We explore the non-existence and existence of transcendental entire solutions of a certain type of partial differential-difference equations in $\mathbb{C}^{n}$ of the type defined by Fermat with finite-order constraint. We also study the existence as well as the form of transcendental entire solutions for a particular class of Fermat's type partial differential-difference equations in $\mathbb{C}^{2}$ with finite-order constraint. The results provided as solutions for these problems have substantially enhanced the theorem that was previously suggested by Xu, Cao, Liu, Wang, Zhang, Zheng. In contrast to earlier articles, this one uses a different technique. By using several examples, it is shown that there are certain conditions and types of transcendental entire solutions of finite order of such equations.

A coproduct on a vector space $A$ is defined as a linear map $\Delta:A\to A\ot A$ satisfying coassociativity $(\Delta\otimes\iota)\Delta=(\iota\otimes\Delta)\Delta$. We use $\iota$ for the identity map. If $G$ is a finite group and if $A$ is the space of all complex functions on $G$, a coproduct on $A$ is defined by $\Delta(f)(p,q)=f(pq)$ where $p,q\in G$. We identify $A\otimes A$ with complex functions on the Cartesian product $G\times G$. Coassociativity follows from the associativity of the product in $G$. Unfortunately, sometimes this usual notion of a coproduct is not the appropriate one. Just think of the above example for an infinite group. For this reason, we consider the case of a coproduct on an algebra $A$, not necessarily unital but with a non-degenerate product. It is a linear map from $A$ to $M(A\otimes A)$, the multiplier algebra of $A\otimes A$. It is no longer possible to express coassociativity in its usual form as the maps $\Delta\otimes\iota$ and $\iota\otimes\Delta$, defined on $A\otimes A$, may no longer be defined on the multiplier algebra $M(A\otimes A)$ (which in general is bigger than the algebra $A\otimes A$). We will see how this problem can be solved in different ways. There is a similar issue when we want to use the coproduct to make the dual space into an associative algebra. Also this problem is discussed. We refer to the literature on multiplier Hopf algebras and related structures. We also include some examples to illustrate these problems and the solutions proposed in this paper.

By incorporating partial fraction decompositions, recurrence relations and hypergeometric series evaluations, an analytic solution is presented for the evaluation problems about a class of infinite series involving squared central binomial coefficients and skew harmonic numbers.

We investigate the spectra of composition operators acting on weighted Dirichlet spaces induced by non-automorphic analytic self-maps with fixed points in the unit disk. We still provide more detailed characterizations of the spectra of composition operators induced by univalent symbols and monomial symbols.

Let $X$ and $Y$ be compact metric spaces and $\alpha,\beta\in (0,1)$. We consider the weighted composition operators between the little Lipschitz algebras $\mathrm{lip}_\alpha(X)$ and $\mathrm{lip}_\beta(Y)$ and give a set of conditions which is necessary and sufficient for these operators to be Fredholm. We completely characterize disjointness preserving Fredholm operators between these algebras.

An open question of Ramírez and Vendramin (IMRN, 2022) concerning the decomposability of a finite involutive non-degenerate solution to the set-theoretic Yang-Baxter equation is answered in the negative. Counterexamples are obtained from a class of singular, non-simple solutions which are not of finite primitive level. The ideals of the corresponding braces are calculated. For some of these braces it is proved that any minimal ideal gives rise to a simple solution to the Yang-Baxter equation. The concept of ``primitive level'' is extended to infinite solutions and shown to be a special case of a general concept of length. A quantitative interpretation of finite primitive level is obtained from a numeric invariant related to the augmentation ideal.

The initial value problem for the nonlinear system of singularly perturbed differential equations, which emerges as a model for chemical kinetics, is considered. In order to solve this problem numerically, a novel fitted difference scheme is constructed by the finite difference method on non-uniform meshes, like the Shishkin mesh and the Bakhvalov mesh, using quadrature rules with the remaining terms in integral form. The scheme is proven to achieve almost first-order convergence in the discrete maximum norm on the Shishkin mesh and first-order convergence on the Bakhvalov mesh. Two numerical examples are considered to illustrate the accuracy and performance of the method. In order to show the advantage of the proposed method we compare our results with those obtained by an implicit linear difference scheme. Comparison shows that the proposed method is fast convergent and highly accurate.