丢番图方程及Browkin的一个猜想

本文首先证明了一些丢番图方程在整数环Z中无解,然后我们部分验证了Browkin在[Lecture Notes in Math.,Vol.966,1-6]中的一个猜想.     

On a q-analogue for Bernoulli numbers

Inspired by Borwein et al. (Am. Math. Mon., 116(5):387–412, 2009), we define a sequence of q-analogues for the Bernoulli numbers under the framework of Strodt operators. We show that they not only satisfy identities similar to those of the q-analogue proposed by Carlitz (Duke Math. J., 15(4):987–1000, 1948), but also interesting analytical properties as functions of q. In particular, we give a simple analytic proof of a generalization of an explicit formula for the Bernoulli numbers given by Woon (Math. Mag., 70(1):51–56, 1997). We also define a set of q-analogues for the Stirling numbers of the second kind within our framework and prove a q-extension of a related, well-known closed form relating Bernoulli and Stirling numbers.     

Moving boundary problem for a one-dimensional crawling nematode sperm cell model

This paper is a continuation of the authors’ investigation of the crawling nematode sperm cell model proposed by Mogilner and Verzi in 2003 (J. Stat. Phys. 110 (2003) 1169). In the earlier papers (J. Math. Biol. 49 (2004) 310 and J. Math. 8 (2004) 399), the first and last authors and Juliet Lee proved the existence of traveling wave solutions for the nematode sperm cell model. In this paper, we prove local existence of solutions to the same model and global existence under certain additional assumption on the initial data. Finally, we also provide numerical evidence that the traveling wave solutions found in (J. Math. Biol., accepted for publication) is globally asymptotically stable.     

求解一类非线性规划的连续同伦方法(英文)

In this paper we present a homotopy continuation method for finding the Karush-Kuhn-Tucker point of a class of nonlinear non-convex programming problems. Two numerical examples are given to show that this method is effective. It should be pointed out that we extend the results of Lin et al. （see Appl. Math. Comput., 80（1996）, 209-224） to a broader class of non-convex programming problems.     

On a question of Gross

Using the notion of weighted sharing of sets we prove two uniqueness theorems which improve the results proved by Fang and Qiu [H. Qiu, M. Fang, A unicity theorem for meromorphic functions, Bull. Malaysian Math. Sci. Soc. 25 (2002) 31-38], Lahiri and Banerjee [I. Lahiri, A. Banerjee, Uniqueness of meromorphic functions with deficient poles, Kyungpook Math. J. 44 (2004) 575-584] and Yi and Lin [H.X. Yi, W.C. Lin, Uniqueness theorems concerning a question of Gross, Proc. Japan Acad. Ser. A 80 (2004) 136-140] and thus provide an answer to the question of Gross [F. Gross, Factorization of meromorphic functions and some open problems, in: Proc. Conf. Univ. Kentucky, Lexington, KY, 1976, in: Lecture Notes in Math., vol. 599, Springer, Berlin, 1977, pp. 51-69], under a weaker hypothesis.     

Results on uniqueness of entire functions whose difference polynomials share a polynomial

In this paper, we use the concept of weighted sharing of values to investigate the uniqueness results when two difference polynomials of entire functions share a nonzero polynomial or a small function with a finite weight.We also investigate the situation when the original functions share 0 CM. The obtained results improve some recent related results of X. Li et al. [Ann. Polon. Math, 102 (2011), 111-127] and that of W. Li et al. [Bull. Malay. Math. Sci. Soc., 39 (2016), 499-515].     

Global Existence of Large Data Weak Solutions for a Simplified Compressible Oldroyd-B Model Without Stress Diffusion

We start with the compressible Oldroyd-B model derived in [2] (J. W. Barrett, Y. Lu, and E. Süli, Existence of large-data finite-energy global weak solutions to a compressible Oldroyd-B model, Commun. Math. Sci., 15 (2017), 1265-1323), where the existence of global-in-time finite-energy weak solutions was shown in two dimensional setting with stress diffusion. In the paper, we investigate the case without stress diffusion. We first restrict ourselves to the corotational setting as in [28] (P. L. Lions, and N. Masmoudi, Global solutions for some Oldroyd models of non-Newtonian flows, Chin. Ann. Math., Ser. B, 21(2) (2000), 131-146). We further assume the extra stress tensor is a scalar matrix and we derive a simplified model which takes a similar form as the multi-component compressible Navier-Stokes equations, where, however, the pressure term related to the scalar extra stress tensor has the opposite sign. By employing the techniques developed in [30,35], we can still prove the global-in-time existence of finite energy weak solutions in two or three dimensions, without the presence of stress diffusion.     

Exclusive traveling waves for competitive reaction-diffusion systems and their stabilities

We study the existence, uniqueness and asymptotic behavior, as well as the stability of a special kind of traveling wave solutions for competitive PDE systems involving intrinsic growth, competition, crowding effects and diffusion. The traveling waves are exclusive in the sense that as the variable goes to positive or negative infinity, different species are close to extinction or carrying capacity. We perform an appropriate affine transformation of the traveling wave equations into monotone form and construct appropriate upper and lower solutions. By this means, we reduce the existence proof to application of well-known theory about monotone traveling wave systems (cf. [A. Leung, Systems of Nonlinear Partial Differential Equations: Applications to Biology and Engineering, MIA, Kluwer, Boston, 1989; J. Wu, X. Zou, Traveling wave fronts of reaction-diffusion systems with delay, J. Dynam. Differential Equations 13 (2001) 651-687] and [I. Volpert, V. Volpert, V. Volpert, Traveling Wave Solutions of Parabolic Systems, Transl. Math. Monogr., vol. 140, Amer. Math. Soc., Providence, RI, 1994]). Then, by using spectral analysis of the linearization over the profile, we prove the orbital stability of the traveling wave in some Banach spaces with exponentially weighted norm. Furthermore, we show that the introduction of some weight is necessary in the sense that, in general, traveling wave solutions with initial perturbations in the (unweighted) space C₀ are unstable (cf. [I. Volpert, V. Volpert, V. Volpert, Traveling Wave Solutions of Parabolic Systems, Transl. Math. Monogr., vol. 140, Amer. Math. Soc., Providence, RI, 1994] and [D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Math., vol. 840, Springer-Verlag, New York, 1981]).     

求解Brouwer不动点问题的改进同伦方法

In this paper,we modify the homotopy method(proposed by Yu and Lin,Appl.Math.Comput.,74(1996),65)and hence make the modified method be able to solve Brouwer fixed-point problems in a broader class of nonconvex subsets in R~n.In addition,a simple example is given to show the effectiveness of the modified method.     

Existence of solutions for models of shallow water in a basin with a degenerate varying bottom

We prove the existence of solutions for the great lake equations. These equations are obtained from the three-dimensional Euler equations in a basin with a free upper surface and a spatially varying bottom topography by taking a low aspect ratio, i.e., low wave speed and small wave amplitude expansion. These equations are rewritten in an abstract form by considering generalized Euler equations as in Levermore et?al. (Indiana Univ Math J 45:479?C510, 1996). This paper is an extension of Levermore et?al. (Indiana Univ Math J 45:479?C510, 1996), where the varying bottom was assumed to be nondegenerate. Here, we discuss the degenerate case and obtain similar results as in Levermore et?al. (Indiana Univ Math J 45:479?C510, 1996).     

Refined a priori estimates for the axisymmetric Navier-Stokes equations

In this paper, we consider the axisymmetric Navier-Stokes equations, and provide a refined a priori estimate for the swirl component of the vorticity. This extends Theorem 2 of [D. Chae, J. Lee, On the regularity of the axisymmetric solutions of the Navier-Stokes equations, Math. Z., 239 (2002), 645--671].     

From a theorem of R. Ger and T. Kochanek to marginal joints of means

We answer in the negative a problem posed in Daróczy (Report on 52nd International Symposium on Functional Equations. Aequat. Math., 2015) by the first author, in connection with a result of Ger and Kochanek (Colloq Math 115:87–99, 2009), and its generalization formulated in Daróczy et al. (Report on 52nd International Symposium on Functional Equations. Aequat. Math., 2015). A further generalization is posed as an open problem. Elaborating an idea of the construction of means presented in Examples 1.2 and 1.4 we come to the notion of marginal joints of means. It provides a pretty wide class of means extending two given means on adjacent intervals.     

Linear convergence of an algorithm for largest singular value of a nonnegative rectangular tensor

An algorithm for finding the largest singular value of a nonnegative rectangular tensor was recently proposed by Chang, Qi, and Zhou [J. Math. Anal. Appl., 2010, 370: 284–294]. In this paper, we establish a linear convergence rate of the Chang-Qi-Zhou algorithm under a reasonable assumption.     

Letter to the Editor: A note on the preconditioned Gauss-Seidel (GS) method for linear systems

Wen Li (J. Comput. Appl. Math., 182 (2005) 81-90) asserted that there are some errors in article by Hiroshi Niki, Kyouji Harada, Munenori Morimoto and Michio Sakakihara (J. Comput. Appl. Math., 164-165 (2004) 587-600). And Li presented a new proof for the corresponding results in H. Niki et al. In this paper, we point out some errors in Li’s assertion. Moreover, we show that a new proof presented by Li is imperfect.     

Corrigendum to a Paper by Charak and Laine

This is a corrigendum to our paper, "On a class of prime entire functions", published in Acta Math. Sin., Engl. Ser., 25, 1647-1652 (2009).     

Global regularity of a class of p-fluid flows in cylinders

We consider the motion of an incompressible non-Newtonian fluid with shear dependent viscosity. We extend and improve the results obtained in the recent paper by Crispo [F. Crispo, Shear thinning viscous fluids in cylindrical domains. Regularity up to the boundary, J. Math. Fluid Mech., in press], concerning the case of the motion between two coaxial cylinders, to the case of a full cylinder. Actually we prove boundary regularity for solutions to the stationary Dirichlet problem with zero boundary data.     

A distributional version of functional equations and their stabilities

Generalizing the results in [J. Math. Anal. Appl. 286 (2003) 177–186; J. Math. Anal. Appl. 295 (2004) 107–114; Arch. Math., to appear; J. Math. Anal. Appl. 299 (2004) 578–586] that consider the Hyers–Ulam stability problems of several functional equations in the spaces of the Schwartz tempered distributions and the Fourier hyperfunctions we consider the stability problems of the functional equations in the space of distributions.     

On Counting Certain Abelian Varieties Over Finite Fields

This paper contains two parts toward studying abelian varieties from the classification point of view. In a series of papers[Doc. Math., 21, 1607-1643 (2016)],[Taiwanese J. Math., 20(4), 723-741 (2016)], etc., the current authors and T. C. Yang obtain explicit formulas for the numbers of superspecial abelian surfaces over finite fields. In this paper, we give an explicit formula for the size of the isogeny class of simple abelian surfaces with real Weil number q. This establishes a key step that extends our previous explicit calculation of superspecial abelian surfaces to those of supersingular abelian surfaces. The second part is to introduce the notion of genera and idealcomplexes of abelian varieties with additional structures in a general setting. The purpose is to generalize the previous work by the second named author[Forum Math., 22(3), 565-582 (2010)] on abelian varieties with additional structures to similitude classes, which establishes more results on the connection between geometrically defined and arithmetically defined masses for further investigations.     

Uniform infinite planar quadrangulations with a boundary

We introduce and study the uniform infinite planar quadrangulation (UIPQ) with a boundary via an extension of the construction of Curien et al. (Curien et al., Lat Am J Probab Math Stat 49 (2013) 340–373). We then relate this object to its simple boundary analog using a pruning procedure. This enables us to study the aperture of these maps, that is, the maximal graph distance between two points on the boundary, which in turn sheds new light on the geometry of the UIPQ. In particular we prove that the self‐avoiding walk on the UIPQ is at most diffusive. © 2014 Wiley Periodicals, Inc. Random Struct. Alg., 47, 30–58, 2015     

Dynamics of a Ramanujan-type continued fraction with cyclic coefficients

We study several generalizations of the AGM continued fraction of Ramanujan inspired by a series of recent articles in which the validity of the AGM relation and the domain of convergence of the continued fraction were determined for certain complex parameters (Borwein et al., Exp. Math. 13, 275–286, 2004, Ramanujan J., in press, 2004; Borwein and Crandall, Exp. Math. 12, 287–296, 2004). A study of the AGM continued fraction is equivalent to an analysis of the convergence of certain difference equations and the stability of dynamical systems. Using the matrix analytical tools developed in 2004, we determine the convergence properties of deterministic difference equations and so divergence of their corresponding continued fractions. Russell Luke’s work was supported in part by a postdoctoral fellowship from the Pacific Institute for the Mathematical Sciences at Simon Fraser University.