On the Stability of Periodic Mercury-type Rotations

We consider the stability of planar periodic Mercury-type rotations of a rigid body around its center of mass in an elliptical orbit in a central Newtonian field of forces. Mercurytype rotations mean that the body makes 3 turns around its center of mass during 2 revolutions of the center of mass in its orbit (resonance 3:2). These rotations can be 1) symmetrical 2π- periodic, 2) symmetrical 4π-periodic and 3) asymmetrical 4π-periodic. The stability of rotations of type 1) was investigated by A.P.Markeev. In our paper we present a nonlinear stability analysis for some rotations of types 2) and 3) in 3rd- and 4th-order resonant cases, in the nonresonant case and at the boundaries of regions of linear stability.     

Vertex Algebras, Lie Algebras, and Superstrings

We construct Lie algebras from vertex superalgebras and study their structure. They are sometimes generalized Kac–Moody algebras. In some special cases we can calculate the multiplicities of the roots.     

The Best Bound on the Rotations in the Stability of Periodic Solutions of a Newtonian Equation

In most cases, the third order approximation of a scalar Newtonianequation can lead to the Lyapunov stability of a periodic solutionthrough the obtaining of a nonzero twist coefficient. Recently,Ortega obtained the twist property of a periodic solution whenthe second order coefficient does not change sign and the thirdone is negative under a crucial limitation to the rotation ofthe linearization equation. The paper finds that the best boundon the limitation of the rotations is .     

Rotations and quantum theory

Summary A classical theory of rotations is presented in a simple way that can easily be taken over to quantum theory. The application to the theory of elementary particles is indicated. To Enrico Bompiani on his scientific Jubilee     

Irrational Rotations and Differentiability

In this paper we investigate first return differentiation when the trajectories are restricted to irrational rigid rotations.     

Strings and Pions

We assume that QCD can be effectively described with stringlike variables and that this string is approximately represented by a Nambu–Goto or Polyakov string in some energy region. But to overcome the long-standing problems with the behavior of hadronic string amplitudes at low energies, the string is built over the correct (chirally noninvariant) QCD vacuum using a boundary interaction with background chiral fields associated with pions. Making this interaction compatible with the conformal symmetry of the string and with the unitarity constraint on chiral fields, we reconstruct the equations of motion for the latter and, furthermore, recover the Lagrangian of a nonlinear sigma model of pion interactions. We obtain the chiral structural constants of Gasser and Leutwyler in the next-to-leading order in the derivative expansion. The estimated coefficients fit the phenomenological values well.     

Nuclear Spacetime Theories,Superstrings, MonsterGroup and Applications

The present paper is basically a collection of notes and remarks on various nuclear-quantum spacetime theories and their possible application in physics. Several subjects are considered including quantum gravity, fractional quantum Hall effect, knot theory, superstrings, noncommutative geometry, partially ordered sets, Cantorian spaces, branched polymers and sporadic 196884 dimensional monster group.     

Spectral Stability for Classical Periodic Waves of the Ostrovsky and Short Pulse Models

We consider the Ostrovsky and short pulse models in a symmetric spatial interval, subject to periodic boundary conditions. For the Ostrovsky case, we rederive the formulas for the classical periodic traveling waves, while for the short pulse model, we explicitly construct traveling waves in terms of Jacobi elliptic functions. In both cases, we show spectral stability, for all values of the parameters. This is achieved by studying the nonstandard eigenvalue problems in the form , where is a Hill operator.     

Rotations of Hypercyclic and Supercyclic Operators

Let T B(X) be a hypercyclic operator and a complex number of modulus 1. Then T is hypercyclic and has the same set of hypercyclic vectors as T. A version of this results gives for a wide class of supercyclic operators that x X is supercyclic for T if and only if the set {tTⁿ x : t > 0, n = 0, 1, ...} is dense in X. This gives answers to several questions studied in the literature.     

Superstrings,Cantorian-fractal Spacetime and Quantum-like Chaos in Atmospheric Flows

Atmospheric flows exhibit long-range spatiotemporal correlations manifested as the fractal geometry to the global cloud cover pattern concomitant with the inverse power law form for spectra of temporal fluctuations. Such non-local connections are ubiquitous to dynamical systems in nature and are identified as signatures of self-organized criticality. A recently developed cell dynamical system model for atmospheric flows predicts the observed self-organized criticality as a natural consequence of quantum-like mechanics governing flow dynamics. The model is based on the concept that spatial integration of enclosed small scale fluctuations results in the formation of large eddy circulations. The model predicts the following: (a) The flow structure consists of an overall logarithmic spiral trajectory with the quasiperiodic Penrose tiling pattern for the internal structure. (b) Conventional power spectrum analysis will resolve such spiral trajectories as a continuum of eddies with progressive increase in phase. (c) Increments in phase are concomitant with increases in period length and also represent the variance, a characteristic of quantum systems identified as Berrys phase. (d) The universal algorithm for self-organized criticality is expressed in terms of the universal Feigenbaum constants, a and d, as 2a²=πd, where the fractional volume intermittency of occurrence πd contributes to the total variance 2a² of fractal structures. (e) The Feigenbaum constants are expressed as functions of the golden mean. ( f) The quantum mechanical constants fine structure constant and ratio of proton mass to electron mass, which are pure numbers and are obtained by experimental observations only, are now derived in terms of the Feigenbaum constant, a. (g) Atmospheric flow structure follows Keplers third law of planetary motion. Therefore, Newtons inverse square law for gravitation also applies to eddy masses. The centripetal acceleration representing the inertial masses (of eddies) are equivalent to gravitational masses. The fractal-Cantorian structure of spacetime can also be visualized as a nested continuum of vortex (eddy) circulations, whose inertial masses obey Newtons inverse square law of gravitation. The model concept resembles a superstring model for subatomic dynamics which incorporates gravitational forces.     

Rotations and Hermitian symmetric spaces

On an almost Hermitian manifold (M, g, J) one considers the naturally defined field of local diffeomorphismsj _m =exp _m J _m exp _m ^–1 ,mM, and in particular, one studies isometric, harmonic, holomorphic and symplecticj _m . This leads to some characterizations of special classes of almost Hermitian manifolds, including the class of Hermitian symmetric spaces. In addition, one treats some intrinsic and extrinsic geometrical properties of geodesic spheres relating to these local diffeomorphisms.Supported by grant 203.01.50 of the C.N.R., Italy.     

Signatures of Strings

The parity of a permutation π can be defined as the parity of the number of inversions in π. The signature ε(π) of π is an encoding of the parity in a multiplicative group of order 2: ε(π) = (?1)^inv(π). It is also well known that half of the permutations of a finite set are even and half are odd. In this paper, we explore the natural notion of parity for larger moduli; that is, we define the m-signature of a permutation π to be the number of inversions of π, reduced modulo m. Unlike with the 2-signatures of permutations of sets, the m-signatures of the permutations of a multiset are not typically equi-distributed among the modulo m residue classes, though the distribution is close to uniform. We present a recursive method of calculating the distribution of m-signatures of permutations of a multiset, develop properties of this distribution, and present sufficient conditions for the distribution to be uniform.     

Qualitative Analysis and Periodic Cusp Waves to a Class of Generalized Short Pulse Equations

In this paper, we qualitatively study periodic cusp waves to a class of generalized short pulse equations, which are of the general form of three special generalized short pulse equations, from the perspective of dynamical systems. We show the existence of smooth periodic waves, periodic cusp wave and compactons, obtain exact expression of periodic cusp wave and illustrate the limiting process of periodic cusp wave from smooth periodic waves.     

Strings of consecutive successes

A sequence of independent Bernoulli trials is conducted. Let Nbe the number of trials required before observing a string of m successes. Using only elementary methods of probability theory, the expected value, variance and probability generating function of N are determined as functions of m and p, the probability of success on any given trial.     

On conjugate complex time III: Superstrings and complex Lorentz transformation

Starting with El Naschie's ideas of complex conjugate time the inverse Lorentz transformation of time rate is extended onto an internal gravity (artificially induced accelerations) and then expanded onto an external gravitational field. This result raises the total number of physically identifiable distinct dimensions to six. It also implies that the physical reality we live in looks like abstract multispatial geometric structure composed of several single 3D spaces. This conclusion suggests that there are few more physical dimensions to be uncovered. In such a multispatial hyperspace any particle pictured as a point in linear vector space could be viewed as a string or even as a surface in a space that is dual to the primary one.     

Rotations by roots of unity and Diophantine approximation

For a fixed integer n, we study the question whether at least one of the numbers \(\mathfrak {R}X\omega ^k\), \(1\le k\le n\), is \(\varepsilon \)-close to an integer, for any possible value of \(X\in \mathbb {C}\), where \(\omega \) is a primitive nth root of unity. It turns out that there is always an X for which the above numbers are concentrated around \(1/2\,\mathrm{mod}\,1\). The shortest possible interval centered at 1 / 2 containing the fractional parts of all numbers \(\mathfrak {R}X\omega ^k\) depends only on the prime factors of n, rather than its magnitude. This is directly related to the so–called “pyjama” problem which was solved recently.