On residue maps for affine curves

We establish several compatibility results between residue maps in étale and Galois cohomology that arise naturally in the analysis of smooth affine algebraic curves having good reduction over discretely valued fields. These results are needed, and in fact have already been used, for the study of finiteness properties of the unramified cohomology of function fields of affine curves over number fields.     

Symmetry reductions and exact solutions of the affine heat equation

Lie symmetry group method is applied to study the affine heat equation for surface. Its symmetry groups and corresponding optimal systems are determined, and group-invariant solutions associated to the symmetries are obtained and classified.     

The Grothendieck conjecture for affine curves

We prove that the isomorphism class of an affine hyperbolic curve defined over a field finitely generated over Q is completely determined by its arithmetic fundamental group. We also prove a similar result for an affine curve defined over a finite field.     

On a functional equation concerning affine transformations

In this paper we prove that, if the functional equation u(Rx + p) = α(R, p) u(x) + β(R, p) has a nonconstant solution u, then α and β have to be of a very special form. Finally, we obtain, under suitable hypotheses, the general solution of this equation.     

On symmetries of the heat equation

Two aims are pursued in this article. The first one is methodological and consists of demonstrating the symmetry calculation techniques based on the commutation relation on the example of the quasi-linear heat equation. The second one consists of investigating the following open question: is the algebra of higher symmetries of the linear heat equation exhausted by those which may be obtained by means of the classical symmetries and well-known recursion operators? A positive solution of this question is given.     

Cohomology of regular differential forms for affine curves

Let C be a complex affine reduced curve, and denote by H¹(C) its first truncated cohomology group, i.e. the quotient of all regular differential 1-forms by exact 1-forms. First we introduce a nonnegative invariant μ^′(C,x) that measures the complexity of the singularity of C at the point x, and we establish the following formula:

     

On some boundary element methods for the heat equation

Summary The object of this paper is to study some boundary element methods for the heat equation. Two approaches are considered. The first, based on the heat potential, has been studied numerically by previous authors. Here the convergence analysis in one space dimension is presented. In the second approach, the heat equation is first descretized in time and the resulting elliptic problem is put in the boundary formulation. A straight forward implicit method and Crank-Nicolson's method are thus studied. Again convergence in one space dimension is proved.     

Some results concerning Abel's equation in the theory of differential equations and consequences for the affine geometry of closed plane curves

Aequationes mathematicae -     

On duality theory for non-convex semidefinite programming

In this paper, with the help of convex-like function, we discuss the duality theory for nonconvex semidefinite programming. Our contributions are: duality theory for the general nonconvex semidefinite programming when Slater’s condition holds; perfect duality for a special case of the nonconvex semidefinite programming for which Slater’s condition fails. We point out that the results of Fan (Appl. Math. Lett. 18:1068–1073, 2005) can be regarded as a special case of our result.     

Global existence for a quasi-linear evolution equation with a non-convex energy

We establish the existence of global in time weak solutions to the initial-boundary value problem related to the dynamics of coherent solid-solid phase transitions in viscoelasticity. The class of the stored energy functionals includes the double well potential, and a general convolution damping term is considered.

     

Neumann heat flow and gradient flow for the entropy on non-convex domains

For large classes of non-convex subsets Y in \(\mathbb R^n\) or in Riemannian manifolds (M, g) or in RCD-spaces (X, d, m) we prove that the gradient flow for the Boltzmann entropy on the restricted metric measure space \((Y,d_Y,m_Y)\) exists—despite the fact that the entropy is not semiconvex—and coincides with the heat flow on Y with Neumann boundary conditions.