Forcing the Least Measurable to Violate GCH

Starting with a model for “GCH + k is k⁺ supercompact”, we force and construct a model for “k is the least measurable cardinal + 2^k = K⁺”. This model has the property that forcing over it with Add(k,k⁺⁺) preserves the fact k is the least measurable cardinal.     

Use of the Szeged index and the revised Szeged index for measuring network bipartivity

We have revisited the Szeged index (Sz) and the revised Szeged index (Sz^∗), both of which represent a generalization of the Wiener number to cyclic structures. Unexpectedly we found that the quotient of the two indices offers a novel measure for characterization of the degree of bipartivity of networks, that is, offers a measure of the departure of a network, or a graph, from bipartite networks or bipartite graphs, respectively. This is because the two indices assume the same values for bipartite graphs and different values for non-bipartite graphs. We have proposed therefore the quotient Sz/Sz^∗ as a measure of bipartivity. In this note we report on some properties of the revised Szeged index and the quotient Sz/Sz^∗ illustrated on a number of smaller graphs as models of networks.     

Paracompact subspaces in the box product topology

In 1975 E. K. van Douwen showed that if is a family of Hausdorff spaces such that all finite subproducts are paracompact, then for each element of the box product the -product is paracompact. He asked whether this result remains true if one considers uncountable families of spaces. In this paper we prove in particular the following result: Let be an infinite cardinal number, and let be a family of compact Hausdorff spaces. Let be a fixed point. Given a family of open subsets of which covers , there exists an open locally finite in refinement of which covers .

We also prove a slightly weaker version of this theorem for Hausdorff spaces with ``all finite subproducts are paracompact" property. As a corollary we get an affirmative answer to van Douwen's question.

     

On the Dirichlet's box principle

In this note, we will focus on several applications on the Dirichlet's box principle in Discrete Mathematics lesson and number theory lesson. In addition, the main result is an innovative game on a triangular board developed by the authors. The game has been used in teaching and learning mathematics in Discrete Mathematics and some high schools in Hong Kong.     

Statistically self-affine sets: Hausdorff and box dimensions

We introduce a class of random sets in ₂ that include McMullen's generalized Sierpinski carpets. We give exact expressions for the Hausdorff and Bouligand-Minkowski (box) dimensions of these sets, and find in particular that typically they are not equal. Our expression for the Hausdorff dimension isnot what one would expect by analogy with McMullen's formula for the Hausdorff dimension of a generalized Sierpinski carpet.Supported by Davis Ross Grant.Supported by NSF Grant DMS-9146252. Department of Statistics, Purdue University.     

Some Remarks on Normal Measures and Measurable Cardinals

We prove two theorems which in a certain sense show that the number of normal measures a measurable cardinal κ can carry is independent of a given fixed behavior of the continuum function on any set having measure 1 with respect to every normal measure over κ . First, starting with a model V ⊨ “ZFC + GCH + o(κ) = δ*” for δ* ≤ κ⁺ any finite or infinite cardinal, we force and construct an inner model N ⊆ V [G] so that N ⊨ “ZF + (∀δ < κ) [DC_δ] + ¬AC_κ + κ carries exactly δ* normal measures + 2^δ = δ⁺⁺ on a set having measure 1 with respect to every normal measure over κ”. There is nothing special about 2^δ = δ here, and other stated values for the continuum function will be possible as well. Then, starting with a modelV ⊨ “ZFC + GCH + κis supercompact”, we force and construct models of AC in which, roughly speaking, regardless of the specified behavior of the continuum function below κ on any set having measure 1 with respect to every normal measure over κ, κ can in essence carry any number of normal measures δ* ≥ κ⁺⁺.     

Phase space analysis on some black hole manifolds

The Schwarzschild and Reissner-Nordstrøm solutions to Einstein's equations describe space-times which contain spherically symmetric black holes. We consider solutions to the linear wave equation in the exterior of a fixed black hole space-time of this type. We show that for solutions with initial data which decay at infinity and at the bifurcation sphere, a weighted L⁶ norm in space decays like . This weight vanishes at the event horizon, but not at infinity. To obtain this control, we require only an ? loss of angular derivatives.     

On a class of Besicovitch functions to have exact box dimension: A necessary and sufficient condition

Let 1 < s < 2, λ_k > 0 with λ_k → ∞ satisfy λ_k+1/λ_k ≥ λ > 1. For a class of Besicovich functions B(t) = sin λ_kt, the present paper investigates the intrinsic relationship between box dimension of their graphs and the asymptotic behavior of {λ_k}. We show that the upper box dimension does not exceed s in general, and equals to s while the increasing rate is sufficiently large. An estimate of the lower box dimension is also established. Then a necessary and sufficient condition is given for this type of Besicovitch functions to have exact box dimensions: for sufficiently large λ, dim _BΓ(B) = dim _BΓ(B) = s holds if and only if lim_n→∞ = 1. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Multisymplecticity of the centred box discretization for hamiltonian PDEs with m ≥ 2 space dimensions

In this letter, it is shown that the centred box discretization for Hamiltonian PDEs with m ≥ 2 space dimensions is multisymplectic in the sense of Bridges and Reich in [1–6]. Multisymplectic discretizations for the generalized KP equation and the wave equation with 2 space dimensions, respectively, are given. A multisymplectically numerical scheme of the wave equation is derived.     

Error analysis of discrete conservation laws for Hamiltonian PDEs under the central box discretizations

The central box scheme has been the most successful of the multisymplectic integrators for Hamiltonian PDEs. In this paper, we investigate conservative properties of the central box scheme for Hamiltonian PDEs and derive the error formulas of discrete local and global conservation laws of energy and momentum. We apply these results to the nonlinear Schrödinger equation and Klein-Gordon equation. Numerical experiments are presented to verify the theoretical predications.     

FRACTIONAL INTEGRALS OF THE WEIERSTRASS FUNCTIONS： THE EXACT BOX DIMENSION

The present paper investigates the fractal structure of fractional integrals of Weierstrass functions. The ezact box dimension for such functions many important cases is established. We need to point out that, although the result itself achieved in the present paper is interesting, the new technique and method should be emphasized. These novel ideas might be useful to establish the box dimension or Hausdorff dimension (especially for the lower bounds) for more general groups of functions.     

Failures of GCH and the level by level equivalence between strong compactness and supercompactness

We force and obtain three models in which level by level equivalence between strong compactness and supercompactness holds and in which, below the least supercompact cardinal, GCH fails unboundedly often. In two of these models, GCH fails on a set having measure 1 with respect to certain canonical measures. There are no restrictions in all of our models on the structure of the class of supercompact cardinals. (© 2003 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Some recurrence formulas for box splines and cone splines

A degree elevation formula for multivariate simplex splines was given by Micchelli and extended to hold ]or multivariate Dirichlet splines in [8]. We report similar formulae for multivariate cone splines and box splsplines andines. To this end, we utilize a relation due to Dahmen and Micchelli that connects box cone splines and a degree reduction formulagiven by Cohen, Lyche, and Riesenfeld in [2].     

Odd‐angulated graphs and cancelling factors in box products

Under what conditions is it true that if there is a graph homomorphism G □ H → G □ T, then there is a graph homomorphism H→ T? Let G be a connected graph of odd girth 2k + 1. We say that G is (2k + 1)‐angulated if every two vertices of G are joined by a path each of whose edges lies on some (2k + 1)‐cycle. We call G strongly (2k + 1)‐angulated if every two vertices are connected by a sequence of (2k + 1)‐cycles with consecutive cycles sharing at least one edge. We prove that if G is strongly (2k + 1)‐angulated, H is any graph, S, T are graphs with odd girth at least 2k + 1, and ?: G□ H→S□T is a graph homomorphism, then either ? maps G□{h} to S□{t_h} for all h∈V(H) where t_h∈V(T) depends on h; or ? maps G□{h} to {s_h}□ T for all h∈V(H) where s_h∈V(S) depends on h. This theorem allows us to prove several sufficient conditions for a cancelation law of a graph homomorphism between two box products with a common factor. We conclude the article with some open questions. © 2008 Wiley Periodicals, Inc. J Graph Theory 58:221‐238, 2008     

Interaction curves for vibration and buckling of thin-walled composite box beams under axial loads and end moments

Interaction curves for vibration and buckling of thin-walled composite box beams with arbitrary lay-ups under constant axial loads and equal end moments are presented. This model is based on the classical lamination theory, and accounts for all the structural coupling coming from material anisotropy. The governing differential equations are derived from the Hamilton’s principle. The resulting coupling is referred to as triply flexural–torsional coupled vibration and buckling. A displacement-based one-dimensional finite element model with seven degrees of freedoms per node is developed to solve the problem. Numerical results are obtained for thin-walled composite box beams to investigate the effects of axial force, bending moment, fiber orientation on the buckling loads, buckling moments, natural frequencies and corresponding vibration mode shapes as well as axial-moment–frequency interaction curves.     

More on the identity of Chaundy and Bullard

In what follows we present a homogeneous identity which implies a more elementary treatment of the Chaundy–Bullard identity with n variables. In a different direction we bring another ramification of the Chaundy–Bullard identity.     

On the limitations of classical benchmark functions for evaluating robustness of evolutionary algorithms

Although evolutionary algorithms (EAs) have some operators which let them explore the whole search domain, still they get trapped in local minima when multimodality of the objective function is increased. To improve the performance of EAs, many optimization techniques or operators have been introduced in recent years. However, it seems that these modified versions exploit some special properties of the classical multimodal benchmark functions, some of which have been noted in previous research and solutions to eliminate them have been proposed.In this article, we show that quite symmetric behavior of the available multimodal test functions is another example of these special properties which can be exploited by some EAs such as covariance matrix adaptation evolution strategy (CMA-ES). This method, based on its invariance properties and good optimization results for available unimodal and multimodal benchmark functions, is considered as a robust and efficient method. However, as far as black box optimization problems are considered, no special trend in the behavior of the objective function can be assumed; consequently this symmetry limits the generalization of optimization results from available multimodal benchmark functions to real world problems. To improve the performance of CMA-ES, the Elite search sub-algorithm is introduced and implemented in the basic algorithm. Importance and effect of this modification is illustrated experimentally by dissolving some test problems in the end.     

More on the inadmissibility of step-up

Cohen and Sackrowitz [Characterization of Bayes procedures for multiple endpoint problems and inadmissibility of the step-up procedure, Ann. Statist. 33 (2005) 145-158] proved that the step-up multiple testing procedure is inadmissible for a multivariate normal model with unknown mean vector and known intraclass covariance matrix. The hypotheses tested are each mean is zero vs. each mean is positive. The risk function is a 2×1 vector where one component is average size and the other component is one minus average power. In this paper, we extend the inadmissibility result to several different models, to two-sided alternatives, and to other risk functions. The models include one-parameter exponential families, independent t-variables, independent χ²-variables, t-tests arising from the analysis of variance, and t-tests arising from testing treatments against a control. The additional risk functions are linear combinations where one component is the false discovery rate (FDR).