We study the functional $I_f\left(u\right)={\int }_{\Omega }f\left(u\left(x\right)\right)dx$, where u=(u₁, ..., uₘ) and each $u_j$ is constant along some subspace $W_j$ of ℝⁿ. We show that if intersections of the $W_j$’s satisfy a certain condition then $I_f$ is weakly lower semicontinuous if and only if f is Λ-convex (see Definition 1.1 and Theorem 1.1). We also give a necessary and sufficient condition on ${W_j}_{j=1,...,m}$ to have the equivalence: $I_f$ is weakly continuous if and only if f is Λ-affine.

We give sufficient conditions for subsets of compact operators to be weakly precompact. Let $L_{w*}(E*,F)$ (resp. $K_{w*}(E*,F)$) denote the set of all w* - w continuous (resp. w* - w continuous compact) operators from E* to F. We prove that if H is a subset of $K_{w*}(E*,F)$ such that H(x*) is relatively weakly compact for each x* ∈ E* and H*(y*) is weakly precompact for each y* ∈ F*, then H is weakly precompact. We also prove the following results: If E has property (wV*) and F has property (V*), then $K_{w*}(E*,F)$ has property (wV*). Suppose...

We introduce the cardinal invariant $\theta$-$a{L}^{\text{'}}\left(X\right)$, related to $\theta$-$aL\left(X\right)$, and show that if $X$ is Urysohn, then $\left|X\right|\le 2^{\theta \text{-}a{L}^{\text{'}}\left(X\right)\chi \left(X\right)}$. As $\theta$-$a{L}^{\text{'}}\left(X\right)\le aL\left(X\right)$, this represents an improvement of the Bella-Cammaroto inequality. We also introduce the classes of firmly Urysohn spaces, related to Urysohn spaces, strongly semiregular spaces, related to semiregular spaces, and weakly $H$-closed spaces, related to $H$-closed spaces.

A matrix $A$ is said to have $𝐗$-simple image eigenspace if any eigenvector $x$ belonging to the interval $𝐗=\{x:\underline{x}\le x\le \overline{x}\}$ is the unique solution of the system $A\otimes y=x$ in $𝐗$. The main result of this paper is a combinatorial characterization of such matrices in the linear algebra over max-min (fuzzy) semiring. The characterized property is related to and motivated by the general development of tropical linear algebra and interval analysis, as well as the notions of simple image set and weak robustness (or weak stability)...

The notion of $\beta$-normality was introduced and studied by Arhangel’skii, Ludwig in 2001. Recently, almost $\beta$-normal spaces, which is a simultaneous generalization of $\beta$-normal and almost normal spaces, were introduced by Das, Bhat and Tartir. We introduce a new generalization of normality, namely weak $\beta$-normality, in terms of $\theta$-closed sets, which turns out to be a simultaneous generalization of $\beta$-normality and $\theta$-normality. A space $X$ is said to be weakly $\beta$-normal (w$\beta$-normal$)$ if for every...

Let $(A_0,A_1)$ be a regular interpolation couple. Under several different assumptions on a fixed $A^{\beta }$, we show that $A^{\theta }=A_{\theta }$ for every $\theta \in (0,1)$. We also deal with assumptions on ${\overline{A}}^{\beta }$, the closure of $A^{\beta }$ in the dual of ${(A_0^{*},A_1^{*})}_{\beta }$.

Let $W$ be the subspace of ${\mathbb{N}}^{*}$ consisting of all weak $P$-points. It is not hard to see that $W$ is a pseudocompact space. In this paper we shall prove that this space has stronger pseudocompact properties. Indeed, it is shown that $W$ is a $p$-pseudocompact space for all $p\in {\mathbb{N}}^{*}$.

We consider the Massera-Schäffer problem for the equation $$-y^{\text{'}}\left(x\right)+q\left(x\right)y\left(x\right)=f\left(x\right),x\in ℝ,$$ where $f\in L_p^{\mathrm{loc}}\left(ℝ\right),$ $p\in [1,\infty )$ and $0\le q\in L_1^{\mathrm{loc}}\left(ℝ\right).$ By a solution of the problem we mean any function $y,$ absolutely continuous and satisfying the above equation almost everywhere in $ℝ.$ Let positive and continuous functions $\mu \left(x\right)$ and $\theta \left(x\right)$ for $x\in ℝ$ be given. Let us introduce the spaces $$\begin{array}{cc}\hfill L_p(ℝ,\mu )& =\left\{f\in L_p^{\mathrm{loc}}{\left(ℝ\right):\parallel f\parallel }_{L_p(ℝ,\mu )}^p={\int }_{-\infty }^{\infty }{\left|\mu \left(x\right)f\left(x\right)\right|}^p\mathrm{d}x<\infty \right\},\\ \hfill L_p(ℝ,\theta )& =\left\{f\in L_p^{\mathrm{loc}}{\left(ℝ\right):\parallel f\parallel }_{L_p(ℝ,\theta )}^p={\int }_{-\infty }^{\infty }{\left|\theta \left(x\right)f\left(x\right)\right|}^p\mathrm{d}x<\infty \right\}.\end{array}$$ We obtain requirements to the functions $\mu$, $\theta$ and $q$ under which (1) for every function $f\in L_p(ℝ,\theta )$ there exists a unique solution $y\in L_p(ℝ,\mu )$ of the above equation; (2) there is an absolute constant...

Seeds of sunflowers are often modelled by $n⟼{\varphi }_{\theta }\left(n\right)=\sqrt{n}{e}^{2i\pi n\theta }$ leading to a roughly uniform repartition with seeds indexed by consecutive integers at angular distance $2\pi \theta$ for $\theta$ the golden ratio. We associate to such a map ${\varphi }_{\theta }$ a geodesic path ${\gamma }_{\theta }:{ℝ}_{\&gt;0}⟶{\mathrm{PSL}}_2\left(\mathbb{Z}\right)\setminus ℍ$ of the modular curve and use it for local descriptions of the image ${\varphi }_{\theta }\left(\mathbb{N}\right)$ of the phyllotactic map ${\varphi }_{\theta }$.

We consider the linear system $|2{\Theta }_0|$ of second order theta functions over the Jacobian $JC$ of a non-hyperelliptic curve $C$. A result by J.Fay says that a divisor $D\in |2{\Theta }_0|$ contains the origin $𝒪\in JC$ with multiplicity $4$ if and only if $D$ contains the surface $C-C=\left\{𝒪\right(p-q)\mid p,q\in C\}\subset JC$. In this paper we generalize Fay’s result and some previous work by R.C.Gunning. More precisely, we describe the relationship between divisors containing $𝒪$ with multiplicity $6$, divisors containing the fourfold $C_2-C_2=\{𝒪(p+q-r-s)\mid p,q,r,s\in C\}$, and divisors singular along $C-C$, using...

A topological space (T,τ) is said to be fragmented by a metric d on T if each non-empty subset of T has non-empty relatively open subsets of arbitrarily small d-diameter. The basic theorem of the present paper is the following. Let (M,ϱ) be a metric space with ϱ bounded and let D be an arbitrary index set. Then for a compact subset K of the product space $M^D$ the following four conditions are equivalent: (i) K is fragmented by $d_D$, where, for each S ⊂ D, $d_S(x,y)=sup\varrho \left(x\right(t),y(t\left)\right):t\in S$. (ii) For each countable subset...