This paper deals with two semilinear parabolic equations involving multi-coupled nonlinearity, where the weighted functions depend on space and time variables. We determine the exponent regions where the nonnegative nontrivial solutions blow up for any initial data. Moreover, some exponent regions on the global solutions are given with some conditions on the initial data and the ten exponents.

Let $C_b(X)$ be the Banach lattice of all bounded continuous real functions on a completely regular Hausdorff space $X$ and let $(E,\|\cdot\|_E)$ be a real Banach space. A linear operator $T:C_b(X)\to E$ is said to be $\sigma$-additive if $\|T(u_n)\|_E\to 0$ whenever $(u_n)$ is a sequence in $C_b(X)$ such that $u_n(x)\downarrow 0$ for each $x\in X$. We characterize $\sigma$-additive nuclear operators $T:C_b(X)\to E$ in terms of their representing Baire vector measures. The formula for the trace of $\sigma$-additive nuclear operators $T:C_b(X)\to C_b(X)$ is derived.

In a normed space we characterize the approximate orthogonality in terms of extreme points of the closed unit ball in dual space. There is also another (and older) definition of an approximate orthogonality and therefore in this paper both approximate orthogonalities are compared. Moreover, we characterize the approximate orthogonality in the class of compact operators on a reflexive Banach space.

Planar analytic constrained differential systems (or impasse systems) are given by $A(x)\dot{x} = F(x), \quad x\in\mathbb{R}^{2}$, where $F$ is a vector field and $A$ is a matrix valued function. This class of systems differs from classical ODE's due to existence of the so called impasse set $\Delta=\{x:{\rm det} A (x) = 0\}$. The dynamics near smooth impasse points is well known in the literature. In this paper, we use Newton polygon and weighted blow ups in order to study its phase portrait near singular points of $\Delta$. Moreover, we apply our results in electric circuit problems.

In this paper, we introduce the generalized notion of generalized discrete bi-polymatriods and the corresponding notion of generalized bi-polymatriodal ideals. Moreover, the behavior of the monomial localizations of such ideals is investigated. It is shown that generalized bi-polymatriodal ideals have the strong persistence property.

In this work we continue with the ongoing search for what are often large algebraic structures of mathematical objects (functions, sequences, etc.) which enjoy certain special properties. This type of study belongs to the recent area of research known as lineability. On this occasion, and among several other results, we shall show that there are large algebraic structures within (i) the set of nets which are weakly convergent, but are not bounded, (ii) nets that are weakly convergent, but are not convergent in norm, or (iii) the set of nets of measurable functions which converge pointwise to a function that is not measurable and that are bounded in $[0,1]$.

Minimal usco/cusco maps are very important in functional analysis, optimization, selection theorems, the study of differentiability of Lipschitz functions, etc. We study topologies of uniform convergence on bornologies on the space of minimal usco and minimal cusco maps. We find sufficient conditions for metrizability and complete metrizability of minimal usco and minimal cusco maps equipped with the topology of uniform convergence on bornologies. We also investigate Fréchet locally convex subspaces of these spaces.