New static regular axially symmetric solutions of SU(2) Euclidean Yang-Mills theory are constructed numerically. They represent calorons having non-trivial Polyakov loop at spacial infinity. The solutions are labeled by two integers m,n. It is shown that besides known, charge one self-dual periodic instanton solution, there are other non-self dual solutions of the Yang-Mills equations naturally...

The equation ∆u + λu + g(λ, u)u = 0 is considered in a bounded domain in R2 with a Signorini condition on a straight part of the boundary and with mixed boundary conditions on the rest of the boundary. It is assumed that g(λ, 0) = 0 for λ ∈ R, λ is a bifurcation parameter. A given eigenvalue of the linearized equation with the same boundary conditions is considered. A smooth local bifurcation b...

We construct explicitly time-dependent exact solutions to the field equations of 6D gauged chiral supergravity, compactified to 4D in the presence of up to two 3-branes situated within the extra dimensions. The solutions we find are scaling solutions, and are plausibly attractors which represent the late-time evolution of a broad class of initial conditions. By matching their near-brane boundar...

We discuss the maximum number of distinct non-trivial solutions that a generalized Fermat equation Ax n + By n = Cz n might possibly have. The abc-conjecture implies that it can never have more than two solutions once n > n 0 (independent of A; B;C); and that it has no solutions for xed A; B; C once n > n A;B;C. On the other hand for any set of pairwise coprime integers p 1 ; p 2 ; :: : ; p k ,...

Here s is the scalar curvature, D2s the Hessian of s, ∆s = trD2s the Laplacian, and ◦ R the action of the curvature tensor R on symmetric bilinear forms, c.f. [B, Ch.4H] for further details. The equation (0.3) is just the trace of (0.2). It is obvious from the trace equation (0.3) that there are no non-flat R2 critical metrics, i.e. solutions of (0.2)-(0.3), on compact manifolds N ; this follow...

In this paper, we first compute the multiple non-trivial solutions of the Schrödinger equation on a square, by using the Liapunov-Schmidt reduction and symmetry-breaking bifurcation theory, combinedwith Legendre pseudospectralmethods. Then, starting from the non-trivial solution branches of the corresponding nonlinear problem, we further obtain the whole positive solution branch with D4 symmetr...