Let $H$ be an infinite-dimensional almost separable Hilbert space. We show that every local automorphism of $\mathcal B(H)$, the algebra of all bounded linear operators on a Hilbert space $H$, is an automorphism.
In stochastic partial differential equations it is important to have pathwise regularity properties of stochastic convolutions. In this note we present a new sufficient condition for the pathwise continuity of stochastic convolutions in Banach spaces.
Using unitary dilations we give a very simple proof of the maximal inequality for a stochastic convolution \[ \int ^t_0 S(t-s)\psi (s)\mathrm{d}W(s) \] driven by a Wiener process $W$ in a Hilbert space in the case when the semigroup $S(t)$ is of contraction type.
We show that there is a model structure in the sense of Quillen on an arbitrary Frobenius category F such that the homotopy category of this model structure is equivalent to the stable category F as triangulated categories. This seems to be well-accepted by experts but we were unable to find a complete proof for it in the literature. When F is a weakly idempotent complete (i.e., every split monomorphism is an inflation) Frobenius category, the model structure we constructed is an exact (closed) model structure in the sense of Gillespie (2011)., Zhi-Wei Li., and Seznam literatury
A variety is called normal if no laws of the form $s=t$ are valid in it where $s$ is a variable and $t$ is not a variable. Let $L$ denote the lattice of all varieties of monounary algebras $(A,f)$ and let $V$ be a non-trivial non-normal element of $L$. Then $V$ is of the form ${\mathrm Mod}(f^n(x)=x)$ with some $n>0$. It is shown that the smallest normal variety containing $V$ is contained in ${\mathrm HSC}({\mathrm Mod}(f^{mn}(x)=x))$ for every $m>1$ where ${\mathrm C}$ denotes the operator of forming choice algebras. Moreover, it is proved that the sublattice of $L$ consisting of all normal elements of $L$ is isomorphic to $L$.
A $k$-ranking of a graph $G=(V,E)$ is a mapping $\varphi \:V \rightarrow \lbrace 1,2,\dots ,k\rbrace $ such that each path with endvertices of the same colour $c$ contains an internal vertex with colour greater than $c$. The ranking number of a graph $G$ is the smallest positive integer $k$ admitting a $k$-ranking of $G$. In the on-line version of the problem, the vertices $v_1,v_2,\dots ,v_n$ of $G$ arrive one by one in an arbitrary order, and only the edges of the induced graph $G[\lbrace v_1,v_2,\dots ,v_i\rbrace ]$ are known when the colour for the vertex $v_i$ has to be chosen. The on-line ranking number of a graph $G$ is the smallest positive integer $k$ such that there exists an algorithm that produces a $k$-ranking of $G$ for an arbitrary input sequence of its vertices. We show that there are graphs with arbitrarily large difference and arbitrarily large ratio between the ranking number and the on-line ranking number. We also determine the on-line ranking number of complete $n$-partite graphs. The question of additivity and heredity is discussed as well.
We consider the stochastic equation \[ X_t=x_0+\int _0^t b(u,X_{u})\mathrm{d}B_u,\quad t\ge 0, \] where $B$ is a one-dimensional Brownian motion, $x_0\in \mathbb{R}$ is the initial value, and $b\:[0,\infty )\times \mathbb{R}\rightarrow \mathbb{R}$ is a time-dependent diffusion coefficient. While the existence of solutions is well-studied for only measurable diffusion coefficients $b$, beyond the homogeneous case there is no general result on the uniqueness in law of the solution. The purpose of the present note is to give conditions on $b$ ensuring the existence as well as the uniqueness in law of the solution.
Let {Xn} be a stationary and ergodic time series taking values from a finite or countably infinite set X and that f(X) is a function of the process with finite second moment. Assume that the distribution of the process is otherwise unknown. We construct a sequence of stopping times λn along which we will be able to estimate the conditional expectation E(f(Xλn+1)|X0,…,Xλn) from the observations (X0,…,Xλn) in a point wise consistent way for a restricted class of stationary and ergodic finite or countably infinite alphabet time series which includes among others all stationary and ergodic finitarily Markovian processes. If the stationary and ergodic process turns out to be finitarily Markovian (in particular, all stationary and ergodic Markov chains are included in this class) then limn→∞nλn>0 almost surely. If the stationary and ergodic process turns out to possess finite entropy rate then λn is upper bounded by a polynomial, eventually almost surely.