El Naschie’s cantorian strings and duality in Weyl–Dirac theory

In the weak-field approximation, some implications of duality in the Weyl–Dirac (WD) theory, using the Gregorash–Papini–Wood approach, are investigated. Any particle is in a permanent interaction with the ‘subquantic level’ (Madelung’s fluid) and, as a result of this interaction, the particle acquires the proper fluctuation curvature and the proper fluctuation energy, respectively. By fixing the fluctuations scale, the quantum fluid orders either by means of bright cnoidal oscillation modes inducing causality, or by means of dark cnoidal oscillation modes inducing acausality, and non-linear effects, respectively. The periodic mode is associated with the undulatory characteristic, and the solitonic one with the corpuscular one. By not fixing the fluctuations scale and keeping the symmetry, the quantum fluid orders like a two-dimensional (2D) lattice of vortices, so that the duality needs coherence. In the compatibility between quantum hydrodynamics in the Madelung’s representation and the wave mechanics, the self-gravitational field of the Weyl–Dirac type physical object is generated. El Naschie’s space–time implies, by means of transfinite heterotic string theory, the masses of nucleons, and, by the gravitational fractional quantum Hall effect, the dispersion of the wave-packet on the particle. The analysis of the fractal dimension of the physical object described by the WD theory shows that the waves, and corpuscle, respectively are 2D projections of a higher dimensional special string in El Naschie’s space–time (El Naschie’s string).     

涉及微分多项式的正规定则（英文）

本文获得如下结果：设φ（z)为区域G内一不恒为零的亚纯函数，a1(z),a2(z),．…，ak(z)为区域G内的全纯函数，F＝{f}为G内一亚纯函数族，若对每一f∈F，在G内恒有f(z)，f(z)≠0,f^(k)(z) a1(z)f^(f-1)(z) … ak(z)f(z)≠φ（z),且与φ（z)没有公共极点，则F在G内正规。     

實向量所成的有序環

<正> 引言 本文討論G.Birkhoff在“格論”第二版中所提出的一個未解决的問題(見参考文獻[1]229頁問題103).這個問題是: 合G為所有賓2維向量(a,b)所成的集合.在向量加法之下,G形成一交換羣.若在G中如下定義順序關係“≧”     

黎曼流形上的法坐標

<正> 1952年P.Hartman和A.Wintner建立了下述定理:設連續函數E(u,ν),F(u,ν),G(u,ν)在點(0,0)的鄰域內满足條件     

线性可移算子与可调和的随机过程

<正> §1.引言T.Onoyama 利用了 Weyl-Stone-Titchmarsh 的特征函数(Eigenfunction)展开公式,对具有二阶矩的实的连续机过程(本文中所用的极限,系指在均方意义下的极限)求得了下列隨机函数方程     

On the Group of p-endotrivial kG-modules

In this paper,we define a group Tp(G) of p-endotrivial kG-modules and a generalized Dade group Dp(G) for a finite group G.We prove that Tp(G) ≌ Tp(H) whenever the subgroup H contains a normalizer of a Sylow p-subgroup of G,in this case,K(G) ≌ K(H).We also prove that the group Dp(G) can be embedded into Tp(G) as a subgroup.     

关于多项式最大公因式的进一步探讨

在［１］中有这样一个结论,对于 Ｐ［ｘ］中任意两个多项式 ｆ（ｘ）、ｇ（ｘ）,在 Ｐ［ｘ］中存在一个 最大公因式 ｄ（ｘ）,且ｄ （ｘ）可以表示成ｆ （ｘ）、ｇ （ｘ）的一个组合,即有 Ｐ［ｘ］中的多项式 ｕ（ｘ）、ｖ（ｘ）,使： ｄ（ｘ）＝ ｕ（ｘ）ｆ（ｘ）＋ ｖ（ｘ）ｇ（ｘ） （１）本文将对ｕ（ｘ）、ｖ（ｘ）作进一步的分析,从而得出有关ｕ（ｘ）、ｖ（ｘ）的一些新的结论,以作为上述结论的补充     

Approximation of solutions of the two-dimensional wave equation with variable velocity and localized right-hand side using some “simple” solutions

Asymptotic solutions based on the characteristics and the modified Maslov canonical operator of the two-dimensional wave equation with variable coefficients and right-hand side corresponding to: (a) an instantaneous source; (b) a rapidly acting, but “time spread,” source, are compared. An algorithm for approximating a (more complicated) solution of problem (b) by linear combinations of the derivatives of the (simpler) solution of problem (a) is proposed. Numerical calculations showing the accuracy of this approximation are presented. The replacement of the solutions of problem (b) by those of problem (a) becomes especially important in the case where the wave equation is considered in the domain with boundary on which the velocity of the wave equation vanishes. Then the characteristics of the problem become singular (nonstandard) and solutions of type (a) generalize to the case referred to above in a much simpler and effective way than solutions of type (b). Such a situation arises in problems where long waves (for example, tsunami waves) are incident on a sloping seashore.     

On the Grassmanian of lines in PG(4,q) and R(1,2) reguli

An R(1,2) regulus is a collection of q+1 mutually skew planes in PG(5,q) with the property that a line meeting three of the planes must meet all the planes. An (l,π)-configuration is the collection of lines in PG(4,q) meeting a line l and a plane π skew to l. A correspondence between (l,π)-configurations in PG(4-,q) and R(1,2) reguli in the associated Grassmanian space G(1,4) is examined. Bose has shown that R(1,2) reguli represent Baer subplanes of a Desarguesian projective plane in a linear representation of the plane. With the purpose of examining the relations between two Baer subplanes of PG(2,q²), the author examines the possible intersections of a 3-flat with an R(1,2) regulus.     

Comments on “A note on “A novel approach to multi attribute group decision making based on trapezoidal interval type-2 fuzzy soft sets””

In a recently published paper “A note on “A novel approach to multi attribute group decision making based on trapezoidal interval type-2 fuzzy soft sets””, Khalil and Hassan pointed out that assertions (3) and (4) of Theorem 3.2 in our previous paper “A novel approach to multi attribute group decision making based on trapezoidal interval type-2 fuzzy soft sets” are not true [2]. Furthermore, they introduced the notions of a generalized trapezoidal interval type-2 fuzzy soft subset and a generalized trapezoidal interval type-2 fuzzy soft equal and used these two notions to correct the flaw in assertions (3) and (4) of Theorem 3.2 in our previous paper. In this paper, we show by a counterexample that Khalil and Hassan's correction is incorrect and provide the modified versions of assertions (3) and (4) of Theorem 3.2, along with a strict proof. In addition, Khalil and Hassan pointed out by a counterexample that assertions (3)–(6) of Theorem 3.5 in our paper are not true and proposed the corrections of those assertions. In this paper, we show that Khalil and Hassan's counterexample and corrections are incorrect and provide a new example to verify the inaccuracies of assertions (3) and (5) of Theorem 3.5 in our paper. Moreover, we offer the modified versions of assertions (3) and (5) of Theorem 3.5 and prove them. Finally, Khalil and Hassan's statement that assertions (4) and (6) of Theorem 3.5 in our previous paper are not true is proven to be incorrect, i.e. assertions (4) and (6) of Theorem 3.5 in our previous paper are true.     

关于CM分担四个公共小函数的亚纯函数结论的改进

关于CM分担四个公共小函数的亚纯函数结论，我们在考虑重值的条件下，改进了李平和杨重骏^[2]的结论：设f(z)与g(z)为非常数亚纯函数，aj(z)(j=1,2,….4)为f(z)与g(z)的四个判别的小函数，若f(z)与g(z)满足Ek)(aj,f)=Ek)(aj,g),(j=1,2,3,4)且k(≥15）是正整数，则f(z)是g(z)的拟分式线性变换。即:存在f(z)与g(z)的四个小函数a(z),b(z),c(z),d(z),使得f=ag b/cg d(ad-bc≠0），（亦称Quasi-Mobuys变换）。     

Constructing finitely presented monoids which have no finite complete presentation

Squier (1987) showed that if a monoid is defined by a finite complete rewriting system, then it satisfies the homological finiteness condition FP₃, and using this fact he gave monoids (groups) which have solvable word problems but cannot be presented by finite complete systems. In the present paper we show that a monoid cannot have a finite complete presentation if it contains certain special elements. This observation enables us to construct monoids without finite complete presentation in a direct and elementary way. We give a finitely presented monoid which has (1) a word problem solvable in linear time and (2) linear growth but (3) no finite complete presentation. We also give a finitely presented monoid which has (1) a word problem solvable in linear time, (2) finite derivation type in the sense of Squier and (3) the property FP_∞, but (4) no finite complete presentation.     

Embeddings of weighted Sobolev spaces and degenerate elliptic problems

New embeddings of some weighted Sobolev spaces with weights a(x)and b(x)are established.The weights a(x)and b(x)can be singular.Some applications of these embeddings to a class of degenerate elliptic problems of the form-div(a(x)?u)=b(x)f(x,u)in?,u=0 on??,where?is a bounded or unbounded domain in RN,N 2,are presented.The main results of this paper also give some generalizations of the well-known Caffarelli-Kohn-Nirenberg inequality.     

New ODE Methods for Equality Constrained Optimization (1) — Equations

To deal with equality constrained optimization problems (ECP), we introduce in this paper "(ECP)-equation", a class of new systems of ordinary differential equations for (ECP), containing a matrix parameter called (ECP)-direction matrix, which plays a central role in it, and a scalar parameter called (ECP)-rate factor. It is shown that by following the trajectory of the equation, a stationary point or hopefully a local solution can be located under very mild conditions. As examples, several schemes of (ECP)-direction matrices and (ECP)-rate factors are given to construct concrete forms of the (ECP)-equation, including almost all the existing projected gradient type versions as special cases. As will be shown in a subsequent paper where the implementation problems are considered in detail, application of an example of these forms results in encouraging performance in experiments.     

A note on biduals of strict (LF)-spaces

Grothendieck asked in 1954 in [1] the following questions. (1) Is the bidual of a strict inductive limit of a sequence of locally convex spaces the inductive limit of the biduals? (2) Is the bidual of a strict (LF)-space again an (LF)-space? (3) Is the bidual of a strict (LF)-space complete? M. Valdivia gave a (negative) answer to the first question in 1979 in [5]. Since his counterexample is not an (LF)-space, problem (2) remained open. The aim of this note is to present a negative solution to questions (2) and (3). The answer to question (2) is negative even if every step of the (LF)-space is distinguished, in which case the strong bidual is complete by a result of Grothendieck. Moreover, we show that the strong dual of a strict (LF)-space need not be countably barrelled.     

Inverse boundary value problems and a theorem of Gel'fand and Levitan

This report concerns two so-called inverse problems of mathematical physics. These are: (i) the problem of determining a second-order differential operator (in a normal form) on the half-axis from its spectral function; and, (ii) the problem of determining a hyperbolic boundary value problem of a special form in a (non-characteristic) half-plane from its response on the boundary to a unit impulse at some reference time t = 0 (boundary value of the Riemann function). We solve problem (ii) by a natural approach, and then indicate how the solution of problem (i) follows from the solution of problem (ii). Our solution of problem (ii) is constructive, and we obtain stability of the solution under perturbation of the data, in a well-defined sense. For problem (i), we obtain the well-known result of Gel'fand and Levitan, in the sharp formulation given by Levitan and Gasymov ([6]).     

随机狄里克莱级数的一些性质

本文研究随机狄里克莱级数的a.s.(几乎必然)收敛性和在a.s.收敛半平面内的a.s.增长性;为此,还研究了狄里克莱级数在收敛半平面内的增长性.这里推进了Valiron G.和Arnold L.的有关结果.文中还证明了在一定条件下,两类随机狄里克莱级数a.s.以其收敛轴上每一点为其Picard点或Borel(R)点.     

An improved error bound for a finite element approximation of a model for phase separation of a multi-component alloy with a concentration dependent mobility matrix

Summary. Using a slightly different discretization scheme in time and adapting the approach in Nochetto et al. (1998) for analysing the time discretization error in the backward Euler method, we improve on the error bounds derived in (i) Barrett and Blowley (1998) and (ii) Barrett and Blowey (1999c) for a fully practical piecewise linear finite element approximation of a model for phase separation of a multi-component alloy with a concentration dependent mobility matrix and (i) a logarithmic free energy, and (ii) a non-smooth free energy (the deep quench limit); respectively. Moreover, the improved error bound in the deep quench limit is optimal. Numerical experiments with three components illustrating the above error bounds are also presented. Received June 28, 1999 / Revised version received December 3, 1999 / Published online November 8, 2000     

映射系统中的零序列

设（Ｅ，ｒ）是局部凸拓扑向量空间，Ω是任意抽象集合，Ｆ∈Ｅ＾Ω，（ｘｊ）∈Ω是映射系统（Ω，Ｆ）中的零序列，即对每个ｆ∈Ｆ，（ｆ（ｘｊ））都是（Ｅ，τ）中收敛于０的序列，本文给出了序列（ｆ（ｘｊ））的Ｆ的子集Ｂ上一致收敛于０的刻划。