arXiv research

New problems defined for maps onto graphs related to Reeb spaces.

New proof for surface groups using Reeb graphs and Morse functions.

Researchers reconstruct algebraic maps onto curves based on prescribed Reeb graphs.

An ensemble technique is characterized by the mechanism that generates the components and by the mechanism that combines them. A common way to achieve the consensus is to enable each component to equally participate in the aggregation process. A problem with this approach is that poor components are likely to negativel…

The paper studies mapping class groups of locally finite graphs and their associated sphere complexes.

We construct examples of free-by-cyclic hyperbolic groups which fiber in infinitely many ways over Z. The construction involves adding a specialized square 2-cell to a non-positively curved, squared 2-complex defined by labeled oriented graphs. The fundamental groups of the resulting complexes are hyperbolic, free-by-c…

New method separates graph structure from node attributes to recover lost signal.

New rays on infinite type surfaces help understand their boundaries.

Proof that certain Anosov flows are almost equivalent.

In this paper, it is shown that every orientable closed 3-manifold maps with nonzero degree onto at most finitely many homeomorphically distinct irreducible non-geometric orientable closed 3-manifolds. Moreover, given any nonzero integer, as a mapping degree up to sign, every orientable closed 3-manifold maps with that…

We prove that a continuum $X$ is tree-like (resp. circle-like, chainable) if and only if for each open cover $\U_4=\{U_1,U_2,U_3,U_4\}$ of $X$ there is a $\U_4$-map $f:X\to Y$ onto a tree (resp. onto the circle, onto the interval). A continuum $X$ is an acyclic curve if and only if for each open cover $\U_3=\{U_1,U_2,U…

We construct harmonic diffeomorphisms from the complex plane $C$ onto any Hadamard surface $M$ whose curvature is bounded above by a negative constant. For that, we prove a Jenkins-Serrin type theorem for minimal graphs in $M\times R$ over domains of $M$ bounded by ideal geodesic polygons and show the existence of a se…

The paper resolves a problem about metric inequivalence and characterizes proper holomorphic maps.

There is a concept in digital topology of a shy map. We define an analogous concept for topological spaces: We say a function is shy if it is continuous and the inverse image of every path-connected subset of its image is path-connected. Some basic properties of such maps are presented. For example, every shy map onto …

Consider a planar, bounded, $m$-connected region $Ω$, and let $\bordΩ$ be its boundary. Let $\mathcal{T}$ be a cellular decomposition of $Ω\cup\bordΩ$, where each 2-cell is either a triangle or a quadrilateral. From these data and a conductance function we construct a canonical pair $(S,f)$ where $S$ is a genus $(m-1)$…

We prove that the existence of a positively defined, invariant Einstein metric $m$ on a connected homogeneous space $G/H$ of a compact Lie group $G$ is the consequence of non-contractibility of some compact set $C=X_{G,H}^Σ$ (Böhm polyhedron) introduced by C.Böhm. There is a natural continuous map of $C$ onto the flag …

Constructs minimal surfaces over Pitot quadrilaterals using harmonic diffeomorphisms.

Study of one-dimensional non-Hausdorff manifolds and their quotient to CW complexes.

In this paper we prove that geodesic mappings of (pseudo-) Riemannian manifolds preserve the class of differentiability \hbox{$(C^r, r\geq1)$}. Also, if the Einstein space $V_n$ admits a non trivial geodesic mapping onto a \hbox{(pseudo-)} Riemannian manifold $\bar V_n\in C^1$, then $\bar V_n$ is an Einstein space. If …

Real-world applications often combine learning and optimization problems on graphs. For instance, our objective may be to cluster the graph in order to detect meaningful communities (or solve other common graph optimization problems such as facility location, maxcut, and so on). However, graphs or related attributes ar…

ProDAG uses variational inference to learn DAGs with uncertainty quantification.

We prove that every endomorphism of the mapping class group of an orientable surface onto a subgroup of finite index is in fact an automorphism.

A novel method compresses point cloud attributes by folding them onto a 2D grid.

Maps can transform non-homeomorphic manifolds into the same space.

Bi-geodesic mappings preserve distances on hyperbolic surfaces with boundaries.

The author studies regions foliated by 1D families of functions and their applications.

We study the existence or not of harmonic diffeomorphisms between certain domains in the Euclidean 2-sphere. In particular, we show harmonic diffeomorphisms from circular domains in the complex plane onto finitely punctured spheres, with at least two punctures. This result follows from a general existence theorem for m…

We present the non-trivial example how to generate non-Euclidean geometries from associative unital algebras. We consider bundles of the sphere of the degenerate non-Eucleadian space and its two models. The first (conformal) model is obtained by the mapping S onto a plane pass through the origin. It is analogous to the…

The mapping class group of a Heegaard splitting is the group of automorphisms of the ambient 3-manifold that take the surface onto itself, modulo isotopies that keep the surface on itself. We characterize the mapping classes that restrict to periodic and reducible automorphisms of the surface.

BiLipschitz mappings can be extended to preserve area.

Suppose that there exists an epimorphism from the knot group of a $2$-bridge knot $K$ onto that of another knot $K'$. In this paper, we study the relationship between their crossing numbers $c(K)$ and $c(K')$. Especially it is shown that $c(K)$ is greater than or equal to $3 c(K')$ and we estimate how many knot groups …

New Grunsky operator for disk maps to complex plane.

An oriented connected closed manifold $M^n$ is called a URC-manifold if for any oriented connected closed manifold $N^n$ of the same dimension there exists a nonzero degree mapping of a finite-fold covering $\widehat{M}^n$ of $M^n$ onto $N^n$. This condition is equivalent to the following: For any $n$-dimensional integ…

Manifold learning has been successfully applied to a variety of medical imaging problems. Its use in real-time applications requires fast projection onto the low-dimensional space. To this end, out-of-sample extensions are applied by constructing an interpolation function that maps from the input space to the low-dimen…

For modelling of various physical processes, geodesic lines and almost geodesic curves serve as a useful tool. Trasformations or mappings between spaces (endowed with a metric or connection) which preserve such curves play an important role in physics, particularly in mechanics, and in geometry as well. Our aim is to c…

In planar algebras, we show how to project certain simple "quadratic" tangles onto the linear space spanned by "linear" and "constant" tangles. We obtain some corollaries about the principal graphs and annular structure of subfactors.

We deform a map into a Riemannian manifold that is horizontal with respect to a submersion onto a non-positively curved manifold and satisfies a Chow condition into a harmonic one through a horizontal homotopy.

We use controlled topology applied to the action of the infinite dihedral group on a partially compactified plane and deduce two consequences for algebraic K-theory. The first is that the family in the K-theoretic Farrell-Jones conjecture can be reduced to only those virtually cyclic groups which admit a surjection wit…

We determine a particular class of Roter type warped product manifolds. We show that every manifold of that class admits a geodesic mapping onto a some Roter type warped product manifold. Moreover, both geodesically related manifolds are pseudosymmetric of constant type.

The main goal of this paper is to prove that a connected bounded geometry complete Kahler manifold which has at least 3 filtered ends admits a proper holomorphic mapping onto a Riemann surface. This also provides a different proof of the theorem of Gromov and Schoen that, for a connected compact Kahler manifold whose f…

We study complete minimal graphs in HxR, which take asymptotic boundary values plus and minus infinity on alternating sides of an ideal inscribed polygon Γ in H. We give necessary and sufficient conditions on the "lenghts" of the sides of the polygon (and all inscribed polygons in Γ) that ensure the existence…

If Pi: M -> B is an onto smooth maximal rank map between complete Riemannian manifolds M and B with bounded geometry, we prove sufficient conditions for M to be roughly isometric to the Riemannian product FxB, where F is a fiber of M.

We study forgetful maps between Deligne-Mostow moduli spaces of weighted points on P^1, and classify the forgetful maps that extend to a map of orbifolds between the stable completions. The cases where this happens include the Livné fibrations and the Mostow/Toledo maps between complex hyperbolic surfaces. They also in…

Maps between certain Lipschitz manifolds are isometries if they preserve volume.

Simplicial sets deformation retract onto transverse simplices.

A Reeb space is defined as the space of all the connected components of inverse images of a smooth map, which is a fundamental tool in studying smooth manifolds using generic smooth maps whose codimensions are not positive such as Morse functions, their higher dimensional versions including fold maps and general stable…

The large-N limit of Segal-Bargmann transform on spheres is studied.

We present a proof due to Duistermaat that the gradient flow of the norm squared of the moment map defines a deformation retract of the appropriate piece of the manifold onto the zero level set of the moment map. Duistermaat's proof is an adaptation of Lojasiewicz's argument for analytic functions to functions which ar…