Volume 2011, Issue 657

Using an involved study of the Kauffman bracket, we give formulas for the second and third coefficient of the Jones polynomial in semiadequate diagrams. As applications, we show that several classes of links, including semiadequate links and Whitehead doubles of semiadequate knots, have non-trivial Jones polynomial. We also prove that there are infinitely many positive knots with no positive minimal crossing diagrams. Some relations to the twist number of a link, Mahler measure and the hyperbolic volume are given, for example explicit upper bounds on the volume for Montesinos and 3-braid links in terms of their Jones polynomial.

Schur studied limits of the arithmetic means A n of zeros for polynomials of degree n with integer coefficients and simple zeros in the closed unit disk. If the leading coefficients are bounded, Schur proved that . We show that A n → 0, and estimate the rate of convergence by generalizing the Erdős–Turán theorem on the distribution of zeros. As an application, we show that integer polynomials have some unexpected restrictions of growth on the unit disk. Schur also studied problems on means of algebraic numbers on the real line. When all conjugate algebraic numbers are positive, the problem of finding the sharp lower bound for was developed further by Siegel and others. We provide a solution of this problem for algebraic numbers equidistributed in subsets of the real line. Potential theoretic methods allow us to consider the distribution of algebraic numbers in or near general sets in the complex plane. We introduce the generalized Mahler measure, and use it to characterize asymptotic equidistribution of algebraic numbers in arbitrary compact sets of capacity one. The quantitative aspects of this equidistribution are also analyzed in terms of the generalized Mahler measure.

Following the approach of B. Roberts, we characterize the non-vanishing of the global theta lift for symplectic-orthogonal dual pairs in terms of its local counterpart. In particular, we replace the temperedness assumption present in Robert's work by a certain weaker assumption, and apply our results to small rank similitude groups. Among our applications is a certain instance of Langlands functorial transfer of a (non-generic) cuspidal automorphic representation of GSp(4) to GL(4).

We consider the class of algebras of rank 4 equipped with a standard involution over an arbitrary base ring. In particular, we characterize quaternion rings, those algebras defined by the construction of the even Clifford algebra.

We prove a simple transcendence criterion suitable for function field arithmetic. We apply it to show the transcendence of special values at non-zero rational arguments (or more generally, at algebraic arguments which generate extension of the rational function field with less than q places at infinity) of the entire hypergeometric functions in the function field (over 𝔽 q ) context, and to obtain a new proof of the transcendence of special values at non-natural p -adic integers of the Carlitz–Goss gamma function. We also characterize in the balanced case the algebraicity of hypergeometric functions, giving an analog of the result of F. R. Villegas, based on Beukers–Heckman results and E. Landau's method in the classical hypergeometric case.

We show that 3-fold terminal flips and divisorial contractions to a curve may be factored by a sequence of weighted blow-ups, flops, blow-downs to a locally complete intersection curve in a smooth 3-fold or divisorial contractions to a point.

Considered herein is the well-posedness of a periodic Degasperis–Procesi equation with linear dispersion modeling for an approximation to the incompressible Euler equation for shallow water waves. The existence of permanent waves is established with certain small initial profiles depending on the linear dispersive parameter in a range of Sobolev spaces. On the other hand, sufficient conditions guaranteeing the development of breaking waves in finite time are established. Moreover, the blow-up rate of breaking waves is obtained.

We prove that all strongly outer -actions on a UHF algebra of infinite type are strongly cocycle conjugate to each other. We also prove that all strongly outer, asymptotically representable -actions on a unital simple AH algebra with real rank zero, slow dimension growth and finitely many extremal tracial states are cocycle conjugate to each other.