The upper bound of the $${{\rm L}_{\mu}^2}$$ spectrum

Let M be an n-dimensional complete non-compact Riemannian manifold, dμ = e ^h (x)dV(x) be the weighted measure and \triangle_m{\triangle_{\mu}} be the weighted Laplacian. In this article, we prove that when the m-dimensional Bakry–émery curvature is bounded from below by Ric _m ≥ −(m − 1)K, K ≥ 0, then the bottom of the L_m²{{\rm L}_{\mu}^2} spectrum λ₁(M) is bounded by

Lipschitz equivalence of generalized {1,3,5}-{1,4,5} self-similar sets

This paper investigates the Lipschitz equivalence of generalized {1,3,5}-{1,4,5} self-similar sets D=(r_1D)∪(r_2D (1 r_1-r_2-r_3)/2)∪(r_3D 1 r_3) and E=(r_1E)∪(r_2E 1-r_2- r_3)∪(r_3E 1-r_3),and proves that D and E are Lipschitz equivalent if and only if there are positive integers m and n such that r_1~m=r_3~n.     

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 )}.      4. Characterizations of Bent and Almost Bent Function on {\mathbb{Z}_p^2}      Xiyong Zhang  Hua Guo  Zongsheng Gao 《Graphs and Combinatorics》2011,27(4):603-620 Bent and almost-bent functions on \mathbbZp2{\mathbb{Z}_p^2} are studied in this paper. By calculating certain exponential sum and using a technique due to Hou (Finite Fields Appl 10:566–582, 2004), we obtain a degree bound for quasi-bent functions, and prove that almost-bent functions on \mathbbZp2{\mathbb{Z}_p^2} are equivalent to a degenerate quadratic form. From the viewpoint of relative difference sets, we also characterize bent functions on \mathbbZp2{\mathbb{Z}_p^2} in two classes of M{\mathcal{M}} ’s and PS{\mathcal{PS}} ’s, and show that the graph set corresponding to a bent function on \mathbbZp2{\mathbb{Z}_p^2} can be written as the sum of a graph set of M{\mathcal{M}} ’s type bent function and another group ring element. By using our characterization and some technique of permutation polynomial, we obtain the result: a bent function must be of M{\mathcal{M}} ’s type if its corresponding set contains more than (p − 3)/2 flats. A problem proposed by Ma and Pott (J Algebra 175:505–525, 1995) is therefore partially answered.      5. A new multivariable <Subscript><Emphasis Type="BoldItalic">6</Emphasis></Subscript><Emphasis Type="BoldItalic">ψ</Emphasis><Subscript><Emphasis Type="BoldItalic">6</Emphasis></Subscript> summation formula      Michael Schlosser 《The Ramanujan Journal》2008,17(3):305-319 By multidimensional matrix inversion, combined with an A r extension of Jackson’s 8 φ 7 summation formula by Milne, a new multivariable 8 φ 7 summation is derived. By a polynomial argument this 8 φ 7 summation is transformed to another multivariable 8 φ 7 summation which, by taking a suitable limit, is reduced to a new multivariable extension of the nonterminating 6 φ 5 summation. The latter is then extended, by analytic continuation, to a new multivariable extension of Bailey’s very-well-poised 6 ψ 6 summation formula. Partly supported by FWF Austrian Science Fund grants P17563-N13, and S9607 (the second is part of the Austrian National Research Network “Analytic Combinatorics and Probabilistic Number Theory”).      6. {0, 1/2}-Chvátal-Gomory cuts      Alberto Caprara  Matteo Fischetti 《Mathematical Programming》1996,74(3):221-235 Given the integer polyhedronP t := conv{x ∈ℤ n :Ax⩽b}, whereA ∈ℤ m × n andb ∈ℤ m , aChvátal-Gomory (CG)cut is a valid inequality forP 1 of the type λτAx⩽⌊λτb⌋ for some λ∈ℝ + m such that λτA∈ℤ n . In this paper we study {0, 1/2}-CG cuts, arising for λ∈{0, 1/2} m . We show that the associated separation problem, {0, 1/2}-SEP, is equivalent to finding a minimum-weight member of a binary clutter. This implies that {0, 1/2}-SEP is NP-complete in the general case, but polynomially solvable whenA is related to the edge-path incidence matrix of a tree. We show that {0, 1/2}-SEP can be solved in polynomial time for a convenient relaxation of the systemAx<-b. This leads to an efficient separation algorithm for a subclass of {0, 1/2}-CG cuts, which often contains wide families of strong inequalities forP 1. Applications to the clique partitioning, asymmetric traveling salesman, plant location, acyclic subgraph and linear ordering polytopes are briefly discussed.      7. 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.      8. The calculation of photoionization angular distribution parameter β of atomic Na      Jinchao Liu  Fengnian Pu  Hankui Liu  Jianjun Guo  Yansong Cheng  Xiangdong Yang 《中国科学A辑(英文版)》1999,42(10):1081-1088 The differential cross sections and angular distribution parameter of the photoionization processes 2p 63s→ 2p 53skl of atomic Na have been calculated by using many-body perturbation theory. In the calculation, the resonant structure of the excitation process 2s→3p has been included. The electron correlation interaction was analyzed by using the effective diagram method. The summation of specific classes of these diagrams is to an infinite order. The results of calculations are compared with experimental data, which are in good agreement with the experiment.      9. 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 .      10. <Emphasis Type="Italic">q</Emphasis>-extensions of Dougall’s bilateral <Subscript>2</Subscript>H<Subscript>2</Subscript>-series      Wenchang?ChuEmail author 《The Ramanujan Journal》2011,25(1):121-139 Transformations for a bilateral 2 ψ 2-series are investigated by means of Abel’s lemma on summation by parts. Two q-extensions of Dougall’s classical identity for bilateral 2H2-sum are established. As by-products, the Rogers–Fine identity is recovered and a new proof is presented for Bailey’s identity of bilateral well-poised 6 ψ 6-series.      11. Saturation of the linear methods of summation of Fourier series in the spaces <Emphasis Type="Italic">S</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript><Subscript>φ</Subscript>      A. L. Shydlich 《Ukrainian Mathematical Journal》2008,60(6):950-966 We consider the problem of saturation of the linear methods of summation of Fourier series in the spaces S p φ specified by arbitrary sequences of functions defined in a certain subset of the space ℂ. Sufficient conditions for the saturation of the indicated methods in these spaces are established. Translated from Ukrains'kyi Matematychnyi Zhurnal, Vol. 60, No. 6, pp. 815–828, June, 2008.      12. 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.      13. Multilateral Inversion of A r , C r , and D r Basic Hypergeometric Series      Michael J. Schlosser 《Annals of Combinatorics》2009,13(3):341-363 In [Electron. J. Combin. 10 (2003), #R10], the author presented a new basic hypergeometric matrix inverse with applications to bilateral basic hypergeometric series. This matrix inversion result was directly extracted from an instance of Bailey’s very-well-poised 6ψ6 summation theorem, and involves two infinite matrices which are not lower-triangular. The present paper features three different multivariable generalizations of the above result. These are extracted from Gustafson’s A r and C r extensions and from the author’s recent A r extension of Bailey’s 6ψ6 summation formula. By combining these new multidimensional matrix inverses with A r and D r extensions of Jackson’s 8ϕ7 summation theorem three balanced verywell- poised 8ψ8 summation theorems associated to the root systems A r and C r are derived.      14. Liftings for Haar measure on {0,1} k      Maxim R. Burke  Winfried Just 《Israel Journal of Mathematics》1991,73(1):33-44 We callE ⊆ {0,1} k projective if for some countableA ⊆κ there is anE A ⊆ {0, 1} A such thatE=E A ×{0,1} k\A andE A is a projective subset of the Cantor set {0, 1} A . We construct a model where Haar measure on {0,1} k has no projective lifting (and in particular no Baire lifting) for anyκ≥ω. Research partially supported by NATO Science Fellowship. The first author would like to thank the Mathematics Department at the University of Essex for its hospitality during the academic year 1988/89 while part of this research was being carried out. This research was initiated while the second author was a postdoctoral fellow at the University of Toronto. Its completion was supported by NSF grant DMS-8505550.      15. The Quasi-Canonical Solution Operator to $${\bar \partial }$$ Restricted to the Fock-Space      Georg Schneider 《Czechoslovak Mathematical Journal》2005,55(4):947-956 We consider the solution operator S: ℱμ,(p,q) → L 2(μ)(p, q) to the -operator restricted to forms with coefficients in ℱμ = {f: f is entire and ∫ℂn |f(z)|2 dμ(z) < ∞}. Here ℱμ,(p,q) denotes (p,q)-forms with coefficients in ℱμ, L 2(μ) is the corresponding L 2-space and μ is a suitable rotation-invariant absolutely continuous finite measure. We will develop a general solution formula S to . This solution operator will have the property Sv ⊥ ℱ(p,q) ∀v ∈ ℱ(p,q+1). As an application of the solution formula we will be able to characterize compactness of the solution operator in terms of compactness of commutators of Toeplitz-operators : ℱμ → L 2(μ).      16. 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.      17. A note on the theory of positive induction, {{\rm ID}^*_1}      Bahareh Afshari  Michael Rathjen 《Archive for Mathematical Logic》2010,49(2):275-281 The article shows a simple way of calibrating the strength of the theory of positive induction, ID*1{{\rm ID}^{*}_{1}} . Crucially the proof exploits the equivalence of S11{\Sigma^{1}_{1}} dependent choice and ω-model reflection for P12{\Pi^{1}_{2}} formulae over ACA 0. Unbeknown to the authors, D. Probst had already determined the proof-theoretic strength of ID*1{{\rm ID}^{*}_{1}} in Probst, J Symb Log, 71, 721–746, 2006.      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. Entropy and uniform distribution of orbits in\mathbb{T}      David Meiri 《Israel Journal of Mathematics》1998,105(1):155-183 We present a class of integer sequences {c n } with the property that for everyp-invariant and ergodic positive-entropy measure μ on L 2 , {c n x (mod 1)} is uniformly distributed for μ-almost everyx. This extends a result of B. Host, who proved this for the sequence {q n }, forq relatively prime top. Our class of sequences includes, for instance, the sequencec n =Мf i (n)q i n , where the numbersq i are distinct and are relatively prime top andf i are any polynomials. More generally, recursion sequences for which the free coefficient of the recursion polynomial is relatively prime top are in this class as well, provided they satisfy a simple irreducibility condition. In the multi-dimensional case we derive sufficient conditions for a pair of endomorphisms (withA diagonal) and anA-invariant and ergodic measure μ, such thatB-orbits of the form {B n ω} are uniformly distributed for μ-almost every .      20. Approximation of the classes C β ψ H ω by generalized de la Valiée-Poussin sums      V. I. Rukasov  O. A. Novikov 《Ukrainian Mathematical Journal》1997,49(4):672-677 On the classes of periodic functions C βψ H β, we study approximating properties of trigonometric polynomials generated by methods for summation of Fourier series. Slavyansk Pedagogical Institute, Slavyansk. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 49, No. 4, pp. 606–610, April, 1997.

Characterizations of Bent and Almost Bent Function on {\mathbb{Z}_p^2}

Bent and almost-bent functions on \mathbbZ_p²{\mathbb{Z}_p^2} are studied in this paper. By calculating certain exponential sum and using a technique due to Hou (Finite Fields Appl 10:566–582, 2004), we obtain a degree bound for quasi-bent functions, and prove that almost-bent functions on \mathbbZ_p²{\mathbb{Z}_p^2} are equivalent to a degenerate quadratic form. From the viewpoint of relative difference sets, we also characterize bent functions on \mathbbZ_p²{\mathbb{Z}_p^2} in two classes of M{\mathcal{M}} ’s and PS{\mathcal{PS}} ’s, and show that the graph set corresponding to a bent function on \mathbbZ_p²{\mathbb{Z}_p^2} can be written as the sum of a graph set of M{\mathcal{M}} ’s type bent function and another group ring element. By using our characterization and some technique of permutation polynomial, we obtain the result: a bent function must be of M{\mathcal{M}} ’s type if its corresponding set contains more than (p − 3)/2 flats. A problem proposed by Ma and Pott (J Algebra 175:505–525, 1995) is therefore partially answered.     

A new multivariable <Subscript><Emphasis Type="BoldItalic">6</Emphasis></Subscript><Emphasis Type="BoldItalic">ψ</Emphasis><Subscript><Emphasis Type="BoldItalic">6</Emphasis></Subscript> summation formula

By multidimensional matrix inversion, combined with an A _r extension of Jackson’s ₈ φ ₇ summation formula by Milne, a new multivariable ₈ φ ₇ summation is derived. By a polynomial argument this ₈ φ ₇ summation is transformed to another multivariable ₈ φ ₇ summation which, by taking a suitable limit, is reduced to a new multivariable extension of the nonterminating ₆ φ ₅ summation. The latter is then extended, by analytic continuation, to a new multivariable extension of Bailey’s very-well-poised ₆ ψ ₆ summation formula. Partly supported by FWF Austrian Science Fund grants P17563-N13, and S9607 (the second is part of the Austrian National Research Network “Analytic Combinatorics and Probabilistic Number Theory”).     

{0, 1/2}-Chvátal-Gomory cuts

Given the integer polyhedronP _t := conv{x ∈ℤ ⁿ :Ax⩽b}, whereA ∈ℤ ^{m × n} andb ∈ℤ ^m , aChvátal-Gomory (CG)cut is a valid inequality forP ₁ of the type λτAx⩽⌊λτb⌋ for some λ∈ℝ ₊ ^m such that λτA∈ℤ ⁿ . In this paper we study {0, 1/2}-CG cuts, arising for λ∈{0, 1/2} ^m . We show that the associated separation problem, {0, 1/2}-SEP, is equivalent to finding a minimum-weight member of a binary clutter. This implies that {0, 1/2}-SEP is NP-complete in the general case, but polynomially solvable whenA is related to the edge-path incidence matrix of a tree. We show that {0, 1/2}-SEP can be solved in polynomial time for a convenient relaxation of the systemAx<-b. This leads to an efficient separation algorithm for a subclass of {0, 1/2}-CG cuts, which often contains wide families of strong inequalities forP ₁. Applications to the clique partitioning, asymmetric traveling salesman, plant location, acyclic subgraph and linear ordering polytopes are briefly discussed.     

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.     

The calculation of photoionization angular distribution parameter β of atomic Na

The differential cross sections and angular distribution parameter of the photoionization processes 2p ⁶3_s→ 2p ⁵3skl of atomic Na have been calculated by using many-body perturbation theory. In the calculation, the resonant structure of the excitation process 2s→3p has been included. The electron correlation interaction was analyzed by using the effective diagram method. The summation of specific classes of these diagrams is to an infinite order. The results of calculations are compared with experimental data, which are in good agreement with the experiment.     

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 .     

<Emphasis Type="Italic">q</Emphasis>-extensions of Dougall’s bilateral <Subscript>2</Subscript>H<Subscript>2</Subscript>-series

Transformations for a bilateral ₂ ψ ₂-series are investigated by means of Abel’s lemma on summation by parts. Two q-extensions of Dougall’s classical identity for bilateral ₂H₂-sum are established. As by-products, the Rogers–Fine identity is recovered and a new proof is presented for Bailey’s identity of bilateral well-poised ₆ ψ ₆-series.     

Saturation of the linear methods of summation of Fourier series in the spaces <Emphasis Type="Italic">S</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript><Subscript>φ</Subscript>

We consider the problem of saturation of the linear methods of summation of Fourier series in the spaces S ^p _φ specified by arbitrary sequences of functions defined in a certain subset of the space ℂ. Sufficient conditions for the saturation of the indicated methods in these spaces are established. Translated from Ukrains'kyi Matematychnyi Zhurnal, Vol. 60, No. 6, pp. 815–828, June, 2008.     

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.     

Multilateral Inversion of A r , C r , and D r Basic Hypergeometric Series

In [Electron. J. Combin. 10 (2003), #R10], the author presented a new basic hypergeometric matrix inverse with applications to bilateral basic hypergeometric series. This matrix inversion result was directly extracted from an instance of Bailey’s very-well-poised ₆ψ₆ summation theorem, and involves two infinite matrices which are not lower-triangular. The present paper features three different multivariable generalizations of the above result. These are extracted from Gustafson’s A _r and C _r extensions and from the author’s recent A _r extension of Bailey’s ₆ψ₆ summation formula. By combining these new multidimensional matrix inverses with A _r and D _r extensions of Jackson’s ₈ϕ₇ summation theorem three balanced verywell- poised ₈ψ₈ summation theorems associated to the root systems A _r and C _r are derived.     

Liftings for Haar measure on {0,1} k

We callE ⊆ {0,1} ^k projective if for some countableA ⊆κ there is anE _A ⊆ {0, 1} ^A such thatE=E _A ×{0,1} ^k\A andE _A is a projective subset of the Cantor set {0, 1} ^A . We construct a model where Haar measure on {0,1} ^k has no projective lifting (and in particular no Baire lifting) for anyκ≥ω. Research partially supported by NATO Science Fellowship. The first author would like to thank the Mathematics Department at the University of Essex for its hospitality during the academic year 1988/89 while part of this research was being carried out. This research was initiated while the second author was a postdoctoral fellow at the University of Toronto. Its completion was supported by NSF grant DMS-8505550.     

The Quasi-Canonical Solution Operator to $${\bar \partial }$$ Restricted to the Fock-Space

We consider the solution operator S: ℱ_μ,(p,q) → L ²(μ)_{(p, q)} to the -operator restricted to forms with coefficients in ℱ_μ = {f: f is entire and ∫_ℂn |f(z)|² dμ(z) < ∞}. Here ℱ_μ,(p,q) denotes (p,q)-forms with coefficients in ℱ_μ, L ²(μ) is the corresponding L ²-space and μ is a suitable rotation-invariant absolutely continuous finite measure. We will develop a general solution formula S to . This solution operator will have the property Sv ⊥ ℱ_(p,q) ∀v ∈ ℱ_(p,q+1). As an application of the solution formula we will be able to characterize compactness of the solution operator in terms of compactness of commutators of Toeplitz-operators : ℱ_μ → L ²(μ).     

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.     

A note on the theory of positive induction, {{\rm ID}^*_1}

The article shows a simple way of calibrating the strength of the theory of positive induction, ID^*₁{{\rm ID}^{*}_{1}} . Crucially the proof exploits the equivalence of S¹₁{\Sigma^{1}_{1}} dependent choice and ω-model reflection for P¹₂{\Pi^{1}_{2}} formulae over ACA ₀. Unbeknown to the authors, D. Probst had already determined the proof-theoretic strength of ID^*₁{{\rm ID}^{*}_{1}} in Probst, J Symb Log, 71, 721–746, 2006.     

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     

Entropy and uniform distribution of orbits in\mathbb{T}

We present a class of integer sequences {c _n } with the property that for everyp-invariant and ergodic positive-entropy measure μ on L ₂ , {c _n x (mod 1)} is uniformly distributed for μ-almost everyx. This extends a result of B. Host, who proved this for the sequence {q ⁿ }, forq relatively prime top. Our class of sequences includes, for instance, the sequencec _n =Мf _i (n)q _i ⁿ , where the numbersq _i are distinct and are relatively prime top andf _i are any polynomials. More generally, recursion sequences for which the free coefficient of the recursion polynomial is relatively prime top are in this class as well, provided they satisfy a simple irreducibility condition. In the multi-dimensional case we derive sufficient conditions for a pair of endomorphisms (withA diagonal) and anA-invariant and ergodic measure μ, such thatB-orbits of the form {B ⁿ ω} are uniformly distributed for μ-almost every .     

Approximation of the classes C β ψ H ω by generalized de la Valiée-Poussin sums

On the classes of periodic functions C _β^ψ H _β, we study approximating properties of trigonometric polynomials generated by methods for summation of Fourier series. Slavyansk Pedagogical Institute, Slavyansk. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 49, No. 4, pp. 606–610, April, 1997.