15 October 2013, Volume 29 Issue 10

    

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  BGP-Reflection Functors and Lusztig’s Symmetries of Modified Quantized Enveloping Algebras

  Jie XIAO, Ming Hui ZHAO

  Acta Mathematica Sinica. 2013, 29(10): 1833-1856. https://doi.org/10.1007/s10114-013-2295-9

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  Let U be a quantized enveloping algebra and   its modified form. Lusztig gives some symmetries on U and . In view of the realization of U by the reduced Drinfeld double of the Ringel-Hall algebra, one can apply the BGP-reflection functors to the double Ringel-Hall algebra to obtain Lusztig's symmetries on U and their important properties, for instance, the braid relations. In this paper, we define a modified form  of the Ringel-Hall algebra and realize the Lusztig's symmetries on  by applying the BGP-reflection functors to .
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  Parameterized Littlewood-Paley Operators and Area Integrals on Weak Hardy Spaces

  Yan LIN, Zong Guang LIU, Dong Lan MAO, Zhen Kai SUN

  Acta Mathematica Sinica. 2013, 29(10): 1857-1870. https://doi.org/10.1007/s10114-013-2440-5

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  In this paper, we establish the boundedness of parameterized Littlewood-Paley operator μ_λ^*,ρ and parameterized area integral μ_Ω,S^ρ with kernel satisfying the logarithmic type Lipschitz condition on the weak Hardy space.
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  New Results on Common Properties of Bounded Linear Operators RS and SR

  Qing Ping ZENG, Huai Jie ZHONG

  Acta Mathematica Sinica. 2013, 29(10): 1871-1884. https://doi.org/10.1007/s10114-013-1758-3

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  Let X, Y be Banach spaces, R:X → Y and S:Y → X be bounded linear operators. When λ≠0, we investigate common properties of λI-SR and λI-RS. This work should be viewed as a continuation of researches of Barnes and Lin et al.. We also apply these results obtained to B-Fredholm theory, extensions and Aluthge transforms.
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  Multidimensional BSDEs with Weak Monotonicity and General Growth Generators

  Sheng Jun FAN, Long JIANG

  Acta Mathematica Sinica. 2013, 29(10): 1885-1906. https://doi.org/10.1007/s10114-013-2128-x

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  This paper aims at solving a multidimensional backward stochastic differential equation (BSDE) whose generator g satisfies a weak monotonicity condition and a general growth condition in y. We first establish an existence and uniqueness result of solutions for this kind of BSDEs by using systematically the technique of the priori estimation, the convolution approach, the iteration, the truncation and the Bihari inequality. Then, we overview some assumptions related closely to the monotonicity condition in the literature and compare them in an effective way, which yields that our existence and uniqueness result really and truly unifies the Mao condition in y and the monotonicity condition with the general growth condition in y, and it generalizes some known results. Finally, we prove a stability theorem and a comparison theorem for this kind of BSDEs, which also improves some known results.
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  Approximation for Counts of 2-runs in a Two State Markov Chain

  Hui CHEN, Mei ZHANG

  Acta Mathematica Sinica. 2013, 29(10): 1907-1916. https://doi.org/10.1007/s10114-013-1709-z

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  We consider a Markov chain X={X_i, i=1, 2,...} with the state space {0, 1}, and define W=∑_i=1ⁿX_iX_i+1, which is the number of 2-runs in X before time n+1. In this paper, we prove that the negative binomial distribution is an appropriate approximation for LW when VarW is greater than EW. The error estimate obtained herein improves the corresponding result in previous literatures.
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  Boundedness for a Class of Generalized Commutators on λ-Central Morrey Space

  Xiao YU, Xiang Xing TAO

  Acta Mathematica Sinica. 2013, 29(10): 1917-1926. https://doi.org/10.1007/s10114-013-2174-4

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  Let 0 ≤ α < n, Ω be a rough kernel, and let A have derivatives of order m-1 in CMO^q,μ₂ with m ≥ 2. We consider a class of generalized commutators T_Ω,α^A of Cohen-Gosselin type, and obtain the boundedness of T_Ω,α^A from the central Morrey spaces E^p,μ₁ to E^r,λ for λ=μ₁+μ₂+α/n and 1/r=1/p+1/q-α/n.
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  Option Pricing by Mean Correcting Method for Non-Gaussian Lévy Processes

  Luo Gen YAO, Gang YANG, Xiang Qun YANG

  Acta Mathematica Sinica. 2013, 29(10): 1927-1938. https://doi.org/10.1007/s10114-013-2446-z

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  For a non-Gaussian Lévy model, it is shown that if the model exists a trivial arbitrage-free interval, option pricing by mean correcting method is always arbitrage-free, and if the arbitrage-free interval is non-trivial, this pricing method may lead to arbitrage in some cases. In the latter case, some necessary and sufficient conditions under which option price is arbitrage-free are obtained.
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  Some Results Associated with the Longest Run in a Strongly Ergodic Markov Chain

  Ya Zhe ZHANG, Xian Yuan WU

  Acta Mathematica Sinica. 2013, 29(10): 1939-1948. https://doi.org/10.1007/s10114-013-2549-6

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  This paper discusses the asymptotic behaviors of the longest run on a countable state Markov chain. Let {X_a}_a∈Z+ be a stationary strongly ergodic reversible Markov chain on countablestate space S={1, 2,...}. Let TS be an arbitrary finite subset of S. Denote by Ln the length of the longest run of consecutive i's for i ∈ T, that occurs in the sequence X₁,...,X_n. In this paper, we obtain a limit law and a week version of an Erdös-Rényi type law for L_n. A large deviation result of L_n is also discussed.
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  Spectral Gap and Convergence Rate for Discrete-time Markov Chains

  Yong Hua MAO, Yan Hong SONG

  Acta Mathematica Sinica. 2013, 29(10): 1949-1962. https://doi.org/10.1007/s10114-013-2594-1

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  Let P be a transition matrix which is symmetric with respect to a measure π. The spectral gap of P in L²(π)-space, denoted by gap(P), is defined as the distance between 1 and the rest of the spectrum of P. In this paper, we study the relationship between gap(P) and the convergence rate of Pⁿ. When P is transient, the convergence rate of Pⁿ is equal to 1-gap(P). When P is ergodic, we give the explicit upper and lower bounds for the convergence rate of Pⁿ in terms of gap(P). These results are extended to L^∞(π)-space.
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  The Cancellation of Fourier Coefficients of Cusp Forms over Different Sparse Sequences

  Hui Xue LAO

  Acta Mathematica Sinica. 2013, 29(10): 1963-1972. https://doi.org/10.1007/s10114-013-2706-y

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  Let λ_f (n) be the n-th normalized Fourier coefficient of a holomorphic Hecke eigenform f(z) ∈S_k(Γ). In this paper, we established nontrivial estimates for  where 1 ≤ i < j ≤ 4.
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  A Note on Holonomy of Gerbes over Orbifolds

  Xiao Qin YIN

  Acta Mathematica Sinica. 2013, 29(10): 1973-1980. https://doi.org/10.1007/s10114-013-2770-3

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  In this paper, we generalize the construction of the inverse transgression map done by Adem, A., Ruan, Y. and Zhang, B. in [A stringy product on twisted orbifold K-theory. Morfismos, 11, 33-64 (2007)] and give a different proof to the statement that the image of the inverse transgression map for a gerbe with connection over an orbifold is an inner local system on its inertia orbifold. Keywords Gerbe, orbifold, Lie groupoid, inner local system, holonomy
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  Picard Values and the Growth of Meromorphic Functions

  Jun Fan CHEN

  Acta Mathematica Sinica. 2013, 29(10): 1981-1992. https://doi.org/10.1007/s10114-013-1353-7

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  The purpose of this paper is to investigate the growth of meromorphic functions concerning Picard values with a radially distributed value. This generalizes a classic result due to Hayman [Hayman, W. K.: Picard values of meromorphic functions and their derivatives. Ann. of Math., 70, 9-42 (1959)].
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  A Note on the Auslander-Reiten Conjecture

  Deng Ming XU

  Acta Mathematica Sinica. 2013, 29(10): 1993-1996. https://doi.org/10.1007/s10114-013-1428-5

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  It is proved that the Auslander-Reiten conjecture is true for local Artin algebras with radical cube zero.
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  A Bombieri-type Theorem for Exponential Sums

  Wei Li YAO

  Acta Mathematica Sinica. 2013, 29(10): 1997-2012. https://doi.org/10.1007/s10114-013-1519-3

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  Let f_k(n) be the characteristic function of n with Ω(n) = k, and  The main purpose of this paper is to establish a Bombieri-type mean-value theorem for T_k(x, α), via using the modified Huxley-Hooley contour and the large-sieve type zero-density estimate for Dirichlet L-functions. The result plays an important role in handling the enlarge major arcs when we try to solve the Waring-Goldbach type problem by the circle method.
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  Approximation by Neural Networks with Sigmoidal Functions

  Dan Sheng YU

  Acta Mathematica Sinica. 2013, 29(10): 2013-2026. https://doi.org/10.1007/s10114-013-1730-2

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  In this paper, we introduce a type of approximation operators of neural networks with sigmodal functions on compact intervals, and obtain the pointwise and uniform estimates of the approximation. To improve the approximation rate, we further introduce a type of combinations of neural networks. Moreover, we show that the derivatives of functions can also be simultaneously approximated by the derivatives of the combinations. We also apply our method to construct approximation operators of neural networks with sigmodal functions on infinite intervals.