In this paper, we first define the coquasitriangular monoidal Hom-Hopf algebras. Secondly, we present a method to construct braided monoidal Hom-Hopf algebras $\overline{B}$ and $\underline{B}$ in Yetter-Drinfeld category $\widetilde{\mathcal{H}}(_{H}\mathcal{YD}_1^{H})$ and $\widetilde{\mathcal{H}}(_{H}\mathcal{YD}_2^{H})$ respectively. As applications, we study some special cases in both module and comodule form for $(H, \xi_H)$ being quasitriangular and for $(H, \xi_H)$ being coquasitriangular respectively. Finally, we give some applications and examples of braided monoidal Hom-Hopf algebras in this article.

Let $R$ be an associative ring with identity, $S$ a multiplicative subset of $R$, and $M$ a right $R$-module. Then $M$ is called an $S$-Noetherian module if for each submodule $N$ of $M$, there exist an element $s \in S$ and a finitely generated submodule $F$ of $M$ such that $Ns \subseteq F \subseteq N$, and $R$ is called a right $S$-Noetherian ring if $R_R$ is an $S$-Noetherian module. In this paper, we study some properties of right $S$-Noetherian rings and $S$-Noetherian modules. Among other things, we study Ore extensions, skew-Laurent polynomial ring extensions, and power series ring extensions of $S$-Noetherian rings.

Hyper-Kloosterman sums of different moduli appear naturally in Voronoi's summation formula for cusp forms for $\operatorname{GL}_m(\mathbb{Z})$. In this paper their square moment is evaluated and their bounds are proved in the case of consecutively dividing moduli. As an application, smooth sums of Fourier coefficients of a Maass form for $\operatorname{SL}_m(\mathbb{Z})$ against an exponential function $e(\alpha n)$ are estimated. These sums are proved to have rapid decay when $\alpha$ is a fixed rational number or a transcendental number with approximation exponent $\tau(\alpha) \gt m$. Non-trivial bounds are proved for these sums when $\tau(\alpha) \gt (m+1)/2$.

We give conditions for when the fixed points by the partial Atkin-Lehner involutions on $X_0(N)$ are Weierstrass points as an extension of the result by Lehner and Newman [18]. Furthermore, we complete their result by determining whether the fixed points by the full Atkin-Lehner involutions on $X_0(N)$ are Weierstrass points or not.

In this paper we consider Lorentzian surfaces in the $4$-dimensional pseudo-Riemannian sphere $\mathbb{S}^4_2(1)$ with index $2$ and curvature one. We obtain the complete classification of minimal Lorentzian surfaces $\mathbb{S}^4_2(1)$ whose Gaussian and normal curvatures are constants. We conclude that such surfaces have the Gaussian curvature $1/3$ and the absolute value of normal curvature $2/3$. We also give some explicit examples.

We study geodesics of the form $\gamma(t) = \pi(\exp(tX) \exp(tY))$, $X, Y \in \mathfrak{g} = \operatorname{Lie}(G)$, in homogeneous spaces $G/K$, where $\pi \colon G \to G/K$ is the natural projection. These curves naturally generalise homogeneous geodesics, that is orbits of one-parameter subgroups of $G$ (i.e., $\gamma(t) = \pi(\exp(tX))$, $X \in \mathfrak{g}$). We obtain sufficient conditions on a homogeneous space implying the existence of such geodesics for $X, Y \in \mathfrak{m} = T_o(G/K)$. We use these conditions to obtain examples of Riemannian homogeneous spaces $G/K$ so that all geodesics of $G/K$ are of the above form. These include total spaces of homogeneous Riemannian submersions endowed with one parameter families of fiber bundle metrics, Lie groups endowed with special one parameter families of left-invariant metrics, generalised Wallach spaces, generalized flag manifolds, and $k$-symmetric spaces with $k$-even, equipped with certain one-parameter families of invariant metrics.

In this paper, based on a Poisson type extension of Triebel-Lizorkin type oscillation spaces $\dot{F}^{\gamma_1, \gamma_2}_{p, q}(\mathbb{R}^{n})$, we establish a bilinear estimate on some new tent spaces $\mathbb{F}^{\gamma_1, \gamma_2}_{p, q, m, m'}$ associated with $\dot{F}^{\gamma_1, \gamma_2}_{p, q}(\mathbb{R}^{n})$. As an application, we get the well-posedness and regularity of the fractional magneto-hydrodynamic equations and quasi-geostrophic equations with initial data in the critical $\dot{F}^{\gamma_1, \gamma_2}_{p, q}(\mathbb{R}^{n})$.

This paper is concerned with a non-autonomous SIR epidemic model, which involves almost periodic incidence rate and saturated treatment function. By using the differential inequality technique and Lyapunov functional method, we obtain the existence and global exponential stability of almost periodic solutions for the addressed SIR model, which improve and supplement existing ones. Also, an example and its numerical simulations are given to demonstrate our theoretical results.

Let $X, Y$ be locally compact Hausdorff spaces, and $V, W$ be JB*-triples such that $W$ is prime. Let $T \colon C_0(X, V) \to C_0(Y, W)$ be an orthogonality preserving linear operator with dense range. We show that $T$ can be written as a weighted composition operator.

A necessary and sufficient condition for a Lau type binary operation defined by two mappings to be an algebra-operation is given in terms of multipliers. Also a characterization of multipliers of a Lau algebra constructed by semisimple commutative Banach algebras is given in terms of multipliers of original Banach algebras.

We provide solution existence results for the convex quadratic programming problems in Hilbert spaces, which the constraint set is defined by finitely many convex quadratic inequalities. In order to obtain our results, we shall use either the properties of the Legendre form or the properties of the finite-rank operator. The existence results are established without requesting neither coercivity of the objective function nor compactness of the constraint set.

We concern the problem of modifying the edge lengths of a tree in minimum total cost so that the prespecified $p$ vertices become the $p$-maxian with respect to the new edge lengths. This problem is called the inverse $p$-maxian problem on trees. Gassner proposed in $2008$ an efficient combinatorial algorithm to solve the inverse $1$-maxian problem on trees. For the case $p \geq 2$, we claim that the problem can be reduced to $O(p^2)$ many inverse $2$-maxian problems. We then develop algorithms to solve the inverse $2$-maxian problem under various objective functions. The problem under $l_1$-norm can be formulated as a linear program and thus can be solved in polynomial time. Particularly, if the underlying tree is a star, the problem can be solved in linear time. We also develop $O(n \log n)$ algorithms to solve the problems under Chebyshev norm and bottleneck Hamming distance, where $n$ is the number of vertices of the tree.