05 May 2007, Volume 23 Issue 5

    

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  Jacobi Structures on Affine Bundles

  J. GRABOWSKI, D. IGLESIAS, J. C. MARRERO, E. PADRÓN, P. URBAŃSKI

  Acta Mathematica Sinica. 2007, 23(5): 769-788. https://doi.org/10.1007/s10114-005-0716-0

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  We study affine Jacobi structures (brackets) on an affine bundle π:A→M,i.e.Jacobi brackets that close on affine functions.We prove that if the rank of A is non-zero,there is a one-toone correspondence between affine Jacobi structures on A and Lie algebroid structures on the vector bundle A⁺=∪_p∈M Aff (A_p,R) of affine functionals.In the case rank A=0,it is shown that there is a one-to-one correspondence between affine Jacobi structures on A and local Lie algebras on A⁺.Some examples and applications,also for the linear case,are discussed.For a special type of affine Jacobi structures which are canonically exhibited (strongly-affine or affine-homogeneous Jacobi structures) over a real vector space of finite dimension,we describe the leaves of its characteristic foliation as the orbits of an affine representation.These affine Jacobi structures can be viewed as an analog of the Kostant-Arnold-Liouville linear Poisson structure on the dual space of a real finite-dimensional Lie algebra.
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  A Construction of Cryptogroups

  Guang Tian SONG, Guo Xin LI, Jian Gang ZHANG

  Acta Mathematica Sinica. 2007, 23(5): 789-798. https://doi.org/10.1007/s10114-005-0802-3

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  In this paper,a cryptogroup is constructed by a band and a family of groups and homomorphisms satisfying some conditions.
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  Pattern Formation and Continuation in a Trineuron Ring with Delays

  Shang Jiang GUO, Li Hong HUANG

  Acta Mathematica Sinica. 2007, 23(5): 799-818. https://doi.org/10.1007/s10114-005-0842-8

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  In this paper,we consider a single-directional ring of three neurons with delays.First,linear stability of the model is investigated by analyzing the associated characteristic transcendental equation.Next,we studied the local Hopf bifurcations and the spatio-temporal patterns of Hopf bifurcating periodic orbits.Basing on the normal form approach and the center manifold theory,we derive the formula for determining the properties of Hopf bifurcating periodic orbit,such as the direction of Hopf bifurcation.Finally,global existence conditions for Hopf bifurcating periodic orbits are derived by using degree theory methods.
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  Some Elements of Finite Order in K₂Q

  Xiao Yun CHENG, Jian Guo XIA, Hou Rong QIN

  Acta Mathematica Sinica. 2007, 23(5): 819-826. https://doi.org/10.1007/s10114-005-0852-6

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  Let K2 be the Milnor functor and let Φn(x) ∈ Q[x] be the n-th cyclotomic polynomial.Let Gn(Q) denote a subset consisting of elements of the form {a,Φn(a)},where a ∈Q*.and {,} denotes the Steinberg symbol in K2Q.J.Browkin proved that G n(Q) is a subgroup of K2Q if n=1,2,3,4 or 6 and conjectured that Gn(Q) is not a group for any other values of n.This conjecture was confirmed for n=2r3s or n=pr,where p≥5 is a prime number such that h(Q(ζp)) is not divisible by p.In this paper we confirm the conjecture for some n,where n is not of the above forms,more precisely,for n=15,21,33,35,60 or 105.
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  Exact Values of Some Geometric Constants of Orlicz Spaces

  Ya Qiang YAN

  Acta Mathematica Sinica. 2007, 23(5): 827-846. https://doi.org/10.1007/s10114-005-0929-2

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  We present the exact values of nonsquare constants,von Neumann-Jordan constants,Jung constants,packing constants,weakly convergent sequence coefficients and normal structure coefficients in a class of reflexive Orlicz function spaces and sequence spaces equipped with Luxemburg norm and Orlicz norm.
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  The Frattini Subalgebra of n-Lie Algebras

  Rui Pu BAI, Liang Yun CHEN, Dao Ji MENG

  Acta Mathematica Sinica. 2007, 23(5): 847-856. https://doi.org/10.1007/s10114-005-0923-8

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  In this paper,we mainly study the structural notions of Frattini subalgebra and Jacobson radical of n-Lie algebras.The properties on Frattini subalgebra and the relationship between the Frattini subalgebra and Jacobson radical are studied.Moreover,we also study them when an n-Lie algebra is strong semi-simple,k-solvable and nilpotent.
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  Commuting Dual Toeplitz Operators on the Polydisk

  Yu Feng LU, Shu Xia SHANG

  Acta Mathematica Sinica. 2007, 23(5): 857-868. https://doi.org/10.1007/s10114-005-0877-x

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  On the polydisk,the commutativity of dual Toeplitz operators is studied.We obtain characterizations of commuting dual Toeplitz operators,essentially commuting dual Toeplitz operators and essentially semi-commuting dual Toeplitz operators.
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  On the Bifurcations of a Hamiltonian Having Three Homoclinic Loops under Z₃ Invariant Quintic Perturbations

  Yu Hai WU, Mao An HAN

  Acta Mathematica Sinica. 2007, 23(5): 869-878. https://doi.org/10.1007/s10114-005-0790-3

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  A cubic system having three homoclinic loops perturbed by Z3 invariant quintic polynomials is considered.By applying the qualitative method of differential equations and the numeric computing method,the Hopf bifurcation,homoclinic loop bifurcation and heteroclinic loop bifurcation of the above perturbed system are studied.It is found that the above system has at least 12 limit cycles and the distributions of limit cycles are also given.
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  Fibonacci Lengths of Certain Nilpotent 2-Groups

  H. DOOSTIE, A. T. ADNANI

  Acta Mathematica Sinica. 2007, 23(5): 879-884. https://doi.org/10.1007/s10114-005-0717-z

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  In this paper,we study two classes of 2-generated 2-groups of nilpotency class 2 classified by Kluempen in 2002 and also a class of finite 2-groups of high nilpotency class for their Fibonacci lengths.Their involvement in certain interesting sequences of Tribonacci numbers gives us some explicit formulas for the Fibonacci lengths and this adds to the small class of finite groups for which the Fibonacci length are known.
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  On F-Abnormal Maximal Subgroups of Finite Groups

  Shi Rong LI

  Acta Mathematica Sinica. 2007, 23(5): 885-888. https://doi.org/10.1007/s10114-005-0872-2

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  For any saturated formation F of finite groups containing all supersolvable groups,the groups in F are characterized by F-abnormal maximal subgroups.
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  Explicit Formula for the Hyperbolic Scattering Determinant

  Muharem AvdispahiĆ, Lejla SmajloviĆ

  Acta Mathematica Sinica. 2007, 23(5): 889-894. https://doi.org/10.1007/s10114-005-0791-2

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  We prove the explicit formula for the hyperbolic scattering determinant in the case of a general subgroup Γ of PSL (2,R).The class of test functions involved (not necessarily odd nor continuous) is much broader than that previously known.As an application of the technique,a new representation of the Millson-Shintani zeta function is obtained.
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  White Noise Approach to The Construction of Φ₄⁴ Quantum Fields (Ⅱ)

  Guang Lin RANG, Zhi Yuan HUANG

  Acta Mathematica Sinica. 2007, 23(5): 895-904. https://doi.org/10.1007/s10114-005-0817-9

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  In this paper a complete proof for the existence of generalized operators satisfying abstract dynamical equations of quantum motions ∂²/∂t²Φ(t,x)+(Δ-m²)Φ(t,x)=-λ:Φ³(t,x),subject to a suitable initial condition,is given under the framework of white noise analysis.Also some important commutation relations related to φ₄⁴ quantum fields are discussed and proved in detail.
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  On Supercontractivity for Markov Semigroups

  Yong Hua MAO

  Acta Mathematica Sinica. 2007, 23(5): 905-914. https://doi.org/10.1007/s10114-005-0830-z

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  In this paper,we study the supercontractivity for Markov semigroups and obtain some sufficient and necessary conditions; especially explicit formulae are obtained for birth-death process and diffusion on the line.Sufficient conditions and necessary conditions in terms of isoperimetric inequalities are also presented.Moreover,we prove that the supercontractivity is equivalent to the compact embedding of Sobolev space into an Orlicz space.
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  Mean Value on the Difference Between a Quadratic Residue and Its Inverse Modulo p

  Hua Ning LIU, Wen Peng ZHANG

  Acta Mathematica Sinica. 2007, 23(5): 915-924. https://doi.org/10.1007/s10114-005-0847-3

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  The main purpose of this paper is to use the generalized Bernoulli numbers,Gauss sums and the mean value theorems of Dirichlet L-functions to study the asymptotic property of the difference between a quadratic residue and its inverse modulo p (a prime),and to give an interesting hybrid mean value formula.
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  Operators on Differential Form Spaces for Riemann Surfaces

  Guang Fu CAO, Xiao Feng WANG

  Acta Mathematica Sinica. 2007, 23(5): 925-934. https://doi.org/10.1007/s10114-005-0846-4

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  In the present paper,a problem of Ioana Mihaila is negatively answered on the invertibility of composition operators on Riemann surfaces,and it is proved that the composition operator C_p is Fredholm if and only if it is invertible if and only if p is invertible for some special cases.In addition,the Toeplitz operators on ∧_2,a¹(M) for Riemann surface M are defined and some properties of these operators are discussed.
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  Pointwise Simultaneous Approximation by Combinations of Baskakov Operators

  Lin Sen XIE, Zhong Rui SHI

  Acta Mathematica Sinica. 2007, 23(5): 935-944. https://doi.org/10.1007/s10114-005-0827-7

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  In this paper,we investigate the relation between the rate of convergence for the derivatives of the combinations of Baskakov operators and the smoothness for the derivatives of the functions approximated.We give some direct and inverse results on pointwise simultaneous approximation by the combinations of Baskakov operators.We also give a new equivalent result on pointwise approximation by these operators.
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  All General Solutions of Post Equations

  Dragić BANKOVIĆ

  Acta Mathematica Sinica. 2007, 23(5): 945-950. https://doi.org/10.1007/s10114-005-0780-5

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  In a previous paper,we have described all reproductive general solutions of a Post equation,supposing that a general solution is known.In this paper we describe all general solutions of Post equation,supposing that a general solution of this equation is known (Theorem 6).As a special case we get the previous characterization of reproductive solutions and a similar result for Boolean equations (Theorem 9).
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  A-Browder’s Theorem and Generalized a-Weyl’s Theorem

  Xiao Hong CAO

  Acta Mathematica Sinica. 2007, 23(5): 951-960. https://doi.org/10.1007/s10114-005-0870-4

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  Two variants of the essential approximate point spectrum are discussed.We find for example that if one of them coincides with the left Drazin spectrum then the generalized a-Weyl's theorem holds,and conversely for a-isoloid operators.We also study the generalized a-Weyl's theorem for Class A operators.