We present a modification of the Kantorovich variant of the operators due to Lupaş and Lupaş which are connected with Pólya distribution and we provide a quantitative Voronovskaya-kind theorem involving modulus of continuity. Also, we observe that a better approximation can be achieved for our modified form of operators. We also estimate the difference of the original operator with its modified form. Lastly, we depict the convergence of the modified operators to some functions using graphical representation.

We use properties of the Gaussian hypergeometric function to prove the following identities for combinatorial polynomials:

$${\displaystyle \sum _{j=0}^n\left(\begin{array}{c}n+\alpha \\ j\end{array}\right)\left(\begin{array}{c}n+\beta \\ n-j\end{array}\right)z^j=\left(\begin{array}{c}n+\alpha \\ n\end{array}\right)}{\displaystyle \sum _{j=0}^n\left(\begin{array}{c}n\\ j\end{array}\right)\frac{\left(\begin{array}{c}n+j+\alpha +\beta \\ j\end{array}\right)}{\left(\begin{array}{c}j+\alpha \\ j\end{array}\right)}{(z-1)}^{n-j}}$$

and

$$\begin{array}{c}m\left(\begin{array}{c}m+n\\ m\end{array}\right){(1-z)}^n{\displaystyle \sum _{k=0}^n\frac{\left(\begin{array}{c}n\\ k\end{array}\right)}{m+k}{\left(\frac{z}{1-z}\right)}^k}-{\displaystyle \sum _{k=0}^n\left(\begin{array}{c}m+n\\ k\end{array}\right){(-z)}^k}={\displaystyle \sum _{k=0}^n\left(\begin{array}{c}m+n\\ k\end{array}\right){(-z)}^{n-k}}-{\displaystyle \sum _{k=0}^n\left(\begin{array}{c}m+n\\ n-k\end{array}\right){(-z)}^{n-k}}.\end{array}$$

These formulas extend two combinatorial identities published by Brereton et al. in 2011.

We give an improved algorithm for the action of the divided power of a Chevalley basis element of an affine Lie algebra of type $A$ on canonical basis elements satisfying an easily checked uniformity condition and compare calculation times for our algorithm against the standard algorithm. For symmetric Kashiwara crystals of affine type $A$ and rank $e=2$, and for the canonical basis elements that we call external, corresponding to weights on the outer skin of the Kashiwara crystal, we construct the canonical basis elements in a nonrecursive manner. In particular, for a symmetric crystal with $\mathrm{\Lambda }=a{\mathrm{\Lambda }}_0+a{\mathrm{\Lambda }}_1$, we give formulae for the canonical basis elements for all the $e$-regular multipartitions with defects either $k(a-k)$ or $k(a-k)+2a$, for $0\le k\le a$.

Let $N_{g,n}$ denote a nonorientable surface of genus $g$ with $n$ punctures and one boundary component. We give an algorithm to calculate the geometric intersection number of an arbitrary multicurve $\mathcal{ℒ}$ with so-called relaxed curves in $N_{g,n}$ making use of measured ${\pi }_1$-train tracks. The algorithm proceeds by the repeated application of three moves which take as input the measures of $\mathcal{ℒ}$ and produces as output a multicurve ${\mathcal{ℒ}}^{\prime }$ which is minimal with respect to each of the relaxed curves. The last step of the algorithm calculates the number of intersections between ${\mathcal{ℒ}}^{\prime }$ and the relaxed curves.

Our main goal is to investigate the finiteness of Ext-indices of ring extensions. We discuss some known related conjectures in the literature and observe the relationships among them within large classes of rings. This allows us to present interesting special cases verifying these conjectures. Finally, we tackle questions on the vanishing of Ext-index of trivial extensions of Artinian rings. We put the new results into use to construct new families of examples subject to the Gorenstein or Cohen–Macaulay conditions with finite Ext-index.

We investigate the nonlinear Langevin equation regarding the fractional derivative of a function depending on another function $\psi$ with generalized nonlocal conditions. We assume that the source function of the problem may have a singularity that appears from the discontinuity of the source function at $t=a$. We construct a new formula solution for the problem via the Mittag-Leffler function with two parameters. Based on the obtained formula solution, we propose some suitable conditions such that the problem has at least one or has a unique mild solution. The desired results are proved by applying some well-known fixed point theorems such as the Schaefer, nonlinear Leray–Schauder alternatives and Banach. Furthermore, we discuss the continuous dependence of mild solutions of the problem on the inputs (fractional orders, friction constant, appropriate function and nonlocal conditions) from which we deduce that the solution of the Langevin equation with Caputo–Katugampola fractional derivative ($\psi (t)=(t^{\rho }-1)∕\rho$) converges to the solution of the Langevin equation with Hadamard fractional derivative ($\psi (t)=\text{ln}t$) as $\rho \to 0^{+}$. Finally, two examples are given to illustrate our theoretical findings.

We show that if $C_p(X\times Z)$ is homeomorphic to $C_p(Y\times Z)$, where $Z$ is compact, and $X$ and $Y$ are of countable netweight, then $C_p(X\times M)$ is homeomorphic to $C_p(Y\times M)$ for some metric compactum $M$. Spaces $X,Y,Z$, and $M$ are assumed nonempty.

Based on the framework of linked partition ideals, we derive some double and triple series of Andrews–Gordon type for partitions in Alladi and Gordon’s extension of Schur’s identity. We also display similar series for such partitions with additional restrictions on the smallest part. Also, an alternative proof of Alladi and Gordon’s extension of Schur’s identity is presented.

We study averages along the integers using the divisor function $d(n)$, defined as

$$K_{N}f(x)=\frac{1}{D(N)}{\displaystyle \sum _{n\le N}d(n)f(x+n)},$$

where $D(N)={\sum }_{n=1}^{N}d(n)$. We shall show that these averages satisfy a uniform, scale free ${\ell }^p$-improving estimate for $p\in (1,2)$, that is

as long as $f$ is supported on $[0,N]$.

We will also show that the associated maximal function $K^{\ast }f={\text{sup}}_N|K_{N}f|$ satisfies $(p,p)$ sparse bounds for $p\in (1,2)$, which implies that $K^{\ast }$ is bounded on ${\ell }^p(w)$ for $p\in (1,\infty )$, for all weights $w$ in the Muckenhoupt $A_p$ class.

We consider the poset $\mathcal{𝒬}(P)$ of all nonempty compact saturated subsets of the Scott space of a poset $P$, equipped with the reverse inclusion order. We also introduce the notion of $T$-lattices, that is, a complete lattice $L$ is called a $T$-lattice if for any $x\in L\setminus \{1_L\}$, $\uparrow x\setminus \{x\}\in \mathcal{𝒬}(L)$. We prove that if $P$ and $Q$ are quasicontinuous domains or $T$-lattices, $P$ is order isomorphic to $Q$ if and only if $(\mathcal{𝒬}(P),\supseteq )$ is order isomorphic to $(\mathcal{𝒬}(Q),\supseteq )$. We also prove that for a compact space $X$, if the Scott space of the open set lattice $\mathcal{𝒪}(X)$ is sober, then $X$ is $SI$-compact. Using this result and the Isbell example of a non-sober complete lattice, we present a compact sober space which is not $SI$-compact. This gives a positive answer to an open problem posed by Zhao and Ho.

By the weighted arithmetic-geometric mean inequality we present multiple-term refinements of one of the most important extensions to Young’s inequalities due to Alzer, Fonseca and Kovačec (Linear Multilinear Algebra 63(3) (2015), 622–635). As applications, we give some related inequalities for operators and matrices.

We devote this article to the Calderón-Zygmund estimate for the minimizers of asymptotically regular variational integrals with multi-phase integrands. A local gradient $L^q$-estimate for such problem under a sharp regular assumption is obtained by approximating the solutions of asymptotically regular problems to the solutions of regular problems based on a new perturbation method, while the gradients of solutions close to infinity.

We study the 2D incompressible nematic liquid crystal equations in a smooth bounded domain, where the velocity $u$ and macroscopic molecular orientation $d$ admit the Dirichlet and Neumann boundary condition, respectively. Under a geometric condition for the initial orientation field, we establish the global existence of strong solutions with large initial data. In particular, the initial density can be allowed to vanish. Furthermore, this result extends the corresponding results of J. Li (2015) and X. Li (2017) to the Neumann boundary condition and removes the smallness conditions on the initial data.

Let $\omega$ be a weight function on a locally compact group $\mathcal{𝒢}$ and let $M{(\mathcal{𝒢},\omega )}_0^{\ast }$ be the subspace of $M{(\mathcal{𝒢},\omega )}^{\ast }$ consisting of all functionals that vanish at infinity. We first introduce an Arens product on ${(M{(\mathcal{𝒢},\omega )}_0^{\ast })}^{\ast }$ under which it is a Banach algebra. We then show that the existence of a nonzero compact right multiplier on ${(M{(\mathcal{𝒢},\omega )}_0^{\ast })}^{\ast }$ is equivalent to compactness of $\mathcal{𝒢}$. We also prove that if ${(M{(\mathcal{𝒢},\omega )}_0^{\ast })}^{\ast }$ is Arens regular, then $\mathcal{𝒢}$ is discrete.

We consider the asymptotic behavior of solutions on the half-line to nonlocal fractional differential equations of the form $k\ast u^{\prime }=Au(t)+f(t,u(t))$, where $k\ast u^{\prime }$ stands for the nonlocal derivative of $u$ in Caputo’s sense corresponding to a singular kernel $k$, $A$ is a positively definite, selfadjoint operator; the nonlinearity $f$ is either locally Lipschitz continuous with respect to the second variable or of sublinear growth for small time and Lipschitz continuous for large time. Our main results show that if $f$ is an asymptotically almost periodic function on $t$ then the problem possesses asymptotically almost periodic solutions.

The volume of a line bundle is defined in terms of a limsup. It is a fundamental question whether this limsup is a limit. It has been shown that this is always the case on generically reduced schemes. We show that volumes are limits in two classes of schemes that are not necessarily generically reduced: codimension one subschemes of projective varieties such that their components of maximal dimension contain normal points and projective schemes whose nilradical squared equals zero.

Inspired by the recent work of Guo, we establish some new $q$-supercongruences including $q$-analogues of some Ramanujan-type supercongruences, by using the Bailey transformation formula and the “creative microscoping” method recently introduced by Guo and Zudilin.

Let $F(z)$ be any function. Suppose that $w$ is a fixed point of $F(z)$, that is, $F(w)=w$. Then the recurrence equation

$$x_{n+1}=F(x_n)$$

for $n=0,1,2,\dots$ has a solution of the form

$$x_n(w)=w+\sum _{i=1}^{\infty }{a}_1^i{A}_i{F}_{•1}{(w)}^{in},$$

where $F_{•1}(z)=dF(z)∕dz$. So, for each $w$ there is a set of complex $x_0$ such that $x_0(w)=x_0$. We assume that $F(z)$ is analytic at $w$. This solution appears to be new, even for such famous examples like the logistic map and the Mandelbrot equation.

We explore the existence and the forms of entire and meromorphic solutions for the partial differential-difference equations with more general forms of

$${\left(\alpha \frac{\partial f(z_1,z_2)}{\partial z_1}+\beta \frac{\partial f(z_1,z_2)}{\partial z_2}\right)}^2+f{(z_1+c_1,z_2+c_2)}^2=e^{g(z_1,z_2)}$$

and

$${\left(\alpha \frac{\partial f(z_1,z_2)}{\partial z_1}+\beta \frac{\partial f(z_1,z_2)}{\partial z_2}\right)}^2+{[f(z_1+c_1,z_2+c_2)-f(z_1,z_2)]}^2=e^{g(z_1,z_2)},$$

where $g(z_1,z_2)$ is a polynomial in ${ℂ}^2$ and $\alpha ,\beta$ are constants in $ℂ$. Some of the results about the forms of solutions for these equations that are obtained are great improvements over previous results. It is important that some of the examples show that there exist some significant differences in the forms of transcendental entire solutions of finite order of the equations between several complex variables and a single complex variable.

We introduce the relative Gorenstein defect category of an abelian category with respect to an admissible subcategory, generalizing the Gorenstein defect categories of P. A. Bergh, D. Jorgensen and S. Oppermann. Under a mild condition of the precovering property for the relative Gorenstein category, we show that the relative Gorenstein defect category is triangle equivalent to the relative singularity category with respect to the relative Gorenstein category.

We also introduce relative Ding projective defect categories and, under a similar condition, relate it to the relative singularity category with respect to the relative Ding projective category. Analogous results for relative Ding injective defect categories are also presented.

Here, we are concerned with the positive continuous entire solutions of the Wolff-type integral system

where $n\ge 1$, , and $\beta \gamma \ne n$. In addition, $C_i(x)(i=1,2)$ are some double bounded functions. When $\beta \gamma \in (0,n)$, the Serrin-type condition is critical for existence of positive solutions for some double bounded functions $C_i(x)$ $(i=1,2)$. Such an integral equation system is related to the study of the $\gamma$-Laplace system and $k$-Hessian system with negative exponents. Estimated by the integral of the Wolff type potential, we obtain the asymptotic rates and the integrability of positive solutions, and study whether the radial solutions exist.

Let $\mathcal{𝒱},\mathcal{𝒲},\mathcal{𝒴},\mathcal{𝒳}$ be four classes of modules. We introduce a notion of $(\mathcal{𝒱},\mathcal{𝒲},\mathcal{𝒴},\mathcal{𝒳})$-Gorenstein modules which unifies a number of well-known Gorenstein modules. Under certain mild assumptions on the classes $\mathcal{𝒱},\mathcal{𝒲},\mathcal{𝒴},\mathcal{𝒳}$, we first study properties of $(\mathcal{𝒱},\mathcal{𝒲},\mathcal{𝒴},\mathcal{𝒳})$-Gorenstein modules. Then, we apply the obtained results to establish the Foxby equivalences associate to $(\mathcal{𝒱},\mathcal{𝒲},\mathcal{𝒴},\mathcal{𝒳})$-Gorenstein modules. Many results on the Gorenstein modules involved in this general setting can be obtained as corollaries.

For a connected graph $G$ of order $n$, the harmonic index of $G$ is the sum of weights

$$\frac{2}{d(u)+d(v)}$$

over all edges $uv$ of $G$, where $d(u)$ and $d(v)$ are the degrees of the vertices $u$ and $v$ in $G$, respectively. We prove that

$$H(G)\ge \frac{{\chi }_{DP}(G)-2}{2}+\frac{2({\chi }_{DP}(G)-1)}{n+{\chi }_{DP}(G)-2}+\frac{2(n-{\chi }_{DP}(G))}{n},$$

and this bound is sharp for all $n$ and $2\le {\chi }_{DP}(G)\le n$, where ${\chi }_{DP}(G)$ is the $DP$-chromatic number of $G$. This generalizes the previous lower bounds on $H(G)$. Moreover, we also determine the tree with minimum harmonic index among trees in ${\mathcal{𝒯}}_{n,l}$, where ${\mathcal{𝒯}}_{n,l}$ is the set of trees of order $n$ with a given segment sequence $l$.

We investigate the Cauchy problem on an infinite interval for the fractional evolution equation with Hilfer fractional derivative, which is a generalization of both Riemann–Liouville and Caputo fractional derivatives. Our methods are based on the generalized Ascoli–Arzelà theorem, Schauder’s fixed point theorem, the Wright function and Kuratowski’s measure of noncompactness. We obtain sufficient conditions of the existence for global mild solutions and attractive solutions when the semigroup associated with an almost sectorial operator is compact as well as noncompact. Two examples are provided to illustrate the results.