Communications in Contemporary Mathematics

We study a semilinear heat equation with strong absorption u_t = Δu - u^p with 0 < p < 1 in R^N. A solution is known to develop dead-core in finite time for a wide class of initial data. We construct specific solutions with exact dead-core rates faster than the one given by the corresponding ODE. They are constructed formally by means of matched asymptotic expansion technique and rigorously by means of topological fixed-point arguments based on a priori estimates. To obtain the a priori estimates, we analyze a certain linearized problem in a new function space .

We study the minimizers of the Ginzburg–Landau free energy functional in the class (u, A) ∈ H¹(Ω; ℂ) × H¹(Ω; ℝ²) with |u| = 1 on ∂Ω, where Ω is a bounded simply connected domain in ℝ². We consider the connected components of this class defined by the prescribed topological degree d of u on the boundary ∂Ω. We show that for d ≠ 0 the minimizers exist if 0 < λ ≤ 1 and do not exist if λ > 1, where λ is the coupling constant ( is the Ginzburg–Landau parameter). We also establish the asymptotic locations of vortices for λ → 1 - 0 (the critical value λ = 1 is known as the Bogomol'nyi integrable case).

We show that for any minuscule or cominuscule homogeneous space X, the Gromov–Witten varieties of degree d curves passing through three general points of X are rational or empty for any d. Applying techniques of Buch and Mihalcea BM to constructions of the authors together with Manivel [5], we deduce that the equivariant K-theoretic three points Gromov–Witten invariants are equal to classical equivariant K-theoretic invariants on auxilliary spaces again homogeneous under Aut(X).

In this paper, we define invariants for primitive Legendrian knots in lens spaces L(p, q), q ≠ 1. The main invariant is a differential graded algebra which is computed from a labeled Lagrangian projection of the pair (L(p, q), K). This invariant is formally similar to a DGA defined by Sabloff which is an invariant for Legendrian knots in smooth S¹-bundles over Riemann surfaces. The second invariant defined for K ⊂ L(p, q) takes the form of a DGA enhanced with a free cyclic group action and can be computed from a cyclic cover of the pair (L(p, q), K).

In this paper, we study the Hochschild cohomology ring of convolution algebras associated to orbifolds, as well as their deformation quantizations. In the first case, the ring structure is given in terms of a wedge product on twisted polyvectorfields on the inertia orbifold. After deformation quantization, the ring structure defines a product on the cohomology of the inertia orbifold. We study the relation between this product and an S¹-equivariant version of the Chen–Ruan product. In particular, we give a de Rham model for this equivariant orbifold cohomology.

This note concerns the paper "On the controllability of a coupled system of two Korteweg–de Vries equations" by Micu et al. [2]. They study a nonlinear coupled system of two Korteweg–de Vries equations and prove that the system is controllable by using four boundary controls. Here, we prove that in some cases it is possible to get the controllablity of the system by using only two controls. This can be done depending on both the spatial domain and the control time.