Let $A$ and $B$ be two Archimedean vector lattices and let ${\left(A^{\text{'}}\right)}_n^{\text{'}}$ and ${\left(B^{\text{'}}\right)}_n^{\text{'}}$ be their order continuous order biduals. If $\Psi :A\times A\to B$ is a positive orthosymmetric bimorphism, then the triadjoint ${\Psi }^{***}:{\left(A^{\text{'}}\right)}_n^{\text{'}}\times {\left(A^{\text{'}}\right)}_n^{\text{'}}\to {\left(B^{\text{'}}\right)}_n^{\text{'}}$ of $\Psi$ is inevitably orthosymmetric. This leads to a new and short proof of the commutativity of almost $f$-algebras.

We show that for percolation on any transitive graph, the triangle condition implies the open triangle condition.

This paper is a continuation of our previous investigations on the truncated matrix trigonometric moment problem in Ukrainian Math. J., 2011, 63, no. 6, 786-797, and Ukrainian Math. J., 2013, 64, no. 8, 1199- 1214. In this paper we shall study the truncated matrix trigonometric moment problem with an additional constraint posed on the matrix measure MT(δ), δ ∈ B(T), generated by the seeked function M(x): MT(∆) = 0, where ∆ is a given open subset of T (called a gap). We present necessary and sufficient...

Let ${\varphi }_1,\cdots ,{\varphi }_n$ be real homogeneous functions in $C^{\infty }({ℝ}^n-\left\{0\right\})$ of degree $k\ge 2$, let $\varphi \left(x\right)=({\varphi }_1\left(x\right),\cdots ,{\varphi }_n\left(x\right))$ and let $\mu$ be the Borel measure on ${ℝ}^{2n}$ given by $$\mu \left(E\right)={\int }_{{ℝ}^n}{\chi }_E(x,\varphi \left(x\right)){\left|x\right|}^{\gamma -n}\mathrm{d}x$$ where $\mathrm{d}x$ denotes the Lebesgue measure on ${ℝ}^n$ and $\gamma >0$. Let $T_{\mu }$ be the convolution operator $T_{\mu }f\left(x\right)=(\mu *f)\left(x\right)$ and let $$E_{\mu }=\{(1/p,1/q)\parallel T_{\mu }{\parallel }_{p,q}<\infty ,1\le p,q\le \infty \}.$$ Assume that, for $x\ne 0$, the following two conditions hold: $det\left({\mathrm{d}}^2\varphi \left(x\right)h\right)$ vanishes only at $h=0$ and $det\left(\mathrm{d}\varphi \right(x\left)\right)\ne 0$. In this paper we show that if $\gamma >n(k+1)/3$ then $E_{\mu }$ is the empty set and if $\gamma \le n(k+1)/3$ then $E_{\mu }$ is the closed segment with endpoints $D=\left(1-\frac{\gamma }{n(k+1)},1-\frac{2\gamma }{n(k+1)}\right)$ and $D^{\text{'}}=\left(\frac{2\gamma }{n(1+k)},\frac{\gamma }{n(1+k)}\right)$. Also, we give some examples.

The theory of Markov processes and the analysis on Lie groups are used to study the eigenvalue asymptotics of Dirichlet forms perturbed by scalar potentials.

The main purpose of this paper is to introduce the concept of $F$-type fuzzy topological spaces. Further variational principle and Caristi’s fixed point theorem have been extended in the $F$-type fuzzy topological spaces.

In a series of recent papers, Nils Dencker proves that condition $\left(\psi \right)$ implies the local solvability of principal type pseudodifferential operators (with loss of $\frac{3}{2}+ϵ$ derivatives for all positive $ϵ$), verifying the last part of the Nirenberg-Treves conjecture, formulated in 1971. The origin of this question goes back to the Hans Lewy counterexample, published in 1957. In this text, we follow the pattern of Dencker’s papers, and we provide a proof of local solvability with a loss of $\frac{3}{2}$ derivatives.

We show that the von Neumann algebras generated by an infinite number of t-deformed free gaussian operators are factors of type $I_{\infty }$.

We characterize Banach lattices on which every positive almost Dunford-Pettis operator is weakly compact.