We prove some special cases of Bergeron’s inequality involving two Gaussian polynomials (or $q$-binomials).

We introduce different variants of partial delay differential operators in ${ℂ}^n$ which conveniently accommodate all the existing operators in the literature under one umbrella. Manipulating the operators, we extend a number of results related to the various forms of finite order transcendental entire solutions of several Fermat-type partial delay differential equations up to ${ℂ}^3$. Our results are improvements of some recent results. In addition, some examples relevant to the content of the paper are exhibited.

Asymptotic study on the partition function $p(n)$ began with the work of Hardy and Ramanujan (1918). Later Rademacher (1937) obtained a convergent series for $p(n)$ and an error bound was given by Lehmer (1937). Despite having this, a full asymptotic expansion for $p(n)$ with an explicit error bound is not known. Recently O’Sullivan (2023) studied the asymptotic expansion of $p^k(n)$-partitions into $k$-th powers, initiated by Wright (1934), and consequently obtained an asymptotic expansion for $p(n)$ along with a concise description of the coefficients involved in the expansion but without any estimation of the error term. Here we consider a detailed and comprehensive analysis on an estimation of the error term obtained by truncating the asymptotic expansion for $p(n)$ at any positive integer $N$. This gives rise to an infinite family of inequalities for $p(n)$ which finally answers to a question proposed by Chen (2010). Our error term estimation predominantly relies on applications of algorithmic methods from symbolic summation.

Consider the following mean curvature equation in the static spacetime:

$$\text{div }\left(\frac{f(x)\text{∇ }u}{\sqrt{1-f^2(x)|\text{∇ }u{|}^2}}\right)+\frac{\text{∇ }u\text{∇ }f(x)}{\sqrt{1-f^2(x)|\text{∇ }u{|}^2}}=\lambda NH$$

with Dirichlet boundary condition on a bounded domain. We investigate the existence and uniqueness of classical solution. By the variational method, we also establish the multiplicity of strong solutions. Moreover, according to the behavior of $H$ near $0$, we obtain the global structure of positive solutions for this problem. The symmetry of positive solutions is also investigated.

We analyze curves of genus two that admit a morphism of degree three to an elliptic curve and we give formulas for the Igusa–Clebsch invariants and for plane models of curves of genus two whose Jacobians are $(3,3)$-isogenous to the product of two given elliptic curves in the Hesse pencil.

Fix $X\ge 2$. Let $f$ be a Hecke newform of prime level $p$. We investigate the general triple correlation sum

$$\sum _{h\ge 1}\sum _{l\ge 1}\sum _{n\ge 1}{\lambda }_f(n){\lambda }_f(n+h){\lambda }_f(n+l)U\left(\frac{n}{X}\right)V\left(\frac{h}{H}\right)R\left(\frac{l}{L}\right)$$

for $H,L\ge 1$ in the level aspect. As a result, we prove a nontrivial bound for any $H,L$ satisfying that and . It can be shown that there exist certain newforms such the nontrivial bound for the triple sum can be achieved, so long as $\text{max }\{H,L\}\ge X^{1∕4+𝜀}$. Particularly, whenever $L=H$, we present a nontrivial estimate for any $p$ such that , and further obtain the more significant cancellations for these sums in the different segments of $H$.

We introduce the concept of a natural transformation on a nearness approximation space. We show that natural transformations and the nearness approximation spaces are categories. Also, we characterize an epimorphism and a monomorphism in these categories. In addition, we prove that the category of a nearness approximation space does not necessarily have a product but for some families of an approximation spaces have a coproduct.

We obtain results, involving easily observable properties of bonding mappings, that ensure indecomposability of inverse limits on graphs. An allowable collection of $n$ arcs in a graph is defined, and two related properties of collections of arcs in a graph are introduced. In an inverse sequence on graphs, if compositions of the bonding mappings are $n$-pass maps on certain allowable collections of $n$ arcs, then the inverse limit space will be indecomposable. We provide examples that illustrate the use of our results.

Let $t\ge 2$ be a fixed positive integer. Let ${ped}_t(n)$ denote the number of $t$-regular partitions of $n$ wherein the even parts are distinct and the odd parts are unrestricted. We establish infinite families of congruences for ${ped}_t(n)$ modulo certain positive integers $M$, for specific values of $t$. We next study the distribution of ${ped}_t(n)$ for $t=3,5,7,9$. We prove that the series ${\sum }_{n=0}^{\infty }{ped}_t(2n+1)q^n$ is lacunary modulo arbitrary powers of $2$ for $t=3,5,9$. We also prove that the series ${\sum }_{n=0}^{\infty }{ped}_7(2n+1)q^n$ is lacunary modulo $2$. We use arithmetic properties of modular forms and Hecke eigenforms to prove our results.

We explain how the theory of sandwich cellular algebras can be seen as a version of cell theory for algebras. We apply this theory to many examples such as Hecke algebras, and various monoid and diagram algebras.

We consider the nonlinear boundary value problem

$$\{\begin{array}{c}{y}^{(4)}(x)+(k_1+k_2)y^{\prime \prime }(x)+k_1{k}_{2}y(x)=\lambda f(x,y(x)),x\in [0,1],\hfill \\ y^{\prime }(0)=y^{\prime }(1)=y^{\prime \prime \prime }(0)=y^{\prime \prime \prime }(1)=0,\hfill \end{array}$$

where $k_1$ and $k_2$ are constants, $\lambda$ is a parameter. Based on this, by using the fixed-point index theory in cones and the upper and lower solution method, the criteria of the existence, multiplicity and nonexistence of positive solutions are established in terms of different values of $\lambda$.

In this paper we study some solution techniques of solving the differential-difference equation

$$y^{\prime }(x)=y\left(x+\frac{1}{2}\right)-y\left(x-\frac{1}{2}\right),$$

first without an initial condition and then with some initial function $h$ defined on the unit interval $\left[-\frac{1}{2},\frac{1}{2}\right]$. We show some sufficient conditions that an initial function $h$ is admissible, i.e., it yields a unique continuous solution on some symmetric interval about $0$.