‎the annihilator graph $ag(r)$ of a commutative ring $r$ is a simple undirected graph with the vertex set $z(r)^*$ and two distinct vertices are adjacent if and only if $ann(x) cup ann(y)$ $ neq $ $ann(xy)$‎. ‎in this paper we give the sufficient condition for a graph $ag(r)$ to be complete‎. ‎we characterize rings for which $ag(r)$ is a regular graph‎, ‎we show that $gamma (ag(r))in {1,2}$ and...

We propose a simple method of explicit description of families of closed geodesics on a triaxial ellipsoid Q that are cut out by algebraic surfaces in R. Such geodesics are either connected components of spatial elliptic curves or of rational curves. Our approach is based on elements of the Weierstrass–Poncaré reduction theory for hyperelliptic tangential covers of elliptic curves, the addition...

Generalizing self-duality on R2×S2 to higher dimensions, we consider the Donaldson-UhlenbeckYau equations on R2n×S2 and their noncommutative deformation for the gauge group U(2). Imposing SO(3) invariance (up to gauge transformations) reduces these equations to vortex-type equations for an abelian gauge field and a complex scalar on R θ . For a special S-radius R depending on the noncommutativi...

(1.1) du/dt = 9u = a{x)uxx + b{x)ux + c{x)u in R x (0, T) where T ^ oo. Here a(x) ^ 0, c(x) S M, l/«(x), b(x)/a(x) and c(x)/a(x) are locally integrable in R, but otherwise the coefficients are unrestricted. The results below extend characterizations of Widder ([17], [18]) for positive solutions of the heat equation (see §1.1). In particular, we find that all positive solutions of (1.1) are of t...

The Singular Asymptotics Lemma by Brüning and Seeley and the Push-Forward Theorem by Melrose lie at the very heart of their respective approaches to singular analysis. We review both and show that they deal with the same basic problem, giving solutions that emphasize different aspects of it. This also points to a possible common extension. 1. An Example As a simple example, consider f : R+ → R+...

we consider the bifurcation of periodic solutions from an equilibrium point of the given equation: x =f(x,?) , where x ? r , ? is a vector of real parameters ? , ? , ... , ? and f:r x r ->r has at least second continuous derivations in variables

We study a porous medium equation with right hand side. The operator has nonlocal diffusion effects given by an inverse fractional Laplacian operator. The derivative in time is also fractional of Caputo-type and which takes into account “memory”. The precise model is D t u− div(u(−∆)−σu) = f, 0 < σ < 1/2. We pose the problem over {t ∈ R+, x ∈ Rn} with nonnegative initial data u(0, x) ≥ 0 as wel...