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.     

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.     

Characterization and algorithms of curve map graphs

A curve map is a planar map obtained by dividing the Euclidean plane into a finite number of regions by a finite set of two-way infinite Jordan curves (every one dividing the plane in two regions) such that no two curves intersect in more than one point. A line map is a curve map obtained by Jordan curves being all straight lines. A graph is called a curve map graph respectively a line map graph if it is the dual of a curve map respectively of a line map.In this paper we give a characterization of the curve map graphs and we describe a polynomial time algorithm for their recognition.     

Bergman Space Structure,Commutative Algebras of Toeplitz Operators,and Hyperbolic Geometry

We exhibit a surprising but natural connection among the Bergman space structure, commutative algebras of Toeplitz operators and pencils of hyperbolic straight lines. The commutative C*-algebras of Toeplitz operators on the unit disk can be classified as follows. Each pencil of hyperbolic straight lines determines the set of symbols consisting of functions which are constant on corresponding cycles, the orthogonal trajectories to lines forming a pencil. It turns out that the C*-algebra generated by Toeplitz operators with this class of symbols is commutative. Submitted: January 15, 2001?Revised: February 7, 2002     

Inversion of degree <Emphasis Type="Italic">n</Emphasis> + 2

By the method of synthetic geometry, we define a seemingly new transformation of a three-dimensional projective space where the corresponding points lie on the rays of the first order, nth class congruence C _n ¹ and are conjugate with respect to a proper quadric Ψ. We prove that this transformation maps a straight line onto an n + 2 order space curve and a plane onto an n + 2 order surface which contains an n-ple (i.e. n-multiple) straight line. It is shown that in the Euclidean space the pedal surfaces of the congruences C _n ¹ can be obtained by this transformation. The analytical approach enables new visualizations of the resulting curves and surfaces with the program Mathematica. They are shown in four examples.      

On the Bennequin Invariant and the Geometry of Wave Fronts

The theory of Arnold's invariants of plane curves and wave fronts is applied to the study of the geometry of wave fronts in the standard 2-sphere, in the Euclidean plane and in the hyperbolic plane. Some enumerative formulae similar to the Plücker formulae in algebraic geometry are given in order to compute the generalized Bennequin invariant J ⁺ in terms of the geometry of the front. It is shown that in fact every coefficient of the polynomial invariant of Aicardi can be computed in this way. In the case of affine wave fronts, some formulae previously announced by S.L. Tabachnikov are proved. This geometric point of view leads to a generalization to generic wave fronts of a result shown by Viro for smooth plane curves. As another application, the Fabricius-Bjerre and Weiner formulae for smooth plane and spherical curves are generalized to wave fronts.     

Irreducible * representations of realC*-algebras

In this paper, we show that a topologically irreducible * representation of a realC*-algebra is also algebraically irreducible. Moreover, the properties of pure real states on a realC*-algebra and their left kernels are discussed.Partially supported by the National Natural Science Foundation of China.     

The smallness problem for -algebras

Akemann showed that any von Neumann algebra with a weak^* separable dual space has a faithful normal representation on a separable Hilbert space. He posed the question: If a C^*-algebra has a weak^* separable state space, must it have a faithful representation on a separable Hilbert space? Wright solved this question negatively and showed that a unital C^*-algebra has the weak^* separable state space if and only if it has a unital completely positive map, into a type I factor on a separable Hilbert space, whose restriction to the self-adjoint part induces an order isomorphism. He called such a C^*-algebra almost separably representable. We say that a unital C^*-algebra is small if it has a unital complete isometry into a type I factor on a separable Hilbert space. In this paper we show that a unital C^*-algebra is small if and only if the state spaces of all n by n matrix algebras over the C^*-algebra are weak^*-separable. It is natural to ask whether almost separably representable algebras are small or not. We settle this question positively for simple C^*-algebras but the general question remains open.     

On orthogonal curvilinear coordinate systems in constant curvature spaces

We describe a method for constructing an n-orthogonal coordinate system in constant curvature spaces. The construction proposed is actually a modification of the Krichever method for producing an orthogonal coordinate system in the n-dimensional Euclidean space. To demonstrate how this method works, we construct some examples of orthogonal coordinate systems on the two-dimensional sphere and the hyperbolic plane, in the case when the spectral curve is reducible and all irreducible components are isomorphic to a complex projective line.     

Dynamic hyperbolic geometry: building intuition and understanding mediated by a Euclidean model

This paper explores a deep transformation in mathematical epistemology and its consequences for teaching and learning. With the advent of non-Euclidean geometries, direct, iconic correspondences between physical space and the deductive structures of mathematical inquiry were broken. For non-Euclidean ideas even to become thinkable the mathematical community needed to accumulate over twenty centuries of reflection and effort: a precious instance of distributed intelligence at the cultural level. In geometry education after this crisis, relations between intuitions and geometrical reasoning must be established philosophically, rather than taken for granted. One approach seeks intuitive supports only for Euclidean explorations, viewing non-Euclidean inquiry as fundamentally non-intuitive in nature. We argue for moving beyond such an impoverished approach, using dynamic geometry environments to develop new intuitions even in the extremely challenging setting of hyperbolic geometry. Our efforts reverse the typical direction, using formal structures as a source for a new family of intuitions that emerge from exploring a digital model of hyperbolic geometry. This digital model is elaborated within a Euclidean dynamic geometry environment, enabling a conceptual dance that re-configures Euclidean knowledge as a support for building intuitions in hyperbolic space—intuitions based not directly on physical experience but on analogies extending Euclidean concepts.     

Spatiotemporal dynamics and feedforward control of an infinite-dimensional,time-variant,thermal-coating process

In this contribution, a model for the spatiotemporal dynamics of a thermal-coating process is derived from first principles (the conservation of energy). The model is a dynamical system consisting of a system of hyperbolic, partial differential equations (PDEs), describing the evolution of the temperature distribution of the substrate. By studying the C ₀- semigroup generated by the system operator we find that in discrete time the infinite-dimensional plant may conveniently be represented by finite-dimensional operators defined on appropriately chosen Euclidean spaces. This representation provides the basis for numerical efficient solution of several optimal feedforward control problems associated with set point changes and launching of the process. Numerical and experimental studies highlight the value of this approach.     

Centers in domains with quadratic growth

Let F be a field, and let R be a finitely-generated F-algebra, which is a domain with quadratic growth. It is shown that either the center of R is a finitely-generated F-algebra or R satisfies a polynomial identity (is PI) or else R is algebraic over F. Let r ∈ R be not algebraic over F and let C be the centralizer of r. It is shown that either the quotient ring of C is a finitely-generated division algebra of Gelfand-Kirillov dimension 1 or R is PI.     

One-Parameter Homothetic Motion in the Hyperbolic Plane and Euler-Savary Formula

In [10] one-parameter planar motion was first introduced and the relations between absolute, relative, sliding velocities (and accelerations) in the Euclidean plane \mathbb E²{{\mathbb E}^2} were obtained. Moreover, the relations between the complex velocities of one-parameter motion in the complex plane were provided by [10]. One-parameter planar homothetic motion was defined in the complex plane, [9]. In this paper, analogous to homothetic motion in the complex plane given by [9], one-parameter planar homothetic motion is defined in the hyperbolic plane. Some characteristic properties about the velocity vectors, the acceleration vectors and the pole curves are given. Moreover, in the case of homothetic scale h identically equal to 1, the results given in [15] are obtained as a special case. In addition, three hyperbolic planes, of which two are moving and the other one is fixed, are taken into consideration and a canonical relative system for one-parameter planar hyperbolic homothetic motion is defined. Euler-Savary formula, which gives the relationship between the curvatures of trajectory curves, is obtained with the help of this relative system.     

The complexity of detecting crossingfree configurations in the plane

The computational complexity of the following type of problem is studied. Given a geometric graphG=(P, S) whereP is a set of points in the Euclidean plane andS a set of straight (closed) line segments between pairs of points inP, we want to know whetherG possesses a crossingfree subgraph of a special type. We analyze the problem of detecting crossingfree spanning trees, one factors and two factors in the plane. We also consider special restrictions on the slopes and on the lengths of the edges in the subgraphs.Klaus Jansen acknowledges support by the Deutsche Forschungsgemeinschaft. Gerhard J. Woeginger acknowledges support by the Christian Doppler Laboratorium für Diskrete Optimierung.     

How to collect balls moving in the Euclidean plane

In this paper, we study how to collect n balls moving with a fixed constant velocity in the Euclidean plane by k robots moving on straight track-lines through the origin. Since all the balls might not be caught by robots, differently from Moving-target TSP, we consider the following 3 problems in various situations: (i) deciding if k robots can collect all n balls; (ii) maximizing the number of the balls collected by k robots; (iii) minimizing the number of the robots to collect all n balls. The situations considered in this paper contain the cases in which track-lines are given (or not), and track-lines are identical (or not). For all problems and situations, we provide polynomial time algorithms or proofs of intractability, which clarify the tractability-intractability frontier in the ball collecting problems in the Euclidean plane.     

Ebene absolute geometrie über Euklidischen Körpern

We give a set of seven postulates for plane absolute geometry based on the geometric notions points and lines and show that every absolute plane is — up to isomorphisms — either a Euclidean or a hyperbolic plane over a Euclidean field.     

Horospheres and Convex Bodies in n-Dimensional Hyperbolic Space

In n-dimensional Euclidean space, the measure of hyperplanes intersecting a convex domain is proportional to the (n–2)-mean curvature integral of its boundary. This question was considered by Santaló in hyperbolic space. In non-Euclidean geometry the totally geodesic hypersurfaces are not always the best analogue to linear hyperplanes. In some situations horospheres play the role of Euclidean hyperplanes.In dimensions n=2 and 3, Santaló proved that the measure of horospheres intersecting a convex domain is also proportional to the (n–2)-mean curvature integral of its boundary.In this paper we show that this analogy does not generalize to higher dimensions. We express the measure of horospheres intersecting a convex body as a linear combination of the mean curvature integrals of its boundary.     

On locally <Emphasis Type="Italic">A</Emphasis>-convex <Emphasis Type="Italic">H</Emphasis>*-algebras

We endow any proper A-convex H*-algebra (E, τ) with a locally pre-C*-topology. The latter is equivalent to that introduced by the pre C*-norm given by Ptàk function when (E, τ) is a Q-algebra. We also prove that the algebra of complex numbers is the unique proper locally A-convex H*-algebra which is barrelled and Q-algebra.