In this paper, we consider $Q_1(n)$ to be the number of partitions of $n$ into odd parts greater than one and provide new combinatorial interpretations for $Q_1(n)$. New linear relations involving Euler's partition function $p(n)$ and the overpartition function $\overline{p}(n)$ are obtained in this context.

For a prime $p \equiv 3 \pmod{4}$ and $m \geq 2$, Romik raised a question about whether the Taylor coefficients around $\sqrt{-1}$ of the classical Jacobi theta function $\theta_3$ eventually vanish modulo $p^m$. This question can be extended to a class of modular forms of half-integral weight on $\Gamma_1(4)$ and CM points; in this paper, we prove an affirmative answer to it for primes $p \geq 5$. Our result is also a generalization of the results of Larson and Smith for modular forms of integral weight on $\operatorname{SL}_2(\mathbb{Z})$.

In this paper, we show the existence of the attached comultiplication structure on $\operatorname{Hom}_{eAe}(eA,D(Ae))$ if an $(eAe,A)$-bimodule $eA$ has the double centraliser property over an algebra $A$ with the idempotent $e$. Then we apply it on gendo-Gorenstein algebras. As applications, we give a sufficient and necessary condition for a gendo-Gorenstein algebra to be Gorenstein, and give a bocs-theoretic characterisation of the double centraliser property.

Let $d \in \{ 1,2,3,\ldots \}$ and $\Omega \subset \mathbb{R}^d$ be open bounded with Lipschitz boundary. Consider the reaction-diffusion parabolic problem \[ \begin{cases} \frac{u_t}{|x|^4} + \Delta^2 u = k(t) |u|^{p-1}u, &(x,t) \in \Omega \times (0,T), \\ u(x,t) = \frac{\partial u}{\partial \nu}(x,t) = 0, &(x,t) \in \partial \Omega \times (0,T), \\ u(x,0) = u_0(x), &x \in \Omega, \end{cases} \] where $T \gt 0$, $p \in (1,\infty)$, $0 \neq u_0 \in H^2_0(\Omega)$ and $\nu$ is the outward normal vector to $\partial \Omega$. We investigate the existence of a global weak solution to the problem together with the decaying and blow-up properties using the potential well method.

The aim of this paper is to investigate a contact problem involving thermo-electro-visco-elastic body with damage and a rigid foundation. The friction is modelled with a subgradient of a locally Lipschitz mapping, and the contact is described by the Signorini's unilateral contact condition. A parabolic differential inclusion for the damage function is used to include the damaging effect in the model. We establish the model's variational formulation using four systems of three hemivariational inequalities and a parabolic equation, we prove an existence and uniqueness result of this problem. The proof is based on a fixed point argument and a recent finding from hemivariational inequality theory. Finally, by employing the finite element approach, we investigate a fully discrete approximation of the model and we derive error estimates on the approximate solution.

It is known that the fractional integral operator $\mathcal{I}_{\alpha}$ based on a symmetric Markovian semigroup with Varopoulos dimension $d$ is bounded from $L^p$ to $L^q$, if $0 \lt \alpha \lt d$, $1 \lt p \lt q \lt \infty$ and $-d/p + \alpha = -d/q$, like the usual fractional integral operator defined on the $d$ dimensional Euclidean space. We introduce generalized fractional integral operators based on symmetric Markovian semigroups and extend the $L^p$-$L^q$ boundedness to Orlicz spaces. We also apply the result to the semigroup associated with the diffusion process generated by the sub-Laplacian on the Heisenberg group. Moreover, we show necessary and sufficient conditions for the boundedness of the generalized fractional integral operator on the space of homogeneous type and apply them to the Heisenberg group.

This paper is concerned with the dynamic characteristics of a class of Nicholson's blowflies system with patch structure and multiple pairs of distinct time-varying delays. We aim to find the influence of the distinct time-varying delays in the same reproductive function on its asymptotic behavior. First, we derive the global existence, positiveness and uniform persistence of solutions for the addressed system. Then, by employing the theory of functional differential equations, the fluctuation lemma and the technique of differential inequalities, we build up some new delay-dependent criteria for the global attractivity of the positive equilibrium point vector, which does not possess the same components. In addition, we exam the effectiveness and feasibility of the theoretical achievements by some numerical simulations.

Nonlinear programming is explicitly analyzed via a novel perspective/method and from a bottom-up manner. The philosophy is based on the recent findings on convex quadratic equation (CQE), which help clarify a geometric interpretation that relates CQE to convex quadratic function (CQF). More specifically, regarding the solvability of CQE, its necessary and sufficient condition as well as a unified parameterization of all the solutions has recently been analytically formulated. Moving forward, the understanding of CQE is utilized to describe the geometric structure of CQF, and the CQE-CQF relation. All these results are shown closely related to a basis in the optimization literature, namely quadratic programming (QP). For the first time from this viewpoint, the QPs subject to equality, inequality, equality-and-inequality, and extended constraints can be algebraically solved in derivative-free closed formulae, respectively. All the results are derived without knowing a feasible point, a priori and any time during the process.