NONLINEAR STABILITY ANALYSIS OF THE EMDEN–FOWLER EQUATION

In this paper, we qualitatively study radial solutions of the semilinear elliptic equation Δu+uⁿ = 0 with u(0) = 1 and u′(0) = 0 on the positive real line, called the Emden–Fowler or Lane–Emden equation. This equation is of great importance in Newtonian astrophysics and the constant n is called the polytropic index.

By introducing a set of new variables, the Emden–Fowler equation can be written as an autonomous system of two ordinary differential equations which can be analyzed using linear and nonlinear stability analysis. We perform the study of stability by using linear stability analysis, the Jacobi stability analysis (Kosambi–Cartan–Chern-theory) and the Lyapunov function method. Depending on the values of n these different methods yield different results. We identify a parameter range for n where all three methods imply stability.     

CONSERVATION LAWS FOR THE SCHRÖDINGER–NEWTON EQUATIONS

In this Letter a first-order Lagrangian for the Schrödinger–Newton equations is derived by modifying a second-order Lagrangian proposed by Christian [Exactly soluble sector of quantum gravity, Phys. Rev. D 56(8) (1997) 4844–4877]. Then Noether's theorem is applied to the Lie point symmetries determined by Robertshaw and Tod [Lie point symmetries and an approximate solution for the Schrödinger–Newton equations, Nonlinearity 19(7) (2006) 1507–1514] in order to find conservation laws of the Schrödinger–Newton equations.     

LEPAGE EQUIVALENTS OF SECOND-ORDER EULER–LAGRANGE FORMS AND THE INVERSE PROBLEM OF THE CALCULUS OF VARIATIONS

In the calculus of variations, Lepage (n + 1)-forms are closed differential forms, representing Euler–Lagrange equations. They are fundamental for investigation of variational equations by means of exterior differential systems methods, with important applications in Hamilton and Hamilton–Jacobi theory and theory of integration of variational equations. In this paper, Lepage equivalents of second-order Euler–Lagrange quasi-linear PDE's are characterised explicitly. A closed (n + 1)-form uniquely determined by the Euler–Lagrange form is constructed, and used to find a geometric solution of the inverse problem of the calculus of variations.     

A YANG–MILLS ELECTRODYNAMICS THEORY ON THE HOLOMORPHIC TANGENT BUNDLE

Considering a complex Lagrange space ([24]), in this paper the complex electromagnetic tensor fields are defined as the sum between the differential of the complex Liouville 1-form and the symplectic 2-form of the space relative to the adapted frames of the Chern–Lagrange complex nonlinear connection. In particular, an electrodynamics theory on a complex Finsler space is obtained.

We show that our definition of the complex electrodynamics tensors has physical meaning and these tensors generate an adequate field theory which offers the opportunity of coupling with the gravitation. The generalized complex Maxwell equations are written.

A gauge field theory of electrodynamics on the holomorphic tangent bundle is put over T′M and the gauge invariance to phase transformations is studied. An extension of the Dirac Lagrangian on T′M coupled with the electrodynamics Lagrangian is studied and it offers the framework for a unified gauge theory of fields.     

2D LOCUS CONFIGURATIONS AND THE TRIGONOMETRIC CALOGERO–MOSER SYSTEM

A central hyperplane arrangement in ?² with multiplicity is called a "locus configuration" if it satisfies a series of "locus equations" on each hyperplane. Following [4], we demonstrate that the first locus equation for each hyperplane corresponds to a force-balancing equation on a related interacting particle system on ?*: the charged trigonometric Calogero-Moser system. When the particles lie on S¹ ? ?*, there is a unique equilibrium for this system. For certain classes of particle weight, this is enough to show that all the locus equations are satisfied, producing explicit examples of real locus configurations. This in turn produces new examples of Schrödinger operators with Baker–Akhiezer functions.     

PHASE SEPARATION IN THE MELT OF POLYPROPYLENE–1-DECENE COPOLYMERS

A propylene homopolymer and three copolymers with 1-decene containing 1.82, 3.55, and 7.83 mol% of comonomer units, respectively, were prepared with metallocene catalyst and the phase behavior in the melt of these polymers was studied using simultaneous synchrotron small-angle x-ray scattering and differential scanning calorimetry. The results show that the phase behavior of the melt varies with comonomer content and the copolymers tend to be phase-separated with increasing comonomer content. The phase separation in the melt of the propylene–1-decene copolymers was further confirmed by the fitting of the experimental data with Teubner–Strey micro-emulsion model and a transition from the phase-separated melt to the disordered melt was observed. We tentatively attribute this phenomenon to the inhomogeneous intra-molecular composition distribution at high levels of comonomer and the incompatibility of propylene sequences with different lengths. Such a finding is consistent with the facts of multiple melting peaks and mixed γ and α crystal phases in the polymer solids. Since the phase behavior depends on temperature, the effect of annealing temperature, from which the copolymers were cooled, on the nonisothermal crystallization of the copolymers was also investigated. It is found that higher degree of phase separation accelerates the crystallization but reduces the crystallization enthalpy.     

HOW REYE'S CONFIGURATION HELPS IN PROVING THE BELL–KOCHEN–SPECKER THEOREM: A CURIOUS GEOMETRICAL TALE

It is shown that the 24 quantum states or rays used by Peres (J. Phys. A 24, 174-8 (1991)) to give a proof of the Bell–Kochen–Specker (BKS) theorem have a close connection with Reye's configuration, a system of twelve points and sixteen lines known to projective geometers for over a century. The interest of this observation stems from the fact that it provides a ready explanation for many of the regularities exhibited by the Peres rays and also permits a systematic construction of all possible non-coloring proofs of the BKS theorem based on these rays. An elementary exposition of the connection between the Peres rays and Reye's configuration is given, following which its applications to the BKS theorem are discussed.     

INVARIANCE OF THE KAUP–KUPERSHMIDT EQUATION AND TRIANGULAR AUTO-BÄCKLUND TRANSFORMATIONS

We report triangular auto-Bäcklund transformations for the solutions of a fifth-order evolution equation, which is a constraint for an invariance condition of the Kaup–Kupershmidt equation derived by E. G. Reyes in his paper titled "Nonlocal symmetries and the Kaup–Kupershmidt equation" [J. Math. Phys. 46 (2005) 073507, 19 pp.]. These auto-Bäcklund transformations can then be applied to generate solutions of the Kaup–Kupershmidt equation. We show that triangular auto-Bäcklund transformations result from a systematic multipotentialization of the Kupershmidt equation.     

EXACT SOLUTIONS OF THE MODIFIED GROSS–PITAEVSKII EQUATION IN "SMART" PERIODIC POTENTIALS IN THE PRESENCE OF EXTERNAL SOURCE

We report wide class of exact solutions of the modified Gross–Pitaevskii equation (GPE) in "smart" Jacobian elliptic potentials, in the presence of external source. Solitonlike solutions, singular solutions, and periodic solutions are found using a recently developed fractional transform in which all the amplitude parameters are nonzero. These results generalize those contained in (T. Paul, K. Richter and P. Schlagheck, Phys. Rev. Lett. 94 (2005) 020404.) for nonzero trapping potential.     

THE AB INITIO CALCULATION FOR THE BONDING STATE OF THE Hn CTUSTERS

The equilibrium geometries of several Hn（n=4,5,6,7,9,13) clsters with certain symmetry have been optimized by the ab inito Murtaug-Sargent(MS) method.We find that the stability of a centered cluster depends on the structure,the spin symmetry and the coordinate number,and the latter plays a more important role.     

A NOTE ON TRAVELING WAVE SOLUTIONS TO THE TWO COMPONENT CAMASSA–HOLM EQUATION

In this paper we show that non-smooth functions which are distributional traveling wave solutions to the two component Camassa–Holm equation are distributional traveling wave solutions to the Camassa–Holm equation provided that the set u^-1(c), where c is the speed of the wave, is of measure zero. In particular there are no new peakon or cuspon solutions beyond those already satisfying the Camassa–Holm equation. However, the two component Camassa–Holm equation has distinct from Camassa–Holm equation smooth traveling wave solutions as well as new distributional solutions when the measure of u^-1(c) is not zero. We provide examples of such solutions.     

THE MACQM CALCULATION OF THE TOTAL ENERGY CURVE OF THE BODY-CENTERED CUBIC STRUCTURE OF THE H_9~- CLUSTER

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ON THE IRREDUCIBLE REPRESENTATIONS OF THE SIMPLE LIE GROUPS I——THE TENSOR BASIS FOR THE INFINITESIMAL GENERATORS OF THE CLASSICAL SIMPLE LIE GROUPS

In this paper, we analyse the commutation relations of the infinitesimal generatorsof all simple classical Lie groups and establish a new basis for these generators, calledthe tensor basis. In tensor basis, the infinitesimal, generators can be written as somescalar operators, some sets of angular momentum operators and some sets of irreducibletensor operators. The commutation relations, of these operators are very simple andhave many regularities. By means of the method that has been used in the earlier papers, "On the irre-ducible representations of the compact simple Lie groups of rank 2, I,II,III" and thetensor basis, all the irreducible representations of the classical simple Lie groups canbe calculated systematically.     

THE MACQM CALCULATION OF THE TOTAL ENERGY CURVE FOR THE ICOSAHEDRAL CENTRAL STRUCTURE OF THE CLUSTER H_13~-

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GLOBAL EXISTENCE AND BLOW-UP PHENOMENA FOR THE PERIODIC HUNTER–SAXTON EQUATION WITH WEAK DISSIPATION

In this paper, we study the periodic Hunter–Saxton equation with weak dissipation. We first establish the local existence of strong solutions, blow-up scenario and blow-up criteria of the equation. Then, we investigate the blow-up rate for the blowing-up solutions to the equation. Finally, we prove that the equation has global solutions.     

THE ACOUSTIC FUNCTIONS OF THE MELON OF THE CHINESE RIVER DOLPHIN(Lipotes vexillifer)

In this paper,the acoustic functions of the melon of the Chinese river dolphin(Lipotes vexillifer)are studied.Based on the comparison between the directivity patterns of the whole head and the skull,it is suggested that the melon plays an important role in beamforming.At the frequencuy of 40 kHz,the—3 dB beamwidth of the emitted vertical beam of the whole head is approximately 21.6°,withthe major axis elevated 10°above the ristrum plane;the—3 dB beamwidth of the horizontal beam ofthe whole head is approximately 20°,with the major axis deviated 12°rightward from the body axis.The vertical and horizontal—3dB beamwidths of the whole head are 9.4°and 13°,narrower than thoseof the skull respectively,and the directions of the major axes are different for both cases.It canbe seen that the melon narrows the width of the main beam and greatly reduces the side lobes of thebeam formed by the skull.The distribution of the sound velocity in the melon tissue is measured andthe distribution pattern is drawn.Fr     

ON THE CONTROL OF RINGING NOISE EXCITED BY IMPACT II:THE ENERGY OF RINGING NOISE AND THE BUFFERING OF THE IMPACT

In the first of these two papers,the expression for the total energy of vibration excited by impacthas been obtained.In this paper the energy equation of ringing noise was derived from the resultof the first paper.This energy equation indicates the approaches to control ringing noise.Thebuffering method was discussed here in detail.The experiment which has been conducted was describedalso.     

THE ι-E PUZZLE AND THE GENERALIZED MOMENT ANALYSIS

We have introduced the photon angular distribution for moments of the process e⁺e^－→J/ψ→γB,B→P₁P₂P₃ to discuss the ι-E puzzle and got an effective criterion to distinguish whether there exist E(1⁺⁺) component in the peak for the process e⁺e^－→J/ψ→γKKπ.     

DOES PARISI'S SOLUTION OF THE SHERRINGTON-KIRKPATRICK MODEL LOCATE ON THE ABSOLUTEMAXIMUM OF THE FREE ENERGY?

In the spin glass phase near the critical temperature we find in aparticular replica symmetry breaking pattern an order parameter with free energy greater than that of the Parisi's.