Toward the classification of biangular harmonic frames

Equiangular tight frames (ETFs) and biangular tight frames (BTFs) – sets of unit vectors with basis-like properties whose pairwise absolute inner products admit exactly one or two values, respectively – are useful for many applications. A well-understood class of ETFs are those which manifest as harmonic frames – vector sets defined in terms of the characters of finite abelian groups – because they are characterized by combinatorial objects called difference sets.This work is dedicated to the study of the underlying combinatorial structures of harmonic BTFs. We show that if a harmonic frame is generated by a divisible difference set, a partial difference set or by a special structure with certain Gauss summing properties – all three of which are generalizations of difference sets that fall under the umbrella term “bidifference set” – then it is either a BTF or an ETF. However, we also show that the relationship between harmonic BTFs and bidifference sets is not as straightforward as the correspondence between harmonic ETFs and difference sets, as there are examples of bidifference sets that do not generate harmonic BTFs. In addition, we study another class of combinatorial structures, the nested divisible difference sets, which yields an example of a harmonic BTF that is not generated by a bidifference set.     

Phenomenologically symmetric local lie groups of transformations of the space <Emphasis Type="Italic">R</Emphasis><Superscript><Emphasis Type="Italic">s</Emphasis></Superscript>

In this paper we define a phenomenologically symmetric local Lie group of transformations of an arbitrary-dimensional space. We take as a basis the axiom scheme of the theory of physical structures. Phenomenologically symmetric groups of transformations are nondegenerate both with respect to coordinates and to parameters. We obtain a multipoint invariant of this group of transformations and relate it with Ward quasigroups. We define a substructure of a physical structure as a certain phenomenologically symmetric subgroup of transformations. We establish a criterion for the phenomenological symmetry of the Lie group of transformations and prove the uniqueness of a structure with the minimal rank. We also introduce the notion of a phenomenologically symmetric product of physical structures.     

Probabilistic compositional models: Solution of an equivalence problem

Probabilistic compositional models, similarly to graphical Markov models, are able to represent multidimensional probability distributions using factorization and closely related concept of conditional independence. Compositional models represent an algebraic alternative to the graphical models. The system of related conditional independencies is not encoded explicitly (e.g. using a graph) but it is hidden in a model structure itself. This paper provides answers to the question how to recognize whether two different compositional model structures are equivalent – i.e., whether they induce the same system of conditional independencies. Above that, it provides an easy way to convert one structure into an equivalent one in terms of some elementary operations on structures, closely related ability to generate all structures equivalent with a given one, and a unique representative of a class of equivalent structures.     

Rings with kernel inclusion quasiorder

Building on previous work for semigroups of functions and binary relations, we axiomatize structures consisting of endomorphisms of abelian groups equipped with composition, the usual pointwise operations, and the quasi-order of kernel inclusion. The resulting structures are associative rings enriched by a quasiorder satisfying a finite set of laws. More generally, we axiomatize the kernel inclusion quasi-order on the ring R induced by a right R-module, and we call the resulting abstract structures rings with ker-order. A characterisation of the possible ker-orders on a fixed ring is given in terms of certain families of its right ideals. The lattice of all ker-orders on the ring of rational integers is described.     

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.     

Associativity, Commutativity and Symmetry in Residuated Structures

In this paper we develop the study of extended-order algebras, recently introduced by C. Guido and P. Toto, which are implicative algebras that generalize all the widely considered integral residuated structures. Particular care is devoted to the requirement of completeness that can be obtained by the MacNeille completion process. Associativity, commutativity and symmetry assumptions are characterized and their role is discussed toward the structure of the algebra and of its completion. As an application, further operations corresponding to the logical connectives of conjunction negation and disjunction are considered and their properties are investigated, either assuming or excluding the additional conditions of associativity, commutativity and symmetry. An overlook is also devoted to the relationship with other similar structures already considered such as implication algebras (in particular Heyting algebras), BCK algebras, quantales, residuated lattices and closed categories.     

The Theory of Linear G-Difference Equations

We introduce the notion of difference equations defined on a structured set. The symmetry group of the structure determines the set of difference operators. All main notions in the theory of difference equations are introduced as invariants of the action of the symmetry group. Linear equations are modules over the skew group algebra, solutions are morphisms relating a given equation to other equations, symmetries of an equation are module endomorphisms, and conserved structures are invariants in the tensor algebra of the given equation.We show that the equations and their solutions can be described through representations of the isotropy group of the symmetry group of the underlying set. We relate our notion of difference equation and solutions to systems of classical difference equations and their solutions and show that out notions include these as a special case.     

A Unified Approach to Combinatorial Formulas for Schubert Polynomials

Schubert polynomials were introduced in the context of the geometry of flag varieties. This paper investigates some of the connections not yet understood between several combinatorial structures for the construction of Schubert polynomials; we also present simplifications in some of the existing approaches to this area. We designate certain line diagrams for permutations known as rc-graphs as the main structure. The other structures in the literature we study include: semistandard Young tableaux, Kohnert diagrams, and balanced labelings of the diagram of a permutation. The main tools in our investigation are certain operations on rc-graphs, which correspond to the coplactic operations on tableaux, and thus define a crystal graph structure on rc-graphs; a new definition of these operations is presented. One application of these operations is a straightforward, purely combinatorial proof of a recent formula (due to Buch, Kresch, Tamvakis, and Yong), which expresses Schubert polynomials in terms of products of Schur polynomials. In spite of the fact that it refers to many objects and results related to them, the paper is mostly self-contained.     

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.     

On Binary Computation Structures

Based on a modification of Moss' and Parikh's topological modal language [8], we study a generalization of a weakly expressive fragment of a certain propositional modal logic of time. We define a bimodal logic comprising operators for knowledge and nexttime. These operators are interpreted in binary computation structures. We present an axiomatization of the set T of theorems valid for this class of semantical domains and prove – as the main result of this paper – its completeness. Moreover, the question of decidability of T is treated.     

Applications of Symmetry and Group Theory for the Investigation of Molecular Vibrations

The application of symmetry and mathematical group theory is a powerful tool for investigating the vibrations of molecules. In this paper, we present an overview of the methods utilized. First we briefly discuss the quantum mechanical nature of vibrations and the experimental methods used. We then present the principal concepts for applying group theory to molecules. The symmetry operations which are used to comprise groups are described and then used to determine the point groups of molecules. The properties of character tables are presented and the method for obtaining a reducible representation for all the motions of a molecule is detailed. This can then be broken down to obtain the irreducible representation which contains the symmetry species of the individual vibrations. The determination of symmetry adapted linear combinations is outlined and the basis for spectroscopic selection rules is presented. The paper concludes by examining how matrix algebra along with symmetry concepts simplifies calculations with molecular force constants.     

Affine Poisson structures

We establish a one-to-one correspondence between the set of all equivalence classes of affine Poisson structures (defined on the dual of a finite dimensional Lie algebra) and the set of all equivalence classes of central extensions of the Lie algebra by ℝ. We characterize all the affine Poisson structures defined on the duals of some lower dimensional Lie algebras. It is shown that under a certain condition every Poisson structure locally looks like an affine Poisson structure. As an application, we show the role played by affine Poisson structures in mechanics. Finally, we prove some involution theorems.     

General constructions of biquandles and their symmetries

Biquandles are algebraic objects with two binary operations whose axioms encode the generalized Reidemeister moves for virtual knots and links. These objects also provide set theoretic solutions of the well-known Yang-Baxter equation. The first half of this paper proposes some natural constructions of biquandles from groups and from their simpler counterparts, namely, quandles. We completely determine all words in the free group on two generators that give rise to (bi)quandle structures on all groups. We give some novel constructions of biquandles on unions and products of quandles, including what we refer as the holomorph biquandle of a quandle. These constructions give a wealth of solutions of the Yang-Baxter equation. We also show that for nice quandle coverings a biquandle structure on the base can be lifted to a biquandle structure on the covering. In the second half of the paper, we determine automorphism groups of these biquandles in terms of associated quandles showing elegant relationships between the symmetries of the underlying structures.     

THE DESIGN AND ANALYSIS OF ALGORITHM OF MINIMUM COST SPANNING TREE

ＴＨＥＤＥＳＩＧＮＡＮＤＡＮＡＬＹＳＩＳＯＦＡＬＧＯＲＩＴＨＭＯＦＭＩＮＩＭＵＭＣＯＳＴＳＰＡＮＮＩＮＧＴＲＥＥ￥（徐绪松，刘大成，吴丽华）ＸｕＸｕｓｏｎｇ；ＬｉｕＤａｃｈｅｎｇ；ＷｕＬｉｈｕａ（ＳｃｈｏｏｌｏｆＭａｎａｇｅｍｅｎｔ，ＷｕｈａｎＵｎｉ...     

Complete prolongation for infinitesimal automorphisms on almost complex manifolds

We study the almost complex manifold and its symmetry algebra, the set of germs of infinitesimal automorphisms. The main result is that there is a certain condition to a torsion bundle under which the dimension of the symmetry algebra can not exceed 4 in the case of complex dimension 2. We will give an example which attains the maximal dimension 4, and another example without non-degeneracy whose symmetry algebra is of infinite dimension.     

Conceptions of a sequence formed in secondary schools

The paper is based on research carried out on secondary school students and students commencing their university studies in mathematics. The basic purpose of the research was to investigate the understanding of the concept of a sequence and to determine the sources of the formation of revealed conceptions. To achieve the objectives an expanded set of selected mathematical situations – simple but not quite standard – were investigated and various other research instruments were used. The students’ conceptions were divided into two groups. In the first group a sequence was perceived as a function, in the second it was associated with ordered elements. The diversity of the last set of conceptions was particularly interesting. The students understood the word ‘ordered’ as some kind of relationship between the terms of a sequence, a certain regularity or harmony. In the paper some ways of correcting the conceptions revealed and introducing the concept of a sequence in schools were also discussed.     

Lattices in the Pseudoeuclidean Plane

Because of the metric structure of the pseudoeuclidean plane, lattices have some remarkable features, e.g., concerning their symmetry groups and concerning isotropic, space- and time-like elements. Especially (p,q,s)-lattices, which are characterized by the fact that their symmetry groups contain hyperbolic rotations, are showing some interesting structures and substructures.     

Finite edge-transitive Cayley graphs and rotary Cayley maps

This paper aims to develop a theory for studying Cayley graphs, especially for those with a high degree of symmetry. The theory consists of analysing several types of basic Cayley graphs (normal, bi-normal, and core-free), and analysing several operations of Cayley graphs (core quotient, normal quotient, and imprimitive quotient). It provides methods for constructing and characterising various combinatorial objects, such as half-transitive graphs, (orientable and non-orientable) regular Cayley maps, vertex-transitive non-Cayley graphs, and permutation groups containing certain regular subgroups.

In particular, a characterisation is given of locally primitive holomorph Cayley graphs, and a classification is given of rotary Cayley maps of simple groups. Also a complete classification is given of primitive permutation groups that contain a regular dihedral subgroup.

     

Structuralist Logic: Implications, Inferences, and Consequences

On a structuralist account of logic, the logical operators, as well as modal operators are defined by the specific ways that they interact with respect to implication. As a consequence, the same logical operator (conjunction, negation etc.) can appear to be very different with a variation in the implication relation of a structure. We illustrate this idea by showing that certain operators that are usually regarded as extra-logical concepts (Tarskian algebraic operations on theories, mereological sum, products and negates of individuals, intuitionistic operations on mathematical problems, epistemic operations on certain belief states) are simply the logical operators that are deployed in different implication structures. That makes certain logical notions more omnipresent than one would think. Mathematics Subject Classification (2000): Primary 03B22; Secondary 03B20, 03B42, 03B60     

Pseudotoric Lagrangian fibrations of toric and nontoric fano varieties

We introduce the notion of a pseudotoric structure on a symplectic manifold, generalizing the notion of a toric structure. We show that such a pseudotoric structure can exist on toric and nontoric symplectic manifolds. For the toric manifolds, it describes deformations of the standard toric Lagrangian fibrations; for the nontoric ones, it gives Lagrangian fibrations with singularities that are very close to the toric fibrations. We present examples of toric manifolds with different pseudotoric structures and prove that certain nontoric manifolds (smooth complex quadrics) have such structures. In the future, introducing this new structure can be useful for generalizing the geometric quantization and mirror symmetry methods that work well in the toric case to a broader class of Fano varieties.