in this paper we propose a descent method for solving variational inequality problems where the underlying operator is nonsmooth, locally Lipschitz, and monotone over a closed, convex feasible set. The idea is to combine a descent method for variational inequality problems whose operators are nonsmooth, locally Lipschitz, and strongly monotone, with the Tikonov-Browder regularization technique....

*Correspondence: [email protected]; [email protected] 2Department of Mathematics, Guangdong University of Education, Guangzhou, Guangdong 510303, P.R. China Full list of author information is available at the end of the article Abstract By using the way of weight functions and Hadamard’s inequality, a half-discrete Hilbert-type inequality similar to Mulholland’s inequality with a best constant...

The classical Bargmann inequality 1 originates from the nonrelativistic quantummechanics and gives an upper bound for the number of bound states produced by a radially symmetric potential in the two-body system. In the subsequent papers, various proofs and reformulations of this inequality have been presented, we refer to 2, Chapter XIII , and to 3–5 for some details. In the language of singula...

In this paper we study in detail some spectral properties of the magnetic discrete Laplacian. We identify its form-domain, characterize the absence of essential spectrum and provide the asymptotic eigenvalue distribution.

The L-theory of solutions of the homogeneous A-harmonic equation d A x, dω 0 for differential forms has been very well developed in recent years. Many L-norm estimates and inequalities, including the Hardy-Littlewood inequalities, Poincaré inequalities, Caccioppoli-type estimates, and Sobolev imbedding inequalities, for solutions of the homogeneous A-harmonic equation have been established; see...

The well-known Schwarz–Pick lemma states that any analytic mapping φ of the unit disk U into itself satisfies the inequality |φ′(z)| ≤ 1− |φ(z)| 2 1− |z|2 , z ∈ U. This estimate remains the same if we restrict ourselves to univalent mappings. The lower estimate is |φ′(z)| ≥ 0 generally or |φ′(z)| > 0 for univalent functions. To make the lower estimate nontrivial we consider univalent functions ...

Finding relational expressions which exist frequently in one class of data while not in the other class of data is an interesting work. In this paper, a relational expression of this kind is defined as a contrast inequality. Gene Expression Programming (GEP) is powerful to discover relations from data and express them in mathematical level. Hence, it is desirable to apply GEP to such mining tas...

Abstract. As the classical (p, q)-Poincaré inequality is known to fail for 0 < p < 1, we introduce the notion of weighted multilinear Poincaré inequality as a natural alternative when m-fold products and 1/m < p are considered. We prove such weighted multilinear Poincaré inequalities in the subelliptic context associated to vector fields of Hörmader type. We do so by establishing multilinear re...

1. Integration on manifolds. 1 2. The extension of the Levi-Civita connection. 4 3. Covariant derivatives. 7 4. The Laplace operator. 9 5. Self-adjoint extension of the Laplace operator. 11 6. The Poincaré Lemma. 13 7. de Rham Theorem. 14 8. Formal Hodge Theorem. 16 9. The Hodge theorem. 17 10. Proof of the Hodge Theorem. 17 11. More about elliptic regularity. 22 12. Semigroups and their genera...

In this article we further develop a perturbation approach to the Rayleigh– Ritz approximations from our earlier work. We both sharpen the estimates and extend the applicability of the theory to nonnegative definite operators . The perturbation argument enables us to solve two problems in one go: We determine which part of the spectrum of the operator is being approximated by the Ritz values an...