We introduce the class of weakly nil clean rings, as rings $R$ in which for every $a \in R$ there existan idempotent $e$ and a nilpotent $q$ such that $a-e-q \in eRa$. Every weakly nil clean ring is exchange. Weakly nil clean rings contain $\pi$-regular rings as a proper subclass, and these two classes coincide in the case when the ring has central idempotents, or has bounded index of nilpotence, or is a PI-ring. Weakly nil clean rings also properly encompass nil clean rings of Diesl [13]. The center of a weakly nil clean ring is strongly $\pi$-regular, and consequently, every weakly nil clean ring is a corner of a clean ring. These results extend Azumaya [3], McCoy [25], and the second author [33] to a wider class of ringsand provide partial answers to some open questions in [13] and [33]. Some other properties are studied and several examples are given as well.

The best degree-based upper bound for the spectral radius is due to Liu and Weng. This paper begins by demonstrating that a (forgotten) upper bound for the spectral radius dating from 1983 is equivalent to their much more recent bound. This bound is then used to compare lower bounds for the clique number. A series of line graph degree-based upper bounds for the Q-index is then proposedand compared experimentally with a graph based bound. Finally a new lower bound for generalised $r$-partite graphs is proved, by extending a result due to Erdős.

We prove that there is a unique $R$-equivalence class on every del Pezzo surface of degree $4$ defined over the Laurent field $K=k((t))$ in one variable over an algebraically closed field $k$ of characteristic not equal to $2$ or $5$. We also prove that given a smooth cubic surface defined over $\mathbb{C}((t))$, if the induced morphism to the GIT compactification of smooth cubic surfaces lies in the stable locus (possibly after a base change), then there is a unique $R$-equivalence class.

On a del Pezzo surface of degree 4 over an infinite $C_{1}$-field of characteristiczero, all rational points are $R$-equivalent. This more generally holds over anyinfinite perfect $C_{1}$-field of characteristic different from 2.

We show that a $3$-fold terminal flip or divisorial contractioncan be factored into a sequence of flops, blow-downs to a smoothcurve in a smooth $3$-fold or divisorial contractions to pointswith minimal discrepancies.

In this paper, we first study the solution to linear matrix inequality $AXB+(AXB)^* \geqslant (\gt ,\;\leqslant,\;\lt )\;C$ when the matrix $G = (A \quad B^*)$ is full row rank, where $C$ is a Hermitian matrix. Furthermore, for the applications, we derive the representations for the Re-nnd $\{1,2,i\}$-inverses for $A$, $i=3,4$, and the Re-nnd solution to $AXB=C$, some special matrix equations are also considered.

The purpose of this article is to study the existence of a coincidence point for two mappings defined on a nonempty set and taking values on a Banach space using the fixed point theory for nonexpansive mappings. Moreover, this type of results will be applied to obtain the existence of solutions for some classes of ordinary differential equations.

In this paper we consider a suitable extension of the weightedHardy inequalities and by applying to second order half-linearequations we establish some oscillation and nonoscillationresults.

In this paper, we study periodic oscillation of seasonally forced epidemiological models with impulse vaccination where periodicity occurs in contact rate. Using the famous Mawhin's coincidence degree method, we get the existence of positive periodic solutions of seasonally forced SIR models with impulse vaccination at fixed time. Some numerical simulations are presented to illustrate the effectiveness of such pulse vaccination strategy.

In this paper, we establish existence results and energy estimatesof weak solutions for an equation involving a $p$-harmonic operator, subject toDirichlet boundary conditions in a bounded smooth open domain of $\mathbb{R}^N$. A critical point result for differentiable functionals is exploited, in order to prove that the problem admits at least one non-trivial weak solution.

In this article, we obtain several remarks on the qualitative questions such as Picone's identity, Morse index and Hardy-Rellich type inequality for biharmonic operators.

In this paper we study a mathematical model of thermal explosion which is described by the boundary value problem \[\begin{cases}-\Delta u = \lambda e^{u^{\alpha}}, &x \in \Omega, \\\mathbf{n} \cdot \nabla u + g(u) u = 0, &x \in \partial \Omega,\end{cases}\]where the constant $\alpha \in (0,2],~ g:[0, \infty)\rightarrow (0, \infty)$ is an nondecreasing $C^1$ function, $\Omega$ is a bounded domain in $\mathbb{R}^2$ with smooth boundary $\partial \Omega$ and $\lambda \gt 0$ is a bifurcation parameter.Using variational methodswe show that there exists $0\lt \Lambda \lt \infty$ such that the problem has at least two positive solutions if $0 \lt \lambda \lt \Lambda,$ no solution if $\lambda \gt \Lambda$ and at least one positive solution when $\lambda =\Lambda.$

Let $b \in \operatorname{BMO}(\mathbb{R}^n)$ and $\mathscr{M}_\Omega$ be the Marcinkiewicz integral operatorwith kernel $\frac{\Omega(x)}{|x|^{n-1}}$, where $\Omega$ is homogeneous of degree zero, integrable and has mean value zero on the unit sphere $S^{n-1}$. In this paper, by means of Fourier transform estimates and approximationto the operator $\mathscr{M}_\Omega$ with integral operators having smooth kernelswe show that if $b \in \operatorname{CMO}(\mathbb{R}^n)$ and $\Omega$ satisfies a certain weak size condition, then the commutator $\mathscr{M}_{\Omega,b} = [b, \mathscr{M}_\Omega]$ generated by $b$ and $\mathscr{M}_\Omega$ is a compact operator on $L^p(\mathbb{R}^n)$ for some $1\lt p\lt \infty$.

In this paper, we are concerned with exponential integrability for logarithmic potentials of functions in generalized Lebesgue spaces $L(\log L)^{q(\cdot)}$ over non-doubling measure spaces. Here $q$ satisfies the loglog-Hölder condition.

In this paper, we study the minimal speed of traveling wave solutions for a diffusive three species competition system. Our main concern is the linear determinacy for the minimal speed. We provide some conditions on the parameters of the competition system such that the linear determinacy is assured. The main idea is by studying the linear determinacy of the corresponding approximated lattice dynamical systems and using the discrete Fourier transform.