Three-dimensional approximate local DtN boundary conditions for prolate spheroid boundaries

We propose a new class of approximate local DtN boundary conditions to be applied on prolate spheroidal-shaped exterior boundaries when solving problems of acoustic scattering by elongated obstacles. These conditions are: (a) exact for the first modes, (b) easy to implement and to parallelize, (c) compatible with the local structure of the computational finite element scheme, and (d) applicable to exterior ellipsoidal-shaped boundaries that are more suitable in terms of cost-effectiveness for surrounding elongated scatterers. We investigate analytically and numerically the effect of the frequency regime and the slenderness of the boundary on the accuracy of these conditions. We also compare their performance to the second-order absorbing boundary condition (BGT2) designed by Bayliss, Gunzburger and Turkel when expressed in prolate spheroid coordinates. The analysis reveals that, in the low-frequency regime, the new second-order DtN condition (DtN2) retains a good level of accuracy regardless of the slenderness of the boundary. In addition, the DtN2 boundary condition outperforms the BGT2 condition. Such superiority is clearly noticeable for large eccentricity values.     

Comments on the height reducing property

A complex number α is said to satisfy the height reducing property if there is a finite subset, say F, of the ring ? of the rational integers such that ?[α] = F[α]. This property has been considered by several authors, especially in contexts related to self affine tilings and expansions of real numbers in non-integer bases. We prove that a number satisfying the height reducing property, is an algebraic number whose conjugates, over the field of the rationals, are all of modulus one, or all of modulus greater than one. Expecting the converse of the last statement is true, we show some theoretical and experimental results, which support this conjecture.     

ABC implies the radicalized Vojta height inequality for curves

The truncated or radicalized counting function of a meromorphic function counts the number of times that f takes a value a, but without multiplicity. By analogy, one also defines this function for numbers. In this sequel to [M. van Frankenhuijsen, The ABC conjecture implies Vojta's height inequality for curves, J. Number Theory 95 (2002) 289-302], we prove the radicalized version of Vojta's height inequality, using the ABC conjecture. We explain the connection with a conjecture of Serge Lang about the different error terms associated with Vojta's height inequality and with the radicalized Vojta height inequality.     

Non-restricted representations of simple Lie superalgebras of special type and Hamiltonian type

Let F be an algebraically closed field of prime characteristic p > 2,and g be a simple Lie superalgebra of special type or Hamiltonian type over F.We construct the simple g-modules with non-singular characters of height more than one,and some simple modules with singular characters of height more than five.Furthermore,for the case of special type Lie superalgebras,we also construct a class of simple modules with regular semisimple characters of height one.All those simple modules mentioned above are proved to be reduced Kac modules.     

Regular variation and probability: The early years

It is a pleasure for Bingham of Bingham, Goldie and Teugels to write in appreciation of Teugels of Bingham, Goldie and Teugels, on the occasion of Jef Teugels’ retirement, and also to remind myself of the promise we made each other—all those years ago, in the early 1970s—to write the book that regular variation so obviously required. The theme has continued to attract my interest, Jef's and that of his pupils since. As for the book (BGT below), it continues to be my most cited work, and to find its place in the working library of probabilists.     

Convergence of discrete schemes for the Perona-Malik equation

We prove the convergence, up to a subsequence, of the spatial semidiscrete scheme for the one-dimensional Perona-Malik equation ut=(?^′x(ux)), , when the initial datum is 1-Lipschitz out of a finite number of jump points, and we characterize the problem satisfied by the limit solution. In the more difficult case when has a whole interval where is negative, we construct a solution by a careful inspection of the behaviour of the approximating solutions in a space-time neighbourhood of the jump points. The limit solution u we obtain is the same as the one obtained by replacing ?(⋅) with the truncated function min(?(⋅),1), and it turns out that u solves a free boundary problem. The free boundary consists of the points dividing the region where |ux|>1 from the region where |ux|?1. Finally, we consider the full space-time discretization (implicit in time) of the Perona-Malik equation, and we show that, if the time step is small with respect to the spatial grid h, then the limit is the same as the one obtained with the spatial semidiscrete scheme. On the other hand, if the time step is large with respect to h, then the limit solution equals , i.e., the standing solution of the convexified problem.     

A construction for partially ordered sets

A construction I(S) is defined which corresponds to the intuitive notion of the set of places in which new elements can be inserted into a given poset S. It is given the minimal possible ordering. It turns out that if the base sets are chains the construction produces the corresponding interval orders. for whose dimensions there exist good estimates. In this paper we make the dual restriction that the height of the underlying set is ?1. Under this assumption we find a bound for the dimension of I(S) in general and a precise value if the set consists of two antichains all the elements of one lying above all those of the other.     

Isometrical Embeddings of Lattices into Geometric Lattices

A lattice is said to be finite height generated if it is complete and every element is the join of some elements of finite height. Extending former results by Grätzer and Kiss (Order 2:351–365, 1986) on finite lattices, we prove that every finite height generated algebraic lattice that has a pseudorank function is isometrically embeddable into a geometric lattice.     

On mean distance and girth

We bound the mean distance in a connected graph which is not a tree in terms of its order n and its girth g. On one hand, we show that the mean distance is at most if g is even and at most if g is odd. On the other hand, we prove that the mean distance is at least unless G is an odd cycle. This resolves two conjectures of AutoGraphiX.     

On the p-adic meromorphy of the function field height zeta function

In this brief note, we will investigate the number of points of bounded height in a projective variety defined over a function field, where the function field comes from a projective variety of dimension greater than or equal to 2. A first step in this investigation is to understand the p-adic analytic properties of the height zeta function. In particular, we will show that for a large class of projective varieties this function is p-adic meromorphic.     

On harmonic typically real mappings

In this paper, we prove that a univalent orientation-preserving harmonic mapping defined on the unit disk U with the normalization f(0)=0, , is a typically real mapping, if f(U) is a starlike domain with respect to the origin or f(U) is convex in one direction.     

Cospectral graphs and the generalized adjacency matrix

Let J be the all-ones matrix, and let A denote the adjacency matrix of a graph. An old result of Johnson and Newman states that if two graphs are cospectral with respect to yJ − A for two distinct values of y, then they are cospectral for all y. Here we will focus on graphs cospectral with respect to yJ − A for exactly one value of y. We call such graphs -cospectral. It follows that is a rational number, and we prove existence of a pair of -cospectral graphs for every rational . In addition, we generate by computer all -cospectral pairs on at most nine vertices. Recently, Chesnokov and the second author constructed pairs of -cospectral graphs for all rational , where one graph is regular and the other one is not. This phenomenon is only possible for the mentioned values of , and by computer we find all such pairs of -cospectral graphs on at most eleven vertices.     

Strong peak points and strongly norm attaining points with applications to denseness and polynomial numerical indices

Using the variational method, it is shown that the set of all strong peak functions in a closed algebra A of Cb(K) is dense if and only if the set of all strong peak points is a norming subset of A. As a corollary we can induce the denseness of strong peak functions on other certain spaces. In case that a set of uniformly strongly exposed points of a Banach space X is a norming subset of , then the set of all strongly norm attaining elements in is dense. In particular, the set of all points at which the norm of is Fréchet differentiable is a dense Gδ subset. In the last part, using Reisner's graph-theoretic approach, we construct some strongly norm attaining polynomials on a CL-space with an absolute norm. Then we show that for a finite dimensional complex Banach space X with an absolute norm, its polynomial numerical indices are one if and only if X is isometric to . Moreover, we give a characterization of the set of all complex extreme points of the unit ball of a CL-space with an absolute norm.     

Nathanson heights in finite vector spaces

Let p be a prime, and let Zp denote the field of integers modulo p. The Nathanson height of a point is the sum of the least nonnegative integer representatives of its coordinates. The Nathanson height of a subspace is the least Nathanson height of any of its nonzero points. In this paper, we resolve a quantitative conjecture of Nathanson [M.B. Nathanson, Heights on the finite projective line, Int. J. Number Theory, in press], showing that on subspaces of of codimension one, the Nathanson height function can only take values about . We show this by proving a similar result for the coheight on subsets of Zp, where the coheight of A⊆Zp is the minimum number of times A must be added to itself so that the sum contains 0. We conjecture that the Nathanson height function has a similar constraint on its range regardless of the codimension, and produce some evidence that supports this conjecture.     

Moduli spaces and multiple polylogarithm motives

In this paper, we give a natural construction of mixed Tate motives whose periods are a class of iterated integrals which include the multiple polylogarithm functions. Given such an iterated integral, we construct two divisors A and B in the moduli spaces of n-pointed stable curves of genus 0, and prove that the cohomology of the pair is a framed mixed Tate motive whose period is that integral. It generalizes the results of A. Goncharov and Yu. Manin for multiple ζ-values. Then we apply our construction to the dilogarithm and calculate the period matrix which turns out to be same with the canonical one of Deligne.     

Iterative roots of piecewise monotonic functions with finite nonmonotonicity height

It is known that any continuous piecewise monotonic function with nonmonotonicity height not less than 2 has no continuous iterative roots of order n greater than the number of forts of the function. In this paper, we consider the problem of iterative roots in the case that the order n is less than or equal to the number of forts. By investigating the trajectory of possible continuous roots, we give a general method to find all iterative roots of those functions with finite nonmonotonicity height.     

Improving on the mle of a bounded location parameter for spherical distributions

For the problem of estimating under squared error loss the location parameter of a p-variate spherically symmetric distribution where the location parameter lies in a ball of radius m, a general sufficient condition for an estimator to dominate the maximum likelihood estimator is obtained. Dominance results are then made explicit for the case of a multivariate student distribution with d degrees of freedom and, in particular, we show that the Bayes estimator with respect to a uniform prior on the boundary of the parameter space dominates the maximum likelihood estimator whenever and d?p. The sufficient condition matches the one obtained by Marchand and Perron (Ann. Statist. 29 (2001) 1078) in the normal case with identity covariance matrix. Furthermore, we derive an explicit class of estimators which, for , dominate the maximum likelihood estimator simultaneously for the normal distribution with identity covariance matrix and for all multivariate student distributions with d degrees of freedom, d?p. Finally, we obtain estimators which dominate the maximum likelihood estimator simultaneously for all distributions in the subclass of scale mixtures of normals for which the scaling random variable is bounded below by some positive constant with probability one.