and Applied Analysis 3 otherwise [H A (x, s(t))] i = x i , where b i ∈ R is the threshold of neuron i and the function 1 is the Heaviside function: 1(u) = 1 for u ≥ 0, otherwise 0, which describes an instantaneous unit pulse. On each subsequent time t = 0, 1, . . ., the network generates a vector of neuronal active states according to (8), resulting in the dynamic flow x(t), t = 0, 1, . . .. Th...

This paper is a result of more than 3 decades of intensive research and development. It exhibits several keys to understanding and conceiving technologies considered or even proved by the standards of modern science as impossible to produce, such as, for instance, the automated synthesis of recursive programs. The paper shows a direction that allows to escape from limits of contemporary science...

Using a new notion of path-derivative, we study the well-posedness backward stochastic differential equation driven by continuous martingale M when f(s,?,y,z) is locally Lipschitz in (y,z): Yt=?(M[0,T])+?tTf(s,M[0,s],Ys?,Zsms)dtr[M,M]s??tTZsdMs?NT+Nt.Here, M[0,t] path from 0 to t and m defined [M,M]t=?0tmsms?dtr[M,M]s. When BSDE one-dimensional, show existence uniqueness solution. On contrary, ...

Let G be a (p,q) graph and A be a group. We denote the order of an element $a in A $ by $o(a).$  Let $ f:V(G)rightarrow A$ be a function. For each edge $uv$ assign the label 1 if $(o(f(u)),o(f(v)))=1 $or $0$ otherwise. $f$ is called a group A Cordial labeling if $|v_f(a)-v_f(b)| leq 1$ and $|e_f(0)- e_f(1)|leq 1$, where $v_f(x)$ and $e_f(n)$ respectively denote the number of vertices labelled w...

Let f : V (G) → {1, 2,..., |V (G)|} be a bijection, and let us denote S = f(u) + f(v) D |f(u) − f(v)| for every edge uv in E(G). f' the induced labeling, by vertex labeling f, defined as E(G) {0, 1} such that any E(G), (uv)=1 if gcd(S, D)=1, (uv)=0 otherwise. ef' (0) (1) number of edges labeled with 0 1 respectively. is SD-prime cordial |ef' (1)| ≤ G graph it admits labeling. In this paper, we ...

Let F be a eld of characteristic p > 0 and G be a smooth nite algebraic group over F . We compute the essential dimension edF (G; p) of G at p. That is, we show that edF (G; p) = { 1, if p divides |G|, and 0, otherwise.