Central polynomials and matrix invariants

LetK be a field, charK=0 andM _n (K) the algebra ofn×n matrices overK. If λ=(λ₁,…,λ _m ) andμ=(μ ₁,…,μ _m ) are partitions ofn ² let wherex ₁,…,x _n ²,y ₁,…,y _n ² are noncommuting indeterminates andS _n ² is the symmetric group of degreen ². The polynomialsF ^{λ, μ} , when evaluated inM _n (K), take central values and we study the problem of classifying those partitions λ,μ for whichF ^{λ, μ} is a central polynomial (not a polynomial identity) forM _n (K). We give a formula that allows us to evaluateF ^{λ, μ} inM(K) in general and we prove that if λ andμ are not both derived in a suitable way from the partition δ=(1, 3,…, 2n−3, 2n−1), thenF ^{λ, μ} is a polynomial identity forM _n (K). As an application, we exhibit a new class of central polynomials forM _n (K). In memory of Shimshon Amitsur Research supported by a grant from MURST of Italy.     

Notes on the Borwein-Choi conjecture of Littlewood cyclotomic polynomials

Borwein and Choi conjectured that a polynomial P（x） with coefficients ±1 of degree N - 1 is cyclotomic iff
P（x）=±Φp1（±x）ΦP2（±x^p1）…Φpr（±x^p1p2…pr-1）,
where N = P1P2 … pτ and the pi are primes, not necessarily distinct. Here Φ（x） ：= （x^p - 1）/（x - 1） is the p-th cyclotomic polynomial. They also proved the conjecture for N odd or a power of 2. In this paper we introduce a so-called E-transformation, by which we prove the conjecture for a wider variety of cases and present the key as well as a new approach to investigate the coniecture.     

Multidimensional analog of the Hardy condition for power series

In the middle of the 20th century Hardy obtained a condition which must be imposed on a formal power series f(x) with positive coefficients in order that the series f ⁻¹(x) = $ \sum\limits_{n = 0}^\infty {b_n x^n } $ \sum\limits_{n = 0}^\infty {b_n x^n } b _n x ⁿ be such that b ₀ > 0 and b _n ≤ 0, n ≥ 1. In this paper we find conditions which must be imposed on a multidimensional series f(x ₁, x ₂, …, x _m ) with positive coefficients in order that the series f ⁻¹(x ₁, x ₂, …, x _m ) = $ \sum i_1 ,i_2 , \ldots ,i_m \geqslant 0^b i_1 ,i_2 , \ldots ,i_m ^{x_1^{i_1 } x_2^{i_2 } \ldots x_m^{i_m } } $ \sum i_1 ,i_2 , \ldots ,i_m \geqslant 0^b i_1 ,i_2 , \ldots ,i_m ^{x_1^{i_1 } x_2^{i_2 } \ldots x_m^{i_m } } satisfies the property b _{0, …, 0} > 0, $ bi_1 ,i_2 , \ldots ,i_m $ bi_1 ,i_2 , \ldots ,i_m ≤ 0, i ₁² + i ₂² + … + i _m ² > 0, which is similar to the one-dimensional case.     

Typical sheaves of generalized CM-modules

Let M be a generalized Cohen-Macaulay module over a noetherian local ring (R,m). Fix a standard system x₁, …, x_d∈m with respect to M and let . We construct a coherent Cohen-Macaulay sheafK over the projective space ℙ _R/I ^d-1 whose cohomological Hilbert functions depend only on the lengths of the local cohomology modules H _m ⁱ (M), (i=0, …, d−1).     

A note on the non-commutative neutrix product of distributions

The distributionF(x ₊, −r) Inx₊ andF(x ₋, −s) corresponding to the functionsx ₊ ^−r lnx₊ andx ₋ ^−s respectively are defined by the equations

On convergence of a generalized Pál interpolation process

Let f∈C _[−1,1] ^″ (r≥1) and R_n(f,α,β,x) be the generalized Pál interpolation polynomials satisfying the conditions R_n(f,α,β,x_k)=f(x_k),R_n ^′(f,α,β,x_k)=f′(x_k)(k=1,2,…,n), where {x_k} are the roots of n-th Jacobi polynomial P_n(α,β,x),α,β>−1 and {x _k ^″ } are the roots of (1−x²)P_n″(α,β,x). In this paper, we prove that holds uniformly on [0,1]. In Memory of Professor M. T. Cheng Supported by the Science Foundation of CSBTB and the Natural Science Foundatioin of Zhejiang.     

A Note on the Mean Value of Numbersof the Solutions of x^α≡ 1（modn）

Let T = T(p, q, α) be the number of solutions of the congruence x^α ≡ 1 (mod p^ηq^θ). Let A and B be sets of primes satisfying x₁ < p ≤ x₂ and y₁ < q ≤ y₂, respectively. A mean value estimation of is given. Supported by National Natural Science Foundation of China (No. 19971024) and Zhejiang Provincial Natural Science Foundation of China (No. 199047)     

Polynomial solutions of quasi-homogeneous partial differential equations

By means of a method of analytic number theory the following theorem is proved. Letp be a quasi-homogeneous linear partial differential operator with degreem,m > 0, w.r.t a dilation given by ( a₁, …, a_n). Assume that either a₁, …, a_n are positive rational numbers or for some Then the dimension of the space of polynomial solutions of the equationp[u] = 0 on ℝⁿ must be infinite     

Convergence of solutions of stochastic differential equations to the Arratia flow

We consider the solution x _ε of the equation

Uniform estimates for order statistics and Orlicz functions

We establish uniform estimates for order statistics: Given a sequence of independent identically distributed random variables ξ ₁, … , ξ _n and a vector of scalars x = (x ₁, … , x _n ), and 1 ≤ k ≤ n, we provide estimates for \mathbb E   k-min_{1 ￡ i ￡ n} |x_ix_i|{\mathbb E \, \, k-{\rm min}_{1\leq i\leq n} |x_{i}\xi _{i}|} and \mathbb E k-max_{1 ￡ i ￡ n}|x_ix_i|{\mathbb E\,k-{\rm max}_{1\leq i\leq n}|x_{i}\xi_{i}|} in terms of the values k and the Orlicz norm ||y_x||_M{\|y_x\|_M} of the vector y _x  = (1/x ₁, … , 1/x _n ). Here M(t) is the appropriate Orlicz function associated with the distribution function of the random variable |ξ ₁|, G(t) = \mathbb P ({ |x₁| ￡ t}){G(t) =\mathbb P \left(\left\{ |\xi_1| \leq t\right\}\right)}. For example, if ξ ₁ is the standard N(0, 1) Gaussian random variable, then G(t) = ?{\tfrac2p}ò₀^t e^-\fracs22ds {G(t)= \sqrt{\tfrac{2}{\pi}}\int_{0}^t e^{-\frac{s^{2}}{2}}ds }  and M(s)=?{\tfrac2p}ò₀^se^-\frac12t2dt{M(s)=\sqrt{\tfrac{2}{\pi}}\int_{0}^{s}e^{-\frac{1}{2t^{2}}}dt}. We would like to emphasize that our estimates do not depend on the length n of the sequence.     

Asymptotic properties of solutions of semilinear second-order elliptic equations in cylindrical domains

The equations under consideration have the following structure:

Higher integrability for minimizers of variational integrals with nonstandard growth

We consider a (possibly) vector-valued function u: Ω→R ^N, Ω⊂R ⁿ, minimizing the integral , whereD _iu=∂u/∂x _i, or some more general functional retaining the same behaviour; we prove higher integrability forDu:D ₁u,…,D_n−1u∈L^q, under suitable assumptions ona _i(x).

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Sunto Consideriamo una funzione u: Ω→R ^N, Ω⊂R ⁿ che minimizzi l'integrale , doveD _iu=∂u/∂x_i, o un funzionale con un comportamento simile; sotto opportune ipotesi sua _i(x), dimostriamo la seguente maggiore integrabilità perDu:D ₁u,…,D_n−1uεL^q.

     

Fourier and Hermite series estimates of regression functions

Summary In the paper we estimate a regressionm(x)=E {Y|X=x} from a sequence of independent observations (X ₁,Y ₁),…, (X _n, Y_n) of a pair (X, Y) of random variables. We examine an estimate of a type , whereN depends onn andϕ _N is Dirichlet kernel and the kernel associated with the hermite series. Assuming, that E|Y|<∞ and |Y|≦γ≦∞, we give condition for to converge tom(x) at almost allx, provided thatX has a density. if the regression hass derivatives, then converges tom(x) as rapidly asO(nC^{−(2s−1)/4s}) in probability andO(n ^{−(2s−1)/4s} logn) almost completely.     

An approximation of analytical functions by local splines

Local splines are presented for the approximation of functions of one and many variables, which are analytic in the domains , where U_i(z_i) is a unit disk in the complex plane C_i,i=1,2,…,l, l=1,2, …. Results are given for functions whose r-order derivatives belong to the Hardy's class H_p,1≤p≤∞. It is shown that the approximation converge to the function at the rate for functions of one variable and An^{−(r−1/p)/(l−1)} for functions of l variables, where n is the number of points of local splines and A and C are positive constants. This work was supported by Russian Foundation of Fundumental Inverstigations     

Shape preserving widths of Sobolev-type classes ofs-monotone functions on a finite interval

LetI be a finite interval andr ∈ ℕ. Denote by △ ₊ ^s L _q the subset of all functionsy ∈L _q such that thes-difference △ _T ^s y(·) is nonnegative onI, ∀τ>0. Further, denote by △ ₊ ^s W _p ^r the class of functionsx onI with the seminorm ‖x ^(r) ‖L _p ≤1, such that △ _T ^s x≥0, τ > 0, τ>0. Fors=3,…,r+1, we obtain two-sided estimates of the shape preserving widths , whereM ⁿ is the set of all linear manifoldsM ⁿ inL _q , dimM ⁿ ≤n, such thatM ⁿ ⋂△ ₊ ^s L _q ≠ 0. Part of this work was done while the first author visited Tel Aviv University in 2001 and part of it while the second author was a member of the Industrial Mathematics Institute (IMI), University of South Carolina.     

Positive Cubature Formulas and Marcinkiewicz–Zygmund Inequalities on Spherical Caps

Let Π _n ^d denote the space of all spherical polynomials of degree at most n on the unit sphere $\mathbb{S}^{d}Let Π _n ^d denote the space of all spherical polynomials of degree at most n on the unit sphere \mathbbS^d\mathbb{S}^{d} of ℝ ^d+1, and let d(x,y) denote the geodesic distance arccos x⋅y between x,y ? \mathbbS^dx,y\in\mathbb{S}^{d} . Given a spherical cap

B(e,a)={x ? \mathbbSd:d(x,e) ￡ a}    (e ? \mathbbSd, a ? (0,p) is bounded awayfrom p),B(e,\alpha)=\big\{x\in\mathbb{S}^{d}:d(x,e)\leq\alpha\big\}\quad \bigl(e\in\mathbb{S}^{d},\ \alpha\in(0,\pi)\ \mbox{is bounded awayfrom}\ \pi\bigr),      17. A general Hobby-Rice Theorem and cake cutting      Daniel Wulbert 《Israel Journal of Mathematics》2001,126(1):363-380 LetX be a Borel subset of a separable Banach spaceE. Letμ be a non-atomic,σ-finite, Borel measure onX. LetG ⊆L 1 (X, Σ,μ) bem-dimensional. Theorem:There is an l ∈ E* and real numbers −∞=x 0<x 1<x 2<…<x n<x n+1=∞with n≤m, such that for all g ∈ G,       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. Rauzy tilings and bounded remainder sets on the torus      V. G. Zhuravlev 《Journal of Mathematical Sciences》2006,137(2):4658-4672 For the two-dimensional torus , we construct the Rauzy tilings d0 ⊃ d1 ⊃ … ⊃ dm ⊃ …, where each tiling dm+1 is obtained by subdividing the tiles of dm. The following results are proved. Any tiling dm is invariant with respect to the torus shift S(x) = x+ mod ℤ2, where ζ−1 > 1 is the Pisot number satisfying the equation x3− x2−x-1 = 0. The induced map is an exchange transformation of Bmd ⊂ , where d = d0 and . The map S(m) is a shift of the torus , which is affinely isomorphic to the original shift S. This means that the tilings dm are infinitely differentiable. If ZN(X) denotes the number of points in the orbit S1(0), S2(0), …, SN(0) belonging to the domain Bmd, then, for all m, the remainder rN(Bmd) = ZN(Bmd) − N ζm satisfies the bounds −1.7 < rN(Bmd) < 0.5. Bibliography: 10 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 322, 2005, pp. 83–106.      20. Lower bound in the Bernstein inequality for the first derivative of algebraic polynomials      A. I. Podvysotskaya 《Ukrainian Mathematical Journal》2009,61(5):847-853 We prove that max |p′(x)|, where p runs over the set of all algebraic polynomials of degree not higher than n ≥ 3 bounded in modulus by 1 on [−1, 1], is not lower than ( n - 1 ) \mathord

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\vphantom ( n - 1 ) ?{1 - x2} ?{1 - x2} {{\left( {n - 1} \right)} \mathord{\left/{\vphantom {{\left( {n - 1} \right)} {\sqrt {1 - {x^2}} }}} \right.} {\sqrt {1 - {x^2}} }} for all x ∈ (−1, 1) such that | x | ? èk = 0[ n \mathord

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\vphantom n 2 2 ] [ cos\frac2k + 12( n - 1 )p, cos\frac2k + 12np ] \left| x \right| \in \bigcup\nolimits_{k = 0}^{\left[ {{n \mathord{\left/{\vphantom {n 2}} \right.} 2}} \right]} {\left[ {\cos \frac{{2k + 1}}{{2\left( {n - 1} \right)}}\pi, \cos \frac{{2k + 1}}{{2n}}\pi } \right]} .

A general Hobby-Rice Theorem and cake cutting

LetX be a Borel subset of a separable Banach spaceE. Letμ be a non-atomic,σ-finite, Borel measure onX. LetG ⊆L ₁ (X, Σ,μ) bem-dimensional. Theorem:There is an l ∈ E* and real numbers −∞=x ₀<x ₁<x ₂<…<x _n<x _n+1=∞with n≤m, such that for all g ∈ G,      

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.     

Rauzy tilings and bounded remainder sets on the torus

For the two-dimensional torus , we construct the Rauzy tilings d⁰ ⊃ d¹ ⊃ … ⊃ d^m ⊃ …, where each tiling d^m+1 is obtained by subdividing the tiles of d^m. The following results are proved. Any tiling d^m is invariant with respect to the torus shift S(x) = x+ mod ℤ², where ζ⁻¹ > 1 is the Pisot number satisfying the equation x³− x²−x-¹ = 0. The induced map is an exchange transformation of B^md ⊂ , where d = d⁰ and . The map S^(m) is a shift of the torus , which is affinely isomorphic to the original shift S. This means that the tilings d^m are infinitely differentiable. If Z_N(X) denotes the number of points in the orbit S¹(0), S²(0), …, S^N(0) belonging to the domain B^md, then, for all m, the remainder r_N(B^md) = Z_N(B^md) − N ζ^m satisfies the bounds −1.7 < r_N(B^md) < 0.5. Bibliography: 10 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 322, 2005, pp. 83–106.     

Lower bound in the Bernstein inequality for the first derivative of algebraic polynomials

We prove that max |p′(x)|, where p runs over the set of all algebraic polynomials of degree not higher than n ≥ 3 bounded in modulus by 1 on [−1, 1], is not lower than ( n - 1 ) \mathord