In this paper, we focus on the minimization problem with , which is challenging due to the norm being non-Lipschizian. In theory, we derive computable lower bounds for nonzero entries of the generalized first-order stationary points of minimization, and hence of its local minimizers. In algorithms, based on three locally Lipschitz continuous -approximation to norm, we design several iterative reweighted and methods to solve those approximation problems. Furthermore, we show that any accumulation point of the sequence generated by these methods is a generalized first-order stationary point of minimization. This result, in particular, applies to the iterative reweighted methods based on the new Lipschitz continuous -approximation introduced by Lu (Math Program 147(1-2):277-307, 2014), provided that the approximation parameter is below a threshold value. Numerical results are also reported to demonstrate the efficiency of the proposed methods.
In the current work we studied Hardy type and Hardy type inequalities in the half-space on the H-type group, where the Hardy inequality in the upper half-space R-+(n) was proved by Tidblom in (J. Funct. Anal. 221:482-495, 2005).
We give criteria for establishing that a measure is L-p-improving. Many Riesz product measures and Cantor measures satisfy this criteria, as well as certain Markov measures.
An oriented graph is a directed graph without any directed cycle of length at most 2. In this article, we characterize the oriented L(p, 1)-labeling span lambda(o)(p,1) ((G) over right arrow of an oriented graph (G) over right arrow using graph homomorphisms. Using this characterization and probabilistic techniques we prove the upper bound of lambda(o)(p,1) (G Delta) <=3D 2.Delta(2).2(Delta) + (p Delta), where G(Delta) is the family of graphs with maximum degree at most Moreover, by proving a lower bound exponential in Delta for the same graph family we conclude that the upper bound is not too far from being optimal. We also settle an open problem given by Sen (DMGT 2014) for the family of outerplanar graphs O by showing lambda(o)(2,1) (O) =3D 10.
We consider the equivalence relations on R-omega x omega induced by the Banach subspaces l(p)(l(q)). We show that if r > p >=3D q >=3D 1, then there is no Borel reduction from the equivalence relation X-omega/l(r) (X), where X is a Banach space, to R-omega x omega/l(p)(l(q)) (C) 2015 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
For each 1 a parts per thousand currency sign p < a, a space of integrable Schwartz distributions L' (p) , is defined by taking the distributional derivative of all functions in L (p) . Here, L (p) is with respect to Lebesgue measure on the real line. If f a L' (p) such that f is the distributional derivative of F a L (p) , then the integral is defined as , where g a L (q) , and 1/p + 1/q =3D1. A norm is aEuro-faEuro- (p) (') =3D aEuro-FaEuro- (p) . The spaces L' (p) and L (p) are isometrically isomorphic. Distributions in L' (p) share many properties with functions in L (p) . Hence, L' (p) is reflexive, its dual space is identified with L (q) , there is a type of Holder inequality, continuity in norm, convergence theorems, Gateaux derivative. It is a Banach lattice and abstract L-space. Convolutions and Fourier transforms are defined. Convolution with the Poisson kernel is well defined and provides a solution to the half plane Dirichlet problem, boundary values being taken on in the new norm. A product is defined that makes L'(1) into a Banach algebra isometrically isomorphic to the convolution algebra on L (1). Spaces of higher order derivatives of L (p) functions are defined. These are also Banach spaces isometrically isomorphic to L (p) .
By a classical result of Kadec and Pelczynski (1962), every normalized weakly null sequence in L-p, p > 2, contains a subsequence equivalent to the unit vector basis of l(2) or to the unit vector basis of l(p). In this paper we investigate the case 1 <=3D p < 2 and show that a necessary and sufficient condition for the first alternative in the Kadec-Pelczynski theorem is that the limit random measure mu of the sequence satisfies integral(R) x(2)d mu(x) is an element of L-p/2.