The Fibonacci code behind super strings and P-Branes. An answer to M. Kaku’s fundamental question

In “Beyond Einstein” the leading string theoretician and notable science writer Michio Kaku referred to what he labelled the ‘strange’ link between the E₈ exceptional Lie group and the various dimensionalities of strings and super string theories and commented on that by saying “If we could understand why the numbers 8, 10 and 26 continually crop up in super string theory, perhaps we could understand why the universe is four dimensional”.In the present work we demonstrate the existence of a Fibonacci code-like connection between the various coupling constants, charges and dimensionalities of super strings and P-Brane theories. This code is based on the Fibonacci numbers and the golden mean and in the final analysis, may be attributed to the deterministically chaotic nature of the hyperbolic Cantorian sets fixing the geometry and topology of quantum spacetime.     

Constraints, gauge symmetries, and noncommutative gravity in two dimensions

After a short introduction, we show how the canonical formalism is constructed in space-time noncommutative theories such that it allows defining the notion of first-class constraints and analyzing gauge symmetries. We use this formalism to noncommutatively deform the two-dimensional string gravity (also known as the Witten black hole). Dedicated to Yu. V. Novozhilov on the occasion of his 80th birthday __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 148, No. 1, pp. 64–79, July, 2006.     

群与图的对称性

首先对平面图形的对称进行分析和利用其对称群进行量化,进而将此推广到考察一般图的广义"对称性"与图自同构群的关系,最后刻画了无平方因子阶局部本原弧传递图的自同构群结构.     

On the structure of run-maximal strings

We present a combinatorial structure consisting of a special cover of a string by squares. We characterize the covering property of run-maximal strings, i.e. strings achieving the maximal number of runs. The covering property leads to a compression scheme which is particularly efficient for run-maximal strings. It also yields a significant speed improvement in the computer search for good run-maximal string candidates. The implementation of the search and preliminary computational results are discussed.     

On symmetries of the heat equation

Two aims are pursued in this article. The first one is methodological and consists of demonstrating the symmetry calculation techniques based on the commutation relation on the example of the quasi-linear heat equation. The second one consists of investigating the following open question: is the algebra of higher symmetries of the linear heat equation exhausted by those which may be obtained by means of the classical symmetries and well-known recursion operators? A positive solution of this question is given.     

On the realization of symmetries in quantum mechanics

The aim of this paper is to give a simple, geometric proof of Wigner’s theorem on the realization of symmetries in quantum mechanics that clarifies its relation to projective geometry. Although several proofs exist already, it seems that the relevance of Wigner’s theorem is not fully appreciated in general. It is Wigner’s theorem which allows the use of linear realizations of symmetries and therefore guarantees that, in the end, quantum theory stays a linear theory. In the present paper, we take a strictly geometrical point of view in order to prove this theorem. It becomes apparent that Wigner’s theorem is nothing else but a corollary of the fundamental theorem of projective geometry. In this sense, the proof presented here is simple, transparent and therefore accessible even to elementary treatments in quantum mechanics.     

On the realization of symmetries in quantum mechanics

The aim of this paper is to give a simple, geometric proof of Wigner’s theorem on the realization of symmetries in quantum mechanics that clarifies its relation to projective geometry. Although several proofs exist already, it seems that the relevance of Wigner’s theorem is not fully appreciated in general. It is Wigner’s theorem which allows the use of linear realizations of symmetries and therefore guarantees that, in the end, quantum theory stays a linear theory. In the present paper, we take a strictly geometrical point of view in order to prove this theorem. It becomes apparent that Wigner’s theorem is nothing else but a corollary of the fundamental theorem of projective geometry. In this sense, the proof presented here is simple, transparent and therefore accessible even to elementary treatments in quantum mechanics.     

On the dynamics of axially moving strings

We investigate an axially moving cable with large sag modelled as string and calculate the steady state solution between two rolls. Alternatively we model the cable as beam with very small bending stiffness and compare this solution with the string solution. The validity of the analytically found boundary layer solution is verified by numerical methods. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

关于一类Fibonacci行列式的计算

研究了一类由Fibonacci数组成的特殊行列式Dn(m,k)的计算问题,并给出了一个有趣的恒等式     

Fibonacci数列倒数的无穷和

利用初等方法以及取整函数的性质研究了Fibonacci数列三次倒数的求和问题,获得了该和式倒数取整后的确切值,也就是给出了一个包含Fibonacci数列有趣的恒等式.     

Fuzzy knot theory interpretation of Yang–Mills instantons and Witten’s 5-Brane model

A knot theory interpretation of ‘tHooft’s instanton based on hyperbolic volume, crossing numbers and exceptional Lie symmetry groups is given. Subsequently it is shown that although instantons and particle-like states of Heterotic super strings may appear to be different concepts, on a very deep fuzzy level they are not.     

On the equivalence of Lie symmetries and group representations

We consider families of linear, parabolic PDEs in n dimensions which possess Lie symmetry groups of dimension at least four. We identify the Lie symmetry groups of these equations with the (2n+1)-dimensional Heisenberg group and SL(2,R). We then show that for PDEs of this type, the Lie symmetries may be regarded as global projective representations of the symmetry group. We construct explicit intertwining operators between the symmetries and certain classical projective representations of the symmetry groups. Banach algebras of symmetries are introduced.     

On the local fractional wave equation in fractal strings

The key aim of the present study is to attain nondifferentiable solutions of extended wave equation by making use of a local fractional derivative describing fractal strings by applying local fractional homotopy perturbation Laplace transform scheme. The convergence and uniqueness of the obtained solution by using suggested scheme is also examined. To determine the computational efficiency of offered scheme, some numerical examples are discussed. The results extracted with the aid of this technique verify that the suggested algorithm is suitable to execute, and numerical computational work is very interesting.     

On the equations for p-adic closed and open strings

A survey of several pure mathematical results concerning the boundary-value problems for nonlinear pseudo-differential equation for closed and open strings in d-dimensional flat spacetime is presented. We obtained some results on existence or nonexistence of solutions. In particular, the absence of almost-periodic solutions was shown. We consider also some numerical approaches to the problems. The text was submitted by the author in English.     

On the dimensions of bivariate spline spaces and the stability of the dimensions

In this paper, the dimensions of bivariate spline spaces are studied using the Smoothing Cofactor-Conformality method. Based on the analysis on the conformality condition at one interior vertex, the stability (or singularity to the contrary) of the dimensions of general spline spaces is discussed in detail. By the aid of directed partition some new results on dimensions are obtained with the corresponding constraints depending on the degree, the smoothness order of the spline spaces and the structure of the partition as well.