A note on sign symmetry for a subclass of efficient,symmetric, and linear values

Symmetry requires that equally productive players should get the same payoff. Sign symmetry is a considerable weakening of symmetry, which requires equally productive players' payoffs to have the same sign. In this note, we apply sign symmetry, as a substitute of symmetry to characterize a subclass of efficient, symmetric, and linear values.     

The symmetry group of a finite frame

We define the symmetry group of a finite frame as a group of permutations on its index set. This group is closely related to the symmetry group of Vale and Waldron (2005) [12] for tight frames: they are isomorphic when the frame is tight and has distinct vectors. The symmetry group is the same for all similar frames, in particular for a frame, its dual and canonical tight frames. It can easily be calculated from the Gramian matrix of the canonical tight frame. Further, a frame and its complementary frame have the same symmetry group. We exploit this last property to construct and classify some classes of highly symmetric tight frames.     

Symmetry reductions and explicit solutions of a (3 + 1)-dimensional PDE

The symmetry of the (3 + 1)-dimensional partial differential equation has been derived via a direct symmetry method and proved to be infinite dimensional non-Virasoro type symmetry algebra. Many kinds of symmetry reductions have been obtained, including the (2 + 1)-dimensional ANNV equation and breaking soliton equation. And some new soliton solutions and complex solutions are obtained due to the Riccati equation method and symbolic computation.     

Distributing points uniformly on the unit sphere under a mirror reflection symmetry constraint

Uniformly distributed point sets on the unit sphere with and without symmetry constraints have been found useful in many scientific and engineering applications. Here, a novel variant of the Thomson problem is proposed and formulated as an unconstrained optimization problem. While the goal of the Thomson problem is to find the minimum energy configuration of N electrons constrained on the surface of the unit sphere, this novel variant imposes a new symmetry constraint – mirror reflection symmetry with the x–y plane as the plane of symmetry. Qualitative features of the two-dimensional projection of the optimal configurations are briefly mentioned and compared to the ground-state configurations of the two dimensional system of charged particles laterally confined by a parabolic potential well.     

Massless Elementary Particles in a Quantum Theory over a Galois Field

We consider massless elementary particles in a quantum theory based on a Galois field (GFQT). We previously showed that the theory has a new symmetry between particles and antiparticles, which has no analogue in the standard approach. We now prove that the symmetry is compatible with all operators describing massless particles. Consequently, massless elementary particles can have only half-integer spin (in conventional units), and the existence of massless neutral elementary particles is incompatible with the spin–statistics theorem. In particular, this implies that the photon and the graviton in the GFQT can only be composite particles.     

Toward an Infinite-Component Field Theory with a Double Symmetry: Interaction of Fields

We complete the first stage of constructing a theory of fields not investigated before; these fields transform according to Lorentz group representations decomposable into an infinite direct sum of finite-dimensional irreducible representations. We consider only those theories that initially have a double symmetry: relativistic invariance and the invariance under the transformations of a secondary symmetry generated by the polar or the axial four-vector representation of the orthochronous Lorentz group. The high symmetry of the theory results in an infinite degeneracy of the particle mass spectrum with respect to spin. To eliminate this degeneracy, we postulate a spontaneous secondary-symmetry breaking and then solve the problems on the existence and the structure of nontrivial interaction Lagrangians.     

Stereochemical studies on cyclic peptides—VIII. Conformational analysis of hydrogen bonded cyclohexaglycyl molecule with a centre of inversion symmetry

A study on the conformational aspects of cyclo-hexaglycyl having inversion symmetry has been made. The cyclic backbone has been assumed to have two internal 4→1 types of NH... O hydrogen bonds. This molecule has been found to take up two types of conformations designated asA* andB* having nearly the same energy values. The theoretical conformations have been compared with the conformations of cyclohexaglycyl hemihydrate observed in the crystal structure. Two molecules with an approximate inversion symmetry are close to the conformation of the typeB* and two other molecules with exact inversino symmetry correspond nearly to the typesB* andA*. comparison with the theoretically possible conformations of cyclohexaglycyl molecule with 2-fold symmetry has been made. The preference of inversion symmetry and preferred ranges ofψ for glycyl molecules is discussed.     

How to share the cost of a public good?

The paper deals with a simple model with only one private and one public good. The core of such an economy is shown to have strong properties, in particular, it is stable in thevon Neumann-Morgenstern sense and complete. A natural candidate for a selection within the core is the concept ofLindahl equilibrium which constitutes a generalization of the concept of competitive equilibrium in an economy with private goods only. Although theLindahl equilibria belong in general to the core, they do not have the same symmetry properties. It is shown that it is possible to select in the core allocations having stronger symmetry properties than theLindahl equilibria.     

Trajectory Planning for a Non-Flat Mechanical System Using Time-Reversal Symmetry

Some mechanical systems have a special discrete symmetry property associated with the inversion of the direction of the time flow. This "time-reversal symmetry" allows one to construct solutions of the equations of motion from other known solutions. In the present contribution we examine to which systems this property can be applied. Furthermore, we demonstrate how it can be used to determine an open loop control for equilibrium point transition by solving a stabilization problem in the negative time direction. As a non-flat mechanical example the popular pole-cart system is considered. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Approximate Noether-type symmetries and conservation laws via partial Lagrangians for PDEs with a small parameter

We show how one can construct approximate conservation laws of approximate Euler-type equations via approximate Noether-type symmetry operators associated with partial Lagrangians. The ideas of the procedure for a system of unperturbed partial differential equations are extended to a system of perturbed or approximate partial differential equations. These approximate Noether-type symmetry operators do not form a Lie algebra in general. The theory is applied to the perturbed linear and nonlinear (1+1) wave equations and the Maxwellian tails equation. We have also obtained new approximate conservation laws for these equations.     

Analytic construction of the near field in a transonic flow near a thin axisymmetric body

A solution of the axisymmetric problem of unsteady transonic flow around thin bodies of revolution is proposed in the form of a double series expansion in powers of the distance to the axis of symmetry and its logarithm in a neighborhood of a given point at the symmetry axis. Chains of recurrence equations are obtained for the coefficients of the series. The convergence of the constructed series is proved by the method of special majorants. The theorem of existence and uniqueness of the solution to the boundary-value problem for a nonlinear partial differential equation with a singularity at the symmetry axis is obtained in the asymptotic model of unsteady transonic flow under consideration. Thereby the application of the proposed series is justified to the problems of unsteady transonic flow around thin axisymmetric bodies with a drift of the nonpenetration condition onto the symmetry axis. Hence, these series can be used in numerical-analytical methods and model computations.     

The dynamics of a non-uniform spheroid on a horizontal plane

The dynamics of a spheroid with a displaced mass centre on a horizontal plane with friction is considered. It is assumed that the mass centre of the spheroid lies on its dynamic symmetry axis. The steady motions of the spheroid are investigated using the general theory of invariant sets of mechanical systems with symmetry, and a geometrical interpretation of the results is given using generalized Smale diagrams.     

函数图像对称性的一种判定方法

揭示了函数导数的奇偶性与函数图像对称性的关系,发现了一种可以用来判定函数图像是否具有轴对称性或中心对称性的有效方法.     

Green's Function of a Hartree-Type Equation with a Quadratic Potential

Using the complex WKB–Maslov method, we consider a solution of the Cauchy problem for a Hartree-type equation with a quadratic potential in the class of semiclassically supported functions. In this class, we obtain the evolution operator explicitly. We find parametric families of symmetry operators of the Hartree-type equation. Using the symmetry operators, we construct a family of exact solutions of this equation.     

A comparative study of approximate symmetry and approximate homotopy symmetry to a class of perturbed nonlinear wave equations

A comparative study of approximate symmetry and approximate homotopy symmetry to a class of perturbed nonlinear wave equations is performed. First, complete infinite-order approximate symmetry classification of the equation is obtained by means of the method originated by Fushchich and Shtelen. An optimal system of one-dimensional subalgebras is derived and used to construct general formulas of approximate symmetry reductions and similarity solutions. Second, we study approximate homotopy symmetry of the equation and construct connections between the two symmetry methods for the first-order and higher-order cases, respectively. The series solutions derived by the two methods are compared.     

Creep damage modeling of a polycrystalline material

A polycrystalline material is investigated under creep conditions within the framework of continuum micromechanics. Geometrical 3D model of a polycrystalline microstructure is represented as a unit cell with grains of random crystallographical orientation and shape. Thickness of the plains, separating neighboring grains in the unit cell, can have non-zero value. Obtained geometry assigns a special zone in the vicinity of grain boundaries, possessing unordered crystalline structure. A mechanical behavior of this zone should allow sliding of the adjacent grains. Within the grain interior crystalline structure is ordered, what prescribes cubic symmetry of a material. The anisotropic material model with the orthotropic symmetry is implemented in ABAQUS and used to assign elastic and creep behavior of both the grain interior and grain boundary material. Appropriate parameters set allows transition from the orthotropy to the cubic symmetry for the grain interior. Material parameters for the grain interior are identified from creep tests for single crystal copper. Model parameters for the grain boundary are set from the physical considerations and numerical model validation according to the experimental data of the grain boundary sliding in a polycrystalline copper [2]. As the result of analysis representative number of grains and grain boundary thickness in the unit cell are recommended. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

The asymptotic behaviour of the number of three-connected triangulations of the disk,with a reflective symmetry in a line

We consider the enumeration of the three-connected triangulations of the disk, with a reflective symmetry about a line. The asymptotic behavior is unlike that observed for rooted maps or for maps having rotational symmetry.     

Testing for symmetry and independence in a bivariate exponential distribution

Several bivariate exponential distributions have been proposed in the literature. A common problem for independent exponentials is to test the quality of the two distributions. The analogous problem for bivariate exponentials is to test for symmetry. For the bivariate exponential model of Freund (1961, Journal of the American Statistical Association 56, 971–977), tests of symmetry and independence are derived and the small sample distributions of the test statistics are found. The power function of the tests are calculated. The efficiency of the tests is found to be high on both an asymptotic and small sample basis.     

Mathematical modeling of matter distribution in a circular cell ensemble

In this paper, a mathematical model describing substance transport in a circular cell ensemble is considered. The model is represented by an autonomous system of equations. With a parameter continuation model, it is shown that the stationary solutions may have various symmetry types represented by closed curves. The periodic solutions have the same property, where the components differ from each other by a simple shift.     

On compact sets with a certain affine invariant

We give a complete characterization of finite-dimensional compact sets with the following property: all of their images under affine operators are symmetric (that is, have symmetry planes of certain dimensions). We also study the noncompact case; namely, we reveal a class of unbounded closed sets with this property and conjecture that this class is complete.