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 )}.      2. On the error terms for representation numbers of quadratic forms      Z. Xu 《Acta Mathematica Hungarica》2010,127(4):301-319 Let f be a primitive positive integral binary quadratic form of discriminant −D, and r f (n) the number of representations of n by f up to automorphisms of f. We first improve the error term E(x) of $ \sum\limits_{n \leqq x} {r_f (n)^\beta } $ \sum\limits_{n \leqq x} {r_f (n)^\beta } for any positive integer β. Next, we give an estimate of ∫1 T |E(x)|2 x −3/2 dx when β = 1.      3. Quasi-convex Mappings on the Unit Polydisk in $\mathbb{C}^{n}$      Xiaosong LIU  Taishun LIU 《数学年刊B辑(英文版)》2011,32(2):241-252 In this paper, the sharp estimates of all homogeneous expansions for f are established, where f(z) = (f 1(z), f 2(z), …, f n (z))′ is a k-fold symmetric quasi-convex mapping defined on the unit polydisk in ℂ n and $ \begin{gathered} \frac{{D^{tk + 1} + f_p \left( 0 \right)\left( {z^{tk + 1} } \right)}} {{\left( {tk + 1} \right)!}} = \sum\limits_{l_1 ,l_2 ,...,l_{tk + 1} = 1}^n {\left| {apl_1 l_2 ...l_{tk + 1} } \right|e^{i\tfrac{{\theta pl_1 + \theta pl_2 + ... + \theta pl_{tk + 1} }} {{tk + 1}}} zl_1 zl_2 ...zl_{tk + 1} ,} \hfill \\ p = 1,2,...,n. \hfill \\ \end{gathered} $ \begin{gathered} \frac{{D^{tk + 1} + f_p \left( 0 \right)\left( {z^{tk + 1} } \right)}} {{\left( {tk + 1} \right)!}} = \sum\limits_{l_1 ,l_2 ,...,l_{tk + 1} = 1}^n {\left| {apl_1 l_2 ...l_{tk + 1} } \right|e^{i\tfrac{{\theta pl_1 + \theta pl_2 + ... + \theta pl_{tk + 1} }} {{tk + 1}}} zl_1 zl_2 ...zl_{tk + 1} ,} \hfill \\ p = 1,2,...,n. \hfill \\ \end{gathered}       4. Recognition by prime graph of $^2 D_{2^m + 1} (3)$      A. Babai  B. Khosravi 《Siberian Mathematical Journal》2011,52(5):788-795 As shown in [1] the simple group 2 D2m + 1 (3)^2 D_{2^m + 1} (3) is recognizable by spectrum. The main result of this paper generalizes the above, stating that 2 D2m + 1 (3)^2 D_{2^m + 1} (3) is recognizable by prime graph. In other words, we show that if G is a finite group satisfying G(G) = G(2 D2m + 1 (3))\Gamma (G) = \Gamma (^2 D_{2^m + 1} (3)) then G @ 2 D2m + 1 (3)G \cong ^2 D_{2^m + 1} (3).      5. Classification of quasifinite\mathcal{W}_\infty -modules-modules      Yucai Su  Bin Xin 《Israel Journal of Mathematics》2006,151(1):223-236 It is proved that an irreducible quasifinite -module is a highest or lowest weight module or a module of the intermediate series; a uniformly bounded indecomposable weight -module is a module of the intermediate series. For a nondegenerate additive subgroup Λ ofF n, whereF is a field of characteristic zero, there is a simple Lie or associative algebraW(Λ,n)(1) spanned by differential operatorsuD 1 m …D 1 m foru ∈F[Γ] (the group algebra), andm i≥0 with , whereD i are degree operators. It is also proved that an indecomposable quasifinite weightW(Λ,n)(1)-module is a module of the intermediate series if Λ is not isomorphic to ℤ. Supported by NSF grant no. 10471091 of China and two grants “Excellent Young Teacher Program” and “Trans-Century Training Programme Foundation for the Talents” from the Ministry of Education of China.      6. Jumping conics on a smooth quadric in $${\mathbb {P}_3}$$      Sukmoon Huh 《Annali di Matematica Pura ed Applicata》2011,190(2):195-208 We investigate the jumping conics of stable vector bundles E of rank 2 on a smooth quadric surface Q with the first Chern class c1 = OQ(-1,-1){c_1= \mathcal{O}_Q(-1,-1)} with respect to the ample line bundle OQ(1,1){\mathcal {O}_Q(1,1)} . We show that the set of jumping conics of E is a hypersurface of degree c 2(E) − 1 in \mathbb P3*{\mathbb {P}_3^{*}} . Using these hypersurfaces, we describe moduli spaces of stable vector bundles of rank 2 on Q in the cases of lower c 2(E).      7. Hypersurfaces in the Lorentz-Minkowski space satisfying Lkψ?=?Aψ?+?b      Pascual Lucas  H. Fabián Ramírez-Ospina 《Geometriae Dedicata》2011,153(1):151-175 We study hypersurfaces in the Lorentz-Minkowski space \mathbbLn+1{\mathbb{L}^{n+1}} whose position vector ψ satisfies the condition L k ψ = Aψ + b, where L k is the linearized operator of the (k + 1)th mean curvature of the hypersurface for a fixed k = 0, . . . , n − 1, A ? \mathbbR(n+1)×(n+1){A\in\mathbb{R}^{(n+1)\times(n+1)}} is a constant matrix and b ? \mathbbLn+1{b\in\mathbb{L}^{n+1}} is a constant vector. For every k, we prove that the only hypersurfaces satisfying that condition are hypersurfaces with zero (k + 1)th mean curvature, open pieces of totally umbilical hypersurfaces \mathbbSn1(r){\mathbb{S}^n_1(r)} or \mathbbHn(-r){\mathbb{H}^n(-r)}, and open pieces of generalized cylinders \mathbbSm1(r)×\mathbbRn-m{\mathbb{S}^m_1(r)\times\mathbb{R}^{n-m}}, \mathbbHm(-r)×\mathbbRn-m{\mathbb{H}^m(-r)\times\mathbb{R}^{n-m}}, with k + 1 ≤ m ≤ n − 1, or \mathbbLm×\mathbbSn-m(r){\mathbb{L}^m\times\mathbb{S}^{n-m}(r)}, with k + 1 ≤ n − m ≤ n − 1. This completely extends to the Lorentz-Minkowski space a previous classification for hypersurfaces in \mathbbRn+1{\mathbb{R}^{n+1}} given by Alías and Gürbüz (Geom. Dedicata 121:113–127, 2006).      8. Real Paley-Wiener type theorems for the Dunkl transform on {\mathcal{S}}'(\mathbb{R}^{d})      Sihem Ayadi  Slaim Ben Farah 《The Ramanujan Journal》2009,19(2):225-236 We prove real Paley-Wiener type theorems for the Dunkl transform ℱ D on the space of tempered distributions. Let T∈S′(ℝ d ) and Δ κ the Dunkl Laplacian operator. First, we establish that the support of ℱ D (T) is included in the Euclidean ball , M>0, if and only if for all R>M we have lim  n→+∞ R −2n Δ κ n T=0 in S′(ℝ d ). Second, we prove that the support of ℱ D (T) is included in ℝ d ∖B(0,M), M>0, if and only if for all R<M, we have lim  n→+∞ R 2n  ℱ D −1(‖y‖−2n ℱ D (T))=0 in S′(ℝ d ). Finally, we study real Paley-Wiener theorems associated with -slowly increasing function.       9. Bispectral and (\mathfrak{gl}_{N},\mathfrak{gl}_{M}) dualities      Evgenii E. Mukhin  Vitaly O. Tarasov  Alexander N. Varchenko 《Functional Analysis and Other Mathematics》2006,1(1):47-69 Let be a space of quasipolynomials of dimension N=N 1+⋅⋅⋅+N n . We define the regularized fundamental operator of V as the polynomial differential operator D=∑ i=0 N A N−i (x)∂ x i annihilating V and such that its leading coefficient A 0 is a polynomial of the minimal possible degree. We apply a suitable integral transformation to V to construct a space of quasipolynomials whose regularized fundamental operator is the differential operator ∑ i=0 N u i A N−i (∂ u ). Our integral transformation corresponds to the bispectral involution on the space of rational solutions (vanishing at infinity) of the KP hierarchy. As a corollary of the properties of the integral transformation, we obtain a correspondence between critical points of the two master functions associated with the -dual Gaudin models and also between the corresponding Bethe vectors. The research of E. M. was supported in part by the NSF (Grant No. DMS-0140460). The research of A. V. was supported in part by the NSF (Grant No. DMS-0244579).      10. 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.      11. I sottogruppi del gruppo modulare con coefficienti del corpo di Jacobi-Eisenstein e un teorema sui gruppi finiti      Giovanni Sansone 《Annali di Matematica Pura ed Applicata》1926,3(1):73-107 Sommario Introduzione — § 1 – 1. L'indice μ(n) dei sottogruppi Гμ(n) del gruppo Γ di sostituzioni lineari unimodulari con coefficienti del campo diJacobi-Eisenstein — 2. Il poliedro fondamentale del sottogruppo Гμ(1−ε) — § 2 – 3. I campi fondamentali dei gruppi Гμ(n) — 4. Impossibilità di limitare con un numero finito di piani e sfere di riflessione i poliedri fondamentali dei gruppi Гμ(n), conn intero razionale pari, diverso da 2 — § 3 – 5. Relazioni fondamentali fra le sostituzioni generatrici del gruppo di sostituzioni lineari con coefficienti del corpo Kε con determinante ±1 — § 4 – 6. Sulla indipendenza delle sostituzioniS,T,U, generatrici del gruppo finito G2μ(n) e sulle loro relazioni caratteristiche nel gruppo G2μ(n) — § 5 – 7. Dimostrazione del teorema fondamentale sui gruppi G2μ(n). Lemmi preliminari — 8, Dimostrazione del teorema fondamentale nel caso di moduli primi con 2(1−ε) — § 6 – 9. Il teorema fondamentale per i modulim(1−ε), 3m, 2m, 2m(1−ε), 6m conm primo con 6 – 10. Immagine geometrica dei gruppi G2μ(1−ε) — § 7 – 11. Il gruppo delle sostituzioni unimodulari , [c/1+4ma]=+1, e il caso eccezionale dei moduli 4m – 12. Il gruppo delle sostituzioni unimodulari [c/1+3m(1−ε)a]3=+1 e il caso eccezionale dei moduli 3(1−ε)m.      12. 泛函方程组的解析解      刘新和 《高校应用数学学报(英文版)》2003,18(2):129-137 § 1　IntroductionThe Feigenbaum functional equation plays an importantrole in the theory concerninguniversal properties of one-parameter families of maps of the interval that has the formf2 (λx) +λf(x) =0 ,0 <λ=-f(1 ) <1 ,f(0 ) =1 ,(1 .1 )where f is a map ofthe interval[-1 ,1 ] into itself.Lanford[1 ] exhibited a computer-assist-ed proof for the existence of an even analytic solution to Eq.(1 .1 ) .It was shown in[2 ]that Eq.(1 .1 ) does not have an entire solution.Si[3] discussed the it…      13. 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.      14. On simultaneous approximation to a differentiable function and its derivative by inverse Pal-Type interpolation polynomials      Bao Yongguang 《分析论及其应用》1995,11(4):15-23 Let ξn −1 < ξn −2 < ξn − 2 < ... < ξ1 be the zeros of the the (n−1)-th Legendre polynomial Pn−1(x) and −1=xn<xn−1<...<x1=1, the zeros of the polynomial . By the theory of the inverse Pal-Type interpolation, for a function f(x)∈C [−1,1] 1 , there exists a unique polynomial Rn(x) of degree 2n−2 (if n is even) satisfying conditions Rn(f, ξk) = f (εk) (1 ⩽ k ⩽ n −1); R1 n(f,xk)=f1(xk)(1≤k≤n). This paper discusses the simultaneous approximation to a differentiable function f by inverse Pal-Type interpolation polynomial {Rn(f, x)} (n is even) and the main result of this paper is that if f∈C [1,1] r , r≥2, n≥r+2, and n is even then |R1 n(f,x)−f1(x)|=0(1)|Wn(x)|h(x)·n3−r·E2n−r−3(f(r)) holds uniformly for all x∈[−1,1], where .      15. A Class of Integral Operators on the Unit Ball of \mathbb{C}^{n}      Osman Kures  Kehe Zhu 《Integral Equations and Operator Theory》2006,56(1):71-82 For real parameters a, b, c, and t, where c is not a nonpositive integer, we determine exactly when the integral operator is bounded on where is the open unit ball in and dvt (z)  =  (1  −  |z| 2) t dv (z) with dv being volume measure on The characterization remains the same if we replace (1  −  〈z, w 〉) c in the integral kernel above by its modulus |1  −  〈z, w〉| c.      16. Examples of factorial threefolds complete intersection in $${\mathbb{P}^5}$$ with many singularities      Pietro Sabatino 《Ricerche di matematica》2009,58(2):285-297 Denote by ν m (d) the maximal integer for which there exists for d >> 0{d \gg 0} a threefold X ì \mathbbP5{X\subset \mathbb{P}^5} complete intersection of hypersurfaces of degree respectively d and d − 1 such that X has only ordinary singularities of order m and |Sing(X)| = ν m (d). We prove that, nm(d) 3 j(d){\nu_m(d)\ge \varphi(d)} where j(d) ~ d5{\varphi(d)\sim d^5} asymptotically. This result extends (Di Gennaro and Franco in Commun Contemp Math 10(5):745–764, 2008, Corollary 2.10).      17. The region of values of the system { f ( z 1),..., f ( z n )} in the class of typically real functions. III      E. G. Goluzina 《Journal of Mathematical Sciences》2007,143(3):3023-3029 The paper studies the region of values Dm,n(T) of the system {f(z1), f(z2),..., f(zm), f(r1), f(r2),..., f(rn)}, where m ≥ 1; n > 1; zj, j = 1, ... m, are arbitrary fixed points of the disk U = {z: |z| < 1} with Im zj ≠ 0, j = 1, 2, ..., m; rj, 0 < rj < 1, j = 1, 2, ..., n, are fixed; f ∈ T, and the class T consists of functions f(z) = z + c2z2 + ... regular in the disk U and satisfying the condition Im f(z) · Im z > 0 for Im z ≠= 0, z ∈ U. An algebraic characterization of the set Dm,n(T) in terms of nonnegative-definite Hermitian forms is provided, and all the boundary functions are described. As an implication, the region of values of f(z1) in the subclass of functions f ∈ T with prescribed values f(rj) (j = 1, 2, 3) is determined. Bibliography: 12 titles. Dedicated to the 100th anniversary of my father’s birthday __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 337, 2006, pp. 23–34.      18. The Besov Space B^{{0,1}}_{1} on Domains      Yi Qing Gui  Shan Zhen Lu  Da Chun Yang 《数学学报(英文版)》2001,17(2):181-196 Let Ω be a bounded Lipschitz domain. Define B 0,1 1, r (Ω) = {f∈L 1 (Ω): there is an F∈B 0,1 1 (ℝ n ) such that F|Ω = f} and B 0,1 1 z (Ω) = {f∈B 0,1 1 (ℝ n ) : f = 0 on ℝ n \}. In this paper, the authors establish the atomic decompositions of these spaces. As by-products, the authors obtained the regularity on these spaces of the solutions to the Dirichlet problem and the Neumann problem of the Laplace equation of ℝ n +. Received June 8, 2000, Accepted October 24, 2000      19. Precise Rates in the Law of Iterated Logarithm for the Moment of I.I.D. Random Variables   总被引：1，自引：0，他引：1   Ye JIANG Li Xin ZHANG 《数学学报(英文版)》2006,22(3):781-792 Let{X,Xn;n≥1} be a sequence of i,i.d, random variables, E X = 0, E X^2 = σ^2 〈 ∞.Set Sn=X1＋X2＋…＋Xn,Mn=max k≤n│Sk│,n≥1.Let an=O（1/loglogn）.In this paper,we prove that,for b〉-1,lim ε→0 →^2（b＋1）∑n=1^∞ （loglogn）^b/nlogn n^1/2 E{Mn-σ（ε＋an）√2nloglogn}＋σ2^-b/（b＋1）（2b＋3）E│N│^2b＋3∑k=0^∞ （-1）k/（2k＋1）^2b＋3 holds if and only if EX=0 and EX^2=σ^2〈∞.      20. New negative Latin square type partial difference sets in nonelementary abelian 2-groups and 3-groups      John Polhill 《Designs, Codes and Cryptography》2008,46(3):365-377 A partial difference set having parameters (n 2, r(n − 1), n + r 2 − 3r, r 2 − r) is called a Latin square type partial difference set, while a partial difference set having parameters (n 2, r(n + 1), − n + r 2 + 3r, r 2 + r) is called a negative Latin square type partial difference set. Nearly all known constructions of negative Latin square partial difference sets are in elementary abelian groups. In this paper, we develop three product theorems that construct negative Latin square type partial difference sets and Latin square type partial difference sets in direct products of abelian groups G and G′ when these groups have certain Latin square or negative Latin square type partial difference sets. Using these product theorems, we can construct negative Latin square type partial difference sets in groups of the form where the s i are nonnegative integers and s 0 + s 1 ≥ 1. Another significant corollary to these theorems are constructions of two infinite families of negative Latin square type partial difference sets in 3-groups of the form for nonnegative integers s i . Several constructions of Latin square type PDSs are also given in p-groups for all primes p. We will then briefly indicate how some of these results relate to amorphic association schemes. In particular, we construct amorphic association schemes with 4 classes using negative Latin square type graphs in many nonelementary abelian 2-groups; we also use negative Latin square type graphs whose underlying sets can be elementary abelian 3-groups or nonelementary abelian 3-groups to form 3-class amorphic association schemes.

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On the error terms for representation numbers of quadratic forms

Let f be a primitive positive integral binary quadratic form of discriminant −D, and r _f (n) the number of representations of n by f up to automorphisms of f. We first improve the error term E(x) of $ \sum\limits_{n \leqq x} {r_f (n)^\beta } $ \sum\limits_{n \leqq x} {r_f (n)^\beta } for any positive integer β. Next, we give an estimate of ∫₁ ^T |E(x)|² x ^−3/2 dx when β = 1.     

Quasi-convex Mappings on the Unit Polydisk in $\mathbb{C}^{n}$

In this paper, the sharp estimates of all homogeneous expansions for f are established, where f(z) = (f ₁(z), f ₂(z), …, f _n (z))′ is a k-fold symmetric quasi-convex mapping defined on the unit polydisk in ℂ ⁿ and

Recognition by prime graph of $^2 D_{2^m + 1} (3)$

As shown in [1] the simple group ² D_{2^m + 1} (3)^2 D_{2^m + 1} (3) is recognizable by spectrum. The main result of this paper generalizes the above, stating that ² D_{2^m + 1} (3)^2 D_{2^m + 1} (3) is recognizable by prime graph. In other words, we show that if G is a finite group satisfying G(G) = G(² D_{2^m + 1} (3))\Gamma (G) = \Gamma (^2 D_{2^m + 1} (3)) then G @ ² D_{2^m + 1} (3)G \cong ^2 D_{2^m + 1} (3).     

Classification of quasifinite\mathcal{W}_\infty -modules-modules

It is proved that an irreducible quasifinite -module is a highest or lowest weight module or a module of the intermediate series; a uniformly bounded indecomposable weight -module is a module of the intermediate series. For a nondegenerate additive subgroup Λ ofF _n, whereF is a field of characteristic zero, there is a simple Lie or associative algebraW(Λ,n)⁽¹⁾ spanned by differential operatorsuD ₁ ^m …D ₁ ^m foru ∈F[Γ] (the group algebra), andm _i≥0 with , whereD _i are degree operators. It is also proved that an indecomposable quasifinite weightW(Λ,n)⁽¹⁾-module is a module of the intermediate series if Λ is not isomorphic to ℤ. Supported by NSF grant no. 10471091 of China and two grants “Excellent Young Teacher Program” and “Trans-Century Training Programme Foundation for the Talents” from the Ministry of Education of China.     

Jumping conics on a smooth quadric in $${\mathbb {P}_3}$$

We investigate the jumping conics of stable vector bundles E of rank 2 on a smooth quadric surface Q with the first Chern class c₁ = O_Q(-1,-1){c_1= \mathcal{O}_Q(-1,-1)} with respect to the ample line bundle O_Q(1,1){\mathcal {O}_Q(1,1)} . We show that the set of jumping conics of E is a hypersurface of degree c ₂(E) − 1 in \mathbb P₃^*{\mathbb {P}_3^{*}} . Using these hypersurfaces, we describe moduli spaces of stable vector bundles of rank 2 on Q in the cases of lower c ₂(E).     

Hypersurfaces in the Lorentz-Minkowski space satisfying Lkψ?=?Aψ?+?b

We study hypersurfaces in the Lorentz-Minkowski space \mathbbLⁿ⁺¹{\mathbb{L}^{n+1}} whose position vector ψ satisfies the condition L _k ψ = Aψ + b, where L _k is the linearized operator of the (k + 1)th mean curvature of the hypersurface for a fixed k = 0, . . . , n − 1, A ? \mathbbR^(n+1)×(n+1){A\in\mathbb{R}^{(n+1)\times(n+1)}} is a constant matrix and b ? \mathbbLⁿ⁺¹{b\in\mathbb{L}^{n+1}} is a constant vector. For every k, we prove that the only hypersurfaces satisfying that condition are hypersurfaces with zero (k + 1)th mean curvature, open pieces of totally umbilical hypersurfaces \mathbbSⁿ₁(r){\mathbb{S}^n_1(r)} or \mathbbHⁿ(-r){\mathbb{H}^n(-r)}, and open pieces of generalized cylinders \mathbbS^m₁(r)×\mathbbR^n-m{\mathbb{S}^m_1(r)\times\mathbb{R}^{n-m}}, \mathbbH^m(-r)×\mathbbR^n-m{\mathbb{H}^m(-r)\times\mathbb{R}^{n-m}}, with k + 1 ≤ m ≤ n − 1, or \mathbbL^m×\mathbbS^n-m(r){\mathbb{L}^m\times\mathbb{S}^{n-m}(r)}, with k + 1 ≤ n − m ≤ n − 1. This completely extends to the Lorentz-Minkowski space a previous classification for hypersurfaces in \mathbbRⁿ⁺¹{\mathbb{R}^{n+1}} given by Alías and Gürbüz (Geom. Dedicata 121:113–127, 2006).     

Real Paley-Wiener type theorems for the Dunkl transform on {\mathcal{S}}'(\mathbb{R}^{d})

We prove real Paley-Wiener type theorems for the Dunkl transform ℱ _D on the space of tempered distributions. Let T∈S′(ℝ ^d ) and Δ _κ the Dunkl Laplacian operator. First, we establish that the support of ℱ _D (T) is included in the Euclidean ball , M>0, if and only if for all R>M we have lim  _n→+∞ R ⁻²ⁿ Δ _κ ⁿ T=0 in S′(ℝ ^d ). Second, we prove that the support of ℱ _D (T) is included in ℝ ^d ∖B(0,M), M>0, if and only if for all R<M, we have lim  _n→+∞ R ²ⁿ  ℱ _D ⁻¹(‖y‖⁻²ⁿ ℱ _D (T))=0 in S′(ℝ ^d ). Finally, we study real Paley-Wiener theorems associated with -slowly increasing function.      

Bispectral and (\mathfrak{gl}_{N},\mathfrak{gl}_{M}) dualities

Let be a space of quasipolynomials of dimension N=N ₁+⋅⋅⋅+N _n . We define the regularized fundamental operator of V as the polynomial differential operator D=∑ _i=0 ^N A _N−i (x)∂ _x ⁱ annihilating V and such that its leading coefficient A ₀ is a polynomial of the minimal possible degree. We apply a suitable integral transformation to V to construct a space of quasipolynomials whose regularized fundamental operator is the differential operator ∑ _i=0 ^N u ⁱ A _N−i (∂ _u ). Our integral transformation corresponds to the bispectral involution on the space of rational solutions (vanishing at infinity) of the KP hierarchy. As a corollary of the properties of the integral transformation, we obtain a correspondence between critical points of the two master functions associated with the -dual Gaudin models and also between the corresponding Bethe vectors. The research of E. M. was supported in part by the NSF (Grant No. DMS-0140460). The research of A. V. was supported in part by the NSF (Grant No. DMS-0244579).     

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.     

I sottogruppi del gruppo modulare con coefficienti del corpo di Jacobi-Eisenstein e un teorema sui gruppi finiti

Sommario Introduzione — § 1 – 1. L'indice μ(n) dei sottogruppi Г_μ(n) del gruppo Γ di sostituzioni lineari unimodulari con coefficienti del campo diJacobi-Eisenstein — 2. Il poliedro fondamentale del sottogruppo Г_μ(1−ε) — § 2 – 3. I campi fondamentali dei gruppi Г_μ(n) — 4. Impossibilità di limitare con un numero finito di piani e sfere di riflessione i poliedri fondamentali dei gruppi Г_μ(n), conn intero razionale pari, diverso da 2 — § 3 – 5. Relazioni fondamentali fra le sostituzioni generatrici del gruppo di sostituzioni lineari con coefficienti del corpo K_ε con determinante ±1 — § 4 – 6. Sulla indipendenza delle sostituzioniS,T,U, generatrici del gruppo finito G_2μ(n) e sulle loro relazioni caratteristiche nel gruppo G_2μ(n) — § 5 – 7. Dimostrazione del teorema fondamentale sui gruppi G_2μ(n). Lemmi preliminari — 8, Dimostrazione del teorema fondamentale nel caso di moduli primi con 2(1−ε) — § 6 – 9. Il teorema fondamentale per i modulim(1−ε), 3m, 2m, 2m(1−ε), 6m conm primo con 6 – 10. Immagine geometrica dei gruppi G_2μ(1−ε) — § 7 – 11. Il gruppo delle sostituzioni unimodulari , [c/1+4ma]=+1, e il caso eccezionale dei moduli 4m – 12. Il gruppo delle sostituzioni unimodulari [c/1+3m(1−ε)a]₃=+1 e il caso eccezionale dei moduli 3(1−ε)m.     

泛函方程组的解析解

§ 1　IntroductionThe Feigenbaum functional equation plays an importantrole in the theory concerninguniversal properties of one-parameter families of maps of the interval that has the formf2 (λx) +λf(x) =0 ,0 <λ=-f(1 ) <1 ,f(0 ) =1 ,(1 .1 )where f is a map ofthe interval[-1 ,1 ] into itself.Lanford[1 ] exhibited a computer-assist-ed proof for the existence of an even analytic solution to Eq.(1 .1 ) .It was shown in[2 ]that Eq.(1 .1 ) does not have an entire solution.Si[3] discussed the it…     

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.     

On simultaneous approximation to a differentiable function and its derivative by inverse Pal-Type interpolation polynomials

Let ξ_{n −1} < ξ_{n −2} < ξ_{n − 2} < ... < ξ₁ be the zeros of the the (n−1)-th Legendre polynomial P_n−1(x) and −1=x_n<x_n−1<...<x₁=1, the zeros of the polynomial . By the theory of the inverse Pal-Type interpolation, for a function f(x)∈C _[−1,1] ¹ , there exists a unique polynomial R_n(x) of degree 2n−2 (if n is even) satisfying conditions R_n(f, ξ_k) = f (ε_k) (1 ⩽ k ⩽ n −1); R¹ _n(f,x_k)=f¹(x_k)(1≤k≤n). This paper discusses the simultaneous approximation to a differentiable function f by inverse Pal-Type interpolation polynomial {R_n(f, x)} (n is even) and the main result of this paper is that if f∈C _[1,1] ^r , r≥2, n≥r+2, and n is even then |R¹ _n(f,x)−f¹(x)|=0(1)|W_n(x)|h(x)·n^3−r·E_2n−r−3(f^(r)) holds uniformly for all x∈[−1,1], where .     

A Class of Integral Operators on the Unit Ball of \mathbb{C}^{n}

For real parameters a, b, c, and t, where c is not a nonpositive integer, we determine exactly when the integral operator

Examples of factorial threefolds complete intersection in $${\mathbb{P}^5}$$ with many singularities

Denote by ν _m (d) the maximal integer for which there exists for d >> 0{d \gg 0} a threefold X ì \mathbbP⁵{X\subset \mathbb{P}^5} complete intersection of hypersurfaces of degree respectively d and d − 1 such that X has only ordinary singularities of order m and |Sing(X)| = ν _m (d). We prove that, n_m(d) 3 j(d){\nu_m(d)\ge \varphi(d)} where j(d) ~ d⁵{\varphi(d)\sim d^5} asymptotically. This result extends (Di Gennaro and Franco in Commun Contemp Math 10(5):745–764, 2008, Corollary 2.10).     

The region of values of the system { f ( z 1),..., f ( z n )} in the class of typically real functions. III

The paper studies the region of values D_m,n(T) of the system {f(z₁), f(z₂),..., f(z_m), f(r₁), f(r₂),..., f(r_n)}, where m ≥ 1; n > 1; z_j, j = 1, ... m, are arbitrary fixed points of the disk U = {z: |z| < 1} with Im z_j ≠ 0, j = 1, 2, ..., m; r_j, 0 < r_j < 1, j = 1, 2, ..., n, are fixed; f ∈ T, and the class T consists of functions f(z) = z + c₂z² + ... regular in the disk U and satisfying the condition Im f(z) · Im z > 0 for Im z ≠= 0, z ∈ U. An algebraic characterization of the set D_m,n(T) in terms of nonnegative-definite Hermitian forms is provided, and all the boundary functions are described. As an implication, the region of values of f(z₁) in the subclass of functions f ∈ T with prescribed values f(r_j) (j = 1, 2, 3) is determined. Bibliography: 12 titles. Dedicated to the 100th anniversary of my father’s birthday __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 337, 2006, pp. 23–34.     

The Besov Space B^{{0,1}}_{1} on Domains

Let Ω be a bounded Lipschitz domain. Define B ^0,1 _{1, r} (Ω) = {f∈L ¹ (Ω): there is an F∈B ^0,1 ₁ (ℝ ⁿ ) such that F|_Ω = f} and B ^0,1 _{1 z} (Ω) = {f∈B ^0,1 ₁ (ℝ ⁿ ) : f = 0 on ℝ ⁿ \}. In this paper, the authors establish the atomic decompositions of these spaces. As by-products, the authors obtained the regularity on these spaces of the solutions to the Dirichlet problem and the Neumann problem of the Laplace equation of ℝ ⁿ ₊. Received June 8, 2000, Accepted October 24, 2000     

Precise Rates in the Law of Iterated Logarithm for the Moment of I.I.D. Random Variables

Let{X,Xn;n≥1} be a sequence of i,i.d, random variables, E X = 0, E X^2 = σ^2 〈 ∞.Set Sn=X1＋X2＋…＋Xn,Mn=max k≤n│Sk│,n≥1.Let an=O（1/loglogn）.In this paper,we prove that,for b〉-1,lim ε→0 →^2（b＋1）∑n=1^∞ （loglogn）^b/nlogn n^1/2 E{Mn-σ（ε＋an）√2nloglogn}＋σ2^-b/（b＋1）（2b＋3）E│N│^2b＋3∑k=0^∞ （-1）k/（2k＋1）^2b＋3 holds if and only if EX=0 and EX^2=σ^2〈∞.     

New negative Latin square type partial difference sets in nonelementary abelian 2-groups and 3-groups

A partial difference set having parameters (n ², r(n − 1), n + r ² − 3r, r ² − r) is called a Latin square type partial difference set, while a partial difference set having parameters (n ², r(n + 1), − n + r ² + 3r, r ² + r) is called a negative Latin square type partial difference set. Nearly all known constructions of negative Latin square partial difference sets are in elementary abelian groups. In this paper, we develop three product theorems that construct negative Latin square type partial difference sets and Latin square type partial difference sets in direct products of abelian groups G and G′ when these groups have certain Latin square or negative Latin square type partial difference sets. Using these product theorems, we can construct negative Latin square type partial difference sets in groups of the form where the s _i are nonnegative integers and s ₀ + s ₁ ≥ 1. Another significant corollary to these theorems are constructions of two infinite families of negative Latin square type partial difference sets in 3-groups of the form for nonnegative integers s _i . Several constructions of Latin square type PDSs are also given in p-groups for all primes p. We will then briefly indicate how some of these results relate to amorphic association schemes. In particular, we construct amorphic association schemes with 4 classes using negative Latin square type graphs in many nonelementary abelian 2-groups; we also use negative Latin square type graphs whose underlying sets can be elementary abelian 3-groups or nonelementary abelian 3-groups to form 3-class amorphic association schemes.