Generalized Electrodynamics With Ternary Internal Structure

In Refs. [2]–[7] we suggested generalized dynamic equations of motion of relativistic charged particles inside electromagnetic fields. The dynamic equations had been formulated in terms of external as well as internal momenta. Evolution equations for external momenta, the Lorentz-force equations, had been derived from evolution equations for internal momenta. In this paper, along with relativistic dynamics we generalize electromagnetic fields within the scope of ternary algebras. The full theory is constructed in 4D euclidean space. This space possesses an advantage to build ternary mappings from three vectors onto one. The dynamics is given by non-linear evolution equations with cubic characteristic polynomial. In polar representation the internal momenta obey the Jacobi equations whereas external momenta obey the Weierstrass equations for elliptic functions. The generalized electromagnetic fields are defined by the triple fields where the first one has properties of the electric field and the other two have properties of the magnetic field. The field equations for the triple fields analogous to the Maxwell equations are suggested.     

Fundamental solutions of the equations of quantum mechanics as distributions and distinctive properties of the Dirac equation solutions

We show that the causal Green’s functions for interacting particles in external fields in both relativistic quantum mechanics (for the Dirac electron) and nonrelativistic quantum mechanics can be obtained as distributions if the free-particle Green’s functions are used and equations for the corresponding test functions are chosen. We study quantum properties of solutions of the Dirac equations. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 151, No. 2, pp. 287–301, May, 2007.     

The BRST-BV approach to massless fields adapted for the AdS/CFT correspondence

In the framework of the BRST-BV approach to the formulation of relativistic mechanics, we consider massless and massive fields of arbitrary spin propagating in a flat space and massless fields propagating in the AdS space. For such fields, we obtain BRST-BV Lagrangians invariant under gauge transformations. The Lagrangians and gauge transformations are constructed in terms of traceless gauge fields and traceless parameters of the gauge transformations. We consider the fields in the AdS space using the Poincaré parameterization of this space, which leads to a simple form of the BRST-BV Lagrangian. We show that in the Siegel gauge, the Lagrangian of the massless AdS fields leads to a decoupling of the equations of motion, and this substantially simplifies the study of the AdS/CFT correspondence. In a conformal algebra basis, we find a realization of the relativistic symmetries of fields and antifields in the AdS space.     

Geometrical and physical interpretation of evolution governed by general complex algebra

In this paper we explore a geometrical and physical matter of the evolution governed by the generator of General Complex Algebra, GC₂. The generator of this algebra obeys a quadratic polynomial equation. It is shown that the geometrical image of the GC₂-number is given by a straight line fixed by two given points on Euclidean plane. In this representation the straight line possesses the norm and the argument. The motion of the straight line conserving the norm of the line is described by evolution equation governed by the generator of the GC₂-algebra. This evolution is depicted on the Euclidean plane as rotational motion of the straight line around the semicircle to which this line is tangent. Physical interpretation is found within the framework of the relativistic dynamics where the quadratic polynomial is formed by mass-shell equation. In this way we come to a new representation for the momenta of the relativistic particle.     

Geometrical Interpretation of General Complex Algebra and its Application in Relativistic Mechanics

In a generalized formulation of the relativistic dynamics with internal conformation an important role is played by a quadratic polynomial, the coefficients and eigenvalues of which are generated by outer and inner momenta of the relativistic particle. This polynomial induces the general complex algebra, GC. In this paper we explore the geometrical and physical aspects of the evolution generated by the algebraic operations of the GC-algebra. It is shown that the geometrical image of the GC-number is given by a straight line passing through two given points in an euclidean plane. In this representation the straight line is characterized by a norm and an argument. The motions of the straight line are described by hyperbolic trigonometry which brings a correspondence between the Euclidean geometry and the hyperbolic one. It is proved that the evolution equation governed by the generator of the GC-algebra describes the energy conservation law of the relativistic particle. This evolution is depicted on the Euclidean plane as a rotational motion of the straight line, tangent to the circle with radius equal to the mass of the particle. In this way we come to new representation for the momenta in relativistic dynamics.     

Extended Dirac factorization: ‘multi-mass’ special cases in clifford form

The hypercomplex approach to Dirac physics is here summarized. The new mass term is shown to give a ψ wave, in the extension of Dirac’s Klein-Gordon factorization (when the usual Dirac mass part equals zero), that is completely within the ‘complexified’ Hamilton-Pauli sub-algebraE. This may help with the physical interpretation of this new mass, with its five possible parts. New, spin 1/2 quanta also seem possible. These are all variations on Dirac’s factorization, where mass is generalized to several parts (multi-parts) or, instead, each new mass part, of the five, can be taken one at a time in nature, for individual wave equations. Lorentz symmetry is also naturally extended from 6 parameters to 8 parameters, and this has sweeping metaphysical implications. Dirac algebra is doubled also in a natural way using quaternions.     

Automorphism groups of some algebras

The automorphism groups of algebras are found in many papers. Using auto-invariance, we find the automorphism groups of the Laurent extension of the polynomial ring and the quantum n-plane (respectively, twisting polynomial ring) in this work. As an application of the results of this work, we can find the automorphism group of a twisting algebra. We define a generalized Weyl algebra and show that the generalized Weyl algebra is simple. We also find the automorphism group of a generalized Weyl algebra. We show that the generalized Weyl algebra A _m,m+n is the universal enveloping algebra of the generalized Witt algebra W(m,m + n). This work was supported by 2007 Research fund of Hanyang University     

Locomotion of Deformable Bodies in an Ideal Fluid: Newtonian versus Lagrangian Formalisms

This paper is concerned with comparing Newtonian and Lagrangian methods in Mechanics for determining the governing equations of motion (usually called Euler–Lagrange equations) for a collection of deformable bodies immersed in an incompressible, inviscid fluid whose flow is irrotational. The bodies can modify their shapes under the action of inner forces and torques and are endowed with thrusters, which means that they can generate fluid jets by sucking and blowing out fluid through some localized parts of their boundaries. These capabilities may allow them to propel and steer themselves. Our first contribution is to prove that under smoothness assumptions on the fluid-bodies interface, Newtonian and Lagrangian formalisms yield the same equations of motion. However, and rather surprisingly, this is no longer true for nonsmooth shaped bodies.     

有限域上n元n次方程只有零解的充要条件

本文研究了有限域上只有零解的n元n次方程的结构问题.利用对有限域上不可约多元多项式在其扩域中的分解特征的刻画,结合Chevalley定理,得到了有限域上n元n次方程只有零解的一个充要条件,并给出这类方程的一些新的具体构造.     

Suboptimal Kronrod extension formulae for numerical quadrature

Summary We consider cases where the Stieltjes polynomial associated with a Gaussian quadrature formula has complex zeros. In such cases a Kronrod extension of the Gaussian rule does not exist. A method is described for modifying the Stieltjes polynomial so that the resulting polynomial has no complex zeros. The modification is performed in such a way that the Kronrod-type extension rule resulting from the addition of then+1 zeros of the modified Stieltjes polynomial to the original knots of the Gaussian rule has only slightly lower degree of precision than normally achieved when the Kronrod extension rule exists. As examples of the use of the method, we present some new formulae extending the classical Gauss-Hermite quadrature rules. We comment on the limited success of the method in extending Gauss-Laguerre rules and show that several modified extensions of the Gauss Gegenbauer formulae exist in cases where the standard Kronrod extension does not.     

On the formulation of initial-value problems for systems consisting of relativistic particles

We discuss questions related to the well-posedness of problems on the motion of relativistic many-body systems. For one-dimensional relativistic motion of N similar charges, we prove that an ordinary Cauchy problem usual in Newton mechanics can be stated; this is done in the framework of microscopic Maxwell-Lorentz electrodynamics (including a model with self-action) or Wheeler-Feynman theory. __________ Translated from Fundamental’naya i Prikladnaya Matematika (Fundamental and Applied Mathematics), Vol. 11, No. 1, Geometry, 2005.     

Polynomial dynamical systems and ordinary differential equations associated with the heat equation

We consider homogeneous polynomial dynamical systems in n-space. To any such system our construction matches a nonlinear ordinary differential equation and an algorithm for constructing a solution of the heat equation. The classical solution given by the Gaussian function corresponds to the case n = 0, while solutions defined by the elliptic theta-function lead to the Chazy-3 equation and correspond to the case n = 2. We explicitly describe the family of ordinary differential equations arising in our approach and its relationship with the wide-known Darboux-Halphen quadratic dynamical systems and their generalizations.     

Dynamical systems on a Riemannian manifold that admit normal shift

Newtonian dynamical systems that admit normal shift on an arbitrary Riemannian manifold are considered. The determining equations for these systems, which constitute the condition of weak normality, are derived. The extension of the algebra of tensor fields to manifolds is considered.Bashkir State University. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 103, No. 2, pp. 256–266, May, 1995.     

Multicomplex algebras on polynomials and generalized Hamilton dynamics

Generator of the complex algebra within the framework of general formulation obeys the quadratic equation. In this paper we explore multicomplex algebra with the generator obeying n-order polynomial equation with real coefficients. This algebra induces generalized trigonometry ((n+1)-gonometry), underlies of the nth order oscillator model and nth order Hamilton equations. The solution of an evolution equation generated by (n×n) matrix is represented via the set of (n+1)-gonometric functions. The general form of the first constant of motion of the evolution equation is established.     

Una struttura sintattica unitaria per le teorie della relatività ristretta e della meccanica classica

A unified axiomatic scheme for both the Newtonian Mechanics and the Special Theory of Relativity is given, by setting two systems of Axioms that differ from each other in only one requirement about the possibility of measuring time-intervals by light reflections. The concept of «observer» is obtained as a derived concept, rather than a primitive one, as in some previous papers by other Authors. The status of Newtonian Mechanics as a «limiting case» of Special Relativity is rigorously deduced as a consequence of the result that the geometric structure of (neo)classical space-time is a limit of a family of relativistic geometric structures for the space-time.     

The <Emphasis Type="Italic">n</Emphasis>-Body Problem in Spaces of Constant Curvature. Part I: Relative Equilibria

We extend the Newtonian n-body problem of celestial mechanics to spaces of curvature κ=constant and provide a unified framework for studying the motion. In the 2-dimensional case, we prove the existence of several classes of relative equilibria, including the Lagrangian and Eulerian solutions for any κ≠0 and the hyperbolic rotations for κ<0. These results lead to a new way of understanding the geometry of the physical space. In the end we prove Saari’s conjecture when the bodies are on a geodesic that rotates elliptically or hyperbolically.     

A generalized Kiepert formula for <Emphasis Type="Italic">C</Emphasis><Subscript><Emphasis Type="Italic">ab</Emphasis></Subscript> curves

Kiepert (1873) and Brioschi (1864) published algebraic equations for the n-division points of an elliptic curve, in terms of the Weierstrass ℘-function and its derivatives with respect to a uniformizing parameter, or another elliptic function, respectively. We generalize both types of formulas for a compact Riemann surface which, outside from one point, has a smooth polynomial equation in the plane, in the sense that we characterize the points whose n-th multiple in the Jacobian belongs to the theta divisor.     

The modern origin of matrices and their applications

This paper deals with the modern development of matrices, linear transformations, quadratic forms and their applications to geometry and mechanics, eigenvalues, eigenvectors and characteristic equations with applications. Included are the representations of real and complex numbers, and quaternions by matrices, and isomorphism in order to show that matrices form a ring in abstract algebra. Some special matrices, including Hilbert’s matrix, Toeplitz’s matrix, Pauli’s and Dirac’s matrices in quantum mechanics, and Einstein’s Pythagorean formula are discussed to illustrate diverse applications of matrix algebra. Included also is a modern piece of information that puts mathematics, science and mathematics education professionals at the forefront of advanced study and research on linear algebra and its applications.     

Mappings by the solutions of second-order elliptic equations

The properties of mappings by the solutions of second-order elliptic partial differential equations in the plane are studied. We obtain conditions on a function, continuous on the unit circle, that are sufficient for the solution of the Dirichlet problem in the open unit disk for the given equation with the given boundary function to be a homeomorphism between the open unit disk and a Jordan simply connected domain. The properties of the zeros of the solutions of the given equations are also studied. In particular, an analog of the main theorem of algebra is proved for polynomial solutions.     

Universal PI-algebras and algebras of generic matrices

Let Ω[ξ] denote the polynomial algebra (with 1) in commutative indeterminates {ie65-1}, 1 ≦i, j ≦n, 1 ≦k < ∞, over a commutative ring Ω. Thealgebra of generic matrices Ω [Y] is defined to be the Ω-subalgebra ofM _n (Ω[ξ]) generated by the matricesY _k=({ie65-2}), 1 ≦i, j ≦n, 1 ≦k < ∞. This algebra has been studied extensively by Amitsur and by Procesi in particular Amitsur has used it to construct a finite dimensional, central division algebra Ω (Y) which is not a crossed product. In this paper we shall prove, for Ω a domain, that Ω(Y) has exponentn in the Brauer group (Amitsur may already know this fact); consequently, for Ω an infinite field andn a multiple of 4, iff(X ₁, …,X _m) is a polynomial linear in all theX _i but one (similar to Formanek’s central polynomials for matrix rings) andf ² is central forM _n (Ω), thenf is central forM _n (Ω). (The existence of a polynomial not central forM _n (Ω), but whose square is central forM _n(Ω) is equivalent to every central division algebra of degreen containing a quadratic extension of its center; well-known theory immediately shows this is the case of 4‖n and 8χn.) Also, information is obtained about Ω(Y) for arbitary Ω, most notably that the Jacobson radical is the set of nilpotent elements. Partial support for this work was provided by National Science Foundation grant NSF-GP 33591.