Inhomogeneous Diophantine approximation on curves and Hausdorff dimension

The goal of this paper is to develop a coherent theory for inhomogeneous Diophantine approximation on curves in Rn akin to the well established homogeneous theory. More specifically, the measure theoretic results obtained generalize the fundamental homogeneous theorems of R.C. Baker (1978) [2], Dodson, Dickinson (2000) [18] and Beresnevich, Bernik, Kleinbock, Margulis (2002) [8]. In the case of planar curves, the complete Hausdorff dimension theory is developed.     

Rational points near planar curves and Diophantine approximation

In this paper, we establish asymptotic formulae with optimal errors for the number of rational points that are close to a planar curve, which unify and extend the results of Beresnevich–Dickinson–Velani [6] and Vaughan–Velani [22]. Furthermore, we complete the Lebesgue theory of Diophantine approximation on weakly non-degenerate planar curves that was initially developed by Beresnevich–Zorin [5] in the divergence case.     

Inhomogeneous Diophantine approximation on planar curves

In this paper we develop the inhomogeneous metric theory of simultaneous Diophantine approximation on planar curves. Our results naturally extend the homogeneous Khintchine and Jarník type theorems established in Beresnevich et al. (Ann Math 166(2):367–426, 2007) and Vaughan and Velani (Invent Math 166:103–124, 2006) and are the first of their kind. The key lies in obtaining essentially the best possible results regarding the distribution of ‘shifted’ rational points near planar curves.     

On The diophantine equation Fn+Fm=2a

In this paper, we find all the solutions of the title Diophantine equation in positive integer variables (n, m, a), where F_k is the kth term of the Fibonacci sequence. The proof of our main theorem uses lower bounds for linear forms in logarithms (Baker's theory) and a version of the Baker-Davenport reduction method in diophantine approximation.     

Diophantine approximation and badly approximable sets

Let (X,d) be a metric space and (Ω,d) a compact subspace of X which supports a non-atomic finite measure m. We consider ‘natural’ classes of badly approximable subsets of Ω. Loosely speaking, these consist of points in Ω which ‘stay clear’ of some given set of points in X. The classical set Bad of ‘badly approximable’ numbers in the theory of Diophantine approximation falls within our framework as do the sets Bad(i,j) of simultaneously badly approximable numbers. Under various natural conditions we prove that the badly approximable subsets of Ω have full Hausdorff dimension. Applications of our general framework include those from number theory (classical, complex, p-adic and formal power series) and dynamical systems (iterated function schemes, rational maps and Kleinian groups).     

Problèmes diophantiens simultanés

Simultaneous Diophantine Approximations. We study a mixed simultaneous diophantine problem, with an approximation condition and a divisibility condition. We solve this problem for quadratic numbers.     

Salem numbers defined by Coxeter transformation

A real algebraic integer α>1 is called a Salem number if all its remaining conjugates have modulus at most 1 with at least one having modulus exactly 1. It is known [J.A. de la Peña, Coxeter transformations and the representation theory of algebras, in: V. Dlab et al. (Eds.), Finite Dimensional Algebras and Related Topics, Proceedings of the NATO Advanced Research Workshop on Representations of Algebras and Related Topics, Ottawa, Canada, Kluwer, August 10-18, 1992, pp. 223-253; J.F. McKee, P. Rowlinson, C.J. Smyth, Salem numbers and Pisot numbers from stars, Number theory in progress. in: K. Gy?ry et al. (Eds.), Proc. Int. Conf. Banach Int. Math. Center, Diophantine problems and polynomials, vol. 1, de Gruyter, Berlin, 1999, pp. 309-319; P. Lakatos, On Coxeter polynomials of wild stars, Linear Algebra Appl. 293 (1999) 159-170] that the spectral radii of Coxeter transformation defined by stars, which are neither of Dynkin nor of extended Dynkin type, are Salem numbers. We prove that the spectral radii of the Coxeter transformation of generalized stars are also Salem numbers. A generalized star is a connected graph without multiple edges and loops that has exactly one vertex of degree at least 3.     

Diophantine approximation for negatively curved manifolds

Let M be a geometrically finite pinched negatively curved Riemannian manifold with at least one cusp. Inspired by the theory of Diophantine approximation of a real (or complex) number by rational ones, we develop a theory of approximation of geodesic lines starting from a given cusp by ones returning to it. We define a new invariant for M, theHurwitz constant of M. It measures how well all geodesic lines starting from the cusp are approximated by ones returning to it. In the case of constant curvature, we express the Hurwitz constant in terms of lengths of closed geodesics and their depths outside the cusp neighborhood. Using the cut locus of the cusp, we define an explicit approximation sequence for a geodesic line starting from the cusp and explore its properties. We prove that the modular once-punctured hyperbolic torus has the minimum Hurwitz constant in its moduli space. Received: 24 October 2000; in final form: 10 November 2001 / Published online: 17 June 2002     

Shrinking the period length of quasi-periodic continued fractions

Extending the work of Burger et al., here we show that every quasi-periodic simple continued fraction α can be transformed into a quasi-periodic non-simple continued fraction having period length one. Moreover, a certain kind of quasi-periodic non-simple continued fraction is equivalent to a quasi-periodic N-continued fraction. The results of this paper follow from arguments of Burger et al. but we apply our version to offer new continued fractions for certain classes of real numbers.     

A uniformization theorem for complete Kähler manifolds with positive holomorphic bisectional curvature

In the theory of complex geometry, one of the famous problems is the following conjecture of Greene and Wu [13] and Yau [33]: Suppose M is a complete noncompact Kähler manifold with positive holomorphic bisectional curvature; then M is biholomorphic to ?ⁿ. In this paper we use the Ricci flow evolution equation to study this conjecture and prove the result that if M has bounded and positive curvature such that the L’ norm of the curvature on geodesic ball is small enough, then the conjecture is true. Our result gives an improvement on the results of Mok et al. [21] and Mok [22].     

Splitting theorem, Poincaré-Hopf theorem and jumping nonlinear problems

In this paper, we establish a Gromoll-Meyer splitting theorem and a shifting theorem for J∈C^2-0(E,R) and by using the finite-dimensional approximation, mollifiers and Morse theory we generalize the Poincaré-Hopf theorem to J∈C¹(E,R) case. By combining the Poincaré-Hopf theorem and the splitting theorem, we study the existence of multiple solutions for jumping nonlinear elliptic equations.     

Well-posedness and existence of bound states for a coupled Schrödinger-gKdV system

We derive some new results concerning the Cauchy problem and the existence of bound states for a class of coupled nonlinear Schrödinger-gKdV systems. In particular, we obtain the existence of strong global solutions for initial data in the energy space H¹(R)×H¹(R), generalizing previous results obtained in Tsutsumi (1993) [11], Corcho and Linares (2007) [13] and Dias et al. (submitted for publication) [14] for the nonlinear Schrödinger-KdV system.     

Équidistribution presque partout modulo 1 de suites oscillantes perturbées III : cas liouvillien multidimensionnel

We extend the results of uniform distribution modulo 1 given in [B. Rittaud, Équidistribution presque partout modulo 1 de suites oscillantes perturbées, Bull. Soc. Math. France 128 (2000) 451-471; B. Rittaud, Équidistribution presque partout modulo 1 de suites oscillantes perturbées, II: Cas Liouvillien unidimensionnel, Colloq. Math. 96 (1) (2003) 55-73], which deal with sequences of the form , where n(hn), and are polynomially increasing sequences, n(εn) a bounded sequence, essentially a C³-function Zd-periodic, Θ an element of Rd and t a real number. We remove the Diophantine hypothesis on Θ needed in [the first of above mentioned articles], and add a technical hypothesis on hn. We apply this result to the convergence of diagonal averages for d×d matrices.     

On Hopf bifurcation in non-smooth planar systems

As we know, for non-smooth planar systems there are foci of three different types, called focus-focus (FF), focus-parabolic (FP) and parabolic-parabolic (PP) type respectively. The Poincaré map with its analytical property and the problem of Hopf bifurcation have been studied in Coll et al. (2001) [3] and Filippov (1988) [6] for general systems and in Zou et al. (2006) [13] for piecewise linear systems. In this paper we also study the problem of Hopf bifurcation for non-smooth planar systems, obtaining new results. More precisely, we prove that one or two limit cycles can be produced from an elementary focus of the least order (order 1 for foci of FF or FP type and order 2 for foci of PP type) (Theorem 2.3), different from the case of smooth systems. For piecewise linear systems we prove that 2 limit cycles can appear near a focus of either FF, FP or PP type (Theorem 3.3).     

L∞ variational problems and weak KAM theory

In the first part of this paper, we extend several results in Crandall, Evans, and Gariepy [11] and Crandall and Evans [10] to absolute minimizers of more general L^∞ functionals. In the second part, we present some interesting connections between L^∞ variational problems and weak KAM theory. As an application, we will advance the main result in Fathi and Siconolfi [17], i.e, the existence of a C¹ subsolution of the Hamilton‐Jacobi equation. Moreover, we will propose a possible approximation of the projected Aubrey set by a variational approach that was first used by Evans in [14]. © 2007 Wiley Periodicals, Inc.     

Approximation of signals of class Lip(α, p) by linear operators

Mittal, Rhoades [5], [6], [7] and [8] and Mittal et al. [9] and [10] have initiated a study of error estimates E_n(f) through trigonometric-Fourier approximation (tfa) for the situations in which the summability matrix T does not have monotone rows. In this paper we continue the work. Here we extend two theorems of Leindler [4], where he has weakened the conditions on {p_n} given by Chandra [2], to more general classes of triangular matrix methods. Our Theorem also partially generalizes Theorem 4 of Mittal et al. [11] by dropping the monotonicity on the elements of matrix rows, which in turn generalize the results of Quade [15].     

Proper forcing axiom and selective separability

We continue the study of Selectively Separable (SS) and, a game-theoretic strengthening, strategically selectively separable spaces (SS⁺) (see Barman, Dow (2011) [1]). The motivation for studying SS⁺ is that it is a property possessed by all separable subsets of Cp(X) for each σ-compact space X. We prove that the winning strategy for countable SS⁺ spaces can be chosen to be Markov. We introduce the notion of being compactlike for a collection of open sets in a topological space and with the help of this notion we prove that there are two countable SS⁺ spaces such that the union fails to be SS⁺, which contrasts the known result about SS spaces. We also prove that the product of two countable SS⁺ spaces is again countable SS⁺. One of the main results in this paper is that the proper forcing axiom, PFA, implies that the product of two countable Fréchet spaces is SS, a statement that was shown in Barman, Dow (2011) [1] to consistently fail. An auxiliary result is that it is consistent with the negation of CH that all separable Fréchet spaces have π-weight at most ω₁.     

Bounds for ratios of the membrane eigenvalues

For a bounded planar region in R², we obtain the ratios of lower order eigenvalues of Laplace operator. Combining our results with the recursive formula in Cheng and Yang (2007) [11], we can obtain better upper bound of the (k+1)-th (k?3) membrane eigenvalues.     

Well-posedness of the martingale problem for some degenerate diffusion processes occurring in dynamics of populations

In this paper we establish the well-posedness in C([0,∞);[0,1]^d), for each starting point x∈[0,1]d, of the martingale problem associated with a class of degenerate elliptic operators which arise from the dynamics of populations as a generalization of the Fleming-Viot operator. In particular, we prove that such degenerate elliptic operators are closable in the space of continuous functions on [0,1]^d and their closure is the generator of a strongly continuous semigroup of contractions.     

An uncertainty principle for function fields

In a recent paper, Granville and Soundararajan (2007) [5] proved an “uncertainty principle” for arithmetic sequences, which limits the extent to which such sequences can be well-distributed in both short intervals and arithmetic progressions. In the present paper we follow the methods of Granville and Soundararajan (2007) [5] and prove that a similar phenomenon holds in Fq[t].