This short note is a continuation of and and its purpose is to show that every simple zeropotent paramedial groupoid containing at least three elements is strongly balanced in the sense of [4].
Existence results are established for the resonant problem $y^{\prime \prime }+\lambda _m \,a\,y=f(t,y)$ a.e. on $[0,1]$ with $y$ satisfying Dirichlet boundary conditions. The problem is singular since $f$ is a Carathéodory function, $a\in L_{{\mathrm loc}}^1(0,1)$ with $a>0$ a.e. on $[0,1]$ and $\int ^1_0 x(1-x)a(x)\,\mathrm{d}x <\infty $.
G. Grätzer and A. Kisielewicz devoted one section of their survey paper concerning $p_n$-sequences and free spectra of algebras to the topic “Small idempotent clones” (see Section 6 of [18]). Many authors, e.g., [8], [14, 15], [22], [25] and [29, 30] were interested in $p_n$-sequences of idempotent algebras with small rates of growth. In this paper we continue this topic and characterize all idempotent groupoids $(G,\cdot )$ with $p_2(G,\cdot )\le 2$ (see Section 7). Such groupoids appear in many papers see, e.g. [1], [4], [21], [26, 27], [25], [28, 30, 31, 32] and [34].
For fairly general open sets it is shown that we can express a solution of the Neumann problem for the Laplace equation in the form of a single layer potential of a signed measure which is given by a concrete series. If the open set is simply connected and bounded then the solution of the Dirichlet problem is the double layer potential with a density given by a similar series.
For ordered (= partially ordered) sets we introduce certain cardinal characteristics of them (some of those are known). We show that these characteristics—with one exception—coincide.
A closed convex set $Q$ in a local convex topological Hausdorff spaces $X$ is called locally nonconical (LNC) if for every $x, y\in Q$ there exists an open neighbourhood $U$ of $x$ such that $(U\cap Q)+\frac{1}{2}(y-x)\subset Q$. A set $Q$ is local cylindric (LC) if for $x,y\in Q$, $x\ne y$, $z\in (x,y)$ there exists an open neighbourhood $U$ of $z$ such that $U\cap Q$ (equivalently: $\mathrm bd(Q)\cap U$) is a union of open segments parallel to $[x,y]$. In this paper we prove that these two notions are equivalent. The properties LNC and LC were investigated in [3], where the implication ${\mathrm LNC}\Rightarrow {\mathrm LC}$ was proved in general, while the inverse implication was proved in case of Hilbert spaces.
We provide a characterization of continuous images of Radon-Nikodým compacta lying in a product of real lines and model on it a method for constructing natural examples of such continuous images.