A critical index algorithm for nearest point problems on simplicial cones

We consider the linear complementarity problem (q, M) in whichM is a positive definite symmetric matrix of ordern. This problem is equivalent to a nearest point problem [; b] in which = {A.₁,, A. _n } is a basis for R ⁿ ,b is a given point in R ⁿ ; and it is required to find the nearest point in the simplicial cone Pos() tob. We develop an algorithm for solving the linear complementarity problem (q, M) or the equivalent nearest point problem [; b]. Computational experience in comparison with an existing algorithm is presented.Research effort partially supported by the Air Force Office of Scientific Research. Air Force Systems Command, USAF, under grant No. AFOSR 78-3646. The United States Government is authorized to reproduce and distribute reprints for governmental purposes, not withstanding any copyright notation hereon.     

Bounds on the dimensions of trivariate spline spaces

We derive upper and lower bounds on the dimensions of trivariate spline spaces defined on tetrahedral partitions. The results hold for general partitions, and for all degrees of smoothness r and polynomial degrees d.      

On the general routing polytope

The General Routing Problem (GRP) consists of finding a minimum length closed walk in an edge-weighted undirected graph, subject to containing certain sets of required nodes and edges. It is related to the Rural Postman Problem and the Graphical Traveling Salesman Problem.We examine the 0/1-polytope associated with the GRP introduced by Ghiani and Laporte [A branch-and-cut algorithm for the Undirected Rural Postman Problem, Math. Program. Ser. A 87 (3) (2000) 467-481]. We show that whenever it is not full-dimensional, the set of equations and facets can be characterized, and the polytope is isomorphic to the full-dimensional polytope associated with another GRP instance which can be obtained in polynomial time. We also offer a node-lifting method. Both results are applied to prove the facet-defining property of some classes of valid inequalities. As a tool, we study more general polyhedra associated to the GRP.     

Generalized halfspaces in dimension groups

In a dimension group, the projection of a finite intersection of generalized halfspaces is a finite intersection of generalized halfspaces. The dimension groups obeying a stronger version of this result, true in dense Archimedean ordered groups, are characterized algebraically and provided with a simple set of axioms.     

Approximate inversion of the Black–Scholes formula using rational functions

The Black–Scholes formula is often used in the backward direction to invert the implied volatility, usually with some solver method. Solver methods, being aesthetically unappealing, are also slower than closed-form approximations. However, closed-form approximations in previous works lack accuracy, often providing option pricing errors well exceeding the bid–ask spreads. We develop a new closed-form method based on the rational approximation. The rational approximation is much faster than typical solver methods and very accurate for both at-the-money and away-from-the-money options. Its accuracy can be further improved by one or two steps of Newton–Raphson iterations.     

Percolation on infinitely ramified fractal networks

We study how fractal features of an infinitely ramified network affect its percolation properties. The fractal attributes are characterized by the Hausdorff ($D_H$), topological Hausdorff ($D_{tH}$), and spectral ($d_s$) dimensions. Monte Carlo simulations of site percolation were performed on pre-fractal standard Sierpiński carpets with different fractal attributes. Our findings suggest that within the universality class of random percolation the values of critical percolation exponents are determined by the set of dimension numbers ($D_H$, $D_{tH}$, $d_s$), rather than solely by the spatial dimension (d). We also argue that the relevant dimension number for the percolation threshold is the topological Hausdorff dimension $D_{tH}$, whereas the hyperscaling relations between critical exponents are governed by the Hausdorff dimension $D_H$. The effect of the network connectivity on the site percolation threshold is revealed.     

Thin Walled Beams with Residual Stress

A one-dimensional variational problem for an anisotropic, partially inhomogeneous, residually stressed, rectangular thin-walled beam is derived, by Γ-convergence, from the three-dimensional theory of linear elasticity with residual stress.      

HHV和HVV耦合的有效拉氏量及费曼规则

首先包含Higgs粒子的六维算符用物理场来表示,然后给出包含HHV和HVV相互作用顶点的有效拉氏量,其中V=A,Z,W^±。最后,推导出每个六维算符和有效拉氏量的费曼规则。     

Quantum Gravity and Phenomenological Philosophy

The central thesis of this paper is that contemporary theoretical physics is grounded in philosophical presuppositions that make it difficult to effectively address the problems of subject-object interaction and discontinuity inherent to quantum gravity. The core objectivist assumption implicit in relativity theory and quantum mechanics is uncovered and we see that, in string theory, this assumption leads into contradiction. To address this challenge, a new philosophical foundation is proposed based on the phenomenology of Maurice Merleau-Ponty and Martin Heidegger. Then, through the application of qualitative topology and hypernumbers, phenomenological ideas about space, time, and dimension are brought into focus so as to provide specific solutions to the problems of force-field generation and unification. The phenomenological string theory that results speaks to the inconclusiveness of conventional string theory and resolves its core contradiction. This article is based on my 2008 book, The Self-Evolving Cosmos, appearing in the Series on Knots and Everything of World Scientific Publishing Company.     

Estimation of dimension functions of band-limited wavelets

The dimension function D_ψ of a band-limited wavelet ψ is bounded by n if its Fourier transform is supported in [−(2ⁿ⁺²/3)π,(2ⁿ⁺²/3)π]. For each and for each , 0<<δ=δ(n), we construct a wavelet ψ with supp

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such that D_ψ>n on a set of positive measure, which proves that [−(2ⁿ⁺²/3)π,(2ⁿ⁺²/3)π] is the largest symmetric interval for estimating the dimension function by n. This construction also provides a family of (uncountably many) wavelet sets each consisting of infinite number of intervals.