An Investigation of New Graph Invariants Related to the Domination Number of Random Proximity Catch Digraphs

Proximity catch digraphs (PCDs) are a special type of proximity graphs based on proximity maps which yield proximity regions. PCDs are defined using the relative allocation of points from two or more classes in a region of interest and have applications in various fields. We introduce some auxiliary tools for PCDs and graph invariants related to the domination number of the PCDs and investigate their probabilistic properties. We consider the cases in which the vertices of the PCDs come from uniform and non-uniform distributions in the region of interest. We also provide some of the newly defined proximity maps as illustrative examples.     

Buchstaber Invariants of Universal Complexes

Davis and Januszkiewicz introduced (real and complex) universal complexes to give an equivalent definition of characteristic maps of simple polytopes,which now can be seen as "colorings".The author derives an equivalent definition of Buchstaber invariants of a simplicial complex K,then interprets the difference of the real and complex Buchstaber invariants of K as the obstruction to liftings of nondegenerate simplicial maps from K to the real universal complex or the complex universal complex.It was proved by Ayzenberg that real universal complexes can not be nondegenerately mapped into complex universal complexes when dimension is 3.This paper presents that there is a nondegenerate map from 3-dimensional real universal complex to 4-dimensional complex universal complex.     

Duality properties of indicatrices of knots

The bridge index and superbridge index of a knot are important invariants in knot theory. We define the bridge map of a knot conformation, which is closely related to these two invariants, and interpret it in terms of the tangent indicatrix of the knot conformation. Using the concepts of dual and derivative curves of spherical curves as introduced by Arnold, we show that the graph of the bridge map is the union of the binormal indicatrix, its antipodal curve, and some number of great circles. Similarly, we define the inflection map of a knot conformation, interpret it in terms of the binormal indicatrix, and express its graph in terms of the tangent indicatrix. This duality relationship is also studied for another dual pair of curves, the normal and Darboux indicatrices of a knot conformation. The analogous concepts are defined and results are derived for stick knots.     

Broccoli curves and the tropical invariance of Welschinger numbers

In this paper we introduce broccoli curves, certain plane tropical curves of genus zero related to real algebraic curves. The numbers of these broccoli curves through given points are independent of the chosen points — for arbitrary choices of the directions of the ends of the curves, possibly with higher weights, and also if some of the ends are fixed. In the toric Del Pezzo case we show that these broccoli invariants are equal to the Welschinger invariants (with real and complex conjugate point conditions), thus providing a proof of the independence of Welschinger invariants of the point conditions within tropical geometry. The general case gives rise to a tropical Caporaso–Harris formula for broccoli curves which suffices to compute all Welschinger invariants of the plane.     

Topological entropy of maps on the real line

The aim of this paper is to introduce a definition of topological entropy for continuous maps such that, at least for continuous real maps, it keeps the following general philosophy: positive topological entropy implies that the map has a complicated dynamical behaviour. Besides, we pursue that our definition keeps some properties which are hold by the classic definition of topological entropy introduced for compact sets.     

On global singularities of Sobolev mappings

We define various invariants for Sobolev mappings defined between manifolds which are stable under perturbation with respect to the strong Sobolev topology. We show that these invariants classify various types of ``global singularities" of Sobolev maps. These invariants are used to give a simple characterization of the strong closure of the set of smooth maps in the Sobolev space.     

Refining invariants for computing autotopism groups of partial Latin rectangles

Prior to using computational tools that find the autotopism group of a partial Latin rectangle (its stabilizer group under row, column and symbol permutations), it is beneficial to find partitions of the rows, columns and symbols that are invariant under autotopisms and are as fine as possible. We look at the lattices formed by these partitions and introduce two invariant refining maps on these lattices. The first map generalizes the strong entry invariant in a previous work. The second map utilizes some bipartite graphs, introduced here, whose structure is determined by pairs of rows (or columns, or symbols). Experimental results indicate that in most cases (ordinarily 99%+), the combined use of these invariants gives the theoretical best partition of the rows, columns and symbols, outperforms the strong entry invariant, which only gives the theoretical best partitions in roughly 80% of the cases.     

Convergence and existence results for best proximity points

We provide a positive answer to a question raised by Eldred and Veeramani [A.A. Eldred, P. Veeramani, Existence and convergence of best proximity points, J. Math. Anal. Appl. 323 (2006) 1001–1006] about the existence of a best proximity point for a cyclic contraction map in a reflexive Banach space. Moreover, we introduce a new class of maps, called cyclic φ$\phi$-contractions, which contains the cyclic contraction maps as a subclass. Convergence and existence results of best proximity points for cyclic φ$\phi$-contraction maps are also obtained.     

Graph Approach to the Computation of the Homology of Continuous Maps

We introduce an efficient algorithm to compute the homomorphism induced in (relative) homology by a continous map. The algorithm is based on a cubical approximation of the map and the theory of multivalued maps. A software implementation of the algorithms introduced in this paper is available at [27].     

Diagram Invariants of Knots and the Kontsevich Integral

This paper is devoted to the solution of the integration problem for higher derivatives of finite-type invariants (so-called weight systems) up to real invariants defined on the space of knots. __________ Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 19, Topology and Noncommutative Geometry, 2004.     

A new version of Bishop frame and an application to spherical images

In this work, we introduce a new version of Bishop frame using a common vector field as binormal vector field of a regular curve and call this frame as “Type-2 Bishop Frame”. Thereafter, by translating type-2 Bishop frame vectors to the center of unit sphere of three-dimensional Euclidean space, we introduce new spherical images and call them as type-2 Bishop spherical images. Frenet-Serret apparatus of these new spherical images are obtained in terms of base curve's type-2 Bishop invariants. Additionally, we express some interesting relations and illustrate two examples of our main results.     

First Order Semi-Local Invariants of Stable Maps of 3-Manifolds into the Plane

In the late 1980s, Vassiliev introduced new graded numericalinvariants of knots, which are now called Vassiliev invariantsor finite-type invariants. Since he made this definition, manypeople have been trying to construct Vassiliev type invariantsfor various mapping spaces. In the early 1990s, Arnold and Goryunovintroduced the notion of first order (local) invariants of stablemaps. In this paper, we define and study first order semi-localinvariants of stable maps and those of stable fold maps of aclosed orientable 3-dimensional manifold into the plane. Weshow that there are essentially eight first order semi-localinvariants. For a stable map, one of them is a constant invariant,six of them count the number of singular fibers of a given typewhich appear discretely (there are exactly six types of suchsingular fibers), and the last one is the Euler characteristicof the Stein factorization of this stable map. Besides theseinvariants, for stable fold maps, the Bennequin invariant ofthe singular value set corresponding to definite fold pointsis also a first order semi-local invariant. Our study of unstablefold maps with codimension 1 provides invariants for the connectedcomponents of the set of all fold maps. 2000 Mathematics SubjectClassification 57R45 (primary), 32S20, 58K15 (secondary).     

The classification of topologically expansive lorenz maps

We show that topologically expansive Lorenz maps can be described up to topological conjugacy by their kneading invariants. We also give a simple condition on pairs of symbol sequences which is satisfied if and only if that pair of sequences is the kneading invariant for some topologically expansive Lorenz map. A simple extension of the theorems to the case of expansive maps of the interval with multiple discontinuities is described.     

Linear maps preserving commutativity

Commutativity-preserving maps on the real space of all real symmetric or complex self-adjoint matrices are characterized. Related results are given for adjoint-preserving maps defined on all n × n matrices. These results are extended to infinite dimensions in the case of invertible maps.     

A correspondence between scene analysis and motions of frameworks

For over 100 years there has been a known correspondence between plane pictures of spherical polyhedra and static stresses in bar and joint frameworks with planar graphs. We define a new map from any plane picture of a general scene to a corresponding plate and bar framework in the plane. This map, and its return map, give a simple isomorphism between the space of scenes over the picture and the space of instantaneous motions of the framework. When two simple equivalence relations on the pictures and on the frameworks are introduced, the maps define an isomorphism of the equivalence clases which permits the full translation of questions, techniques and theorems between the two fields of study. A direct geometric interpretation of this correspondence is described and an extension to systems with occlusion and tensegrity (systems of inequalities) is presented. The analogous patterns in higher dimensions are described, yielding a correspondence between motions of bar and body frameworks in 3-space and 6-dimensional scenes over 5-dimensional pictures.     

Seeking invariants for blow-analytic equivalence

We introduce some blow-analytic invariants of real analytic function-germsand discuss their properties. As a consequence, we obtain, for instance, the multiplicity of function-germs is a blow-analytic invariant.     

Generalized symplectic Cayley transforms and a higher order formula for the Conley–Zehnder index of symplectic paths

Using a generalized notion of symplectic Cayley transform in the symplectic group, we introduce a sequence of integer valued invariants (higher order signatures) associated with a degeneracy instant of a smooth path of symplectomorphisms. In the real analytic case, we give a formula for the Conley–Zehnder index in terms of the higher order signatures.     

Topological invariance of intersection homology without sheaves

In (1) Goresky and MacPherson defined intersection homology groups for triangulable pseudomanifolds and showed they were PL invariants. Then in [2] they generalized these groups to any pseudomanifold and showed they were topological invariants. These groups have generated a great deal of interest. However, [2] is difficult for many mathematicians (including this author) because it requires a familiarity with a great deal of hefty sheaf-theoretic machinery. This is too bad, because the basic ideas behind intersection homology (elucidated in [1]) are very geometric.In this paper we give a sheafless definition of intersection homology groups for an arbitrary stratified set and we give an elementary sheafless proof that they are topological invariants, i.e. independent of the stratification.In doing so, we find some new perversities whose intersection homology groups are topological invariants. Unfortunately, these new perverse intersection homology classes do not seem to intersect with anything (which is probably why they were ignored by Goresky and MacPherson). But in any case these groups are invariants of singular spaces which might be of some interest.     

Topological Persistence for Circle-Valued Maps

We study circle-valued maps and consider the persistence of the homology of their fibers. The outcome is a finite collection of computable invariants which answer the basic questions on persistence and in addition encode the topology of the source space and its relevant subspaces. Unlike persistence of real-valued maps, circle-valued maps enjoy a different class of invariants called Jordan cells in addition to bar codes. We establish a relation between the homology of the source space and of its relevant subspaces with these invariants and provide a new algorithm to compute these invariants from an input matrix that encodes a circle-valued map on an input simplicial complex.     

Totally positive extensions and weakly isotropic forms

The aim of this article is to analyse a new field invariant, relevant to (formally) real fields, defined as the supremum of the dimensions of all anisotropic, weakly isotropic quadratic forms over the field. This invariant is compared with the classical u-invariant and with the Hasse number. Furthermore, in order to be able to obtain examples of fields where these invariants take certain prescribed values, totally positive field extensions are studied.