Automorphism group of exceptional symmetric domains RVI

Here we give the definition of the exceptional symmetric Siegel domain R_VI(27) in C²⁷, and compute the exceptional symmetric domain ℛ_VI(27) = τ(R_VI(27)), where t is the Bergman mapping of the Siegel domainR _VI (27). Moreover, we present the holomorphical automorphism group Aut (ℛ_VI(27)) of the exceptional symmetric domain (ℛ_VI(27)).     

Automorphism group of exceptional symmetric domainsR V

The exceptional symmetric Siegel domainR _v(16) in ?¹⁶ is defined. The exceptional classical domain ?_v(16) = t(R_v(16)) is computed, where t is the Bergman mapping of the Siegel domain R_v(16). And holomorphical automorphism group Aut (R_v(16)) of the exceptional symmetric Siegel domainR _v(16) is presented     

Automorphism group of exceptional symmetric domains RVI

Here we give the definition of the exceptional symmetric Siegel domain R_VI(27) in C²⁷, and compute the exceptional symmetric domain ?_VI(27) = τ(R_VI(27)), where t is the Bergman mapping of the Siegel domainR _VI (27). Moreover, we present the holomorphical automorphism group Aut (?_VI(27)) of the exceptional symmetric domain (?_VI(27)).     

Automorphism group of exceptional symmetric domains RV

The exceptional symmetric Siegel domain RV(16) in C16 is defined. The exceptional classical domain (R)v(16)=τ(RV(16)) is computed, where τ is the Bergman mapping of the Siegel domain RV(16). And holomorphical automorphism group Aut (RV(16)) of the exceptional symmetric Siegel domain RV(16) is presented.     

On quasi-similarity of subnormal operators

For a compact subset K in the complex plane, let Rat(K) denote the set of the rational functions with poles off K. Given a finite positive measure with support contained in K, let R2(K,v) denote the closure of Rat(K) in L2(v) and let Sv denote the operator of multiplication by the independent variable z on R2(K, v), that is, Svf = zf for every f∈R2(K, v). SupposeΩis a bounded open subset in the complex plane whose complement has finitely many components and suppose Rat(Ω) is dense in the Hardy space H2(Ω). Letσdenote a harmonic measure forΩ. In this work, we characterize all subnormal operators quasi-similar to Sσ, the operators of the multiplication by z on R2(Ω,σ). We show that for a given v supported onΩ, Sv is quasi-similar to Sσif and only if v/■Ω■σ and log(dv/dσ)∈L1(σ). Our result extends a well-known result of Clary on the unit disk.     

Strong approximation ofGSBV functions by piecewise smooth functions

Let Ω be an open and bounded subset ofR ⁿ with locally Lipschitz boundary. We prove that the functionsv∈SBV(Ω,R ^m ) whose jump setS _vis essentially closed and polyhedral and which are of classW ^{k, ∞} (S _v,R ^m) for every integerk are strongly dense inGSBV ^p(Ω,R ^m ), in the sense that every functionu inGSBV ^p(Ω,R ^m ) is approximated inL ^p(Ω,R ^m ) by a sequence of functions {v _k{_j∈N with the described regularity such that the approximate gradients ∇v _jconverge inL ^p(Ω,R ^nm ) to the approximate gradient ∇u and the (n−1)-dimensional measure of the jump setsS _{v j} converges to the (n−1)-dimensional measure ofS _u. The structure ofS _v can be further improved in casep≤2.

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Sunto Sia Ω un aperto limitato diR ⁿ con frontiera localmente Lipschitziana. In questo lavoro si dimostra che le funzioniv∈SBV(Ω,R ^m ) con insieme di saltoS _v essenzialmente chiuso e poliedrale che sono di classeW ^{k, ∞} (S _v,R ^m ) per ogni interok sono fortemente dense inGSBV ^p(Ω,R ^m ), nel senso che ogni funzioneu∈GSBV ^p(Ω,R ^m ) è approssimata inL ^p(Ω,R ^m ) da una successione di funzioni {v _j}_j∈N con la regolaritá descritta tali che i gradienti approssimati ∇v _jconvergono inL ^p(Ω,R ^nm ) al gradiente approssimato ∇u e la misura (n−1)-dimensionale degli insiemi di saltoS _{v j}converge alla misura (n−1)-dimensionale diS _u. La struttura diS _vpuó essere migliorata nel caso in cuip≤2.

     

Admissible wavelets on the Siegel domain of type one

LetSp(n, R) be the sympletic group, and letK _n * be its maximal compact subgroup. ThenG=Sp(n,R)/K _n * can be realized as the Siegel domain of type one. The square-integrable representation ofG gives the admissible wavelets AW and wavelet transform. The characterization of admissibility condition in terms of the Fourier transform is given. The Bergman kernel follows from the viewpoint of coherent state. With the Laguerre polynomials, Hermite polynomials and Jacobi polynomials, two kinds of orthogonal bases for AW are given, and they then give orthogonal decompositions ofL ²-space on the Siegel domain of type one ℒ(ℋ _n , |y| *dxdy). Project supported in part by the National Natural Science Foundation of China (Grant No. 19631080).     

Weights andL Φ-boundedness of the Poisson integral operator

In this paper, we get a necessary and sufficient condition on the weights (μ,v) for the Poisson integral operator to be bounded fromL _Φ(R ⁿ, v(x)dx) to weak-L _Φ(R ₊ ⁿ⁺¹ ,dμ), where Φ is anN-function satisfying the Δ₂-condition. We also find a necessary and sufficient condition on the weights (μ,v) for the Poisson integral operator to be bounded fromL _Φ(R ⁿ,v(x)dx) toL _Φ(R ₊ ⁿ⁺¹ ,dμ) under some additional condition. Partially supported by NNSF of P.R. China     

Theoremes de Viabilite Pour les Inclusions Differentielles du Second Ordre

This paper gives conditions ensuring the existence for an initial value (x ₀,v ₀) of a solution to the second order differential inclusionx″(t) ∈F[x(t),x′(t)],x(0)=x ₀,x′(0)=v ₀ such thatx(t) ∈K for allt whereK is a nonempty given subset ofR ⁿ .      

Functions of ann-dimensional Brownian motion that are Markovian

Letf be a continuous function fromR ⁿ toR and letX(t)=(X ₁ (t), …, X _n (t)) be a Brownian motion onR ⁿ . The explicit form off necessary in order to makef(X(t)) a Markov process is determined.     

Potential space estimates in local Hardy spaces for Green potentials in convex domains

Let Ω be a bounded co.nvex domain in Rn(n≥3) and G(x,y) be the Green function of the Laplace operator -△ on Ω. Let hrp(Ω) = {f ∈ D'(Ω) :(E)F∈hp(Rn), s.t. F|Ω = f}, by the atom characterization of Local Hardy spaces in a bounded Lipschitz domain, the bound of f→(△)2(Gf) for every f ∈ hrp(Ω) is obtained, where n/(n 1)＜p≤1.     

Sharp Estimates in Bergman and Besov Spaces on Bounded Symmetric Domains

Sharp estimates of the point-evaluation functional in weighted Bergman spaces L ^p _a (Ω, dν _α) and for the point-evaluation derivalive functional in Besov spaces B ^p (Ω) are obtained for bounded symmetric domains Ω in ℂ ⁿ . Received October 25, 1999, Accepted December 6, 2000     

Archimedean L -factors on GL ( n ) × GL ( n ) and generalized Barnes integrals

The Rankin-Selberg method associates, to each local factorL(s, π _v × π _v ^′ ) of an automorphicL-function onGL(n) ×GL(n), a certain local integral of Whittaker functions for π _v and _v ^′ . In this paper we show that, if ν is archimedean, and π _v and _v ^′ are spherical principal series representations with trivial central character, then the localL-factor and local integral are, in fact, equal. This result verifies a conjecture of Bump, which predicts that the archimedean situation should, in the present context, parallel the nonarchimedean one. We also derive, as prerequisite to the above result, some identities for generalized Barnes integrals. In particular, we deduce a new transformation formula for certain single Barnes integrals, and a multiple-integral analog of the classical Barnes’ Lemma.     

The lifetime of diffusion processes with application to conditioned diffusions

Let τ _D . denote the lifetime of a diffusion process on domainD ⊂R ^d. This paper presents a sufficient condition for the exponential moment of τ _D to be finite. Here, both of the domain and the diffusion operator are general. As an application, the main result of Gao (1995) for conditioned diffusions is improved on. Research supported in part by NNSFC (19631060) and Beijing Normal University     

A nodal basis of Cμ-rational spline functions on triangulations

Let ρ be a triangulation of a polygonal domain D⊂R² with vertices V={v_i:l≤i≤N_v} and RS^k(D, ρ)={u∈C^k(D): ≠ T∈ρ, u/_T is a rational function}. The purpose of this paper is to study the existence and construction of C^μ-rational spline functions on any triangulation ρ for CAGD. The Hermite problem H^μ(V,U)={find u∈U: D^αu(v_i)=D^αf(v_i),|α|≤μ} is solved by the generalized wedge function method in rational spline function family, i.e. U=RS^μ. this solution needs only the knowledge of partial derivatives of order≤μ at v_i. The explicit repesentations of all C^μ-GWF(generalized wedge functions)and the interpolating operator with degree of precision at least 2μ+1 for any triangulation are given.     

Measure of central symmetry for fields of convex figures and three-dimensional convex bodies

Let g₂³:E₂( \mathbbR³ ) ? G₂( \mathbbR³ ) \gamma_2^3:{E_2}\left( {{\mathbb{R}^3}} \right) \to {G_2}\left( {{\mathbb{R}^3}} \right) be the tautological vector bundle over the Grassmann manifold of 2-planes in \mathbbR³ {\mathbb{R}^3} , where the fiber over a plane is the plane itself regarded as a two-dimensional subspace of \mathbbR³ {\mathbb{R}^3} . A field of convex figures is given in γ₂³ if a convex figure is distinguished in each fiber so that the figure continuously depends on the fiber. It is proved that each field of convex figures in γ₂³ contains a figure K containing a centrally symmetric convex figure of area ( 4 + 16?2 ) \left( {4 + 16\sqrt {2} } \right) S(K)/31 > 0.858 S(K) (S(K) denotes the area of K), and a figure K′ that is contained in a centrally symmetric convex figure of area ( 12?2 - 8 ) \left( {12\sqrt {2} - 8} \right) S(K′)/7 < 1.282 S(K′). It is also proved that each three-dimensional convex body K is contained in a centrally symmetric convex cylinder of volume ( 36?2 - 24 ) \left( {36\sqrt {2} - 24} \right) V(K)/7 < 3.845 V(K). (Here, V(K) denotes the volume of K.) Bibliography: 5 titles.     

环域上一类拟线性椭圆型方程的正径向解的存在性

The existence of positive radial solutions of the equation -din( |Du|p-2Du)=f(u) is studied in annular domains in Rn,n≥2. It is proved that if f(0)≥0, f is somewherenegative in (0,∞), limu→0^ f‘ (u)=0 and limu→∞ (f(u)/u^p-1)=∞, then there is alarge positive radial solution on all annuli. If f(0)≤0 and satisfies certain conditions, then the equation has no radial solution if the annuli are too wide.     

The blow-up rate for a system of heat equations with neumann boundary conditions

This paper deals with the blow-up properties of solutions to a system of heat equations u _t=Δu, v _t=Δv in B _R×(0, T) with the Neumann boundary conditions εu/εη=e ^v, εv/εη=e ^u on S _R×[0, T). The exact blow-up rates are established. It is also proved that the blow-up will occur only on the boundary. This work is supported by the National Natural Science Foundation of China     

Lp-decay rates to nonlinear diffusion waves for p-system with nonlinear damping

In this paper, we study the L ^p (2 ⩽ p ⩽ +∞) convergence rates of the solutions to the Cauchy problem of the so-called p-system with nonlinear damping. Precisely, we show that the corresponding Cauchy problem admits a unique global solution (v(x,t), u(x,t)) and such a solution tends time-asymptotically to the corresponding nonlinear diffusion wave (ῡ(x,t), ū(x,t)) governed by the classical Darcys’s law provided that the corresponding prescribed initial error function (w ₀(x), z ₀(x)) lies in (H ³ × H ²) (ℝ) and |v ₊ − v ₋| + ∥w ₀∥₃ + ∥z ₀∥₂ is sufficiently small. Furthermore, the L ^p (2 ⩽ p ⩽ +∞) convergence rates of the solutions are also obtained.     

The extreme core

For a Siegel modular cusp formf of weightk letv(f) be the closure of the convex ray hull of the support of the Fourier series inside the cone of semidefinite forms. We show the existence of the extreme core,C _ext, which satisfiesv(f) ⊇k C_ext for all cusp forms. This is a generalization of the Valence Inequality to Siegel modular cusp forms. We give estimations of the extreme core for general n. For n ≤5 we use noble forms to improve these estimates. Forn = 2 we almost specify the extreme core but fall short. We supply improved estimates for all relevant constants and show optimality in some cases. The techniques are mainly from the geometry of numbers but we also use IGUSA’s generators for the ring of Siegel modular forms in degree two.