We prove that the classical Prandtl, Ishlinskii and Preisach hysteresis operators are continuous in Sobolev spaces $W^{1,p}(0,T)$ for $1\le p<+\infty$, (localy) Lipschitz continuous in $W^{1,1}(0,T)$ and discontinuous in $W^{1,\infty }(0,T)$ for arbitrary $T>0$. Examples show that this result is optimal.

The recent development of mathematical methods of investigation of problems with hysteresis has shown that the structure of the hysteresis memory plays a substantial role. In this paper we characterize the hysteresis operators which exhibit a memory effect of the Preisach type (memory preserving operators). We investigate their properties (continuity, invertibility) and we establish some relations between special classes of such operators (Preisach, Ishlinskii and Nemytskii operators). For a general...

In this short note we utilize the Borsuk-Ulam Anitpodal Theorem to present a simple proof of the following generalization of the “Ham Sandwich Theorem”: Let $A_1,...,A_m\subseteq {ℝ}^n$ be subsets with finite Lebesgue measure. Then, for any sequence $f_0,...,f_m$ of $ℝ$-linearly independent polynomials in the polynomial ring $ℝ[X_1,...,X_n]$ there are real numbers ${\lambda }_0,...,{\lambda }_m$, not all zero, such that the real affine variety $\{x\in {ℝ}^n;{\lambda }_0{f}_0\left(x\right)+\cdots +{\lambda }_m{f}_m\left(x\right)=0\}$ simultaneously bisects each of subsets $A_k$, $k=1,...,m$. Then some its applications are studied.