On the shape of the moduli of spherical minimal immersions

The DoCarmo-Wallach moduli space parametrizing spherical minimal immersions of a Riemannian manifold is a compact convex body in a linear space of tracefree symmetric endomorphisms of an eigenspace of . In this paper we define and study a sequence of metric invariants , , associated to a compact convex body with base point in the interior of . The invariant measures how lopsided is in dimension with respect to . The results are then appplied to the DoCarmo-Wallach moduli space. We also give an efficient algorithm to calculate for convex polytopes.

     

A new majorization between functions, polynomials, and operator inequalities

Let P₊ be the set of all non-negative operator monotone functions defined on [0,∞), and put . Then and . For a function and a strictly increasing function h we write if is operator monotone. If and and if and , then . We will apply this result to polynomials and operator inequalities. Let and be non-increasing sequences, and put for t≧a₁ and for t≧b₁. Then v₊?u₊ if m≦n and : in particular, for a sequence of orthonormal polynomials, (p_n-1)₊?(p_n)₊. Suppose 0<r,p and s=0 or 1≦s≦1+p/r. Then 0≦A≦B implies for 0<α≦r/(p+r).     

Is the Approximation Rate for European Pay-offs in the Black–Scholes Model Always 1/ \sqrt{n}

For a Borel-function , we consider the approximation of a random variable f(W ₁) with by stochastic integrals with respect to the Brownian motion and the geometric Brownian motion, where the integrands are piecewise constant within certain deterministic time intervals. In earlier papers it has been shown that under certain regularity conditions the optimal approximation rate is 1/ , if one optimizes over deterministic time-nets of cardinality n. We will show the existence of random variables f(W ₁) such that the approximation error tends as slowly to zero as one wishes.     

ON NEWMAN-TYPE RATIONAL INTERPOLATION TO ｜x｜ AT THE CHEBYSHEV NODES OF THE SECOND KIND

Recently Brutman and Passow considered Newman-type rational interpolation to ｜x｜ induced by arbitrary set of symmetric nodes in [-1,1] and gave the general estimation of the approximation error.By their methods one could establish the exact order of approximation for some special nodes. In the present paper we consider the special case where the interpolation nodes are the zeros of the Chebyshev polynomial of the second kind and prove that in this case the exact order of approximation is O（1/n｜nn）     

The number of k-colorings of a graph on a fixed surface

We prove that, for every fixed surface S, there exists a largest positive constant c such that every 5-colorable graph with n vertices on S has at least c·2ⁿ distinct 5-colorings. This is best possible in the sense that, for each sufficiently large natural number n, there is a graph with n vertices on S that has precisely c·2ⁿ distinct 5-colorings. For the sphere the constant c is , and for each other surface, it is a finite problem to determine c. There is an analogous result for k-colorings for each natural number k>5.     

Hall-Littlewood polynomials and fixed point enumeration

We resolve affirmatively some conjectures of Reiner, Stanton, and White (2004) [12] regarding enumeration of transportation matrices which are invariant under certain cyclic row and column rotations. Our results are phrased in terms of the bicyclic sieving phenomenon introduced by Barcelo, Reiner, and Stanton (2009) [1]. The proofs of our results use various tools from symmetric function theory such as the Stanton-White rim hook correspondence (Stanton and White (1985) [18]) and results concerning the specialization of Hall-Littlewood polynomials due to Lascoux, Leclerc, and Thibon (1994, 1997) [5] and [6].     

Uniform asymptotics of some q-orthogonal polynomials

Uniform asymptotic formulas are obtained for the Stieltjes-Wigert polynomial, the q⁻¹-Hermite polynomial and the q-Laguerre polynomial as the degree of the polynomial tends to infinity. In these formulas, the q-Airy polynomial, defined by truncating the q-Airy function, plays a significant role. While the standard Airy function, used frequently in the uniform asymptotic formulas for classical orthogonal polynomials, behaves like the exponential function on one side and the trigonometric functions on the other side of an extreme zero, the q-Airy polynomial behaves like the q-Airy function on one side and the q-Theta function on the other side. The last two special functions are involved in the local asymptotic formulas of the q-orthogonal polynomials. It seems therefore reasonable to expect that the q-Airy polynomial will play an important role in the asymptotic theory of the q-orthogonal polynomials.     

A recursive algorithm for constructing complicated Dixon matrices

Whether the determinant of the Dixon matrix equals zero or not is used for determining if a system of n + 1 polynomial equations in n variables has a common root, and is a very efficient quantifier elimination approach too. But for a complicated polynomial system, it is not easy to construct the Dixon matrix. In this paper, a recursive algorithm to construct the Dixon matrix is proposed by which some problems that cannot be tackled by other methods can be solved on the same computer platform. A dynamic programming algorithm based on the recursive formula is developed and compared for speed and efficiency to the recursive algorithm.     

The polynomial invariants of twisted links

A twisted link is a generalization of a virtual link, which is related to a link diagram on a closed, possibly non-orientable surface. In this paper we generalize the Miyazawa polynomial invariant of a virtual link to an invariant of a twisted link in two formulae one of which is introduced by A. Ishii and the other by the author.     

The Miyazawa polynomial for long virtual knots

We introduce a two-variable polynomial invariant of a long virtual knot, which dominates the Kauffman f-polynomial and the Miyazawa polynomial of the closure. Our invariant satisfies a product formula for the concatenation product of long virtual knots. It describes a formula of the Miyazawa polynomial of a ‘connected sum’ of two virtual knots. It also gives lower bounds for the real crossing number and the virtual crossing number of a long virtual knot.