generators of the matrix algebra $$L_{0,n}(\mathbb {R})$$ that is isomorphic with the Clifford algebra $$C\ell _{0,n}$$. Moreover, we investigate the existence of solutions for some linear equations in $$C\ell _{0, 3}$$ by means of real matrix representation.

In this paper, we study canal surfaces in the Euclidean 3-space $$\mathbb {E}^{3}$$ in terms of their Gauss map $$\mathbb {G}$$ . We obtain a complete classification of canal surfaces whose Gauss maps are of the so-called pointwise 1-type, i.e., the Gauss map $$\mathbb {G}$$ satisfies $$\Delta \mathbb {G}=f(\mathbb {G}+C)$$ for a nonzero smooth function f and a constant vector C, where $$\Delta $$ denotes the Laplace operator.

Let G be a finite group and c(G) denote the number of cyclic subgroups of G. The group G is called an n-cyclic group if $$c(G) = n$$ . In an earlier paper, finite n-cyclic groups with $$n \le 8$$ are classified and a characterization of the alternating group on five symbols based on the number of cyclic subgroups is given. The aim of this article is to continue this work by presenting a characterization of the simple group PSL(2, 7), by the number of cyclic subgroups. It is also proved that G is a 9-cyclic group if and only if $$G \cong D_{14}$$ , $$Z_5:Z_4$$ , $$Z_7:Z_3$$ , $$Z_3:Z_8$$ , $$Z_7 \times Z_7$$ , $$Z_{p^2q^2}$$ and $$Z_{p^8}$$ , where p and q are different primes, and G is 10-cyclic group if and only if $$G \cong D_{12}$$ , $$SD_{16}$$ , $$Z_4:Z_4$$ , $$Z_4 \times Z_4$$ , $$Z_{16}:Z_2$$ , $$Z_{16} \times Z_2$$ , $$Z_2 \times Q_8$$ , $$Z_{3r} \times Z_3$$ , $$Z_r \times S_3$$ , $$Z_p \times Q_8$$ and $$Z_{p^4q}$$ , where p, q, r are primes and $$r \ne 3$$ .

In this paper, we present some inequalities for Riemann–Liouville and Caputo q-fractional derivatives related to the q-Leibniz rule. We used one of the inequalities to prove the maximum principle for a time fractional q-Cauchy differential equation.

In this paper, we introduce graphic contraction mappings in b-metric spaces and establish some fixed point theorems in the aforesaid space. Appropriate graphs are presented to visualize related definitions and concepts. On the other hand, some examples are presented to verify the authenticity and effectiveness of the established results. Our results generalize and extend some pioneer results existing in the literature. Moreover, applications to existence of solution for ordinary differential equations and for integral equations are also established to show the adequacy of our results.

This paper deals with the Ulam stability of linear differential equations by using the method of variation of parameters, which provides a unified method to study the Ulam stability problem of linear differential equations of n-th order with constant and nonconstant coefficients. As an application of the main results, we also obtain the Hyers–Ulam stability of the Cauchy–Euler differential equations of second order, third order and n-th order. Our results make up to some deficiencies in the relevant literature.

In this paper, we prove the following result. Let $$m\ge 1,n\ge 1$$ be some fixed integers and let R be a prime ring with $$\mathrm{char}(R)=0$$ or $$(m+n)^2

In this paper, we investigate the flow through a thin corrugated domain filled with fluid-saturated porous medium. The porous medium flow is described by the nonlinear Darcy–Lapwood–Brinkman model acknowledging the viscous shear and the inertial effects. The thickness of the domain is assumed to be of the same small order $$\varepsilon $$ as the period of the oscillating boundaries. Depending on the magnitude of the permeability with respect to $$\varepsilon $$ , we rigorously derive different asymptotic models and compare the results with the non-oscillatory case. We employ a homogenization technique based on the adaption of the unfolding method and deduce the influence of the porous structure and boundary oscillations on the effective flow.

In this paper, we begin a systematic study of modified Rota–Baxter algebras, as an associative analogue of the modified classical Yang–Baxter equation. We construct free commutative modified Rota–Baxter algebras by a variation of the shuffle product and describe the structure both recursively and explicitly. We then provide these algebras with a Hopf algebra structure by applying a Hochschild cocycle.

In this paper, we investigate weaving g-frames and weaving fusion frames and we show that they share many properties with weaving frames. We show that they are stable under invertible operators and small perturbations. We also show that woven g-frames and weakly woven g-frames coincide. Also some more results are obtained.