Reflection Symmetry

Nearly a thousand years ago the Dine people,ancestors of today's Navajo in the Southwest United States,learned to weave on looms from the neighboring Pueblo Indians:If you fold a blanket along a horizontal line through its center,the top and bottom halves match.     

The Degree of Symmetry of Certain Compact Smooth Manifolds Ⅱ

We give the sharp estimates for the degree of symmetry and the semi-simple degree of symmetry of certain compact fiber bundles with non-trivial four dimensional fibers in the sense of cobordism, by virtue of the rigidity theorem of harmonic maps due to Schoen and Yau (Topology, 18, 1979, 361-380). As a corollary of this estimate, we compute the degree of symmetry and the semi-simple degree of symmetry of CP2×V, where V is a closed smooth manifold admitting a real analytic Riemannian metric of non-positive curvature. In addition, by the Albanese map, we obtain the sharp estimate of the degree of symmetry of a compact smooth manifold with some restrictions on its one dimensional cohomology.     

Symmetry Reductions of the Combined KdV—mKdV Equation

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Symmetry analysis of an integrable reaction–diffusion equation

In this paper, we investigate the invariance and integrability properties of an integrable two-component reaction–diffusion equation. We perform Painlevé analysis for both the reaction–diffusion equation modelled by a coupled nonlinear partial differential equations and its general similarity reduced ordinary differential equation and confirm its integrability. Further, we perform Lie symmetry analysis for this model. Interestingly our investigations reveals a rich variety of particular solutions, which have not been reported in the literature, for this model.     

Symmetry and Classification of the Dirac–Fock Equation

We consider the properties of the Dirac–Fock equation with differential operators of the first-order symmetry. For a relativistic particle in an electromagnetic field, we describe the covariant properties of the Dirac equation in an arbitrary Riemannian space V₄ with the signature (?1,?1,?1, 1). We present a general form of the differential operator with a first-order symmetry and characterize the pair of such commuting operators. We list the spaces where the free Dirac equation admits at least one differential operator with a first-order symmetry. We perform a symmetry classification of electromagnetic field tensors and construct complete sets of symmetry operators.     

Symmetry algebra of the AdS4×ℂℙ3 superstring

By direct calculation in the classical theory, we derive the central extension of the off-shell symmetry algebra for a string propagating in AdS ₄ ×?? ³ . It turns out to be the same as in the case of the AdS ₅ ×S ⁵ string. We consider the choice of the κ-symmetry gauge in detail and also explain how this gauge can be chosen without breaking the bosonic symmetries.     

Symmetry properties of Cayley graphs of small valencies on the alternating group A_5

The normality of symmetry property of Cayley graphs of valencies 3 and 4 on the alternating group A5 is studied. We prove that all but four such graphs are normal; that A5 is not 5-CI. A complete classification of all arc-transitive Cayley graphs on A5 of valencies 3 and 4 as well as some examples of trivalent and tetravalent GRRs of A5 is given.     

Invariant Measures and Symmetry Property of Lévy Type Operators

Second order elliptic integro-differential operators (Lévy type operators) are investigated. The notion of regular (infinitesimal) invariant probability measures for such operators is posed. Sufficient conditions for the existence of such regular infinitesimal invariant probability measures are obtained and the symmetrization problem is discussed.     

Symmetry Reduction and Exact Solutions of a Hyperbolic Monge-Ampère Equation

By means of the classical symmetry method,a hyperbolic Monge-Ampère equation is investigated.The symmetry group is studied and its corresponding group invariant solutions are constructed.Based on the a...     

Exponents, Symmetry Groups and Classification of?Operator Fractional Brownian Motions

Operator fractional Brownian motions (OFBMs) are zero mean, operator self-similar (o.s.s.) Gaussian processes with stationary increments. They generalize univariate fractional Brownian motions to the multivariate context. It is well-known that the so-called symmetry group of an o.s.s. process is conjugate to subgroups of the orthogonal group. Moreover, by a celebrated result of Hudson and Mason, the set of all exponents of an operator self-similar process can be related to the tangent space of its symmetry group.     

Symmetry and regularity of extremals of an integral equation related to the Hardy�CSobolev inequality

Let ?? be a real number satisfying 0?<????<?n, ${0\leq t<\alpha, \alpha{^\ast}(t)=\frac{2(n-t)}{n-\alpha}}$ . We consider the integral equation $$u(x)=\int\limits_{{\mathbb{R}^n}}\frac{u^{{\alpha{^\ast}(t)}-1}(y)}{|y|^t|x-y|^{n-\alpha}}\,dy,\quad\quad\quad\quad\quad\quad\quad(1)$$ which is closely related to the Hardy?CSobolev inequality. In this paper, we prove that every positive solution u(x) is radially symmetric and strictly decreasing about the origin by the method of moving plane in integral forms. Moreover, we obtain the regularity of solutions to the following integral equation $$u(x)=\int\limits_{{\mathbb{R}^n}}\frac{|u(y)|^{p}u(y)}{|y|^t|x-y|^{n-\alpha}}\, dy\quad\quad\quad\quad\quad\quad\quad(2)$$ that corresponds to a large class of PDEs by regularity lifting method.     

Symmetry Properties of Positive Solutions of Parabolic Equations on ℝ N : II. Entire Solutions

We consider nonautonomous quasilinear parabolic equations satisfying certain symmetry conditions. We prove that each positive bounded solution u on ? ^N  × (?∞, T) decaying to zero at spatial infinity uniformly with respect to time is radially symmetric around some origin in ? ^N . The origin depends on the solution but is independent of time. We also consider the linearized equation along u and prove that each bounded (positive or not) solution is a linear combination of a radially symmetric solution and (nonsymmetric) spatial derivatives of u. Theorems on reflectional symmetry are also given.     

Comment on “Symmetry breaking of systems of linear second-order ordinary differential equations with constant coefficients”

The present paper corrects the way of using Jordan canonical forms for studying the symmetry structures of systems of linear second-order ordinary differential equations with constant coefficients applied in [1]. The approach is demonstrated for a system consisting of two equations.     

Quantum Cohomology of a Pfaffian Calabi–Yau Variety: Verifying Mirror Symmetry Predictions

We formulate a generalization of Givental–Kim's quantum hyperplane principle. This is applied to compute the quantum cohomology of a Calabi–Yau 3-fold defined as the rank 4 locus of a general skew-symmetric 7×7 matrix with coefficients in P ⁶. The computation verifies the mirror symmetry predictions of Rødland [25].     

Binary Nonlinearization of the Nonlinear Schr?dinger Equation Under an Implicit Symmetry Constraint

By modifying the procedure of binary nonlinearization for the AKNS spectral problem and its adjoint spectral problem under an implicit symmetry constraint,we obtain a finite dimensional system from the Lax pair of the nonlinear Schr¨odinger equation.We show that this system is a completely integrable Hamiltonian system.     

Symmetry and monotonicity of positive solutions to Schr?dinger systems with fractional p-Laplacians

In this paper,we first establish narrow region principle and decay at infinity theorems to extend the direct method of moving planes for general fractional p-Laplacian systems.By virtue of this method,we investigate the qualitative properties of positive solutions for the following Schrodinger system with fractional p-Laplacian{(-△)^s_pu+au^p-1=f(u,v),(-△)^t_pv+bv(p-1)=g(u,v),where 0N(N≥2),the monotonicity in the parabolic domain and the nonexistence on the half space for positive solutions to the above system under some suitable conditions on f and g,respectively.     

Symmetry properties of two-dimensional Ciarlet–Mooney–Rivlin constitutive models in nonlinear elastodynamics

Nonlinear dynamic equations for isotropic homogeneous hyperelastic materials are considered in the Lagrangian formulation. An explicit criterion of existence of a natural state for a given constitutive law is presented, and is used to derive natural state conditions for some common constitutive relations.For two-dimensional planar motions of Ciarlet–Mooney–Rivlin solids, equivalence transformations are computed that lead to a reduction of the number parameters in the constitutive law. Point symmetries are classified in a general dynamical setting and in traveling wave coordinates. A special value of traveling wave speed is found for which the nonlinear Ciarlet–Mooney–Rivlin equations admit an additional infinite set of point symmetries. A family of essentially two-dimensional traveling wave solutions is derived for that case.     

Symmetry of some entire solutions to the Allen–Cahn equation in two dimensions

In this paper, we prove even symmetry and monotonicity of certain solutions of Allen–Cahn equation in a half plane. We also show that entire solutions with finite Morse index and four ends must be evenly symmetric with respect to two orthogonal axes. A classification scheme of general entire solutions with finite Morse index is also presented using energy quantization.     

Symmetry Breaking and Restoration in the Ginzburg–Landau Model of Nematic Liquid Crystals

In this paper we study qualitative properties of global minimizers of the Ginzburg–Landau energy which describes light–matter interaction in the theory of nematic liquid crystals near the Fréedericksz transition. This model depends on two parameters: \(\epsilon >0\) which is small and represents the coherence scale of the system and \(a\ge 0\) which represents the intensity of the applied laser light. In particular, we are interested in the phenomenon of symmetry breaking as a and \(\epsilon \) vary. We show that when \(a=0\) the global minimizer is radially symmetric and unique and that its symmetry is instantly broken as \(a>0\) and then restored for sufficiently large values of a. Symmetry breaking is associated with the presence of a new type of topological defect which we named the shadow vortex. The symmetry breaking scenario is a rigorous confirmation of experimental and numerical results obtained earlier in Barboza et al. (Phys Rev E 93(5):050201, 2016).