Instability of standing waves for a weakly coupled non-linear Schrödinger system

This article discusses the weakly coupled non-linear Schrödinger equations. With the variational characterization of the ground state solutions, the potential well argument and the concavity method, we derive a sharp condition for blow-up and global existence to the solutions of the Cauchy problem. At the same time, we also prove the instability of standing waves.     

Minimal Blocking Sets in PG(n, 2) and Covering Groups by Subgroups

In this article we prove that a set of points B of PG(n, 2) is a minimal blocking set if and only if ?B? = PG(d, 2) with d odd and B is a set of d + 2 points of PG(d, 2) no d + 1 of them in the same hyperplane. As a corollary to the latter result we show that if G is a finite 2-group and n is a positive integer, then G admits a ? _n+1-cover if and only if n is even and G? (C ₂) ⁿ , where by a ? _m -cover for a group H we mean a set 𝒞 of size m of maximal subgroups of H whose set-theoretic union is the whole H and no proper subset of 𝒞 has the latter property and the intersection of the maximal subgroups is core-free. Also for all n < 10 we find all pairs (m,p) (m > 0 an integer and p a prime number) for which there is a blocking set B of size n in PG(m,p) such that ?B? = PG(m,p).     

Character Tables of Parabolic Subgroups of the Chevalley Groups of Type G 2

The main aim of this paper is to construct the character tables of the parabolic subgroups of the Chevalley groups G ₂(q), where q is a power of a prime p > 3.     

New Constructions of Groups Without Semiregular Subgroups

An elusive permutation group is a transitive permutation group with no fixed point free elements of prime order, or equivalently, no nontrivial semiregular subgroups. We provide several new constructions of elusive groups, some of which enable us to build elusive groups with new degrees.     

On SS-Quasinormal Subgroups of Finite Groups

A subgroup of a group G is said to be Sylow-quasinormal (S-quasinormal) in G if it permutes with every Sylow subgroup of G. A subgroup H of a group G is said to be Supplement-Sylow-quasinormal (SS-quasinormal) in G if there is a supplement B of H to G such that H is permutable with every Sylow subgroup of B. In this article, we investigate the influence of SS-quasinormal of maximal or minimal subgroups of Sylow subgroups of the generalized Fitting subgroup of a finite group.     

On the “Shidov Property” and a Formation Satisfying It

If G is a nonsoluble finite group, the intersection of the maximal subgroups of G which are not nilpotent is the Frattini subgroup of G. This was proved by Shidov (1971 Shidov , L. I. ( 1971 ). On maximal subgroups of finite groups . Sibirsk. Mat. Zh. 12 ( 3 ): 682 – 683 . [Google Scholar]). The authors here present a new formation ? larger than the formation of the nilpotent groups for which the analogous of the theorem of Shidov holds. The theorem makes use of the classification of finite simple groups.     

Blow-up for a weakly dissipative modified two-component Camassa–Holm system

In this paper, we study the Cauchy problem of a weakly dissipative modified two-component Camassa–Holm (MCH2) system. We first derive the precise blow-up scenario and then give several criteria guaranteeing the blow-up of the solutions. We finally discuss the blow-up rate of the blowing-up solutions.     

Least-squares regularized regression with dependent samples and q-penalty

Least-squares regularized learning algorithms for regression were well-studied in the literature when the sampling process is independent and the regularization term is the square of the norm in a reproducing kernel Hilbert space (RKHS). Some analysis has also been done for dependent sampling processes or regularizers being the qth power of the function norm (q-penalty) with 0?q?≤?2. The purpose of this article is to conduct error analysis of the least-squares regularized regression algorithm when the sampling sequence is weakly dependent satisfying an exponentially decaying α-mixing condition and when the regularizer takes the q-penalty with 0?q?≤?2. We use a covering number argument and derive learning rates in terms of the α-mixing decay, an approximation condition and the capacity of balls of the RKHS.     

Bounds for the Number of Conjugacy Classes of Non-Normal Subgroups in a Finite p-Group

Let ν(G) be the number of conjugacy classes of non-normal subgroups of a finite group G. We obtain two new lower bounds for ν(G) when G is a non-abelian finite p-group and p is odd. More precisely, if |G| =p ⁿ , exp Z(G) = p ^e , and exp G/G′ =p ^f , let us define λ(G) = n ? e and κ(G) = n ? f. Then we prove that ν(G) ≥ p(λ(G) ?3) +2 and ν(G) ≥ p(κ(G) ?3) +2. The first bound improves the bound ν(G) ≥ λ(G) ?1 given by [10 La Haye , R. , Rhemtulla , A. ( 1999 ). Groups with a bounded number of conjugacy classes of non-normal subgroups . J. Algebra 214 : 41 – 63 .[Crossref], [Web of Science ®] , [Google Scholar]], and almost in every case, the second one improves the bound ν(G) ≥ p(k ? 1) +1 obtained by [6 Fernández-Alcober , G. A. , Legarreta , L. ( 2008 ). Conjugacy classes of non-normal subgroups in finite nilpotent groups . J. Group Theory 11 ( 3 ): 381 – 397 .[Crossref] , [Google Scholar]], where k is defined by the condition that |G′| =p ^k .