Circular demonstration of the commutative law

The associative, commutative and distributive laws are taught as axioms. Proceeding in a necessarily cautious fashion, this article shows that the commutative law (as representative) cannot be demonstrated over negative integers. Demonstration is only possible at the cost of assuming ‘negative x negative = positive’ as ‘true without proof before proceeding. It is noted that there is then no point in proving ’negative x negative = positive’ by the use of the associative, commutative and distributive laws after it has already been so assumed true without proof to establish those very laws. Nevertheless, the circle of proof can be broken, and a suitable demonstration of the commutative law is given. However, this first of all requires a new circular theory of number. Such a theory exists and this article is in support of its findings.     

A Note on the History of the Four‐Colour Conjecture

The four‐colour conjecture was brought to public attention in 1854, most probably by Francis or Frederick Guthrie. This moves back by six years the date of the earliest known publication.     

An alternative proof of Wigner theorem on quantum transformations based on elementary complex analysis

According to Wigner theorem, transformations of quantum states which preserve the probabilities are either unitary or antiunitary. This Letter presents an elementary proof of this theorem that significantly departs from the numerous ones already existing in the literature. The main line of the argument remains valid even in quantum field theory where Hilbert spaces are non-separable.     

Wave equations with time-dependent dissipation II. Effective dissipation

This article is intended to present a construction of structural representations of solutions to the Cauchy problem for wave equations with time-dependent dissipation above scaling. These representations are used to give estimates of the solution and its derivatives based on Lq(Rn), q?2.The article represents the second part within a series. In [Jens Wirth, Wave equations with time-dependent dissipation I. Non-effective dissipation, J. Differential Equations 222 (2) (2006) 487-514] weak dissipations below scaling were discussed.     

Expansions and polynomial limit representations of weighted pseudoinverses

Expansions of weighted pseudoinverses with positive definite or singular weights in matrix power series or power products with negative exponents and arbitrary positive parameters are proposed and analyzed. Based on these expansions, polynomial limit representations of weighted pseudoinverses are obtained. Issues related to the construction of direct and iterative methods for calculating weighted pseudoinverses and weighted normal pseudosolutions, as well as solving constrained least squares problems, are examined.     

UNIFORM ESTIMATES OF THE RESOLVENT OF THE LAPLACE –BELTRAMI OPERATOR ON INFINITE VOLUME RIEMANNIAN MANIFOLDS WITH CUSPS

Abstract

In this article we extract characteristic points from scattering theories of the Lax–Phillips type and the Wilcox one, and in view of those points we show that the both settings can be translated into each other. Furthermore, we construct concrete translation representations of the Lax–Phillips type for the elastic equation in the half-space, in order to show the good selection from admitted constructions in the abstract arguments.     

一类新的非线性光学材料的研究——Fe2O3氧化物超微粒溶胶

80年代以来,超微粒材料的光学非线性引起了广泛的注意,其大的三阶光学非线性系数和快的响应,可能在光电子学中获得应用。这种非线性,是由具有量子尺寸效应(quantum size effect)的超微粒中的载流子限域运动(carrier confinement)和载流子间相互作用引起的,局域场效应(local field effect)可增强这一非线性过程^[1]。     

Novel materials from bis(arene)metal‐containing polyacrylonitrile

Novel bis(arene)metal‐containing polyacrylonitrile materials have been prepared by the polycyanoethylation reaction between acrylonitrile and (arene)₂M (M = Cr or V; arene = PhH, C₆H₄Et₂ or mesitylene) in the absence of solvent. The resulting star‐shaped molecules consist of a central (arene)₂M species with up to four polyacrylonitrile arms covalently bonded to the arene ligands. The materials are readily soluble and films can be cast from solutions in acetonitrile. The IR and solid state ¹³C NMR spectra (or EPR spectrum for the oxidized chromium‐containing polymer) are consistent with the presence of a metal–arene bond and confirm the persistence of the sandwich structure. The properties of the thermolysed materials are consistent with the formation of conjugated naphthyridine‐type structures. The value of |n₂| determined by the degenerate four‐wave mixing technique at 1064 nm with a 6 ns pulse duration for a solution in conc. H₂SO₄ (1 g l⁻¹) of the chromium‐containing polymer pyrolysed at 350 °C was found to be 0.8 × 10⁻¹³ cm² W⁻¹ corresponding to |lRe _χ⁽³⁾| = 0.4 × 10⁻¹¹ esu.     

On the use of symmetry-adapted crystalline orbitals in SCF-LCAO periodic calculations. I. The construction of the symmetrized orbitals

A computational procedure for generating space-symmetry-adapted Bloch functions (BF) is presented. The case is discussed when BF are built from a basis of local functions (atomic orbitals [AOs]). The method, which is completely general in the sense that it applies to any space group and AOs of any quantum number, is based on the diagonalization of Dirac characters. For its implementation, it does not require as an input character tables or related data, since this information is automatically generated starting from the space group symbol and the AO basis set. Formal aspects of the method, not available in textbooks, are discussed. © 1998 John Wiley & Sons, Inc. Int J Quant Chem 67: 299–309, 1998