Algebras of multiplace functions for signatures containing antidomain

We define antidomain operations for algebras of multiplace partial functions. For all signatures containing composition, the antidomain operations and any subset of intersection, preferential union and fixset, we give finite equational or quasiequational axiomatisations for the representation class. We do the same for the question of representability by injective multiplace partial functions. For all our representation theorems, it is an immediate corollary of our proof that the finite representation property holds for the representation class. We show that for a large set of signatures, the representation classes have equational theories that are coNP-complete.     

Signatures and higher signatures of S^1-quotients

We define and study the signature, -genus and higher signatures of the quotient space of an -action on a closed oriented manifold. We give applications to questions of positive scalar curvature and to an Equivariant Novikov Conjecture. Received: 13 January 1999     

Schur dynamics of the Schur processes

We construct discrete time Markov chains that preserve the class of Schur processes on partitions and signatures.One application is a simple exact sampling algorithm for qvolume-distributed skew plane partitions with an arbitrary back wall. Another application is a construction of Markov chains on infinite Gelfand–Tsetlin schemes that represent deterministic flows on the space of extreme characters of the infinite-dimensional unitary group.     

Graded Steinberg algebras and partial actions

Given a graded ample Hausdorff groupoid, we realise its graded Steinberg algebra as a partial skew inverse semigroup ring. We use this to show that for a partial action of a discrete group on a locally compact Hausdorff topological space which is totally disconnected, the Steinberg algebra of the associated groupoid is graded isomorphic to the corresponding partial skew group ring. We show that there is a one-to-one correspondence between the open invariant subsets of the topological space and the graded ideals of the partial skew group ring. We also consider the algebraic version of the partial $C^{?}$-algebra of an abelian group and realise it as a partial skew group ring via a partial action of the group on a topological space. Applications to the theory of Leavitt path algebras are given.     

Bicompleting weightable quasi-metric spaces and partial metric spaces

We show that the bicompletion of a weightable quasi-metric space is a weightable quasi-metric space. From this result we deduce that any partial metric space has an (up to isometry) unique partial metric bicompletion. Some other consequences are derived. In particular, applications to two interesting examples of partial metric spaces which appear in Computer Science, as the domain of words and the complexity space, are given. The second and third listed authors are partially supported by grants from Spanish Ministry of Science and Technology, BFM 2000-III, and Polytechnical University of Valencia.     

Monotone normality and neighborhood assignments

We prove that every neighborhood assignment for a monotonically normal space has a kernel which is homeomorphic to some subspace of an ordinal. As a corollary, every monotone neighborhood assignment for a monotonically normal space has a discrete kernel, which gives a partial answer to a question posed in Buzyakova et al. (2007) [5]. We also give an example of a regular space which has a neighborhood assignment with no kernels homeomorphic to any subspace of an ordinal.     

Completions of partial metrics into value lattices

In this paper we investigate some notions of completion of partial metric spaces, including the bicompletion, the Smyth completion, and a new “spherical completion”. Given an auxiliary relation, we show that it arises from a totally bounded partial metric space, and the spherical completion of such a space is its round ideal completion. We also give an example of a totally bounded partial metric space whose bicompletion and Smyth completion are not continuous posets. Finally, we present an example of a totally bounded partial metric giving rise to the Scott and lower topologies of a continuous poset, but whose spherical completion is not a continuous poset.     

Extending partial isometries

We show that a finite metric spaceA admits an extension to a finite metric spaceB so that each partial isometry ofA extends to an isometry ofB. We also prove a more precise result on extending a single partial isometry of a finite metric space. Both these results have consequences for the structure of the isometry groups of the rational Urysohn metric space and the Urysohn metric space. Research supported by NSF grant DMS-0400931. I would like to thank Ward Henson and Alekos Kechris for conversations and emails regarding the paper. Part of this work was done when I visited Caltech in May 2004. I thank the mathematics department there for support     

Pauli theorem in the description of n-dimensional spinors in the Clifford algebra formalism

We discuss a generalized Pauli theorem and its possible applications for describing n-dimensional (Dirac, Weyl, Majorana, and Majorana-Weyl) spinors in the Clifford algebra formalism. We give the explicit form of elements that realize generalizations of Dirac, charge, and Majorana conjugations in the case of arbitrary space dimensions and signatures, using the notion of the Clifford algebra additional signature to describe conjugations. We show that the additional signature can take only certain values despite its dependence on the matrix representation     

Polygon Recognition and Symmetry Detection

We introduce an approach based on moving frames for polygon recognition and symmetry detection. We present detailed algorithms for the recognition of polygons in R² modulo the special Euclidean, Euclidean, equi-affine, skewed-affine, and similarity Lie groups. We also solve the case of polygons in the Poincar\'e half-plane under the action of SL(2) and explain a method applicable to Lie group actions in general. The time complexity of our algorithms is linear in the number of vertices and they are noise resistant. The signatures used allow the detection of partial, as well as approximate, equivalences.     

Arithmetic Laplacians

We develop an arithmetic analogue of elliptic partial differential equations. The role of the space coordinates is played by a family of primes, and that of the space derivatives along the various primes are played by corresponding Fermat quotient operators subjected to certain commutation relations. This leads to arithmetic linear partial differential equations on algebraic groups that are analogues of certain operators in analysis constructed from Laplacians. We classify all such equations on one-dimensional groups, and analyze their spaces of solutions.     

On the stability of abstract monotone impulsive differential equations in terms of two measures

We consider differential equations in a Banach space subjected to an impulsive influence at fixed times. It is assumed that a partial ordering is introduced in the Banach space by using a normal cone and that the differential equations are monotone with respect to the initial data. We propose a new approach to the construction of comparison systems in finite-dimensional spaces without using auxiliary Lyapunov-type functions. On the basis of this approach, we establish sufficient conditions for the stability of this class of differential equations in terms of two measures. In this case, a Birkhoff measure is chosen as the measure of initial displacements, and the norm in the given Banach space is used as the measure of current displacements. We present some examples of investigations of the impulsive systems of differential equations in the critical cases and linear impulsive systems of partial differential equations.     

Bifurcation analysis of some forced Lu systems

Bifurcation behaviour of a forced Lu system is analyzed as the system parameter c and a forcing parameter F are varied. The Lu system belongs to a family of generalized Lorenz system. Members of this family are known to exhibit different types of chaotic attractors. Some of these attractors have been named Lorenz type L, Lu or Transition type T, Chen type T and Transverse 8 Type S. These different types of chaotic attractors are visually distinct when the parameters are widely separated. However, there is a need for identifying the precise point where transition from one type of chaotic attractor to another takes place. We identified signatures in the return map, which could be used for determining the point of transition and classifying the different types of chaotic attractors. These signatures helped to identify the point in coordinate space associated with such transitions. We find that such transitions take place when a chaotic attractor comes very close to a one-dimensional manifold on which the time derivatives of two of the variables is zero. We also find that just before coming to this point in coordinate space associated with the transition, the trajectory had approached, very closely, the equilibrium point at the origin.     

Controllability of Second Order Impulsive Neutral Functional Differential Inclusions with Infinite Delay

This paper is concerned with controllability of a partial neutral functional differential inclusion of second order with impulse effect and infinite delay. We introduce a new phase space to prove the controllability of an inclusion which consists of an impulse effect with infinite delay. We claim that the phase space considered by different authors is not correct. We establish the controllability of mild solutions using a fixed point theorem for contraction multi-valued maps and without assuming compactness of the family of cosine operators.     

Stratifications on the Ran Space

We describe a partial order on finite simplicial complexes. This partial order provides a poset stratification of the product of the Ran space of a metric space and the nonnegative real numbers, through the ?ech simplicial complex. We show that paths in this product space respecting its stratification induce simplicial maps between the endpoints of the path.

     

Algebraic signatures of convex and non-convex codes

A convex code is a binary code generated by the pattern of intersections of a collection of open convex sets in some Euclidean space. Convex codes are relevant to neuroscience as they arise from the activity of neurons that have convex receptive fields. In this paper, we develop algebraic methods to determine if a code is convex. Specifically, we use the neural ideal of a code, which is a generalization of the Stanley–Reisner ideal. Using the neural ideal together with its standard generating set, the canonical form, we provide algebraic signatures of certain families of codes that are non-convex. We connect these signatures to the precise conditions on the arrangement of sets that prevent the codes from being convex. Finally, we also provide algebraic signatures for some families of codes that are convex, including the class of intersection-complete codes. These results allow us to detect convexity and non-convexity in a variety of situations, and point to some interesting open questions.     

Well-posed solvability of volterra integro-differential equations in Hilbert space

We study the well-posed solvability of initial value problems for abstract integrodifferential equations with unbounded operator coefficients in a Hilbert space. These equations are an abstract form of linear partial integro-differential equations that arise in the theory of viscoelasticity and have a series of other important applications. We obtain results on the wellposed solvability of the considered integro-differential equations in weighted Sobolev spaces of vector functions defined on the positive half-line and ranging in a Hilbert space.     

Diagonal structures on topological spaces

We investigate when a topological space admits a partial product operation satisfying some rather weak continuity restrictions and almost nothing else-the only algebraic requirement is that some element e of X is a left and a right identity with respect to this multiplication. The operation is called partial diagonalization of X at e. Several sufficient conditions for a space to be partially diagonalizable are established. On the other hand, it is shown that certain deep results about the topological structure of compact topological groups can be extended to partially diagonalizable compact spaces. We also discover that partial diagonalizability plays an important role in the theory of cardinal invariants, in the study of homogeneous spaces, and in such classical topics of general topology as the theory of Stone–Čech compactification and the theory of Hewitt–Nachbin compactification. The notions of a Moscow space and of a C-embedding are instrumental in our study.