Dual Approach in the Application of Geometric Interpretation of Linear Programming on the Organisation of Goods Distribution

The topic of the paper is the application of dual approach in formulation and resolution of goods distribution tasks problems. The gap in previous goods distribution research is the absence of the methodologies and goods transportation calculation methods for manufacturing companies with not too large amount of goods distribution whereby goods distribution is not the core activity. The goal of this paper is to find a solution for transportation in such companies. In such cases it is not rational to procure a specific software for the improvement of goods transportation but rather apply the calculation presented in this paper. The aim of this paper from mathematical aspect is to show the convenience of switching from the basic geometric interpretation of linear programming applied on transportation tasks to dual approach for the companies with too many costs limitations per transport task but not enough available transportation means. Recent research studies that use dual approach in linear programming are generally not applied to transportation tasks although such approach is very convenient. The goal of the paper is also to resolve transportation tasks by both primal and dual approach in order to prove the correctness of the method.

Embeddings of locally compact abelian p-groups in Hawaiian groups

We show that locally compact abelian p-groups can be embedded in the first Hawaiian group on a compact path connected subspace of the Euclidean space of dimension four. This result gives a new geometric interpretation for the classification of locally compact abelian groups which are rich in commuting closed subgroups. It is then possible to introduce the idea of an algebraic topology for topologically modular locally compact groups via the geometry of the Hawaiian earring. Among other things, we find applications for locally compact groups which are just noncompact.

Affine connection representation of gauge fields

There are two ways to unify gravitational field and gauge field. One is to represent gravitational field asprincipal bundle connection, and the other is to represent gauge field as affine connection. Poincaré gauge theoryand metric-affine gauge theory adopt the first approach. This paper adopts the second. In this approach:(i) Gauge field and gravitational field can both be represented by affine connection; they can be described by aunified spatial frame.(ii) Time can be regarded as the total metric with respect to all dimensions of internal coordinate space andexternal coordinate space. On-shell can be regarded as gradient direction. Quantum theory can be regarded as ageometric theory of distribution of gradient directions. Hence, gauge theory, gravitational theory, and quantumtheory all reflect intrinsic geometric properties of manifold.(iii) Coupling constants, chiral asymmetry, PMNS mixing and CKM mixing arise spontaneously as geometricproperties in affine connection representation, so they are not necessary to be regarded as direct postulates in theLagrangian anymore.(iv) The unification theory of gauge fields that are represented by affine connection can avoid the problem thata proton decays into a lepton in theories such as SU(5).(v) There exists a geometric interpretation to the color confinement of quarks.In the affine connection representation, we can get better interpretations to the above physical properties,therefore, to represent gauge fields by affine connection is probably a necessary step towards the ultimate theory ofphysics.

Roots of Elliptic Scator Numbers

The Victoria equation, a generalization of De Moivre’s formula in 1+n dimensional scator algebra, is inverted to obtain the roots of a scator. For the qth root in S1+n of a real or a scator number, there are qn possible roots. For n=1, the usual q complex roots are obtained with their concomitant cyclotomic geometric interpretation. For n≥2, in addition to the previous roots, new families arise. These roots are grouped according to two criteria: sets satisfying Abelian group properties under multiplication and sets catalogued according to director conjugation. The geometric interpretation is illustrated with the roots of unity in S1+2.

The noncommutative geometry of the Landau Hamiltonian: Differential aspects

In this work we study the differential aspects of the noncommutative geometry for the magnetic C*-algebra which is a 2-cocycle deformation of the group C*-algebra of R2. This algebra is intimately related to the study of the Quantum Hall Effect in the continuous, and our results aim to provide a new geometric interpretation of the related Kubo's formula. Taking inspiration from the ideas developed by Bellissard during the 80's, we build an appropriate Fredholm module for the magnetic C*-algebra based on the magnetic Dirac operator which is the square root (à la Dirac) of the quantum harmonic oscillator. Our main result consist of establishing an important piece of Bellissard's theory, the so-called second Connes' formula. In order to do so, we establish the equality of three cyclic 2-cocycles defined on a dense subalgebra of the magnetic C*-algebra. Two of these 2-cocycles are new in the literature and are defined by Connes' quantized differential calculus, with the use of the Dixmier trace and the magnetic Dirac operator.

The elements of General Relativity

This chapter is a survey of central ideas and equations in general relativity. The basic equations are written down with a view to seeing where we are heading in the book, and in order to present both the field theory and the geometric interpretation of gravity. The central role of the metric is introduced, and the equivalence principle is discussed. It is emphasized that spacetime interval is both a mathematical and a physical idea. It is explained how gravity works “behind the scenes” by modifying equations which otherwise look like familiar equations of electromagnetism. The sense in which acceleration is in some respects a relative and in some respects an absolute concept is explained. It is shown why the motion of matter, not just its mass, must influence gravitation. The stress-energy tensor is introduced and defined.

The Pauli Problem for Gaussian Quantum States: Geometric Interpretation

We solve the Pauli tomography problem for Gaussian signals using the notion of Schur complement. We relate our results and method to a notion from convex geometry, polar duality. In our context polar duality can be seen as a sort of geometric Fourier transform and allows a geometric interpretation of the uncertainty principle and allows to apprehend the Pauli problem in a rather simple way.