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I prove, under mild assumptions, that solutions to linear evolution equations admit sectorial solutions. The size of the sector depends on the regularity of the initial data. If it is regular enough the solution is holomorphic and unique otherwise it is sectorial. I also prove that the result is optimal for many partial differential systems (which includes KdV and other examples).
Let $G$ be a finite group and let $mathfrak{F}$ be a hereditary saturated formation. We denote by $mathbf{Z}_{mathfrak{F}}(G)$ the product of all normal subgroups $N$ of $G$ such that every chief factor $H/K$ of $G$ below $N$ is $mathfrak{F}$-central
We prove a version of the Cauchy-Davenport theorem for general linear maps. For subsets $A,B$ of the finite field $mathbb{F}_p$, the classical Cauchy-Davenport theorem gives a lower bound for the size of the sumset $A+B$ in terms of the sizes of the
We present a well-structured detailed exposition of a well-known proof of the following celebrated result solving Hilberts 13th problem on superpositions. For functions of 2 variables the statement is as follows. Kolmogorov Theorem. There are conti
This note contains a representation formula for positive solutions of linear degenerate second-order equations of the form $$ partial_t u (x,t) = sum_{j=1}^m X_j^2 u(x,t) + X_0 u(x,t) qquad (x,t) in mathbb{R}^N times, ]- infty ,T[,$$ proved by a func
We outline a simple proof of Hulanickis theorem, that a locally compact group is amenable if and only if the left regular representation weakly contains all unitary representations. This combines some elements of the literature which have not appeared together, before.