Formal foundations for the origins of human consciousness

In the framework of p-adic analysis (the simplest version of analysis on trees in which hierarchic structures are presented through ultrametric distance) applied to formalize psychic phenomena, we would like to propose some possible first hypotheses about the origins of human consciousness centered on the basic notion of time symmetry breaking as meant according to quantum field theory of infinite systems. Starting with Freud’s psychophysical (hydraulic) model of unconscious and conscious flows of psychic energy based on the three-orders mental representation, the emotional order, the thing representation order, and the word representation order, we use the p-adic (treelike) mental spaces to model transition from unconsciousness to preconsciousness and then to consciousness. Here we explore theory of hysteresis dynamics: conscious states are generated as the result of integrating of unconscious memories. One of the main mathematical consequences of our model is that trees representing unconscious and consciousmental states have to have different structures of branching and distinct procedures of clustering. The psychophysical model of Freud in combination with the p-adic mathematical representation gives us a possibility to apply (for a moment just formally) the theory of spontaneous symmetry breaking of infinite dimensional field theory, to mental processes and, in particular, to make the first step towards modeling of interrelation between the physical time (at the level of the emotional order) and psychic time at the levels of the thing and word representations. Finally, we also discuss some related topological aspects of the human unconscious, following Jacques Lacan’s psychoanalytic concepts.     

Linear representations of formal loops

A representation of an object in a category is an abelian group in the corresponding comma category. In this paper, we derive the formulas describing linear representations of objects in the category of formal loops and formal loop homomorphisms and apply them to obtain a new approach to the representation theory of formal Moufang loops and Malcev algebras based on Moufang elements. Certain ‘non-associative Moufang symmetry’ of groups is revealed.     

Symmetry in semidefinite programs

This paper is a tutorial in a general and explicit procedure to simplify semidefinite programs which are invariant under the action of a symmetry group. The procedure is based on basic notions of representation theory of finite groups. As an example we derive the block diagonalization of the Terwilliger algebra of the binary Hamming scheme in this framework. Here its connection to the orthogonal Hahn and Krawtchouk polynomials becomes visible.     

Tight Frames and Their Symmetries

The aim of this paper is to investigate the symmetry properties of tight frames, with a view to constructing tight frames of orthogonal polynomials in several variables which share the symmetries of the weight function, and other similar applications. This is achieved by using representation theory to give methods for constructing tight frames as orbits of groups of unitary transformations acting on a given finite-dimensional Hilbert space. Along the way, we show that a tight frame is determined by its Gram matrix and discuss how the symmetries of a tight frame are related to its Gram matrix. We also give a complete classification of those tight frames which arise as orbits of an abelian group of symmetries.     

Group theory for tetraammineplatinum(II) withC 2v andC 4v point group in the non-rigid system

The non-rigid molecule group theory (NRG) in which the dynamical symmetry operations are defined as physical operations is a new field of chemistry. Smeyers in a series of papers applied this notion to determine the character table of restricted NRG of some molecules. In this work, a simple method is described, by means of which it is possible to calculate character tables for the symmetry group of molecules consisting of a number of NH3 groups attached to a rigid framework. We study the full non-rigid group (f-NRG) of tetraammineplatinum (II) with two separate symmetry groups C2v and C4v. We prove that they are groups of order 216 and 5184 with 27 and 45 conjugacy classes, respectively. Also, we will compute the character tables of these groups.     

Using Mathematica to compute conjugacy classes

The conjugacy classes of finite groups play an important role in the representation theory of those groups, and it is useful to be able to compute the conjugacy classes quickly. A procedure is developed and then implemented with Mathematica to discover these conjugacy classes. The computations make use of the Cayley table in its regular form for the group. The conjugacy classes for C_4v, the point symmetry group of the square, are displayed.     

Complex group algebras of finite groups: Brauer's Problem 1

Brauer's Problem 1 asks the following: What are the possible complex group algebras of finite groups? It seems that with the present knowledge of representation theory it is not possible to settle this question. The goal of this paper is to present a partial solution to this problem. We conjecture that if the complex group algebra of a finite group does not have more than a fixed number m of isomorphic summands, then its dimension is bounded in terms of m. We prove that this is true for every finite group if it is true for the symmetric groups. The problem for symmetric groups reduces to an explicitly stated question in number theory or combinatorics.     

Discrete symmetries and chirality invariance in quantum field dynamics

The aim of the present paper is to discuss systematically the discrete symmetry operations on a quantized field in interaction; and to base the introduction of the new quantum number “chirality” for spinor fields on these symmetry properties. In the course of this investigation, several general results on the group of symmetry operations are proved and relation between certain sets of discrete symmetry operations and the spinor representation of the rotation group in 3 and 4 dimensions is established. An attempt has been made to present clearly the connection between additive and multiplicative quantum numbers, gauge transformations, unitary transformations and invariance laws. The chirality invariance of spinor fields in interaction is discussed in some detail. The emphasis throughout is on the systematic development rather than on details of application. The paper is divided into two parts, the first dealing with the general theory of discrete symmetry operations and the second concerned with chirality invariance for spinor fields.     

Symmetric spaces and Lie triple systems in numerical analysis of differential equations

A remarkable number of different numerical algorithms can be understood and analyzed using the concepts of symmetric spaces and Lie triple systems, which are well known in differential geometry from the study of spaces of constant curvature and their tangents. This theory can be used to unify a range of different topics, such as polar-type matrix decompositions, splitting methods for computation of the matrix exponential, composition of selfadjoint numerical integrators and dynamical systems with symmetries and reversing symmetries. The thread of this paper is the following: involutive automorphisms on groups induce a factorization at a group level, and a splitting at the algebra level. In this paper we will give an introduction to the mathematical theory behind these constructions, and review recent results. Furthermore, we present a new Yoshida-like technique, for self-adjoint numerical schemes, that allows to increase the order of preservation of symmetries by two units. The proposed techniques has the property that all the time-steps are positive.     

Laplacian and vibrational spectra for homogeneous graphs

A homogeneous graph is a graph togerther with a group that acts transitively on vertices as symmertries of the graph. We consider Laplacians of homogeneous graphs and generalizations of Laplacians whose eigenvalues can be associated with various equilibria of forces in molecules (such as vibrational modes of buckyballs). Methods are given for calculating such eigenvalues by combining concepts and techniques in group representation theory, gauge theory and graph theory.     

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.     

The Quiver of an Algebra Associated to the Mantaci-Reutenauer Descent Algebra and the Homology of Regular Semigroups

We develop the homology theory of the algebra of a regular semigroup, which is a particularly nice case of a quasi-hereditary algebra in good characteristic. Directedness is characterized for these algebras, generalizing the case of semisimple algebras studied by Munn and Ponizovksy. We then apply homological methods to compute (modulo group theory) the quiver of a right regular band of groups, generalizing Saliola’s results for a right regular band. Right regular bands of groups come up in the representation theory of wreath products with symmetric groups in much the same way that right regular bands appear in the representation theory of finite Coxeter groups via the Solomon-Tits algebra of its Coxeter complex. In particular, we compute the quiver of Hsiao’s algebra, which is related to the Mantaci-Reutenauer descent algebra.     

Moving frames and singularities of prolonged group actions

The prolongation of a transformation group to jet bundles forms the geometric foundation underlying Lie's theory of symmetry groups of differential equations, the theory of differential invariants, and the Cartan theory of moving frames. Recent developments in the moving frame theory have necessitated a detailed understanding of the geometry of prolonged transformation groups. This paper begins with a basic review of moving frames, and then focuses on the study of both regular and singular prolonged group orbits. Highlights include a corrected version of the basic stabilization theorem, a discussion of "totally singular points," and geometric and algebraic characterizations of totally singular submanifolds, which are those that admit no moving frame. In addition to applications to the method of moving frames, the paper includes a generalized Wronskian lemma for vector-valued functions, and methods for the solution to Lie determinant equations.     

Localization phenomena in structural dynamics

A survey of vibration localization phenomena in the context of structural dynamics and vibrations is presented. The review covers the more common and relevant cases where mode localization and vibration confinement are likely to occur in engineering structures. Examples considered include periodic or nearly periodic multi-span beams and multi-bay trusses, large space structures, space antennas, and almost periodic (a.p.) structures with circular symmetry, e.g., bladed disks in turbomachines. Both analytical and numerical methods for analyzing and predicting localization in finite and infinite systems are discussed. In this paper, we show how the problem of mode localization and vibration confinement can be formulated as a problem in the theory of stability of differential equations with a.p. coefficients. Using stability theory, new definitions of mode localization can be established for both linear and nonlinear structures. The possibility of stabilizing certain nonconservative fluid-structure systems using structural disorder is demonstrated, and stability theorems are given for aeroelastic systems governed by normal operators. We also illustrate how the results from localization theory and the associated stability theory can be applied to the vibration control problem, by triggering vibration confinement by active or passive means.     

Homotopy equivalent group representations and Picard groups op the burnside ring and the character ring

We are concerned with the homotopy theory of group representations and its relation to character theory and the theory of the Burnside ring. We combine the methods of tom Dieck — Petrie [4] and torn Dieck [3] to show that the canonical map from the J-group jO(G), a subquotient of the representation ring RO(G), into the Picard group of the rational representation ring is injective for p-groups G. Moreover we compute the order of the cokernel of this map. We show that the Picard group of the rational representation ring is a direct summand in the Picard group of the Burnside ring. Finally we compute the Picard groups if G is abelian and indicate a computation for general G.     

A decomposition for the space of games with externalities

The main goal of this paper is to present a different perspective than the more ‘traditional’ approaches to study solutions for games with externalities. We provide a direct sum decomposition for the vector space of these games and use the basic representation theory of the symmetric group to study linear symmetric solutions. In our analysis we identify all irreducible subspaces that are relevant to the study of linear symmetric solutions and we then use such decomposition to derive some applications involving characterizations of classes of solutions.