We consider a family of self-adjoint 2 × 2-block operator matrices Ãϑ in the space L2(0, 1) ⊕ L2(0, 1), depending on the real parameter ϑ. If Ã0 has an eigenvalue that is embedded in the essential spectrum, then it is shown that for ϑ ≠ 0 this eigenvalue in general disappears, but the resolvent of Ãϑ has a pole on the unphysical sheet of the Riemann surface. Such a pole is called a resonance pole. The unphysical sheet arises from analytic continuation from the upper half-plane C+ across the essential spectrum. Furthermore, the asymptotic behaviour of this resonance pole for small ϑ is investigated. The results are proved by considering a certain λ-rational Sturm–Liouville problem and its Titchmarsh–Weyl coefficient.

We compute critical groups in semilinear elliptic boundary-value problems in which the nonlinear term may fail to have asymptotic limits at zero and at infinity. As applications, we prove several new existence results.

We introduce what are called regular materials for which, by definition, the corresponding solution of the classical periodic homogenization problem remains bounded in . We give examples of two types of such materials depending on whether the coefficients representing them belong to W1,∞ or not. A complete characterization is obtained in the former case.

We approximate functionals depending on the gradient of u and on the behaviour of u near the discontinuity points by families of non-local functionals where the gradient is replaced by finite differences. We prove pointwise convergence, Γ-convergence and a compactness result, which implies, in particular, the convergence of minima and minimizers.

We prove that, contrary to the L1-Nash inequality, there exists a second best constant for the L2-Nash inequality on any smooth compact Riemannian manif

In this paper, we study the asymptotic decay properties in both spatial and temporal variables for a class of weak and strong solutions, by constructing the weak and strong solutions in corresponding weighted spaces. It is shown that, for the strong solution, the rate of temporal decay depends on the rate of spatial decay of the initial data. Such rates of decay are optimal.

The Brezis–Nirenberg equation and the scalar field equation on the three-dimensional unit ball are studied. Under the Robin condition, we show the existence and uniqueness of radial solutions in a unified way. In particular, it is shown that the global structure of solutions changes qualitatively when a parameter in the boundary condition exceeds a certain critical value.

Lower semicontinuity and relaxation results in BV are obtained for multiple integrals where the energy density f satisfies lower semicontinuity conditions with respect to (x, u) and is not subjected to coercivity hypotheses. These results call for methods recently developed in the calculus of variations.

We discuss the three-space problem on discreteness for the Jacobson topology on the spectrum of a C*-algebra in detail. As an application, it is shown that a C*-algebra A is a dual C*-algebra if and only if a closed ideal I of A and the quotient A/I are dual C*-algebras and the open central projection, in the second dual of A, corresponding to I is a multiplier for A.

We consider Dirichlet problems of the form −|x|αΔu = λu + g(u) in Ω, u = 0 on ∂Ω, where α, λ ∈ R, g ∈ C(R) is a superlinear and subcritical function, and Ω is a domain in R2. We study the existence of positive solutions with respect to the values of the parameters α and λ, and according that 0 ∈ Ω or 0 ∈ ∂Ω, and that Ω is an exterior domain or not.

The Brezis-Nirenberg equation and the scalar field equation on the three-dimensional unit ball are studied. Under the Robin condition, we show the existence and uniqueness of radial solutions in a unified way. In particular, it is shown that the global structure of solutions changes qualitatively when a parameter in the boundary condition exceeds a certain critical value.

We compute critical groups in semilinear elliptic boundary-value problems in which the nonlinear term may fail to have asymptotic limits at zero and at infinity. As applications, we prove several new existence results.

Lower semicontinuity and relaxation results in BV are obtained for multiple integralswhere the energy density f satisfies lower semicontinuity conditions with respect to (x, u) and is not subjected to coercivity hypotheses. These results call for methods recently developed in the calculus of variations.

In this paper, we study the asymptotic decay properties in both spatial and temporal variables for a class of weak and strong solutions, by constructing the weak and strong solutions in corresponding weighted spaces. It is shown that, for the strong solution, the rate of temporal decay depends on the rate of spatial decay of the initial data. Such rates of decay are optimal.

We consider a family of self-adjoint 2 × 2-block operator matrices Ãϑ in the space L2(0, 1) ⊕ L2(0, 1), depending on the real parameter ϑ. If Ã0 has an eigenvalue that is embedded in the essential spectrum, then it is shown that for ϑ ≠ 0 this eigenvalue in general disappears, but the resolvent of Ãϑ has a pole on the unphysical sheet of the Riemann surface. Such a pole is called a resonance pole. The unphysical sheet arises from analytic continuation from the upper half-plane ℂ+ across the essential spectrum. Furthermore, the asymptotic behaviour of this resonance pole for small ϑ is investigated. The results are proved by considering a certain λ-rational Sturm-Liouville problem and its Titchmarsh–Weyl coefficient.

We consider Dirichlet problems of the form −|x|αΔu = λu + g(u) in Ω, u = 0 on ∂Ω, where α, λ ∈ ℝ, g ∈ C(ℝ) is a superlinear and subcritical function, and Ω is a domain in ℝ2. We study the existence of positive solutions with respect to the values of the parameters α and λ, and according that 0 ∈ Ω or 0 ∈ ∂Ω, and that Ω is an exterior domain or not.

We prove limiting imbeddings of spaces with dominating mixed derivatives into the spaces of almost Lipschitz continuous functions.

We approximate functionals depending on the gradient of u and on the behaviour of u near the discontinuity points by families of non-local functionals where the gradient is replaced by finite differences. We prove pointwise convergence, Γ-convergence and a compactness result, which implies, in particular, the convergence of minima and minimizers.

Fix q and let Mn be an n × n matrix with entries drawn independently from the finite field Fq according to some distribution μn. It is shown that, except in certain pathological cases, the probability that Mn is nonsingular is asymptotically the same as for uniform entries; that is,

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It has been conjectured by Alspach [2] that given integers n and m1, …, mt with 3 [les ] mi [les ] n and [sum ]ti=1mi = (n2) (n odd) or [sum ]ti=1mi = (n2) − n/2 (n even), then one can pack Kn (n odd) or Kn minus a 1-factor (n even) with cycles of lengths m1, …, mt. In this paper we show that if the cycle lengths mi are bounded by some linear function of n and n is sufficiently large then this conjecture is true.