Symmetric Vector Research Articles

ON THE CENTER CONDITIONS FOR ANALYTIC MONODROMIC DEGENERATE SINGULARITIES

In this paper, we present two methods for detecting centers of monodromic degenerate singularities of planar analytic vector fields. These methods use auxiliary symmetric vector fields and can be applied independently so that the singularity is algebraic solvable or not, or has characteristic directions or not. We remark that these are the first methods which allow to study monodromic degenerate singularities with characteristic directions.

Bootstrap confidence intervals for isotonic estimators in a stereological problem

Let $\mathbf{X}=(X_1,X_2,X_3)$ be a spherically symmetric random vector of which only $(X_1,X_2)$ can be observed. We focus attention on estimating F, the distribution function of the squared radius $Z:=X_1^2+X_2^2+X_3^2$, from a random sample of $(X_1,X_2)$. Such a problem arises in astronomy where $(X_1,X_2,X_3)$ denotes the three dimensional position of a star in a galaxy but we can only observe the projected stellar positions $(X_1,X_2)$. We consider isotonic estimators of F and derive their limit distributions. The results are nonstandard with a rate of convergence $\sqrt{n/{\log n}}$. The isotonized estimators of F have exactly half the limiting variance when compared to naive estimators, which do not incorporate the shape constraint. We consider the problem of constructing point-wise confidence intervals for F, state sufficient conditions for the consistency of a bootstrap procedure, and show that the conditions are met by the conventional bootstrap method (generating samples from the empirical distribution function).

Design of an 11 kW power factor correction and 10 kW ZVS DC/DC converter for a high-efficiency battery charger in electric vehicles

Usually a battery charger for the electric vehicle consists of a power factor correction (PFC) and a DC/DC converter, both of which require the high-efficiency operation. In this study, a three-phase 11 kW full-bridge PFC is designed to transform the grid voltage to 400 VDC and reach the grid-side power factor equal to 1. This circuit adopts the symmetric space vector pulse width modulation, which effectively decreases the electrical stress of switches compared with other algorithms. The generated 400 VDC directly drives a CoolMOS-based 10 kW full-bridge DC/DC converter, that utilises the zero-voltage-switching technique to maintain the high efficiency. Experimental results validate that (i) the PFC part could reach 97–98% efficiency from 5 to 10 kW and (ii) the efficiency of the DC/DC part could reach ∼97%at10 kW for 200–400 V output and 95.9% for 400–450 V output. Fast reverse recovery diodes and SiC Schottky diodes are also experimentally compared in this prototype.

Global generic dynamics close to symmetry

Our object of study is the dynamics that arises in generic perturbations of an asymptotically stable heteroclinic cycle in S3. The cycle involves two saddle-foci of different type and is structurally stable within the class of (Z2⊕Z2)-symmetric vector fields. The cycle contains a two-dimensional connection that persists as a transverse intersection of invariant surfaces under symmetry-breaking perturbations.Gradually breaking the symmetry in a two-parameter family we get a wide range of dynamical behaviour: an attracting periodic trajectory; other heteroclinic trajectories; homoclinic orbits; n-pulses; suspended horseshoes and cascades of bifurcations of periodic trajectories near an unstable homoclinic cycle of Shilnikov type. We also show that, generically, the coexistence of linked homoclinic orbits at the two saddle-foci has codimension 2 and takes place arbitrarily close to the symmetric cycle.

Propagation of vectorial laser beams

The angular spectrum of a vectorial laser beam is expressed in terms of an intrinsic coordinate system instead of the usual Cartesian laboratory coordinates. This switch leads to simple, elegant, and new expressions, such as for the angular spectrum of the Hertz vectors corresponding to the electromagnetic fields. As an application of this approach, we consider axially symmetric vector beams, showing nondiffracting properties of these beams, without invoking the paraxial approximation.

On invariant structures of black hole charges

Abstract We study “minimal degree” complete bases of duality- and “horizontal”- invariant homogeneous polynomials in the flux representation of two-centered black hole solutions in two classes of D = 4 Einstein supergravity models with symmetric vector multiplets’ scalar manifolds. Both classes exhibit an SL(2, $ \mathbb{R} $ ) “horizontal” symmetry which mixes the two centers. The first class encompasses $ \mathcal{N} = {2} $ and $ \mathcal{N} = {4} $ matter-coupled theories, with semisimple U-duality given by SL(2, $ \mathbb{R} $ ) × SO(m,n); the analysis is carried out in the so-called Calabi-Vesentini symplectic frame (exhibiting maximal manifest covariance) and until order six in the fluxes included. The second class, exhibiting a non-trivial “horizontal” stabilizer SO(2), includes $ \mathcal{N} = {2} $ minimally coupled and $ \mathcal{N} = 3 $ matter coupled theories, with U-duality given by the pseudounitary group U(r,s) (related to complex flux representations). Finally, we comment on the formulation of special Kähler geometry in terms of “generalized” groups of type E 7.

Existence and Stability of Solutions for Generalized Symmetric Strong Vector Quasi-equilibrium Problems

In this paper, a class of generalized symmetric strong vector quasi-equilibrium problems in real locally convex Hausdorff topological vector spaces is studied. By using the Kakutani-Fan-Glicksberg fixed point theorem, an existence theorem of solutions for the generalized symmetric strong vector quasi-equilibrium problems is obtained. Moreover, the closedness of the solution set and a stability result of solutions for such problem are also derived

猫眼腔激光器光束合成轴对称线偏振矢量光束

猫眼腔激光器光束合成轴对称线偏振矢量光束

Invariant sets forced by symmetry

Given a linear (algebraic) group $G$ acting on real or complex $n$-space, we determine all the common invariant sets of $G$-symmetric vector fields. It turns out that the investigation of certain algebraic varieties is sufficient to characterize these invariant sets forced by symmetry. Toral, compact and reductive groups are discussed in some detail, and examples, including a Couette-Taylor system, are presented.

EWPD constraints on flavor symmetric vector fields

Electroweak precision data constraints on flavor symmetric vector fields are determined. The flavor multiplets of spin one that we examine are the complete set of fields that couple to quark bi-linears at tree level while not initially breaking the quark global flavor symmetry group. Flavor safe vector masses proximate to, and in some cases below, the electroweak symmetry breaking scale are found to be allowed. Many of these fields provide a flavor safe mechanism to explain the $ t\overline t $ forward backward anomaly, and can simultaneously significantly raise the allowed values of the Standard Model Higgs mass consistent with electroweak precision data.

Anisotropic perturbations in three-dimensional O(N)-symmetric vector models

We investigate the effects of anisotropic perturbations in three-dimensional O($N$)-symmetric vector models. In order to assess their relevance for the critical behavior, we determine the renormalization-group dimensions of the anisotropic perturbations associated with the first few spin values of the representations of the O($N$) group, because the lowest spin values give rise to the most important effects. In particular, we determine them up to spin 4 for $N=2,\phantom{\rule{0.222222em}{0ex}}3,\phantom{\rule{0.222222em}{0ex}}4$, by finite-size analyses of Monte Carlo simulations of lattice O($N$) models, achieving a significant improvement of their accuracy. These results are relevant for several physical systems, such as density-wave systems, magnets with cubic symmetry, and multicritical phenomena arising from the competition of different order parameters.

Well-posedness for convex symmetric vector quasi-equilibrium problems

In this paper, the notion of a generalized Levitin–Polyak well-posedness is defined for symmetric vector quasi-equilibrium problems. Sufficient conditions are given for the generalized Levitin–Polyak well-posedness. Moreover, it is shown that the results can be refined in the convex case.

Symmetric strong vector quasiequilibrium problems in Hausdorff locally convex spaces

Abstract In this article, a new symmetric strong vector quasiequilibrium problem in real locally convex Hausdorff topological vector spaces is introduced and studied. An existence theorem of solutions for the symmetric strong vector quasiequilibrium problem by using Kakutani-Fan-Glicksberg fixed point theorem is obtained. Moreover, the closedness of the solution set for this problem is derived. The results presented in this article improve and extend some known results according to Long et al. [Math. Comput. Model. 47, 445-451 (2008)], Somyot and Kanokwan [Fixed Point Theory Appl. doi:10.1155/2011/475121], and Wang et al. [Bull. Malays. Math. Sci. Soc. http://www.emis.de/journals/BMMSS/pdf/acceptedpapers/2009-11-022_R1.pdf]. 2000 MSC: 49J40; 90C29.

KNOTTED PERIODIC ORBITS AND CHAOTIC BEHAVIOR OF A CLASS OF THREE-DIMENSIONAL FLOWS

This paper considers a class of three-dimensional systems constructed by rotating some planar symmetric polynomial vector fields. It shows that this class of systems has infinitely many distinct types of knotted periodic orbits, which lie on a family of invariant torus. For two three-dimensional systems, exact explicit parametric representations of the knotted periodic orbits are given. For their perturbed systems, the chaotic behavior is discussed by using two different methods.

On homaloidal polynomial functions of degree 3 and prehomogeneous vector spaces

In this paper we consider homaloidal polynomial functions f such that their multiplicative Legendre transform f *, defined as in Etingof et al. (Sel. Math. (N.S.) 8(1):27–66, 2002), Section 3.2 is again polynomial. Following Dolgachev (Michigan Math. J. 48:191–202, 2000), we call such polynomials EKP-homaloidal. We prove that every EKP-homaloidal polynomial function of degree three is a relative invariant of a symmetric prehomogeneous vector space. This provides a complete proof of Etingof et al. (Sel. Math. (N.S.) 8(1):27–66, 2002), Theorem 3.10, p. 39. Our argument may suggest a way to attack the more general problem raised in Etingof et al. (Sel. Math. (N.S.) 8(1):27–66, 2002), Section 3.4 of EKP-homaloidal polynomials of arbitrary degree.

Semi-Invariants of Symmetric Quivers of Tame Type

A symmetric quiver (Q, σ) is a finite quiver without oriented cycles Q = (Q 0, Q 1) equipped with a contravariant involution σ on $Q_0\sqcup Q_1$ . The involution allows us to define a nondegenerate bilinear form $\langle -,-\rangle_V$ on a representation V of Q. We shall say that V is orthogonal if $\langle -,-\rangle_V$ is symmetric and symplectic if $\langle -,-\rangle_V$ is skew-symmetric. Moreover, we define an action of products of classical groups on the space of orthogonal representations and on the space of symplectic representations. So we prove that if (Q, σ) is a symmetric quiver of tame type then the rings of semi-invariants for this action are spanned by the semi-invariants of determinantal type c V and, when the matrix defining c V is skew-symmetric, by the Pfaffians pf V . To prove it, moreover, we describe the symplectic and orthogonal generic decomposition of a symmetric dimension vector.

On constitutive equations for thermoelastic dielectric continuum in terms of invariants

Electro-thermomechanical behavior of a thermoelastic dielectric body subject to external loading has been investigated theoretically in the present analysis. The theory is formulated in the context of continuum electrodynamics. The reaction of the body subject to external loads is expressed in symmetric stress, electrical polarization and heat flux. The solid medium is assumed to be linear, dielectric, isotropic, incompressible and dependent on temperature gradient. It has been observed that, as a result of thermodynamic constraints, the stress potential function is dependent on the deformation tensor, the electric field vector and the temperature, while the heat flux vector function is dependent on the deformation tensor, the electric field vector, the temperature and temperature gradient. To determine arguments of the stress potential and the heat flux vector functionals, findings of the theory of invariants have been used as a method because of that isotropy constraint is imposed on the material. As a result, constitutive equations of symmetric stress, polarization field and heat flux vector have been obtained in both material and spatial coordinates and asymmetric stress has been found using the expressions of symmetric stress and polarization field.

On linear vector program and vector matrix game equivalence

We define a vector matrix game, which is a vector version of the usual matrix game and consists of more than two skew symmetric matrices and vector ordering. Using vector optimization techniques, we characterize solutions for the vector matrix game. We establish equivalent relations between a linear vector optimization problem and its corresponding vector matrix game.

Control of planar Bautin bifurcation

On a versal deformation of the Bautin bifurcation it is possible to find dynamical systems that undergo Hopf or non-hyperbolic limit cycle bifurcations. Our paper concerns a nonlinear control system in the plane whose nominal vector field has a pair of purely imaginary eigenvalues. We find conditions to control the Hopf and Bautin bifurcation using the symmetric multilinear vector functions that appear in the Taylor expansion of the vector field around the equilibrium. The control law designed by us depends on two bifurcation parameters and four control parameters, which establish the stability of the equilibrium point and the orientation and stability of the limit cycles. Two examples are given.

A symmetry reduction scheme of the Dirac algebra without dimensional defects

In relating the Dirac algebra to homogeneous coordinates of a projective geometry, we present a simple geometric scheme which allows to identify various Lie algebras and Lie groups well-known from classical physics as well as from quantum field theory. We introduce a 1 -point-compactification and quaternionic Mobius transformations, and we use SU* (4) and a symmetry reduction scheme without dimensional defects to identify transformations and particle representations thoroughly. As such, two subsequent nonlinear σ models SU*(4)/U Sp(4) and U Sp(4)/SU(2) × U(1) emerge as well as a possible double coset decomposition of SU*(4) with respect to SU(2) × U(1). Whereas the first model leads to equivalence classes of hyperbolic manifolds and naturally introduces coordinates and velocities, the second coset model leads to a Hermitian symmetric (vector) space (Kahlerian space) of real dimension 6, i.e., to a 3-dimensional complex space with a global symplectic and a local SU(2) × U(1) symmetry which allows to identify the (local) gauge group of electroweak interactions as well as under certain assumptions it admits compact SU(3) transformations as automorphisms of this 3-dimensional (hyper)complex vector space. In the limit of low energies, this geometric SU*(4) scheme naturally yields the (compact) group SU(4) to describe “chiral symmetry” and conserved isospin of hadrons as well as the low-dimensional hadron representations. Last not least, with respect to some of the SU*(4) generators we find a multiplication table which (up to signs) is identical with the octonions represented in the Fano plane.