Measures of axial symmetry for ovals

A measure of axial symmetry for ovals is defined, and eleven particular measures are studied. Lower bounds for these measures are determined on the classes of arbitrary ovals, centrally symmetric ovals, and ovals of constant breadth. The proofs of these results make use only of elementary geometry and the properties of convexity.     

On replica symmetry of large deviations in random graphs

The following question is due to Chatterjee and Varadhan (2011). Fix and take , the Erd?s‐Rényi random graph with edge density p, conditioned to have at least as many triangles as the typical . Is G close in cut‐distance to a typical ? Via a beautiful new framework for large deviation principles in , Chatterjee and Varadhan gave bounds on the replica symmetric phase, the region of where the answer is positive. They further showed that for any small enough p there are at least two phase transitions as r varies. We settle this question by identifying the replica symmetric phase for triangles and more generally for any fixed d‐regular graph. By analyzing the variational problem arising from the framework of Chatterjee and Varadhan we show that the replica symmetry phase consists of all such that lies on the convex minorant of where is the rate function of a binomial with parameter p. In particular, the answer for triangles involves rather than the natural guess of where symmetry was previously known. Analogous results are obtained for linear hypergraphs as well as the setting where the largest eigenvalue of is conditioned to exceed the typical value of the largest eigenvalue of . Building on the work of Chatterjee and Diaconis (2012) we obtain additional results on a class of exponential random graphs including a new range of parameters where symmetry breaking occurs. En route we give a short alternative proof of a graph homomorphism inequality due to Kahn (2001) and Galvin and Tetali (2004). © 2014 Wiley Periodicals, Inc. Random Struct. Alg., 47, 109–146, 2015     

Buckling and vibration of the two-dimensional quasicrystal cylindrical shells under axial compression

In this paper, the buckling and the free vibration of the quasicrystal cylindrical shells under axial compression are investigated. Three quasi-periodicity cases of quasicrystal cylindrical shells are considered. The first-order shear displacement theory of the cylindrical shells is utilized to obtain the equations of motion and the boundary conditions. Numerical results for simply supported cylindrical shells at the two ends are calculated. The effects of the geometry, in-plane phonon and phason loads, and half-wave number of the quasicrystal cylindrical shells on both the buckling loads and the frequency are demonstrated.     

On instability under small periodic perturbation of two-dimensional hyperbolic systems

Under consideration is a mixed problem for a strictly hyperbolic linear first-order system in the half-strip Π = {(x, t): 0 < x < 1, t > 0} generating a group of unitary operators. In the case of a periodic perturbation a method is proposed for finding the frequencies for which the perturbed system develops parametric resonance. The method is illustrated with a system of two equations.     

Portfolio diversification under local and moderate deviations from power laws

This paper analyzes portfolio diversification for nonlinear transformations of heavy-tailed risks. It is shown that diversification of a portfolio of convex functions of heavy-tailed risks increases the portfolio’s riskiness if expectations of these risks are infinite. In contrast, for concave functions of heavy-tailed risks with finite expectations, the stylized fact that diversification is preferable continues to hold. The framework of transformations of heavy-tailed risks includes many models with Pareto-type distributions that exhibit local or moderate deviations from power tails in the form of additional slowly varying or exponential factors. The class of distributions under study is therefore extended beyond the stable class.     

Q-conditional symmetry of a nonlinear two-dimensional heat-conduction equation

We investigate theQ-conditional symmetry of a nonlinear two-dimensional heat-conduction equation. By using ansatzes, we obtain reduced equations. Poltava University, Poltava. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 52, No. 6, pp.846–849, June, 2000.     

On probabilities of small deviations for compound cox processes

We derive logarithmic asymtotics for probabilities of small deviations for compound Cox processes. We show that under appropriate conditions, these asymptotics are the same as those for sums of independent random variables and processes with independent increments. When these conditions do not hold, the asymptotics of small deviations for compound Cox processes are quite different. Bibliography: 9 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 339, 2006, pp. 163–175.     

Abelian Lie symmetry algebras of two-dimensional quasilinear evolution equations

We carry out the classification of abelian Lie symmetry algebras of two-dimensional second-order nondegenerate quasilinear evolution equations. It is shown that such an equation is linearizable if it admits an abelian Lie symmetry algebra that is of dimension greater than or equal to 5 or of dimension greater than or equal to 3 with rank-one.     

Isotropic model of quadrupole glass. II. Breaking of replica symmetry

In a model of an isotropic quadrupole glass with two order parameters, a macroscopically large number of metastable states is found at zero temperature. The replica symmetry is broken in the immediate vicinity of the transition to the glass state, and it is shown that the obtained solution is stable.L. F. Vershchagin Institute of High Pressure Physics, USSR Academy of Sciences. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 91, No. 1, pp. 157–167, April, 1992.     

The small free oscillations of a strip

The refined equations of the free oscillations of a rod-strip, constructed previously in a first approximation by reducing the two-dimensional equations to one-dimensional equations by using trigonometric basis functions and satisfying the static boundary conditions on the boundary surfaces are analysed. These equations, the solutions of which are obtained for the case of hinge-supported end sections of the rod, are split into two independent systems of equations. The first of these describe non-classical fixed longitudinal-transverse forms of free oscillations, which are accompanied by a distortion of the plane form of the cross section. It is shown that the oscillation frequencies corresponding to them depend considerably on Poisson's ratio and the modulus of elasticity in the transverse direction, while for a rod of average thickness for the same value of the frequency parameter (the tone) they may be considerably lower than the frequencies corresponding to the classical longitudinal forms of free oscillations, which are performed while preserving the plane form of the cross sections. The second system of equations describes transverse flexural-shear forms of free oscillations, whose frequencies decrease as the transverse shear modulus decreases. They are practically equivalent in quality and content to the similar equations of well-known versions of the refined theories, but, unlike them, when the number of the tone increases and the relative thickness parameter decreases they lead to the solutions of the classical theory of rods.     

Quark propagator and spontaneous breaking of chiral symmetry in axial Gauge

An effective nonperturbative model of QCD in the axial gauge based on singular behavior of the gluon propagator of the type 1/k ⁴ is studied. An approximate solution for the quark propagator corresponding to spontaneous breaking of the chiral symmetry is found. The chiral quark condensate is calculated by means of the obtained solutions.Institute of High Energy Physics, Serpukhov. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 90, No. 1, pp. 65–74, January, 1992.     

Rigid-plastic model of a fibrous composite: Problems involving axial symmetry

A rigid-plastic transversally isotropic medium with the additional kinematic assumption of inextensibility along the symmetry axis was examined as a model of a fibrous composite with a metal matrix. A complete system of equations for the axially symmetric deformation of such a body was obtained. In contrast to planar deformation examined in our previous work [5, 6], the axial symmetry problem is not locally statically determined, which prevents the separate analysis of the stress and velocity fields in the general case. The principal simplifications in formulating the equation system may be obtained for special cases of the stress state. One such simplifying assumption is extension of the axial conditions _r= or _rz=0 over the entire body. This hypothesis yields a separate system of equations for stresses. On the other hand, the common assumption of complete plasticity for an isotropic body when two principal stresses are equal does not provide fundamental simplification due to the lack of coincidence between the directions of the major stresses and the symmetry axes. An analytical solution was obtained for the axially symmetric problem of a plate with a round hole. A special feature of this problem is that the type of system of equations changes in the plastic region. The stress distribution in the circular hole was found to depend significantly on the type of equation system.M. V. Lomonosov Moscow State University, Russia. Translated from Mekhanika Kompozitnykh Materialov, Vol. 33, No. 3, pp. 300–311, May–June, 1997.     

High-order numerical solution of the nonlinear Helmholtz equation with axial symmetry

The nonlinear Helmholtz (NLH) equation models the propagation of intense laser beams in a Kerr medium. The NLH takes into account the effects of nonparaxiality and backward scattering that are neglected in the more common nonlinear Schrödinger model. In [G. Fibich, S. Tsynkov, High-order two-way artificial boundary conditions for nonlinear wave propagation with backscattering, J. Comput. Phys., 171 (2001) 632–677] and [G. Fibich, S. Tsynkov, Numerical solution of the nonlinear Helmholtz equation using nonorthogonal expansions, J. Comput. Phys., 210 (2005) 183–224], a novel high-order numerical method for solving the NLH was introduced and implemented in the case of a two-dimensional Cartesian geometry. The NLH was solved iteratively, using the separation of variables and a special nonlocal two-way artificial boundary condition applied to the resulting decoupled linear systems. In the current paper, we propose a major improvement to the previous method. Instead of using LU decomposition after the separation of variables, we employ an efficient summation rule that evaluates convolution with the discrete Green's function. We also extend the method to a three-dimensional setting with cylindrical symmetry, under both Dirichlet and Sommerfeld-type transverse boundary conditions.