All 4-Edge-Connected HHD-Free Graphs are {\mathbb{Z}_3}-Connected

An undirected graph G = (V, E) is called \mathbbZ₃{\mathbb{Z}_3}-connected if for all b: V ? \mathbbZ₃{b: V \rightarrow \mathbb{Z}_3} with ?_{v ? V}b(v)=0{\sum_{v \in V}b(v)=0}, an orientation D = (V, A) of G has a \mathbbZ₃{\mathbb{Z}_3}-valued nowhere-zero flow f: A? \mathbbZ₃-{0}{f: A\rightarrow \mathbb{Z}_3-\{0\}} such that ?_{e ? d⁺(v)}f(e)-?_{e ? d^-(v)}f(e)=b(v){\sum_{e \in \delta^+(v)}f(e)-\sum_{e \in \delta^-(v)}f(e)=b(v)} for all v ? V{v \in V}. We show that all 4-edge-connected HHD-free graphs are \mathbbZ₃{\mathbb{Z}_3}-connected. This extends the result due to Lai (Graphs Comb 16:165–176, 2000), which proves the \mathbbZ₃{\mathbb{Z}_3}-connectivity for 4-edge-connected chordal graphs.     

Dimension Free Estimates for Riesz Transforms Associated to the Harmonic Oscillator on \mathbb{R}^{n}

Let L=?Δ+|ξ|² be the harmonic oscillator on $\mathbb{R}^{n}Let L=−Δ+|ξ|² be the harmonic oscillator on \mathbbRⁿ\mathbb{R}^{n} , with the associated Riesz transforms R_2j−1=(∂/∂ξ_j)L^−1/2,R_2j=ξ_jL^−1/2. We give a shorter proof of a recent result of Harboure, de Rosa, Segovia, Torrea: For 1<p<∞ and a dimension free constant C_p,

||(?k=12n|Rk(f)|2)1/2||Lp(\mathbbRn,dx)\leqslant Cp||f||Lp(\mathbbRn,dx).\bigg\Vert \bigg(\sum_{k=1}^{2n}\vert R_{k}(f)\vert ^{2}\bigg)^{{1}/{2}}\bigg\Vert _{L^{p}(\mathbb{R}^{n},\mathrm{d}\xi )}\leqslant C_{p}\Vert f\Vert _{L^{p}(\mathbb{R}^{n},\mathrm{d}\xi )}.      3. On the dimension of invariant measures of endomorphisms of $${\mathbb{C}\mathbb{P}^k}$$      Christophe Dupont 《Mathematische Annalen》2011,349(3):509-528 Let f be an endomorphism of \mathbbC\mathbbPk{\mathbb{C}\mathbb{P}^k} and ν be an f-invariant measure with positive Lyapunov exponents (λ 1, . . . , λ k ). We prove a lower bound for the pointwise dimension of ν in terms of the degree of f, the exponents of ν and the entropy of ν. In particular our result can be applied for the maximal entropy measure μ. When k = 2, it implies that the Hausdorff dimension of μ is estimated by dimHm 3 [(log d)/(l1)] + [(log d)/(l2)]{{\rm dim}_\mathcal{H}\mu \geq {{\rm log} d \over \lambda_1} + {{\rm log} d \over \lambda_2}}, which is half of the conjectured formula. Our method for proving these results consists in studying the distribution of the ν-generic inverse branches of f n in \mathbbC\mathbbPk{\mathbb{C}\mathbb{P}^k} . Our tools are a volume growth estimate for the bounded holomorphic polydiscs in \mathbbC\mathbbPk{\mathbb{C}\mathbb{P}^k} and a normalization theorem for the ν-generic inverse branches of f n .      4. Linear Forms and Higher-Degree Uniformity for Functions On {\mathbb{F}^{n}_{p}}      W. T. Gowers  J. Wolf 《Geometric And Functional Analysis》2011,21(1):36-69 In [GW1] we began an investigation of the following general question. Let L 1, . . . , L m be a system of linear forms in d variables on Fnp{F^n_p}, and let A be a subset of Fnp{F^n_p} of positive density. Under what circumstances can one prove that A contains roughly the same number of m-tuples L 1(x 1, . . . , x d ), . . . , L m (x 1, . . . , x d ) with x1,?, xd ? \mathbb Fnp{x_1,\ldots, x_d \in {\mathbb F}^n_p} as a typical random set of the same density? Experience with arithmetic progressions suggests that an appropriate assumption is that ||A - d1||Uk{||A - \delta 1||_{U{^k}}} should be small, where we have written A for the characteristic function of the set A, δ is the density of A, k is some parameter that depends on the linear forms L 1, . . . , L m , and || ·||Uk{|| \cdot ||_U{^k}} is the kth uniformity norm. The question we investigated was how k depends on L 1, . . . , L m . Our main result was that there were systems of forms where k could be taken to be 2 even though there was no simple proof of this fact using the Cauchy–Schwarz inequality. Based on this result and its proof, we conjectured that uniformity of degree k − 1 is a sufficient condition if and only if the kth powers of the linear forms are linearly independent. In this paper we prove this conjecture, provided only that p is sufficiently large. (It is easy to see that some such restriction is needed.) This result represents one of the first applications of the recent inverse theorem for the U k norm over Fnp{F^n_p} by Bergelson, Tao and Ziegler [TZ2], [BTZ]. We combine this result with some abstract arguments in order to prove that a bounded function can be expressed as a sum of polynomial phases and a part that is small in the appropriate uniformity norm. The precise form of this decomposition theorem is critical to our proof, and the theorem itself may be of independent interest.      5. Singular oscillatory integrals on {\mathbb{R}^n}      M. Papadimitrakis  I. R. Parissis 《Mathematische Zeitschrift》2010,266(1):169-179 Let ${\mathcal{P}_{d,n}}Let Pd,n{\mathcal{P}_{d,n}} denote the space of all real polynomials of degree at most d on \mathbbRn{\mathbb{R}^n} . We prove a new estimate for the logarithmic measure of the sublevel set of a polynomial P ? Pd,1{P\in \mathcal{P}_{d,1}} . Using this estimate, we prove that supP ? Pd,n| p.v.ò\mathbbRneiP(x)\fracW(x/|x|)|x|ndx| ￡ c log d (||W||L logL(Sn-1)+1),\mathop{\rm sup}\limits_ {P \in \mathcal{P}_{d,n}}\left| p.v.\int_{\mathbb{R}^{n}}{e^{iP(x)}}{\frac{\Omega(x/|x|)}{|x|^n}dx}\right | \leq c\,{\rm log}\,d\,(||\Omega||_L \log L(S^{n-1})+1),      6. An Inverse Theorem for the Uniformity Seminorms Associated with the Action of {{\mathbb {F}^{\infty}_{p}}}      Vitaly Bergelson  Terence Tao  Tamar Ziegler 《Geometric And Functional Analysis》2010,19(6):1539-1596 Let ${\mathbb {F}}Let \mathbb F{\mathbb {F}} a finite field. We show that the universal characteristic factor for the Gowers–Host–Kra uniformity seminorm U k (X) for an ergodic action (Tg)g ? \mathbb Fw{(T_{g})_{{g} \in \mathbb {F}^{\omega}}} of the infinite abelian group \mathbb Fw{\mathbb {F}^{\omega}} on a probability space X = (X, B, m){X = (X, \mathcal {B}, \mu)} is generated by phase polynomials f: X ? S1{\phi : X \to S^{1}} of degree less than C(k) on X, where C(k) depends only on k. In the case where k ￡ char(\mathbb F){k \leq {\rm char}(\mathbb {F})} we obtain the sharp result C(k) = k. This is a finite field counterpart of an analogous result for \mathbb Z{\mathbb {Z}} by Host and Kra [HK]. In a companion paper [TZ] to this paper, we shall combine this result with a correspondence principle to establish the inverse theorem for the Gowers norm in finite fields in the high characteristic case k ￡ char(\mathbb F){k \leq {\rm char}(\mathbb {F})} , with a partial result in low characteristic.      7. Truncated Complex Moment Problems with a $ \widetilde{zz} $ Relation      Lawrence A. Fialkow 《Integral Equations and Operator Theory》2003,45(4):405-435 We solve the truncated complex moment problem for measures supported on the variety K o \mathcal{K}\equiv { z ? \in C: z [(z)\tilde]\widetilde{z} = A+Bz+C [(z)\tilde]\widetilde{z} +Dz 2 ,D 1 \neq 0}. Given a doubly indexed finite sequence of complex numbers g o g(2n):g00,g01,g10,?,g0,2n,g1,2n-1,?,g2n-1,1,g2n,0 \gamma\equiv\gamma^{(2n)}:\gamma_{00},\gamma_{01},\gamma_{10},\ldots,\gamma_{0,2n},\gamma_{1,2n-1},\ldots,\gamma_{2n-1,1},\gamma_{2n,0} , there exists a positive Borel measure m\mu supported in K \mathcal{K} such that gij=ò[`(z)]izj dm (0 ￡ 1+j ￡ 2n) \gamma_{ij}=\int\overline{z}^{i}z^{j}\,d\mu\,(0\leq1+j\leq2n) if and only if the moment matrix M(n)( g\gamma ) is positive, recursively generated, with a column dependence relation Z [(Z)\tilde]\widetilde{Z} = A1+BZ +C [(Z)\tilde]\widetilde{Z} +DZ 2, and card V(g) 3\mathcal{V}(\gamma)\geq rank M(n), where V(g)\mathcal{V}(\gamma) is the variety associated to g \gamma . The last condition may be replaced by the condition that there exists a complex number gn,n+1 \gamma_{n,n+1} satisfying gn+1,n o [`(g)]n,n+1=Agn,n-1+Bgn,n+Cgn+1,n-1+Dgn,n+1 \gamma_{n+1,n}\equiv\overline{\gamma}_{n,n+1}=A\gamma_{n,n-1}+B\gamma_{n,n}+C\gamma_{n+1,n-1}+D\gamma_{n,n+1} . We combine these results with a recent theorem of J. Stochel to solve the full complex moment problem for K \mathcal{K} , and we illustrate the connection between the truncated and full moment problems for other varieties as well, including the variety z k = p(z, [(Z)\tilde] \widetilde{Z} ), deg p < k.      8. Analytic cohomology groups in top degrees of Zariski open sets in {\mathbb{P}^n}      Gabriel Chiriacescu  Mihnea Colţoiu  Cezar Joiţa 《Mathematische Zeitschrift》2010,264(3):671-677 We show that if A is a closed analytic subset of \mathbbPn{\mathbb{P}^n} of pure codimension q then Hi(\mathbbPn\ A,F){H^i(\mathbb{P}^n{\setminus} A,{\mathcal F})} are finite dimensional for every coherent algebraic sheaf F{{\mathcal F}} and every i 3 n-[\fracn-1q]{i\geq n-\left[\frac{n-1}{q}\right]} . If n-1 3 2q we show that Hn-2(\mathbbPn\ A,F)=0{n-1\geq 2q\,{\rm we show that}\, H^{n-2}(\mathbb{P}^n{\setminus} A,{\mathcal F})=0} .      9. On cyclic codes of length $${2^{2^r}-1}$$ with two zeros whose dual codes have three weights      Tao Feng 《Designs, Codes and Cryptography》2012,62(3):253-258 In 1976, Helleseth conjectured that two binary m-sequences of length 2 m − 1 can not have a three-valued crosscorrelation function when m is a power of 2. We show that this conjecture is true when −1 is a correlation value. In other words, if C1,k{{\mathcal{C}}_{1,k}} is the cyclic code of length 2 m − 1 with two zeros α, α k , where α is a primitive element of \mathbbF2m{{\mathbb{F}}_{2^m}} and gcd(k, 2 m − 1) = 1, then its dual C1,k^{{\mathcal{C}}_{1,k}^{\perp}} can not have three weights when m is a power of 2.      10. Holomorphic Extension Theorems in Lipschitz Domains of {\mathbb{C}}^{2}      Ricardo Abreu Blaya  Juan Bory Reyes  Dixan Peña Peña  Frank Sommen 《Advances in Applied Clifford Algebras》2010,20(1):1-12 The holomorphic functions of several complex variables are closely related to the continuously differentiable solutions $f : {\mathbb{R}}^{2n} \mapsto {\mathbb{C}}_{n}$f : {\mathbb{R}}^{2n} \mapsto {\mathbb{C}}_{n} of the so called isotonic system ?x1 + i [(f)\tilde] ?x 2 = 0\partial _{\underbar{x}_1 } + i \tilde{f} \mathop{\partial _{\underbar{x} _2 } = 0}      11. Absolutely convergent fourier series. An improvement of the Beurling-Helson theorem      V. V. Lebedev 《Functional Analysis and Its Applications》2012,46(2):121-132 We consider the space A(\mathbbT)A(\mathbb{T}) of all continuous functions f on the circle \mathbbT\mathbb{T} such that the sequence of Fourier coefficients [^(f)] = { [^(f)]( k ), k ? \mathbbZ }\hat f = \left\{ {\hat f\left( k \right), k \in \mathbb{Z}} \right\} belongs to l 1(ℤ). The norm on A(\mathbbT)A(\mathbb{T}) is defined by || f ||A(\mathbbT) = || [^(f)] ||l1 (\mathbbZ)\left\| f \right\|_{A(\mathbb{T})} = \left\| {\hat f} \right\|_{l^1 (\mathbb{Z})}. According to the well-known Beurling-Helson theorem, if f:\mathbbT ? \mathbbT\phi :\mathbb{T} \to \mathbb{T} is a continuous mapping such that || einf ||A(\mathbbT) = O(1)\left\| {e^{in\phi } } \right\|_{A(\mathbb{T})} = O(1), n ∈ ℤ then φ is linear. It was conjectured by Kahane that the same conclusion about φ is true under the assumption that || einf ||A(\mathbbT) = o( log| n | )\left\| {e^{in\phi } } \right\|_{A(\mathbb{T})} = o\left( {\log \left| n \right|} \right). We show that if $\left\| {e^{in\phi } } \right\|_{A(\mathbb{T})} = o\left( {\left( {{{\log \log \left| n \right|} \mathord{\left/ {\vphantom {{\log \log \left| n \right|} {\log \log \log \left| n \right|}}} \right. \kern-\nulldelimiterspace} {\log \log \log \left| n \right|}}} \right)^{1/12} } \right)$\left\| {e^{in\phi } } \right\|_{A(\mathbb{T})} = o\left( {\left( {{{\log \log \left| n \right|} \mathord{\left/ {\vphantom {{\log \log \left| n \right|} {\log \log \log \left| n \right|}}} \right. \kern-\nulldelimiterspace} {\log \log \log \left| n \right|}}} \right)^{1/12} } \right), then φ is linear.      12. Dyadic Bivariate Wavelet Multipliers in L^2（R@2）      Zhong Yan LI  Xian Liang SHI 《数学学报(英文版)》2011,27(8):1489-1500 The single 2 dilation wavelet multipliers in one-dimensional case and single A-dilation (where A is any expansive matrix with integer entries and |detA| = 2) wavelet multipliers in twodimensional case were completely characterized by Wutam Consortium (1998) and Li Z., et al. (2010). But there exist no results on multivariate wavelet multipliers corresponding to integer expansive dilation matrix with the absolute value of determinant not 2 in L 2(ℝ2). In this paper, we choose $2I_2 = \left( {{*{20}c} 2 & 0 \\ 0 & 2 \\ } \right)$2I_2 = \left( {\begin{array}{*{20}c} 2 & 0 \\ 0 & 2 \\ \end{array} } \right) as the dilation matrix and consider the 2I 2-dilation multivariate wavelet Φ = {ψ 1, ψ 2, ψ 3}(which is called a dyadic bivariate wavelet) multipliers. Here we call a measurable function family f = {f 1, f 2, f 3} a dyadic bivariate wavelet multiplier if Y1 = { F - 1 ( f1 [^(y1 )] ),F - 1 ( f2 [^(y2 )] ),F - 1 ( f3 [^(y3 )] ) }\Psi _1 = \left\{ {\mathcal{F}^{ - 1} \left( {f_1 \widehat{\psi _1 }} \right),\mathcal{F}^{ - 1} \left( {f_2 \widehat{\psi _2 }} \right),\mathcal{F}^{ - 1} \left( {f_3 \widehat{\psi _3 }} \right)} \right\} is a dyadic bivariate wavelet for any dyadic bivariate wavelet Φ = {ψ 1, ψ 2, ψ 3}, where [^(f)]\hat f and F −1 denote the Fourier transform and the inverse transform of function f respectively. We study dyadic bivariate wavelet multipliers, and give some conditions for dyadic bivariate wavelet multipliers. We also give concrete forms of linear phases of dyadic MRA bivariate wavelets.      13. Spectral theorem for multipliers on {L_{\omega}^{2}(\mathbb{R})}      Violeta Petkova 《Archiv der Mathematik》2009,93(4):357-368 We study the spectrum σ(M) of the multipliers M which commute with the translations on weighted spaces ${L_{\omega}^{2}(\mathbb{R})}We study the spectrum σ(M) of the multipliers M which commute with the translations on weighted spaces Lw2(\mathbbR){L_{\omega}^{2}(\mathbb{R})} For operators M in the algebra generated by the convolutions with f ? Cc(\mathbb R){\phi \in {C_c(\mathbb {R})}} we show that [`(m(W))] = s(M){\overline{\mu(\Omega)} = \sigma(M)}, where the set Ω is determined by the spectrum of the shift S and μ is the symbol of M. For the general multipliers M we establish that [`(m(W))]{\overline{\mu(\Omega)}} is included in σ(M). A generalization of these results is given for the weighted spaces L2w(\mathbb Rk){L^2_{\omega}(\mathbb {R}^{k})} where the weight ω has a special form.      14. The Hermite Property of a Causal Wiener Algebra Used in Control Theory      Amol Sasane 《Complex Analysis and Operator Theory》2010,4(1):97-107 Let \mathbbC+ : = {s ? \mathbbC    |     Re(s) 3 0}{{\mathbb{C}}}_{+} := \{s \in {{\mathbb{C}}}\quad | \quad {\rm Re}(s) \geq 0\} and let A\mathcal{A} denote the Banach algebra A = { s( ? \mathbbC+ ) ? [^(f)]a (s) + ?k = 0￥ fk e - stk | lfa ? L1 (0,￥),(fk )k 3 0 ? l1, 0 = t0 < t1 < t2 < ? }{{{\mathcal{A}}}} = \left\{ s( \in {{{\mathbb{C}}}}_ + ) \mapsto \hat{f}_a (s) + \sum\limits_{k = 0}^\infty {f_k e^{ - st_k }}\bigg | \bigg.{\begin{array}{l}{f_a \in L^1 (0,\infty ),(f_k )_{k \geq 0} \in \ell^{1}, } \cr {{0 = t_0 < t_1 < t_2 < \ldots}} \end{array}} \right\}      15. On approximation of periodic functions by Fourier sums      V. V. Zhuk 《Journal of Mathematical Sciences》2010,166(2):167-185 Let L p , 1 ≤ p< ∞, be the space of 2π-periodic functions f with the norm || f ||p = ( ò - pp | f |p )1 \mathord / \vphantom 1 p p {\left\| f \right\|_p} = {\left( {\int\limits_{ - \pi }^\pi {{{\left| f \right|}^p}} } \right)^{{1 \mathord{\left/{\vphantom {1 p}} \right.} p}}} , and let C = L ∞ be the space of continuous 2π-periodic functions with the norm || f ||￥ = || f || = maxe ? \mathbbR | f(x) | {\left\| f \right\|_\infty } = \left\| f \right\| = \mathop {\max }\limits_{e \in \mathbb{R}} \left| {f(x)} \right| . Let CP be the subspace of C with a seminorm P invariant with respect to translation and such that P(f) \leqslant M|| f || P(f) \leqslant M\left\| f \right\| for every f ∈ C. By ?k = 0￥ Ak (f) \sum\limits_{k = 0}^\infty {{A_k}} (f) denote the Fourier series of the function f, and let l = { lk }k = 0￥ \lambda = \left\{ {{\lambda_k}} \right\}_{k = 0}^\infty be a sequence of real numbers for which ?k = 0￥ lk Ak(f) \sum\limits_{k = 0}^\infty {{\lambda_k}} {A_k}(f) is the Fourier series of a certain function f λ ∈ L p . The paper considers questions related to approximating the function f λ by its Fourier sums S n (f λ) on a point set and in the spaces L p and CP. Estimates for || fl - Sn( fl ) ||p {\left\| {{f_\lambda } - {S_n}\left( {{f_\lambda }} \right)} \right\|_p} and P(f λ − S n (f λ)) are obtained by using the structural characteristics (the best approximations and the moduli of continuity) of the functions f and f λ. As a rule, the essential part of deviation is estimated with the use of the structural characteristics of the function f. Bibliography: 11 titles.      16. Closed-Range Composition Operators on {\mathbb{A}^2} and the Bloch Space      John R. Akeroyd  Pratibha G. Ghatage  Maria Tjani 《Integral Equations and Operator Theory》2010,68(4):503-517 For any analytic self-map j{\varphi} of {z : |z| <  1} we give four separate conditions, each of which is necessary and sufficient for the composition operator Cj{C_{\varphi}} to be closed-range on the Bloch space B{\mathcal{B}} . Among these conditions are some that appear in the literature, where we provide new proofs. We further show that if Cj{C_{\varphi}} is closed-range on the Bergman space \mathbbA2{\mathbb{A}^2} , then it is closed-range on B{\mathcal{B}} , but that the converse of this fails with a vengeance. Our analysis involves an extension of the Julia-Carathéodory Theorem.      17. On $\frak {F}$-normal Fitting classes of finite soluble groups      S. Reifferscheid 《Archiv der Mathematik》2000,75(3):164-172 Let \frak X, \frak F,\frak X\subseteqq \frak F\frak {X}, \frak {F},\frak {X}\subseteqq \frak {F}, be non-trivial Fitting classes of finite soluble groups such that G\frak XG_{\frak {X}} is an \frak X\frak {X}-injector of G for all G ? \frak FG\in \frak {F}. Then \frak X\frak {X} is called \frak F\frak {F}-normal. If \frak F=\frak Sp\frak {F}=\frak {S}_{\pi }, it is known that (1) \frak X\frak {X} is \frak F\frak {F}-normal precisely when \frak X*=\frak F*\frak {X}^{\ast }=\frak {F}^{\ast }, and consequently (2) \frak F í \frak X\frak N\frak {F}\subseteq \frak {X}\frak {N} implies \frak X*=\frak F*\frak {X}^{\ast }=\frak {F}^{\ast }, and (3) there is a unique smallest \frak F\frak {F}-normal Fitting class. These assertions are not true in general. We show that there are Fitting classes \frak F\not = \frak Sp\frak {F}\not =\frak {S}_{\pi } filling property (1), whence the classes \frak Sp\frak {S}_{\pi } are not characterized by satisfying (1). Furthermore we prove that (2) holds true for all Fitting classes \frak F\frak {F} satisfying a certain extension property with respect to wreath products although there could be an \frak F\frak {F}-normal Fitting class outside the Lockett section of \frak F\frak {F}. Lastly, we show that for the important cases \frak F=\frak Nn, n\geqq 2\frak {F}=\frak {N}^{n},\ n\geqq 2, and \frak F=\frak Sp1?\frak Spr, pi \frak {F}=\frak {S}_{p_{1}}\cdots \frak {S}_{p_{r}},\ p_{i} primes, there is a unique smallest \frak F\frak {F}-normal Fitting class, which we describe explicitly.      18. Ring of invariants of general linear group over local ring \mathbb{Z}_{p^m }      Jizhu Nan  Yin Chen 《Frontiers of Mathematics in China》2011,6(5):887-899 Let \mathbbZpm \mathbb{Z}_{p^m } be the ring of integers modulo p m , where p is a prime and m ⩾ 1. The general linear group GL n ( \mathbbZpm \mathbb{Z}_{p^m } ) acts naturally on the polynomial algebra A n := \mathbbZpm \mathbb{Z}_{p^m } [x 1, …, x n ]. Denote by AnGL2 (\mathbbZpm ) A_n^{GL_2 (\mathbb{Z}_{p^m } )} the corresponding ring of invariants. The purpose of the present paper is to calculate this invariant ring. Our results also generalize the classical Dickson’s theorem.      19. Approximation by the Bernstein–Durrmeyer Operator on a Simplex      Feng Dai  Hongwei Huang  Kunyang Wang 《Constructive Approximation》2010,31(3):289-308 For the Jacobi-type Bernstein–Durrmeyer operator M n,κ on the simplex T d of ℝ d , we proved that for f∈L p (W κ ;T d ) with 1<p<∞, K2,\varPhi(f,n-1)k,p ￡ c||f-Mn,kf||k,p ￡ c￠K2,\varPhi(f,n-1)k,p+c￠n-1||f||k,p,K_{2,\varPhi}\bigl(f,n^{-1}\bigr)_{\kappa,p}\leq c\|f-M_{n,\kappa}f\|_{\kappa,p}\leq c'K_{2,\varPhi}\bigl(f,n^{-1}\bigr)_{\kappa ,p}+c'n^{-1}\|f\|_{\kappa,p},      20. A new class of integrable systems      A. Lesfari 《Archiv der Mathematik》2001,77(4):347-353 This paper concerns the integrability of Hamiltonian systems with two degrees of freedom whose Hamiltonian has the form¶ H=1/2(x12+x22) +V(y1,y2) H={1\over2}(x_{1}^{2}+x_{2}^{2}) +V(y_{1},y_{2}) where¶¶ V(y1,y2)=1/2(a1y12+a2y22) + 1/4b1y14 + 1/4b2y24 + 1/2b3y12y22 + ?k=13gk(y12+y22) k+2 V(y_{1},y_{2})={1\over2}\big(\alpha _{1}y_{1}^{2}+\alpha_{2}y_{2}^{2}\big) + {1\over4}\beta _{1}y_{1}^{4} + {1\over4}\beta_{2}y_{2}^{4} + {1\over2}\beta _{3}y_{1}^{2}y_{2}^{2} + \sum_{k=1}^{3}\gamma_{k}\big(y_{1}^{2}+y_{2}^{2}\big) ^{k+2} ¶¶ which, constitues a generalization of some well-known integrable systems. We give new values of the vector (a1,a2,b1,b2,b3,g1,g2,g3) (\alpha _{1},\alpha_{2},\beta _{1},\beta _{2},\beta _{3},\gamma _{1},\gamma _{2},\gamma _{3}) for which this system is completely integrable and we show that the system is linearized in the Jacobian variety Jac(G \Gamma ) of a smooth genus 2 hyperelliptic Riemann surface G \Gamma .      设为首页 | 免责声明 | 关于勤云 | 加入收藏 Copyright©北京勤云科技发展有限公司  京ICP备09084417号

On the dimension of invariant measures of endomorphisms of $${\mathbb{C}\mathbb{P}^k}$$

Let f be an endomorphism of \mathbbC\mathbbP^k{\mathbb{C}\mathbb{P}^k} and ν be an f-invariant measure with positive Lyapunov exponents (λ ₁, . . . , λ _k ). We prove a lower bound for the pointwise dimension of ν in terms of the degree of f, the exponents of ν and the entropy of ν. In particular our result can be applied for the maximal entropy measure μ. When k = 2, it implies that the Hausdorff dimension of μ is estimated by dim_Hm 3 [(log d)/(l₁)] + [(log d)/(l₂)]{{\rm dim}_\mathcal{H}\mu \geq {{\rm log} d \over \lambda_1} + {{\rm log} d \over \lambda_2}}, which is half of the conjectured formula. Our method for proving these results consists in studying the distribution of the ν-generic inverse branches of f ⁿ in \mathbbC\mathbbP^k{\mathbb{C}\mathbb{P}^k} . Our tools are a volume growth estimate for the bounded holomorphic polydiscs in \mathbbC\mathbbP^k{\mathbb{C}\mathbb{P}^k} and a normalization theorem for the ν-generic inverse branches of f ⁿ .     

Linear Forms and Higher-Degree Uniformity for Functions On {\mathbb{F}^{n}_{p}}

In [GW1] we began an investigation of the following general question. Let L ₁, . . . , L _m be a system of linear forms in d variables on Fⁿ_p{F^n_p}, and let A be a subset of Fⁿ_p{F^n_p} of positive density. Under what circumstances can one prove that A contains roughly the same number of m-tuples L ₁(x ₁, . . . , x _d ), . . . , L _m (x ₁, . . . , x _d ) with x₁,?, x_d ? \mathbb Fⁿ_p{x_1,\ldots, x_d \in {\mathbb F}^n_p} as a typical random set of the same density? Experience with arithmetic progressions suggests that an appropriate assumption is that ||A - d1||_U^k{||A - \delta 1||_{U{^k}}} should be small, where we have written A for the characteristic function of the set A, δ is the density of A, k is some parameter that depends on the linear forms L ₁, . . . , L _m , and || ·||_U^k{|| \cdot ||_U{^k}} is the kth uniformity norm. The question we investigated was how k depends on L ₁, . . . , L _m . Our main result was that there were systems of forms where k could be taken to be 2 even though there was no simple proof of this fact using the Cauchy–Schwarz inequality. Based on this result and its proof, we conjectured that uniformity of degree k − 1 is a sufficient condition if and only if the kth powers of the linear forms are linearly independent. In this paper we prove this conjecture, provided only that p is sufficiently large. (It is easy to see that some such restriction is needed.) This result represents one of the first applications of the recent inverse theorem for the U ^k norm over Fⁿ_p{F^n_p} by Bergelson, Tao and Ziegler [TZ2], [BTZ]. We combine this result with some abstract arguments in order to prove that a bounded function can be expressed as a sum of polynomial phases and a part that is small in the appropriate uniformity norm. The precise form of this decomposition theorem is critical to our proof, and the theorem itself may be of independent interest.     

Singular oscillatory integrals on {\mathbb{R}^n}

Let ${\mathcal{P}_{d,n}}Let P_d,n{\mathcal{P}_{d,n}} denote the space of all real polynomials of degree at most d on \mathbbRⁿ{\mathbb{R}^n} . We prove a new estimate for the logarithmic measure of the sublevel set of a polynomial P ? P_d,1{P\in \mathcal{P}_{d,1}} . Using this estimate, we prove that

supP ? Pd,n| p.v.ò\mathbbRneiP(x)\fracW(x/|x|)|x|ndx| ￡ c log d (||W||L logL(Sn-1)+1),\mathop{\rm sup}\limits_ {P \in \mathcal{P}_{d,n}}\left| p.v.\int_{\mathbb{R}^{n}}{e^{iP(x)}}{\frac{\Omega(x/|x|)}{|x|^n}dx}\right | \leq c\,{\rm log}\,d\,(||\Omega||_L \log L(S^{n-1})+1),      6. An Inverse Theorem for the Uniformity Seminorms Associated with the Action of {{\mathbb {F}^{\infty}_{p}}}      Vitaly Bergelson  Terence Tao  Tamar Ziegler 《Geometric And Functional Analysis》2010,19(6):1539-1596 Let ${\mathbb {F}}Let \mathbb F{\mathbb {F}} a finite field. We show that the universal characteristic factor for the Gowers–Host–Kra uniformity seminorm U k (X) for an ergodic action (Tg)g ? \mathbb Fw{(T_{g})_{{g} \in \mathbb {F}^{\omega}}} of the infinite abelian group \mathbb Fw{\mathbb {F}^{\omega}} on a probability space X = (X, B, m){X = (X, \mathcal {B}, \mu)} is generated by phase polynomials f: X ? S1{\phi : X \to S^{1}} of degree less than C(k) on X, where C(k) depends only on k. In the case where k ￡ char(\mathbb F){k \leq {\rm char}(\mathbb {F})} we obtain the sharp result C(k) = k. This is a finite field counterpart of an analogous result for \mathbb Z{\mathbb {Z}} by Host and Kra [HK]. In a companion paper [TZ] to this paper, we shall combine this result with a correspondence principle to establish the inverse theorem for the Gowers norm in finite fields in the high characteristic case k ￡ char(\mathbb F){k \leq {\rm char}(\mathbb {F})} , with a partial result in low characteristic.      7. Truncated Complex Moment Problems with a $ \widetilde{zz} $ Relation      Lawrence A. Fialkow 《Integral Equations and Operator Theory》2003,45(4):405-435 We solve the truncated complex moment problem for measures supported on the variety K o \mathcal{K}\equiv { z ? \in C: z [(z)\tilde]\widetilde{z} = A+Bz+C [(z)\tilde]\widetilde{z} +Dz 2 ,D 1 \neq 0}. Given a doubly indexed finite sequence of complex numbers g o g(2n):g00,g01,g10,?,g0,2n,g1,2n-1,?,g2n-1,1,g2n,0 \gamma\equiv\gamma^{(2n)}:\gamma_{00},\gamma_{01},\gamma_{10},\ldots,\gamma_{0,2n},\gamma_{1,2n-1},\ldots,\gamma_{2n-1,1},\gamma_{2n,0} , there exists a positive Borel measure m\mu supported in K \mathcal{K} such that gij=ò[`(z)]izj dm (0 ￡ 1+j ￡ 2n) \gamma_{ij}=\int\overline{z}^{i}z^{j}\,d\mu\,(0\leq1+j\leq2n) if and only if the moment matrix M(n)( g\gamma ) is positive, recursively generated, with a column dependence relation Z [(Z)\tilde]\widetilde{Z} = A1+BZ +C [(Z)\tilde]\widetilde{Z} +DZ 2, and card V(g) 3\mathcal{V}(\gamma)\geq rank M(n), where V(g)\mathcal{V}(\gamma) is the variety associated to g \gamma . The last condition may be replaced by the condition that there exists a complex number gn,n+1 \gamma_{n,n+1} satisfying gn+1,n o [`(g)]n,n+1=Agn,n-1+Bgn,n+Cgn+1,n-1+Dgn,n+1 \gamma_{n+1,n}\equiv\overline{\gamma}_{n,n+1}=A\gamma_{n,n-1}+B\gamma_{n,n}+C\gamma_{n+1,n-1}+D\gamma_{n,n+1} . We combine these results with a recent theorem of J. Stochel to solve the full complex moment problem for K \mathcal{K} , and we illustrate the connection between the truncated and full moment problems for other varieties as well, including the variety z k = p(z, [(Z)\tilde] \widetilde{Z} ), deg p < k.      8. Analytic cohomology groups in top degrees of Zariski open sets in {\mathbb{P}^n}      Gabriel Chiriacescu  Mihnea Colţoiu  Cezar Joiţa 《Mathematische Zeitschrift》2010,264(3):671-677 We show that if A is a closed analytic subset of \mathbbPn{\mathbb{P}^n} of pure codimension q then Hi(\mathbbPn\ A,F){H^i(\mathbb{P}^n{\setminus} A,{\mathcal F})} are finite dimensional for every coherent algebraic sheaf F{{\mathcal F}} and every i 3 n-[\fracn-1q]{i\geq n-\left[\frac{n-1}{q}\right]} . If n-1 3 2q we show that Hn-2(\mathbbPn\ A,F)=0{n-1\geq 2q\,{\rm we show that}\, H^{n-2}(\mathbb{P}^n{\setminus} A,{\mathcal F})=0} .      9. On cyclic codes of length $${2^{2^r}-1}$$ with two zeros whose dual codes have three weights      Tao Feng 《Designs, Codes and Cryptography》2012,62(3):253-258 In 1976, Helleseth conjectured that two binary m-sequences of length 2 m − 1 can not have a three-valued crosscorrelation function when m is a power of 2. We show that this conjecture is true when −1 is a correlation value. In other words, if C1,k{{\mathcal{C}}_{1,k}} is the cyclic code of length 2 m − 1 with two zeros α, α k , where α is a primitive element of \mathbbF2m{{\mathbb{F}}_{2^m}} and gcd(k, 2 m − 1) = 1, then its dual C1,k^{{\mathcal{C}}_{1,k}^{\perp}} can not have three weights when m is a power of 2.      10. Holomorphic Extension Theorems in Lipschitz Domains of {\mathbb{C}}^{2}      Ricardo Abreu Blaya  Juan Bory Reyes  Dixan Peña Peña  Frank Sommen 《Advances in Applied Clifford Algebras》2010,20(1):1-12 The holomorphic functions of several complex variables are closely related to the continuously differentiable solutions $f : {\mathbb{R}}^{2n} \mapsto {\mathbb{C}}_{n}$f : {\mathbb{R}}^{2n} \mapsto {\mathbb{C}}_{n} of the so called isotonic system ?x1 + i [(f)\tilde] ?x 2 = 0\partial _{\underbar{x}_1 } + i \tilde{f} \mathop{\partial _{\underbar{x} _2 } = 0}      11. Absolutely convergent fourier series. An improvement of the Beurling-Helson theorem      V. V. Lebedev 《Functional Analysis and Its Applications》2012,46(2):121-132 We consider the space A(\mathbbT)A(\mathbb{T}) of all continuous functions f on the circle \mathbbT\mathbb{T} such that the sequence of Fourier coefficients [^(f)] = { [^(f)]( k ), k ? \mathbbZ }\hat f = \left\{ {\hat f\left( k \right), k \in \mathbb{Z}} \right\} belongs to l 1(ℤ). The norm on A(\mathbbT)A(\mathbb{T}) is defined by || f ||A(\mathbbT) = || [^(f)] ||l1 (\mathbbZ)\left\| f \right\|_{A(\mathbb{T})} = \left\| {\hat f} \right\|_{l^1 (\mathbb{Z})}. According to the well-known Beurling-Helson theorem, if f:\mathbbT ? \mathbbT\phi :\mathbb{T} \to \mathbb{T} is a continuous mapping such that || einf ||A(\mathbbT) = O(1)\left\| {e^{in\phi } } \right\|_{A(\mathbb{T})} = O(1), n ∈ ℤ then φ is linear. It was conjectured by Kahane that the same conclusion about φ is true under the assumption that || einf ||A(\mathbbT) = o( log| n | )\left\| {e^{in\phi } } \right\|_{A(\mathbb{T})} = o\left( {\log \left| n \right|} \right). We show that if $\left\| {e^{in\phi } } \right\|_{A(\mathbb{T})} = o\left( {\left( {{{\log \log \left| n \right|} \mathord{\left/ {\vphantom {{\log \log \left| n \right|} {\log \log \log \left| n \right|}}} \right. \kern-\nulldelimiterspace} {\log \log \log \left| n \right|}}} \right)^{1/12} } \right)$\left\| {e^{in\phi } } \right\|_{A(\mathbb{T})} = o\left( {\left( {{{\log \log \left| n \right|} \mathord{\left/ {\vphantom {{\log \log \left| n \right|} {\log \log \log \left| n \right|}}} \right. \kern-\nulldelimiterspace} {\log \log \log \left| n \right|}}} \right)^{1/12} } \right), then φ is linear.      12. Dyadic Bivariate Wavelet Multipliers in L^2（R@2）      Zhong Yan LI  Xian Liang SHI 《数学学报(英文版)》2011,27(8):1489-1500 The single 2 dilation wavelet multipliers in one-dimensional case and single A-dilation (where A is any expansive matrix with integer entries and |detA| = 2) wavelet multipliers in twodimensional case were completely characterized by Wutam Consortium (1998) and Li Z., et al. (2010). But there exist no results on multivariate wavelet multipliers corresponding to integer expansive dilation matrix with the absolute value of determinant not 2 in L 2(ℝ2). In this paper, we choose $2I_2 = \left( {{*{20}c} 2 & 0 \\ 0 & 2 \\ } \right)$2I_2 = \left( {\begin{array}{*{20}c} 2 & 0 \\ 0 & 2 \\ \end{array} } \right) as the dilation matrix and consider the 2I 2-dilation multivariate wavelet Φ = {ψ 1, ψ 2, ψ 3}(which is called a dyadic bivariate wavelet) multipliers. Here we call a measurable function family f = {f 1, f 2, f 3} a dyadic bivariate wavelet multiplier if Y1 = { F - 1 ( f1 [^(y1 )] ),F - 1 ( f2 [^(y2 )] ),F - 1 ( f3 [^(y3 )] ) }\Psi _1 = \left\{ {\mathcal{F}^{ - 1} \left( {f_1 \widehat{\psi _1 }} \right),\mathcal{F}^{ - 1} \left( {f_2 \widehat{\psi _2 }} \right),\mathcal{F}^{ - 1} \left( {f_3 \widehat{\psi _3 }} \right)} \right\} is a dyadic bivariate wavelet for any dyadic bivariate wavelet Φ = {ψ 1, ψ 2, ψ 3}, where [^(f)]\hat f and F −1 denote the Fourier transform and the inverse transform of function f respectively. We study dyadic bivariate wavelet multipliers, and give some conditions for dyadic bivariate wavelet multipliers. We also give concrete forms of linear phases of dyadic MRA bivariate wavelets.      13. Spectral theorem for multipliers on {L_{\omega}^{2}(\mathbb{R})}      Violeta Petkova 《Archiv der Mathematik》2009,93(4):357-368 We study the spectrum σ(M) of the multipliers M which commute with the translations on weighted spaces ${L_{\omega}^{2}(\mathbb{R})}We study the spectrum σ(M) of the multipliers M which commute with the translations on weighted spaces Lw2(\mathbbR){L_{\omega}^{2}(\mathbb{R})} For operators M in the algebra generated by the convolutions with f ? Cc(\mathbb R){\phi \in {C_c(\mathbb {R})}} we show that [`(m(W))] = s(M){\overline{\mu(\Omega)} = \sigma(M)}, where the set Ω is determined by the spectrum of the shift S and μ is the symbol of M. For the general multipliers M we establish that [`(m(W))]{\overline{\mu(\Omega)}} is included in σ(M). A generalization of these results is given for the weighted spaces L2w(\mathbb Rk){L^2_{\omega}(\mathbb {R}^{k})} where the weight ω has a special form.      14. The Hermite Property of a Causal Wiener Algebra Used in Control Theory      Amol Sasane 《Complex Analysis and Operator Theory》2010,4(1):97-107 Let \mathbbC+ : = {s ? \mathbbC    |     Re(s) 3 0}{{\mathbb{C}}}_{+} := \{s \in {{\mathbb{C}}}\quad | \quad {\rm Re}(s) \geq 0\} and let A\mathcal{A} denote the Banach algebra A = { s( ? \mathbbC+ ) ? [^(f)]a (s) + ?k = 0￥ fk e - stk | lfa ? L1 (0,￥),(fk )k 3 0 ? l1, 0 = t0 < t1 < t2 < ? }{{{\mathcal{A}}}} = \left\{ s( \in {{{\mathbb{C}}}}_ + ) \mapsto \hat{f}_a (s) + \sum\limits_{k = 0}^\infty {f_k e^{ - st_k }}\bigg | \bigg.{\begin{array}{l}{f_a \in L^1 (0,\infty ),(f_k )_{k \geq 0} \in \ell^{1}, } \cr {{0 = t_0 < t_1 < t_2 < \ldots}} \end{array}} \right\}      15. On approximation of periodic functions by Fourier sums      V. V. Zhuk 《Journal of Mathematical Sciences》2010,166(2):167-185 Let L p , 1 ≤ p< ∞, be the space of 2π-periodic functions f with the norm || f ||p = ( ò - pp | f |p )1 \mathord / \vphantom 1 p p {\left\| f \right\|_p} = {\left( {\int\limits_{ - \pi }^\pi {{{\left| f \right|}^p}} } \right)^{{1 \mathord{\left/{\vphantom {1 p}} \right.} p}}} , and let C = L ∞ be the space of continuous 2π-periodic functions with the norm || f ||￥ = || f || = maxe ? \mathbbR | f(x) | {\left\| f \right\|_\infty } = \left\| f \right\| = \mathop {\max }\limits_{e \in \mathbb{R}} \left| {f(x)} \right| . Let CP be the subspace of C with a seminorm P invariant with respect to translation and such that P(f) \leqslant M|| f || P(f) \leqslant M\left\| f \right\| for every f ∈ C. By ?k = 0￥ Ak (f) \sum\limits_{k = 0}^\infty {{A_k}} (f) denote the Fourier series of the function f, and let l = { lk }k = 0￥ \lambda = \left\{ {{\lambda_k}} \right\}_{k = 0}^\infty be a sequence of real numbers for which ?k = 0￥ lk Ak(f) \sum\limits_{k = 0}^\infty {{\lambda_k}} {A_k}(f) is the Fourier series of a certain function f λ ∈ L p . The paper considers questions related to approximating the function f λ by its Fourier sums S n (f λ) on a point set and in the spaces L p and CP. Estimates for || fl - Sn( fl ) ||p {\left\| {{f_\lambda } - {S_n}\left( {{f_\lambda }} \right)} \right\|_p} and P(f λ − S n (f λ)) are obtained by using the structural characteristics (the best approximations and the moduli of continuity) of the functions f and f λ. As a rule, the essential part of deviation is estimated with the use of the structural characteristics of the function f. Bibliography: 11 titles.      16. Closed-Range Composition Operators on {\mathbb{A}^2} and the Bloch Space      John R. Akeroyd  Pratibha G. Ghatage  Maria Tjani 《Integral Equations and Operator Theory》2010,68(4):503-517 For any analytic self-map j{\varphi} of {z : |z| <  1} we give four separate conditions, each of which is necessary and sufficient for the composition operator Cj{C_{\varphi}} to be closed-range on the Bloch space B{\mathcal{B}} . Among these conditions are some that appear in the literature, where we provide new proofs. We further show that if Cj{C_{\varphi}} is closed-range on the Bergman space \mathbbA2{\mathbb{A}^2} , then it is closed-range on B{\mathcal{B}} , but that the converse of this fails with a vengeance. Our analysis involves an extension of the Julia-Carathéodory Theorem.      17. On $\frak {F}$-normal Fitting classes of finite soluble groups      S. Reifferscheid 《Archiv der Mathematik》2000,75(3):164-172 Let \frak X, \frak F,\frak X\subseteqq \frak F\frak {X}, \frak {F},\frak {X}\subseteqq \frak {F}, be non-trivial Fitting classes of finite soluble groups such that G\frak XG_{\frak {X}} is an \frak X\frak {X}-injector of G for all G ? \frak FG\in \frak {F}. Then \frak X\frak {X} is called \frak F\frak {F}-normal. If \frak F=\frak Sp\frak {F}=\frak {S}_{\pi }, it is known that (1) \frak X\frak {X} is \frak F\frak {F}-normal precisely when \frak X*=\frak F*\frak {X}^{\ast }=\frak {F}^{\ast }, and consequently (2) \frak F í \frak X\frak N\frak {F}\subseteq \frak {X}\frak {N} implies \frak X*=\frak F*\frak {X}^{\ast }=\frak {F}^{\ast }, and (3) there is a unique smallest \frak F\frak {F}-normal Fitting class. These assertions are not true in general. We show that there are Fitting classes \frak F\not = \frak Sp\frak {F}\not =\frak {S}_{\pi } filling property (1), whence the classes \frak Sp\frak {S}_{\pi } are not characterized by satisfying (1). Furthermore we prove that (2) holds true for all Fitting classes \frak F\frak {F} satisfying a certain extension property with respect to wreath products although there could be an \frak F\frak {F}-normal Fitting class outside the Lockett section of \frak F\frak {F}. Lastly, we show that for the important cases \frak F=\frak Nn, n\geqq 2\frak {F}=\frak {N}^{n},\ n\geqq 2, and \frak F=\frak Sp1?\frak Spr, pi \frak {F}=\frak {S}_{p_{1}}\cdots \frak {S}_{p_{r}},\ p_{i} primes, there is a unique smallest \frak F\frak {F}-normal Fitting class, which we describe explicitly.      18. Ring of invariants of general linear group over local ring \mathbb{Z}_{p^m }      Jizhu Nan  Yin Chen 《Frontiers of Mathematics in China》2011,6(5):887-899 Let \mathbbZpm \mathbb{Z}_{p^m } be the ring of integers modulo p m , where p is a prime and m ⩾ 1. The general linear group GL n ( \mathbbZpm \mathbb{Z}_{p^m } ) acts naturally on the polynomial algebra A n := \mathbbZpm \mathbb{Z}_{p^m } [x 1, …, x n ]. Denote by AnGL2 (\mathbbZpm ) A_n^{GL_2 (\mathbb{Z}_{p^m } )} the corresponding ring of invariants. The purpose of the present paper is to calculate this invariant ring. Our results also generalize the classical Dickson’s theorem.      19. Approximation by the Bernstein–Durrmeyer Operator on a Simplex      Feng Dai  Hongwei Huang  Kunyang Wang 《Constructive Approximation》2010,31(3):289-308 For the Jacobi-type Bernstein–Durrmeyer operator M n,κ on the simplex T d of ℝ d , we proved that for f∈L p (W κ ;T d ) with 1<p<∞, K2,\varPhi(f,n-1)k,p ￡ c||f-Mn,kf||k,p ￡ c￠K2,\varPhi(f,n-1)k,p+c￠n-1||f||k,p,K_{2,\varPhi}\bigl(f,n^{-1}\bigr)_{\kappa,p}\leq c\|f-M_{n,\kappa}f\|_{\kappa,p}\leq c'K_{2,\varPhi}\bigl(f,n^{-1}\bigr)_{\kappa ,p}+c'n^{-1}\|f\|_{\kappa,p},      20. A new class of integrable systems      A. Lesfari 《Archiv der Mathematik》2001,77(4):347-353 This paper concerns the integrability of Hamiltonian systems with two degrees of freedom whose Hamiltonian has the form¶ H=1/2(x12+x22) +V(y1,y2) H={1\over2}(x_{1}^{2}+x_{2}^{2}) +V(y_{1},y_{2}) where¶¶ V(y1,y2)=1/2(a1y12+a2y22) + 1/4b1y14 + 1/4b2y24 + 1/2b3y12y22 + ?k=13gk(y12+y22) k+2 V(y_{1},y_{2})={1\over2}\big(\alpha _{1}y_{1}^{2}+\alpha_{2}y_{2}^{2}\big) + {1\over4}\beta _{1}y_{1}^{4} + {1\over4}\beta_{2}y_{2}^{4} + {1\over2}\beta _{3}y_{1}^{2}y_{2}^{2} + \sum_{k=1}^{3}\gamma_{k}\big(y_{1}^{2}+y_{2}^{2}\big) ^{k+2} ¶¶ which, constitues a generalization of some well-known integrable systems. We give new values of the vector (a1,a2,b1,b2,b3,g1,g2,g3) (\alpha _{1},\alpha_{2},\beta _{1},\beta _{2},\beta _{3},\gamma _{1},\gamma _{2},\gamma _{3}) for which this system is completely integrable and we show that the system is linearized in the Jacobian variety Jac(G \Gamma ) of a smooth genus 2 hyperelliptic Riemann surface G \Gamma .      设为首页 | 免责声明 | 关于勤云 | 加入收藏 Copyright©北京勤云科技发展有限公司  京ICP备09084417号

An Inverse Theorem for the Uniformity Seminorms Associated with the Action of {{\mathbb {F}^{\infty}_{p}}}

Let ${\mathbb {F}}Let \mathbb F{\mathbb {F}} a finite field. We show that the universal characteristic factor for the Gowers–Host–Kra uniformity seminorm U ^k (X) for an ergodic action (T_g)_{g ? \mathbb F^w}{(T_{g})_{{g} \in \mathbb {F}^{\omega}}} of the infinite abelian group \mathbb F^w{\mathbb {F}^{\omega}} on a probability space X = (X, B, m){X = (X, \mathcal {B}, \mu)} is generated by phase polynomials f: X ? S¹{\phi : X \to S^{1}} of degree less than C(k) on X, where C(k) depends only on k. In the case where k ￡ char(\mathbb F){k \leq {\rm char}(\mathbb {F})} we obtain the sharp result C(k) = k. This is a finite field counterpart of an analogous result for \mathbb Z{\mathbb {Z}} by Host and Kra [HK]. In a companion paper [TZ] to this paper, we shall combine this result with a correspondence principle to establish the inverse theorem for the Gowers norm in finite fields in the high characteristic case k ￡ char(\mathbb F){k \leq {\rm char}(\mathbb {F})} , with a partial result in low characteristic.     

Truncated Complex Moment Problems with a $ \widetilde{zz} $ Relation

We solve the truncated complex moment problem for measures supported on the variety K o \mathcal{K}\equiv { z ? \in C: z [(z)\tilde]\widetilde{z} = A+Bz+C [(z)\tilde]\widetilde{z} +Dz ² ,D 1 \neq 0}. Given a doubly indexed finite sequence of complex numbers g o g⁽²ⁿ⁾:g₀₀,g₀₁,g₁₀,?,g_0,2n,g_1,2n-1,?,g_2n-1,1,g_2n,0 \gamma\equiv\gamma^{(2n)}:\gamma_{00},\gamma_{01},\gamma_{10},\ldots,\gamma_{0,2n},\gamma_{1,2n-1},\ldots,\gamma_{2n-1,1},\gamma_{2n,0} , there exists a positive Borel measure m\mu supported in K \mathcal{K} such that g_ij=ò[`(z)]<sup>i</sup>z<sup>j</sup> dm (0 ￡ 1+j ￡ 2n) \gamma_{ij}=\int\overline{z}^{i}z^{j}\,d\mu\,(0\leq1+j\leq2n) if and only if the moment matrix M(n)( g\gamma ) is positive, recursively generated, with a column dependence relation Z [(Z)\tilde]\widetilde{Z} = A1+BZ +C [(Z)\tilde]\widetilde{Z} +DZ <sup>2</sup>, and card V(g) 3\mathcal{V}(\gamma)\geq rank M(n), where V(g)\mathcal{V}(\gamma) is the variety associated to g \gamma . The last condition may be replaced by the condition that there exists a complex number g<sub>n,n+1</sub> \gamma_{n,n+1} satisfying g<sub>n+1,n</sub> o [`(g)]_n,n+1=Ag_n,n-1+Bg_n,n+Cg_n+1,n-1+Dg_n,n+1 \gamma_{n+1,n}\equiv\overline{\gamma}_{n,n+1}=A\gamma_{n,n-1}+B\gamma_{n,n}+C\gamma_{n+1,n-1}+D\gamma_{n,n+1} . We combine these results with a recent theorem of J. Stochel to solve the full complex moment problem for K \mathcal{K} , and we illustrate the connection between the truncated and full moment problems for other varieties as well, including the variety z ^k = p(z, [(Z)\tilde] \widetilde{Z} ), deg p < k.     

Analytic cohomology groups in top degrees of Zariski open sets in {\mathbb{P}^n}

We show that if A is a closed analytic subset of \mathbbPⁿ{\mathbb{P}^n} of pure codimension q then Hⁱ(\mathbbPⁿ\ A,F){H^i(\mathbb{P}^n{\setminus} A,{\mathcal F})} are finite dimensional for every coherent algebraic sheaf F{{\mathcal F}} and every i 3 n-[\fracn-1q]{i\geq n-\left[\frac{n-1}{q}\right]} . If n-1 3 2q we show that H^n-2(\mathbbPⁿ\ A,F)=0{n-1\geq 2q\,{\rm we show that}\, H^{n-2}(\mathbb{P}^n{\setminus} A,{\mathcal F})=0} .     

On cyclic codes of length $${2^{2^r}-1}$$ with two zeros whose dual codes have three weights

In 1976, Helleseth conjectured that two binary m-sequences of length 2 ^m − 1 can not have a three-valued crosscorrelation function when m is a power of 2. We show that this conjecture is true when −1 is a correlation value. In other words, if C_1,k{{\mathcal{C}}_{1,k}} is the cyclic code of length 2 ^m − 1 with two zeros α, α ^k , where α is a primitive element of \mathbbF_2^m{{\mathbb{F}}_{2^m}} and gcd(k, 2 ^m − 1) = 1, then its dual C_1,k^{^}{{\mathcal{C}}_{1,k}^{\perp}} can not have three weights when m is a power of 2.     

Holomorphic Extension Theorems in Lipschitz Domains of {\mathbb{C}}^{2}

The holomorphic functions of several complex variables are closely related to the continuously differentiable solutions $f : {\mathbb{R}}^{2n} \mapsto {\mathbb{C}}_{n}$f : {\mathbb{R}}^{2n} \mapsto {\mathbb{C}}_{n} of the so called isotonic system

Absolutely convergent fourier series. An improvement of the Beurling-Helson theorem

We consider the space A(\mathbbT)A(\mathbb{T}) of all continuous functions f on the circle \mathbbT\mathbb{T} such that the sequence of Fourier coefficients [^(f)] = { [^(f)]( k ), k ? \mathbbZ }\hat f = \left\{ {\hat f\left( k \right), k \in \mathbb{Z}} \right\} belongs to l ¹(ℤ). The norm on A(\mathbbT)A(\mathbb{T}) is defined by || f ||_A(\mathbbT) = || [^(f)] ||_{l¹ (\mathbbZ)}\left\| f \right\|_{A(\mathbb{T})} = \left\| {\hat f} \right\|_{l^1 (\mathbb{Z})}. According to the well-known Beurling-Helson theorem, if f:\mathbbT ? \mathbbT\phi :\mathbb{T} \to \mathbb{T} is a continuous mapping such that || e^inf ||_A(\mathbbT) = O(1)\left\| {e^{in\phi } } \right\|_{A(\mathbb{T})} = O(1), n ∈ ℤ then φ is linear. It was conjectured by Kahane that the same conclusion about φ is true under the assumption that || e^inf ||_A(\mathbbT) = o( log| n | )\left\| {e^{in\phi } } \right\|_{A(\mathbb{T})} = o\left( {\log \left| n \right|} \right). We show that if $\left\| {e^{in\phi } } \right\|_{A(\mathbb{T})} = o\left( {\left( {{{\log \log \left| n \right|} \mathord{\left/ {\vphantom {{\log \log \left| n \right|} {\log \log \log \left| n \right|}}} \right. \kern-\nulldelimiterspace} {\log \log \log \left| n \right|}}} \right)^{1/12} } \right)$\left\| {e^{in\phi } } \right\|_{A(\mathbb{T})} = o\left( {\left( {{{\log \log \left| n \right|} \mathord{\left/ {\vphantom {{\log \log \left| n \right|} {\log \log \log \left| n \right|}}} \right. \kern-\nulldelimiterspace} {\log \log \log \left| n \right|}}} \right)^{1/12} } \right), then φ is linear.     

Dyadic Bivariate Wavelet Multipliers in L^2（R@2）

The single 2 dilation wavelet multipliers in one-dimensional case and single A-dilation (where A is any expansive matrix with integer entries and |detA| = 2) wavelet multipliers in twodimensional case were completely characterized by Wutam Consortium (1998) and Li Z., et al. (2010). But there exist no results on multivariate wavelet multipliers corresponding to integer expansive dilation matrix with the absolute value of determinant not 2 in L ²(ℝ²). In this paper, we choose $2I_2 = \left( {{*{20}c} 2 & 0 \\ 0 & 2 \\ } \right)$2I_2 = \left( {\begin{array}{*{20}c} 2 & 0 \\ 0 & 2 \\ \end{array} } \right) as the dilation matrix and consider the 2I ₂-dilation multivariate wavelet Φ = {ψ ₁, ψ ₂, ψ ₃}(which is called a dyadic bivariate wavelet) multipliers. Here we call a measurable function family f = {f ₁, f ₂, f ₃} a dyadic bivariate wavelet multiplier if Y₁ = { F ^{- 1} ( f₁ [^(y₁ )] ),F ^{- 1} ( f₂ [^(y₂ )] ),F ^{- 1} ( f₃ [^(y₃ )] ) }\Psi _1 = \left\{ {\mathcal{F}^{ - 1} \left( {f_1 \widehat{\psi _1 }} \right),\mathcal{F}^{ - 1} \left( {f_2 \widehat{\psi _2 }} \right),\mathcal{F}^{ - 1} \left( {f_3 \widehat{\psi _3 }} \right)} \right\} is a dyadic bivariate wavelet for any dyadic bivariate wavelet Φ = {ψ ₁, ψ ₂, ψ ₃}, where [^(f)]\hat f and F ⁻¹ denote the Fourier transform and the inverse transform of function f respectively. We study dyadic bivariate wavelet multipliers, and give some conditions for dyadic bivariate wavelet multipliers. We also give concrete forms of linear phases of dyadic MRA bivariate wavelets.     

Spectral theorem for multipliers on {L_{\omega}^{2}(\mathbb{R})}

We study the spectrum σ(M) of the multipliers M which commute with the translations on weighted spaces ${L_{\omega}^{2}(\mathbb{R})}We study the spectrum σ(M) of the multipliers M which commute with the translations on weighted spaces L_w²(\mathbbR){L_{\omega}^{2}(\mathbb{R})} For operators M in the algebra generated by the convolutions with f ? C_c(\mathbb R){\phi \in {C_c(\mathbb {R})}} we show that [`(m(W))] = s(M){\overline{\mu(\Omega)} = \sigma(M)}, where the set Ω is determined by the spectrum of the shift S and μ is the symbol of M. For the general multipliers M we establish that [`(m(W))]{\overline{\mu(\Omega)}} is included in σ(M). A generalization of these results is given for the weighted spaces L²_w(\mathbb R^k){L^2_{\omega}(\mathbb {R}^{k})} where the weight ω has a special form.     

The Hermite Property of a Causal Wiener Algebra Used in Control Theory

Let \mathbbC₊ : = {s ? \mathbbC    |     Re(s) 3 0}{{\mathbb{C}}}_{+} := \{s \in {{\mathbb{C}}}\quad | \quad {\rm Re}(s) \geq 0\} and let A\mathcal{A} denote the Banach algebra

On approximation of periodic functions by Fourier sums

Let L _p , 1 ≤ p< ∞, be the space of 2π-periodic functions f with the norm || f ||_p = ( ò _{- p}^p | f |^p )^{1 \mathord}

Closed-Range Composition Operators on {\mathbb{A}^2} and the Bloch Space

For any analytic self-map j{\varphi} of {z : |z| <  1} we give four separate conditions, each of which is necessary and sufficient for the composition operator C_j{C_{\varphi}} to be closed-range on the Bloch space B{\mathcal{B}} . Among these conditions are some that appear in the literature, where we provide new proofs. We further show that if C_j{C_{\varphi}} is closed-range on the Bergman space \mathbbA²{\mathbb{A}^2} , then it is closed-range on B{\mathcal{B}} , but that the converse of this fails with a vengeance. Our analysis involves an extension of the Julia-Carathéodory Theorem.     

On $\frak {F}$-normal Fitting classes of finite soluble groups

Let \frak X, \frak F,\frak X\subseteqq \frak F\frak {X}, \frak {F},\frak {X}\subseteqq \frak {F}, be non-trivial Fitting classes of finite soluble groups such that G_{\frak X}G_{\frak {X}} is an \frak X\frak {X}-injector of G for all G ? \frak FG\in \frak {F}. Then \frak X\frak {X} is called \frak F\frak {F}-normal. If \frak F=\frak S_p\frak {F}=\frak {S}_{\pi }, it is known that (1) \frak X\frak {X} is \frak F\frak {F}-normal precisely when \frak X^*=\frak F^*\frak {X}^{\ast }=\frak {F}^{\ast }, and consequently (2) \frak F í \frak X\frak N\frak {F}\subseteq \frak {X}\frak {N} implies \frak X^*=\frak F^*\frak {X}^{\ast }=\frak {F}^{\ast }, and (3) there is a unique smallest \frak F\frak {F}-normal Fitting class. These assertions are not true in general. We show that there are Fitting classes \frak F\not = \frak S_p\frak {F}\not =\frak {S}_{\pi } filling property (1), whence the classes \frak S_p\frak {S}_{\pi } are not characterized by satisfying (1). Furthermore we prove that (2) holds true for all Fitting classes \frak F\frak {F} satisfying a certain extension property with respect to wreath products although there could be an \frak F\frak {F}-normal Fitting class outside the Lockett section of \frak F\frak {F}. Lastly, we show that for the important cases \frak F=\frak Nⁿ, n\geqq 2\frak {F}=\frak {N}^{n},\ n\geqq 2, and \frak F=\frak S_p1?\frak S_pr, p_i \frak {F}=\frak {S}_{p_{1}}\cdots \frak {S}_{p_{r}},\ p_{i} primes, there is a unique smallest \frak F\frak {F}-normal Fitting class, which we describe explicitly.     

Ring of invariants of general linear group over local ring \mathbb{Z}_{p^m }

Let \mathbbZ_p^m \mathbb{Z}_{p^m } be the ring of integers modulo p ^m , where p is a prime and m ⩾ 1. The general linear group GL _n ( \mathbbZ_p^m \mathbb{Z}_{p^m } ) acts naturally on the polynomial algebra A _n := \mathbbZ_p^m \mathbb{Z}_{p^m } [x ₁, …, x _n ]. Denote by A_n^{GL₂ (\mathbbZ_pm )} A_n^{GL_2 (\mathbb{Z}_{p^m } )} the corresponding ring of invariants. The purpose of the present paper is to calculate this invariant ring. Our results also generalize the classical Dickson’s theorem.     

Approximation by the Bernstein–Durrmeyer Operator on a Simplex

For the Jacobi-type Bernstein–Durrmeyer operator M _n,κ on the simplex T ^d of ℝ ^d , we proved that for f∈L ^p (W _κ ;T ^d ) with 1<p<∞,

A new class of integrable systems

This paper concerns the integrability of Hamiltonian systems with two degrees of freedom whose Hamiltonian has the form¶ H=¹/₂(x₁²+x₂²) +V(y₁,y₂) H={1\over2}(x_{1}^{2}+x_{2}^{2}) +V(y_{1},y_{2}) where¶¶ V(y₁,y₂)=¹/₂(a₁y₁²+a₂y₂²) + ¹/₄b₁y₁⁴ + ¹/₄b₂y₂⁴ + ¹/₂b₃y₁²y₂² + ?_k=1³g_k(y₁²+y₂²) ^k+2 V(y_{1},y_{2})={1\over2}\big(\alpha _{1}y_{1}^{2}+\alpha_{2}y_{2}^{2}\big) + {1\over4}\beta _{1}y_{1}^{4} + {1\over4}\beta_{2}y_{2}^{4} + {1\over2}\beta _{3}y_{1}^{2}y_{2}^{2} + \sum_{k=1}^{3}\gamma_{k}\big(y_{1}^{2}+y_{2}^{2}\big) ^{k+2} ¶¶ which, constitues a generalization of some well-known integrable systems. We give new values of the vector (a₁,a₂,b₁,b₂,b₃,g₁,g₂,g₃) (\alpha _{1},\alpha_{2},\beta _{1},\beta _{2},\beta _{3},\gamma _{1},\gamma _{2},\gamma _{3}) for which this system is completely integrable and we show that the system is linearized in the Jacobian variety Jac(G \Gamma ) of a smooth genus 2 hyperelliptic Riemann surface G \Gamma .