Let G be a simple, compact Lie group and let be the gauge group of the principal G-bundle over S4 with second Chern class k. McGibbon classified the groups G that are homotopy commutative when localized at a prime p. We show that in many cases the homotopy commutativity of G, or its failure, determines that of .

In this paper we study the analytical obstructions preventing the germ of a generic analytic family of elliptic ordinary differential equations from being isochronous. The formal obstructions are purely arithmetical and can be set in terms of commutators. The analytical obstructions arise out of the non-convergence of the normalizing coordinates in the Poincaré–Dulac normalization process. The temporal part of the invariant is the obstruction to isochronicity in a particular context.

We study the existence and multiplicity of positive solutions for the following nonlinear Schrödinger–Poisson equations: where 2 < p ≤ 3 or 4 ≤ p < 6, λ > 0 and Q ∈ C(ℝ3). We show that the number of positive solutions is dependent on the profile of Q(x).

We prove the following theorem: if w is a quasi-conformal mapping of the unit disc onto itself satisfying elliptic partial differential inequality , then w is Lipschitz continuous. This result extends some recent results where, instead of an elliptic differential operator, only the Laplace operator is considered.

On any spacelike surface in a light cone of four-dimensional Lorentz–Minkowski space, a distinguished smooth function is considered. We show how both extrinsic and intrinsic geometry of such a surface are codified by this function. The existence of a local maximum is assumed to decide when the spacelike surface must be totally umbilical, deriving a Liebmann-type result. Two remarkable families of examples of spacelike surfaces in a light cone are explicitly constructed. Finally, several results that involve the first eigenvalue of the Laplace operator of a compact spacelike surface in a light cone are obtained.

In this paper, we study the global existence of positive solutions to the norm-preserving non-local heat flow of the porous-media type equations on the compact Riemannian manifold (M, g) with the Cauchy data u0 > 0 on M, where r ≥ 1, p > 1 and λ(t) is chosen to make the L2-norm of the solution u (or a power of u) constant. We show that the limit is an eigenfunction for the Laplacian operator. We use some tricky estimates through the Sobolev imbedding theorem and the Moser iteration method.

We study the semiclassical limit of the least energy solutions to the nonlinear Dirac equation for x ∈ ℝ3. We prove that the equation has least energy solutions for all ħ > 0 small, and, in addition, that the solutions converge in a certain sense to the least energy solution of the associated limit problem as ħ → 0.

For a real-valued function V of the Faddeev–Marchenko class, we prove the norm-resolvent convergence, as ε → 0, of a family Sε of one-dimensional Schrödinger operators on the line of the form

Under certain conditions, the functions ε−2V (x/ε) converge in the sense of distributions as ε → 0 to δ′ (x), and then the limit S0 of Sε may be considered as a ‘physically motivated’ interpretation of the one-dimensional Schrödinger operator with potential δ′.

We consider the positive solutions to singular boundary-value problems of the form where λ > 0, β ∈ (0,1) and Ω is a bounded domain in ℝN, N ≥ 1, with smooth boundary ∂Ω. Here, we assume that f: [0, ∞) → (0, ∞) is a C1 non-decreasing function and f(s)/sβ is decreasing for s large. We establish the uniqueness of the positive solution when λ is large.

Let n ≥ 3 and let ψλ0 be the radially symmetric solution of Δ log ψ + 2βψ + βx · ∇ψ = 0 in ℝn, ψ(0) = λ(0), for some constants λ0 > 0, β > 0. Suppose u0 ≥ 0 satisfies u0 − ψλ0 ∈ L1 (ℝn) and u0 (x) ≈ (2(n − 2)/β)(log∣x∣/∣x∣2) as ∣x∣ → ∞. We prove that the rescaled solution ũ(x,t) = e2βtu(eβtx, t) of the maximal global solution u of the equation ut = Δ log u in ℝn × (0, ∞), u(x, 0) = u0 (x) in ℝn, converges uniformly on every compact subset of ℝn and in L1 (ℝn) to ψλ0 as t → ∞. Moreover, ∥ũ(·, t) − ψλ0∥L1(ℝn) ≤ e−(n−2)βt∥u0 − ψλ0∥L1(ℝn) for all t ≥ 0.

We present new results in the theory of the classical theta functions of Jacobi: series expansions and defining ordinary differential equations (ODEs). The proposed dynamical systems turn out to be Hamiltonian and define fundamental differential properties of theta functions; they also yield an exponential quadratic extension of the canonical θ-series. An integrability condition of these ODEs explains the appearance of the modular ϑ-constants and differential properties thereof. General solutions to all the ODEs are given. For completeness, we also solve the Weierstrassian elliptic modular inversion problem and consider its consequences.