Integration by parts results concerning Stieltjes integrals for functions with values in Banach spaces are presented. The background of the theory is the Kurzweil approach to integration based on Riemann type integral sums, which leads to the Perron integral.
Recall that a space X is a c-semistratifiable (CSS) space, if the compact sets of X are Gδ-sets in a uniform way. In this note, we introduce another class of spaces, denoting it by k-c-semistratifiable (k-CSS), which generalizes the concept of c-semistratifiable. We discuss some properties of k-c-semistratifiable spaces. We prove that a T2-space X is a k-c-semistratifiable space if and only if X has a g function which satisfies the following conditions: (1) For each x ∈ X, {x} = ∩ {g(x, n): n ∈ ℕ} and g(x, n + 1) ⊆ g(x, n) for each n ∈ N. (2) If a sequence {xn}n∈N of X converges to a point x ∈ X and yn ∈ g(xn, n) for each n ∈ N, then for any convergent subsequence {ynk }k∈N of {yn}n∈N we have that {ynk }k∈N converges to x. By the above characterization, we show that if X is a submesocompact locally k-csemistratifiable space, then X is a k-c-semistratifible space, and the countable product of k-c-semistratifiable spaces is a k-c-semistratifiable space. If X = ∪ {Int(Xn): n ∈ N} and Xn is a closed k-c-semistratifiable space for each n, then X is a k-c-semistratifiable space. In the last part of this note, we show that if X = ∪ {Xn : n ∈ N} and Xn is a closed strong β-space for each n ∈ ℕ, then X is a strong β-space.
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.