arXiv research

New binary matrices improve compressed sensing with faster and less storage requirements.

This paper defines girth for knots and links, linking it to Khovanov homology.

Paper constructs graphs with girth four and distinct properties.

The study finds conditions for groups acting on CAT(0) cube complexes to have infinite girth.

Proves Girth Alternative for some HNN extensions, finds counterexamples.

Research shows finiteness in triangulations with girth constraints.

The girth of a finitely generated group G is the supremum of the girth of Cayley graphs for G over all finite generating sets. Let G be a finitely generated subgroup of the mapping class group Mod(S), where S is a compact orientable surface. Then, either G is virtually abelian or it has infinite girth; moreover, if we …

We construct simple curves from immersed curves in the setting of handlebodies and Heegaard splittings. We define a measure of complexity we call girth for closed curves in a handlebody. We extend this complexity to Heegaard splittings and pose a conjecture about all Heegaard splittings. We prove a test case of this co…

Ancient Ricci flows with bounded girth found in 3D and higher.

Study on Lin-Lu-Yau curvature and diameter of amply regular graphs.

In this note we introduce a natural Finsler structure on convex surfaces, referred to as the projective Finsler structure, which is dual in a sense to the obvious inclusion of a convex surface in a normed space. It has an associated projective girth, which is similar to the notion of girth defined by Schäffer. We prove…

The CD inequalities and CDE inequalities are useful in the estimate of curvature on graphs. This article is based on the ufinite graph with large girth, and finally concludes some curvature estimate in CD and CDE.

We introduce a new way to tabulate knots by representing knot diagrams using a pair of planar trees. This pair of trees have their edges labeled by integers, they have no valence 2 vertices, and they have the same number of valence 1 vertices. The number of valence 1 vertices of the trees is called the girth of the kno…

The study shows that certain Artin groups cannot contain hyperbolic manifold groups.

Ricci curvature was proposed by Ollivier in a general framework of metric measure spaces, and it has been studied extensively in the context of graphs in recent years. In this paper we prove upper bounds for Ollivier's Ricci curvature for bipartite graphs and for the graphs with girth at least 5. We also prove a genera…

Study on modified Ricci curvature on graphs, proving rigidity and deriving formulas.

The systole of a hyperbolic surface is bounded by a logarithmic function of its genus. This bound is sharp, in that there exist sequences of surfaces with genera tending to infinity that attain logarithmically large systoles. These are constructed by taking congruence covers of arithmetic surfaces. In this article we p…

The paper solves curvature problems on graphs using a special flow.

In this paper, the second of a series of two, we continue the study of higher index theory for expanders. We prove that if a sequence of graphs has girth tending to infinity, then the maximal coarse Baum-Connes assembly map is an isomorphism for the associated metric space $X$. As discussed in the first paper in this s…

In this paper, the first of a series of two, we continue the study of higher index theory for expanders. We prove that if a sequence of graphs is an expander and the girth of the graphs tends to infinity, then the coarse Baum-Connes assembly map is injective, but not surjective, for the associated metric space $X$. Exp…

The Ollivier Ricci flow with prescribed curvature on infinite graphs.

We prove that a dense subgroup of $\mathrm{Homeo}_{+}(I)$ is not elementary amenable. We also show that the topological group $\mathrm{Homeo}_{+}(I)$ does not satisfy the Stability of the Generators Property, moreover, any finitely generated subgroup of $\mathrm{Homeo}_{+}(I)$ admits a faithful discrete representation …

In [13], it is proved that any subgroup of $\mathrm{Diff}_{+}^{ω}(I)$ (the group of orientation preserving analytic diffeomorphisms of the interval) is either metaabelian or does not satisfy a law. A stronger question is asked whether or not the Girth Alternative holds for subgroups of $\mathrm{Diff}_{+}^{ω}(I)$. In th…

Curvature formulas on regular graphs identified bone idle edges and graphs.

New graphs with maximum degree 4 found to be Ricci-flat.

Khovanov homology of a link and chromatic graph homology are known to be isomorphic in a range of homological gradings that depend on the girth of a graph. We discuss patterns shared by these two homology theories. In particular, we improve the bounds for the homological span of chromatic homology by Helme-Guizon, Przy…

We examine the large systole problem, which concerns compact hyperbolic Riemannian surfaces whose systole, the length of the shortest noncontractible loops, grows logarithmically in genus. The generalization of a construction of Buser and Sarnak by Katz, Schaps, and Vishne, which uses principal "congruence" subgroups o…

The paper studies Ricci flow on graphs with prescribed curvature.

We introduce a new variant of the coarse Baum-Connes conjecture designed to tackle coarsely disconnected metric spaces called the boundary coarse Baum-Connes conjecture. We prove this conjecture for many coarsely disconnected spaces that are known to be counterexamples to the coarse Baum-Connes conjecture. In particula…

We prove six-dimensional nearly Kähler manifolds are Einstein.

Recent research in coarse geometry revealed similarities between certain concepts of analysis, large scale geometry, and topology. Property A of G.Yu is the coarse analog of amenability for groups and its generalization (exact spaces) was later strengthened to be the large scale analog of paracompact spaces using parti…

The paper introduces and studies a new type of submersion in Riemannian geometry.

In this note, the geography problem in dimension four is reviewed and then its extension to dimension six for the symplectic case is explained. Finally some examples in dimension six are provided.

Sharp bounds on diameter and eigenvalues for amply regular graphs.

Study of multi-moment maps on specific six-manifolds.

Fefferman and Graham showed some time ago that four dimensional conformal geometries could be analyzed in terms of six dimensional, ambient, Riemannian geometries admitting a closed homothety. Recently it was shown how conformal geometry provides a description of physics manifestly invariant under local choices of unit…

It is known that there are 34 classes of isomorphic connected simply connected six-dimensional nilpotent Lie groups. Of these, only 26 classes suppose left-invariant symplectic structures \cite{Goze-Khakim-Med}. In \cite{CFU2} it is shown that 14 classes of symplectic six-dimensional nilpotent Lie groups suppose compat…

The six nondegeneracy conditions of geometric nature that are satisfied by the only six possibly existing nondegenerate general classes I, II, III-1, III-2, IV-1, IV-2 of 5-dimensional CR manifolds are shown to be readable instantaneously from their elementarily normalized respective defining graphed equations, without…

Global obstructions found for conformally Einstein metrics in 6D.

Proof of existence of a complex structure on the six-sphere, followed by an explicit computation of its underlying integrable almost complex tensor by the aid of inner automorphisms of the octonions, is exhibited. Both are elementary and self-contained however the size and complexity of the emerging almost complex tens…

Study on non-flat two-plectic geometry of six-sphere and its Hamiltonian dynamics.

Paper proves non-existence of certain balanced metrics on six-manifolds.

Study almost complex structures on six-manifolds using twistor spaces.

Study para-complex structures on specific Lie groups, finding explicit forms and properties.

Study of complex structures on specific solvmanifolds, proving existence and non-existence results.

Local supertwistors help study 6D conformal supergravity.

We discuss the theorem on the existence of six points on a convex closed plane curve in which the curve has a contact of order six with the osculating conic. (This is the ``projective version'' of the well known four vertices theorem for a curve in the Euclidean plane.) We obtain this classical fact as a corollary of s…

In the present paper we study six dimensional solvable Lie algebras with special emphasis on those admitting a symplectic structure. We list all the symplectic structures that they admit and we compute their Betti numbers finding some properties about the codimension of the nilradical. Next, we consider the conjecture …