Abstract Let $\varphi _1,\ldots ,\varphi _r\in {\mathbb Z}[z_1,\ldots z_k]$ be integral linear combinations of elementary symmetric polynomials with $\text {deg}(\varphi _j)=k_j\ (1\le j\le r)$ , where $1\le k_1<k_2<\cdots <k_r=k$ . Subject to the condition $k_1+\cdots +k_r\ge \tfrac {1}{2}k(k-~1)+2$ we show that there is a paucity nondiagonal solutions Diophantine system _j({\mathbf x...

While most approaches in formal methods address system correctness, ensuring robustness has remained a challenge. In this paper we present and study the logic rLTL which provides means to formally reason about both correctness design. Furthermore, identify large fragment of for verification problem can be efficiently solved, i.e., done by using an automaton, recognizing behaviors described form...

For a convex body $K$ in $\mathbb{R}^n$, we introduce and study the extremal general affine surface areas, defined by \[ {\rm IS}_{\varphi}(K):=\sup_{K^\prime\subset K}{\rm as}_{\varphi}(K),\quad os}_{\psi}(K):=\inf_{K^\prime\supset as}_{\psi}(K) \] where ${\rm as}_{\varphi}(K)$ as}_{\psi}(K)$ are $L_\varphi$ $L_\psi$ area of $K$, respectively. We prove that there exist bodies achieve supremum ...