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.
A distinction is introduced between itemized and non-itemized plural predication. It is argued that a full-fledged system of plural logic is not necessary in order to account for the validity of inferences concerning itemized collective predication. Instead, it is shown how this type of inferences can be adequately dealt with in a first-order logic system, after small modifications on the standard treatment. The proposed system, unlike plural logic, has the advantage of preserving completeness. And as a result, inferences such as ''Dick and Tony emptied the bottle, hence Tony and Dick emptied the bottle'' are shown to be first-order., Rozlišování se zavádí mezi rozčleněným množstvím a nerozdělením množných predikcí. Tvrdí se, že plnohodnotný systém pluralitní logiky není nutný pro to, aby bylo možné zohlednit platnost závěrů týkajících se skupinového predikce. Místo toho je ukázáno, jak lze tento typ závěrů adekvátně řešit v logickém systému prvního řádu, po malých úpravách standardního zpracování. Navrhovaný systém, na rozdíl od množné logiky, má výhodu zachování úplnosti. Výsledkem je, že závěry jako ,,Dick a Tony vyprázdnili láhev, proto Tony a Dick vyprázdnili láhev''., and Gustavo Fernandéz Díez
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.
The radio antipodal number of a graph G is the smallest integer c such that there exists an assignment f : V (G) → {1, 2, . . . , c} satisfying |f(u) − f(v)| ≥ D − d(u, v) for every two distinct vertices u and v of G, where D is the diameter of G. In this note we determine the exact value of the antipodal number of the path, thus answering the conjecture given in [G. Chartrand, D. Erwin and P. Zhang, Math. Bohem. 127 (2002), 57– 69]. We also show the connections between this colouring and radio labelings.
We prove that a rank ≥3 Dowling geometry of a group H is partition representable if and only if H is a Frobenius complement. This implies that Dowling group geometries are secret-sharing if and only if they are multilinearly representable., František Matúš and Aner Ben-Efraim., and Obsahuje bibliografické odkazy
When a system of one-sided max-plus linear equations is inconsistent, its right-hand side vector may be slightly modified to reach a consistent one. It is handled in this note by minimizing the sum of absolute deviations in the right-hand side vector. It turns out that this problem may be reformulated as a mixed integer linear programming problem. Although solving such a problem requires much computational effort, it may propose a solution that just modifies few elements of the right-hand side vector, which is a desired property in some practical situations.
Let $R$ be an associative ring with identity and let $J(R)$ denote the Jacobson radical of $R$. $R$ is said to be semilocal if $R/J(R)$ is Artinian. In this paper we give necessary and sufficient conditions for the group ring $RG$, where $G$ is an abelian group, to be semilocal.
A graph X, with a group G of automorphisms of X, is said to be (G, s)-transitive, for some s\geq 1, if G is transitive on s-arcs but not on (s + 1)-arcs. Let X be a connected (G, s)-transitive graph of prime valency s\geq 5, and Gv the vertex stabilizer of a vertex v \in V (X). Suppose that Gv is solvable. Weiss (1974) proved that |Gv | p(p−1)^{2}. In this paper, we prove that Gv\cong (\mathbb{Z}_{p}\rtimes \mathbb{Z}_{m})× \mathbb{Z}_{n} for some positive integers m and n such that n | m and m | p − 1., Song-Tao Guo, Hailong Hou, Yong Xu., and Obsahuje seznam literatury