15 July 2015, Volume 31 Issue 7

    

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  Differential Equations and Singular Vectors in Verma Modules over sl(n,C)

  Wei XIAO

  Acta Mathematica Sinica. 2015, 31(7): 1057-1066. https://doi.org/10.1007/s10114-015-4640-7

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  Xu introduced a system of partial differential equations to investigate singular vectors in the Verma module of highest weight λ over sl(n,C). He gave a differential-operator representation of the symmetric group S_n on the corresponding space of truncated power series and proved that the solution space of the system is spanned by {σ(1) | σ ∈ S_n}. It is known that S_n is also the Weyl group of sl(n,C) and generated by all reflections s_α with positive roots α. We present an explicit formula of the solution s_α(1) for every positive root α and show directly that s_α(1) is a polynomial if and only if <λ + ρ, α> is a nonnegative integer. From this, we can recover a formula of singular vectors given by Malikov et al..
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  Poincaré and Sobolev Inequalities for Vector Fields Satisfying Hörmander's Condition in Variable Exponent Sobolev Spaces

  Xia LI, Guo Zhen LU, Han Li TANG

  Acta Mathematica Sinica. 2015, 31(7): 1067-1085. https://doi.org/10.1007/s10114-015-4488-x

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  In this paper, we will establish Poincaré inequalities in variable exponent non-isotropic Sobolev spaces. The crucial part is that we prove the boundedness of the fractional integral operator on variable exponent Lebesgue spaces on spaces of homogeneous type. We obtain the first order Poincaré inequalities for vector fields satisfying Hörmander's condition in variable non-isotropic Sobolev spaces. We also set up the higher order Poincaré inequalities with variable exponents on stratified Lie groups. Moreover, we get the Sobolev inequalities in variable exponent Sobolev spaces on whole stratified Lie groups. These inequalities are important and basic tools in studying nonlinear subelliptic PDEs with variable exponents such as the p(x)-subLaplacian. Our results are only stated and proved for vector fields satisfying Hörmander's condition, but they also hold for Grushin vector fields as well with obvious modifications.
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  Free Completely J^(ι)-simple Semigroups

  Jun Ying GUO, Xiao Jiang GUO, Juan Ying DING

  Acta Mathematica Sinica. 2015, 31(7): 1086-1096. https://doi.org/10.1007/s10114-015-4117-8

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  A semigroup is called completely J^(ι)-simple if it is isomorphic to some Rees matrix semigroup over a left cancellative monoid and each entry of whose sandwich matrix is in the group of units of the left cancellative monoid. It is proved that completely J^(ι)-simple semigroups form a quasivarity. Moreover, the construction of free completely J^(ι)-simple semigroups is given. It is found that a free completely J^(ι)-simple semigroup is just a free completely J ^*-simple semigroup and also a full subsemigroup of some completely simple semigroups.
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  Partial Regularity for the 3D Magneto-hydrodynamics System with Hyper-dissipation

  Wei REN, Gang WU

  Acta Mathematica Sinica. 2015, 31(7): 1097-1112. https://doi.org/10.1007/s10114-015-4498-8

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  We prove that for the 3D MHD equations with hyper-dissipations (-Δ)^α (1 <α <5/4) the Hausdorff dimension of singular set at the first blowing up time is at most 5 - 4α, by means of physical and frequency localization, Bony's paraproduct and Littlewood-Paley theory.
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  Trivial and Simple Spectrum for SL(d,R) Cocycles with Free Base and Fiber Dynamics

  Mário BESSA, Paulo VARANDAS

  Acta Mathematica Sinica. 2015, 31(7): 1113-1122. https://doi.org/10.1007/s10114-015-4417-z

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  Let AC_D(M,SL(d,R)) denote the pairs (f,A) so that f ∈ A ⊂ Diff¹(M) is a C¹-Anosov transitive diffeomorphisms and A is an SL(d,R) cocycle dominated with respect to f. We prove that open and densely in AC_D(M,SL(d,R)), in appropriate topologies, the pair (f,A) has simple spectrum with respect to the unique maximal entropy measure μ_f. Then, we prove prevalence of trivial spectrum near the dynamical cocycle of an area-preserving map and also for generic cocycles in AutLeb(M) × L^p(M,SL(d,R)).
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  Explicit Estimates for Solutions of Nonlinear Radiation-type Problems

  Luisa CONSIGLIERI

  Acta Mathematica Sinica. 2015, 31(7): 1123-1140. https://doi.org/10.1007/s10114-015-4419-x

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  We establish the existence of weak solutions of a nonlinear radiation-type boundary value problem for elliptic equation on divergence form with discontinuous leading coefficient. Quantitative estimates play a crucial role on the real applications. Our objective is the derivation of explicit expressions of the involved constants in the quantitative estimates, the so-called absolute or universal bounds. The dependence on the leading coefficient and on the size of the spatial domain is precise. This work shows that the expressions of those constants are not so elegant as we might expect.
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  On a Problem of Hornik

  Ting Fan XIE, Fei Long CAO

  Acta Mathematica Sinica. 2015, 31(7): 1141-1148. https://doi.org/10.1007/s10114-015-3759-x

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  In 1991, Hornik proved that the collection of single hidden layer feedforward neural networks (SLFNs) with continuous, bounded, and non-constant activation function σ is dense in C(K) where K is a compact set in R^s (see Neural Networks, 4(2), 251-257 (1991)). Meanwhile, he pointed out “Whether or not the continuity assumption can entirely be dropped is still an open quite challenging problem”. This paper replies in the affirmative to the problem and proves that for bounded and continuous almost everywhere (a.e.) activation function σ on R, the collection of SLFNs is dense in C(K) if and only if σ is un-constant a.e..
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  The Equivariant Family Index Theorem in Odd Dimensions

  Kai Hua BAO, Jian WANG, Yong WANG

  Acta Mathematica Sinica. 2015, 31(7): 1149-1162. https://doi.org/10.1007/s10114-015-3637-6

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  In this paper, we prove a local odd dimensional equivariant family index theorem which generalizes Freed's odd dimensional index formula. Then we extend this theorem to the noncommutative geometry framework. As a corollary, we get the odd family Lichnerowicz vanishing theorem and the odd family Atiyah-Hirzebruch vanishing theorem.
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  Some Properties of β-wordlength Pattern for Four-level Designs

  Wei Wei SHENG, Xia Ming LI, Yu TANG

  Acta Mathematica Sinica. 2015, 31(7): 1163-1170. https://doi.org/10.1007/s10114-015-3616-y

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  Fractional factorial designs have played a prominent role in the theory and practice of experimental design. For designs with qualitative factors under an ANOVA model, the minimum aberration criterion has been frequently used; however, for designs with quantitative factors, a polynomial regression model is often established, thus the β-wordlength pattern can be employed to compare different fractional factorial designs. Although the β-wordlength pattern was introduced in 2004, its properties have not been investigated extensively. In this paper, we will present some properties of β-wordlength pattern for four-level designs. These properties can help find better designs with quantitative factors.
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  Invariance for Stochastic Differential Systems with Time-dependent Constraining Sets

  Marius APETRII, Mihaela-Hanako MATCOVSCHI, Octavian PĂSTRĂVANU, Eduard ROTENSTEIN

  Acta Mathematica Sinica. 2015, 31(7): 1171-1188. https://doi.org/10.1007/s10114-015-3562-8

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  The first part of this article presents invariance criteria for a stochastic differential equation whose state evolution is constrained by time-dependent security tubes. The key results of this section are derived by considering an equivalent problem where the square of distance function represents a viscosity solution to an adequately defined partial differential equation. The second part of the paper analyzes the broader context when solutions are constrained by more general time-dependent convex domains. The approach relies on forward stochastic variational inequalities with oblique reflection, the generalized subgradients acting as a reacting process that operates only when the solution reaches the boundary of the domain.
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  Strong Law of Large Number for Branching Hunt Processes

  Li WANG

  Acta Mathematica Sinica. 2015, 31(7): 1189-1202. https://doi.org/10.1007/s10114-015-3413-7

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  In this paper, we prove that, under certain conditions, a strong law of large number holds for a class of branching particle systems X corresponding to the parameters (Y, β, ψ), where Y is a Hunt process and ψ is the generating function for the offspring. The main tool of this paper is the spine decomposition and we only need an L logL condition.
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  On Two Classes of Finite Inseparable p-Groups

  Joseph KIRTLAND

  Acta Mathematica Sinica. 2015, 31(7): 1203-1214. https://doi.org/10.1007/s10114-015-3309-6

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  A finite group is inseparable, it does not split over any proper nontrivial normal subgroup; that is, if it has no nontrivial semidirect product decompositions. This paper investigates two classes of finite inseparable p-groups and, for p ≥ 3, establishes a necessary and sufficient condition for inseparability.