关于Witt代数基本子代数的一点注记

设g=W_1是特征p3的代数闭域k上的Witt代数.本文确定了g的极大基本子代数.进一步具体给出了最大维数的基本子代数的G共轭类,这里G是g的自同构群.从而证明了最大维数为(p-1)/2的基本子代数射影簇E((p-1)/2,g)是不可约的且是一维的.更进一步,证明了E(1,g)是不可约的,具有维数p-2,而E(2,g)是等维的,共有(p-3)/2个不可约分支,且每个不可约分支的维数是p-4.而当3≤r≤(p-3)/2时,E(r,g)是可约的.给出了E(r,g)(3≤r≤(p-3)/2)维数的一个下界.     

Microlocal Analysis of the Doppler Transform on R<Superscript>3</Superscript>

The Doppler transform of a vector field $F = (f_1,f_2,f_3)$ on $\mathbb{R}^3$ is defined by \[\displaystyle\mathcal{D}F(x,\omega) = \sum_j\int_\mathbb{R} \omega_j f_j(x+t\omega)\, dt~,\] where $x\in \mathbb{R}^3$ and $\omega \in S^2$ specifies the direction of a line passing through $x$. In practical applications, $\mathcal{D}F$ is known only for a small subset of lines in $\mathbb{R}^3$. In this article, we deal with the case of $\mathcal{D}F$ restricted to all lines passing through a fixed smooth curve. Using techniques from microlocal analysis, we study the problem of recovering the wavefront set of $\mbox{curl}(F)$ from that of the restricted Doppler transform of $F$.     

The q-Variation of Functions and Spectral Integration from Dominated Ergodic Estimates

For $1\leq q < \infty$, let $\mathfrak{M}_{q}\left( \mathbb{T}\right)$, (respectively, $\mathfrak{M}_{q}\left( \mathbb{R}\right) $) denote the Banach algebra consisting of the bounded complex-valued functions having uniformly bounded $q$-variation on the dyadic arcs of the unit circle, (respectively, on the dyadic intervals of the real line). Suppose that $(\Omega,\mu)$ is a $\sigma$-finite measure space, $1< p < \infty$, and $T:L^{p}(\mu)\rightarrow L^{p}(\mu)$ is a bounded, invertible, separation-preserving linear operator such that the two-sided ergodic means of the linear modulus of $T$ are uniformly bounded in norm. We show that there is a real number $q_{_{0}} > 1$ such that whenever $1\leq q < q_{_{0}}$, $T $ has a norm-continuous functional calculus associated with $\mathfrak{M}_{q}\left(\mathbb{T}\right) $. Our approach is rooted in a dominated ergodic theorem of Mart\{\i}n--Reyes and de la Torre which assigns $T$ a canonical family of bilateral $A_{p}$ weight sequences. We first establish the relevant multiplier properties of $\mathfrak{M}_{q}\left( \mathbb{R}\right) $ classes in weighted settings, transfer the outcome to $\mathfrak{M}_{q}\left(\mathbb{T}\right) $, and then apply the consequent $\mathfrak{M}_{q}\left(\mathbb{T}\right) $ multiplier theorem for weighted settings to the spectral decomposition of $T$. The desired $\mathfrak{M}_{q}\left(\mathbb{T}\right)$-functional calculus for $T$ then results from an extension criterion for spectral integration obtained in the general Banach space setting. The multiplier result for $\mathfrak{M}_{q}\left( \mathbb{R}\right) $ shown at the outset of this process expands the scope of the weighted Marcinkiewicz multiplier theorem from $q=1$ to appropriate values of $q > 1$     

On $\mathfrak{F}_\tau$-$s$-Supplemented Subgroups of Finite Groups

Let $\mathfrak{F}$ be a non-empty formation of groups, $\tau$ a subgroup functor and $H$ a $p$-subgroup of a finite group $G.$ Let $\overline{G}=G/H_G$ and $\overline{H} =H/H_G.$ We say that $H$ is $\mathfrak{F}_\tau$-$s$-supplemented in $G$ if for some subgroup $\overline{T}$ and some $\tau$-subgroup $\overline{S}$ of $\overline{G}$ contained in $\overline{H},$ $\overline{H}\overline{T}$ is subnormal in $\overline{G}$ and $\overline{H} ∩ \overline{T} ≤ \overline{S}Z_{\mathfrak{F}}(\overline{G}).$ In this paper, we investigate the influence of $\mathfrak{F}_\tau$-$s$-supplemented subgroups on the structure of finite groups. Some new characterizations about solubility of finite groups are obtained.     

The Kowalevski top as a reduction of a Hamiltonian system on

We show that the Kowalevski top and Kowalevski gyrostat are obtained as a reduction of a Hamiltonian system on . Therefore the Lax-pair representations for the Kowalevski top and Kowalevski gyrostat are obtained via a direct method by transforming the canonical Lax-pair representation of a system on . Also we show that the nontrivial integral of motion of the Kowalevski top comes from a Casimir function of the Lie-Poisson algebra .

     

On a semilattice of numberings

We study some properties of a $ \mathfrak{c} $ \mathfrak{c} -universal semilattice $ \mathfrak{A} $ \mathfrak{A} with the cardinality of the continuum, i.e., of an upper semilattice of m-degrees. In particular, it is shown that the quotient semilattice of such a semilattice modulo any countable ideal will be also $ \mathfrak{c} $ \mathfrak{c} -universal. In addition, there exists an isomorphism $ \mathfrak{A} $ \mathfrak{A} such that $ {\mathfrak{A} \mathord{\left/ {\vphantom {\mathfrak{A} {\iota \left( \mathfrak{A} \right)}}} \right. \kern-\nulldelimiterspace} {\iota \left( \mathfrak{A} \right)}} $ {\mathfrak{A} \mathord{\left/ {\vphantom {\mathfrak{A} {\iota \left( \mathfrak{A} \right)}}} \right. \kern-\nulldelimiterspace} {\iota \left( \mathfrak{A} \right)}} will be also $ \mathfrak{c} $ \mathfrak{c} -universal. Furthermore, a property of the group of its automorphisms is obtained. To study properties of this semilattice, the technique and methods of admissible sets are used. More exactly, it is shown that the semilattice of mΣ-degrees $ L_{m\Sigma }^{\mathbb{H}\mathbb{F}\left( S \right)} $ L_{m\Sigma }^{\mathbb{H}\mathbb{F}\left( S \right)} on the hereditarily finite superstructure $ \mathbb{H}\mathbb{F} $ \mathbb{H}\mathbb{F} (S) over a countable set S will be a $ \mathfrak{c} $ \mathfrak{c} -universal semilattice with the cardinality of the continuum.     

Maximal Graded Subalgebras of Loop Toroidal Lie Algebras

Let be a central simple Lie algebra over a field . We study the maximal ℤⁿ-graded subalgebra of . Mathematics Subject Classifications (2000) 17B70, 17B65, 17B67.     

Frobenius distributions of Drinfeld modules over finite fields

We express the weighted class number of Drinfeld -modules of rank two with given characteristic polynomial over the finite field , where as an infinite product of local terms. Some auxiliary results of independent interest about characteristic polynomials of Drinfeld modules are given.

     

Approximation by M. Riesz Kernels in L p for p<1

Let α > 0. We consider the linear span $\mathfrak{X}_\alpha \left( {\mathbb{R}^n } \right)$ of scalar Riesz's kernels $\left\{ {\tfrac{1}{{\left| {x - a} \right|^\alpha }}} \right\}_{a \in \mathbb{R}^n }$ and the linear span $\mathfrak{Y}_\alpha \left( {\mathbb{R}^n } \right)$ of vector Riesz's kernels $\left\{ {\tfrac{1}{{\left| {x - a} \right|^{\alpha + 1} }}\left( {x - a} \right)} \right\}_{a \in \mathbb{R}^n }$ . We study the following problems. (1) When is the intersection $\mathfrak{X}_\alpha \left( {\mathbb{R}^n } \right) \cap L^p \left( {\mathbb{R}^n } \right)$ dense in L^p(?ⁿ)? (2) When is the intersection $\mathfrak{Y}_\alpha \left( {\mathbb{R}^n } \right) \cap L^p \left( {\mathbb{R}^n ,\mathbb{R}^n } \right)$ dense in L^p(?ⁿ, ?ⁿ)? Bibliography: 15 titles.     

On the images of Sobolev spaces under the heat kernel transform on the Heisenberg group

The aim of this paper is to obtain certain characterizations for the image of a Sobolev space on the Heisenberg group under the heat kernel transform. We give three types of characterizations for the image of a Sobolev space of positive order $H^m(\mathbb {H}^n), m\in \mathbb {N}^n,$ under the heat kernel transform on $\mathbb {H}^n,$ using direct sum and direct integral of Bergmann spaces and certain unitary representations of $\mathbb {H}^n$ which can be realized on the Hilbert space of Hilbert‐Schmidt operators on $L^2(\mathbb {R}^n).$ We also show that the image of Sobolev space of negative order $H^{-s}(\mathbb {H}^n), s(>0) \in \mathbb {R}$ is a direct sum of two weighted Bergman spaces. Finally, we try to obtain some pointwise estimates for the functions in the image of Schwartz class on $\mathbb {H}^n$ under the heat kernel transform.     

与截面相关的Morrey空间与奇异积分

In this paper, we define the Morrey spaces M_F~(p,q) (Rn) and the Campanato spaces E_F~(p,q) (R~n) associated with a family F of sections and a doubling measure μ, where F is closely related to the Monge-Amp`ere equation. Furthermore, we obtain the boundedness of the Hardy-Littlewood maximal function associated to F, Monge-Amp`ere singular integral operators and fractional integrals on M_F~(p,q)(R~n). We also prove that the Morrey spaces M_F~(p,q) (R~n)and the Campanato spaces E_F~(p,q) (R~n) are equivalent with 1 ≤ q ≤ p ∞.     

Uncertainty Inequalities on Laguerre Hypergroup

In this paper, we give analogues of local uncertainty inequality on ${\mathbb{R}^n}$ for stratified Laguerre hypergroup, connected with the spectral analysis of a given homogeneous sublaplacian L, also indicate how local uncertainty inequalities imply global uncertainty inequalities. It would be interesting to note that we deduce the local uncertainty inequalities for the radial functions on the Heisenberg group. Finally, we extend Heisenberg-Pauli-Weyl uncertainty inequality by ultracontractive properties of the semigroups generated by the differential operator and on the estimate on the heat kernel.     

全矩阵代数的抛物子代数上具有可导性的非线性映射

Let F be a field of characteristic 0, Mn（F） the full matrix algebra over F, t the subalgebra of Mn（F） consisting of all upper triangular matrices. Any subalgebra of Mn（F） containing t is called a parabolic subalgebra of Mn（F）. Let P be a parabolic subalgebra of Mn（F）. A map φ on P is said to satisfy derivability if φ（x·y） = φ（x）·y＋x·φ（y） for all x,y ∈ P, where φ is not necessarily linear. Note that a map satisfying derivability on P is not necessarily a derivation on P. In this paper, we prove that a map φ on P satisfies derivability if and only if φ is a sum of an inner derivation and an additive quasi-derivation on P. In particular, any derivation of parabolic subalgebras of Mn（F） is an inner derivation.     

Some new integral inequalities of Hermite-Hadamard type for $\boldsymbol{(\alpha,m;P)}$-convex functions on co-ordinates

In the paper, the authors introduce the notion ``$(\alpha,m.P)$-convex function on co-ordinates'' and establish new integral inequalities of Hermite--Hadamard type for $(\alpha,m.P)$-convex functions on co-ordinates in a rectangle from the plane $\mathbb{R}_0\times\mathbb{R}$.     

阶极小渐近基

Let N denote the set of all nonnegative integers and A be a subset of N.Let W be a nonempty subset of N.Denote by F~*(W) the set of all finite,nonempty subsets of W.Fix integer g≥2,let A_g(W) be the set of all numbers of the form sum f∈Fa_fg~f where F∈F~*(W)and 1≤a_f≤g-1.For i=0,1,2,3,let W_i = {n∈N|n≡ i(mod 4)}.In this paper,we show that the set A = U_i~3=0 A_g(W_i) is a minimal asymptotic basis of order four.     

On the generalization of Faltings’ Annihilator Theorem

Let R be a commutative Noetherian ring, and let n be a non-negative integer. In this article, by using the theory of Gorenstein dimensions, it is shown that whenever R is a homomorphic image of a Noetherian Gorenstein ring, then the invariants ${\inf\{i \in \mathbb{N}_0|\, \rm{dim\, Supp}(\mathfrak{b}^t H_{\mathfrak{a}}^i(M)) \geq n\, \rm{for\, all}\, t \in \mathbb{N}_0\}}$ and ${\inf\{\lambda_{\mathfrak{a} R_{\mathfrak{p}}}^{\mathfrak{b} R_{\mathfrak{p}}}(M_{\mathfrak{p}})|\, \mathfrak{p} \in {\rm Spec} \, R\, \rm{and\, dim}\, R/ \mathfrak{p} \geq n\}}$ are equal, for every finitely generated R-module M and for all ideals ${\mathfrak{a}, \mathfrak{b}}$ of R with ${\mathfrak{b}\subseteq \mathfrak{a}}$ . This generalizes Faltings’ Annihilator Theorem (see [6]).     

Inversion Formula for the Dunkl-Wigner Transform and Compactness Property for the Dunkl-Weyl Transforms

We define and study the Fourier-Wigner transform associated with the Dunkl operators,and we prove for this transform an inversion formula.Next,we introduce and study the Weyl transforms W_σ associated with the Dunkl operators,where cr is a symbol in the Schwartz space S(R~d×R~d).An integral relation between the precedent Weyl and Wigner transforms is given.At last,we give criteria in terms of σ for boundedness and compactness of the transform W_σ.     

Endpoint Estimates for Hardy Operator’s Conjugate Operator with Power Weight on $n$-Dimensional Space

In this paper, we establish two integral inequalities for Hardy operator's conjugate operator at the endpoint on $n$-dimensional space. The operator $H^*_n$ is bounded from $L^1_{x^\alpha}(\mathbb{G}^n)$ to $L^q_{x^\beta}(\mathbb{G}^n)$ with the bound explicitly worked out and the similar result holds for $\mathcal{H}^\ast_n$.     

两矩形上的全偏差

考虑如下极值问题的存在性和唯一性:■,其中h代表从矩形Q_1到矩形Q_2并保持端点且具有有限偏差的所有同胚映射的集合.