arXiv research

Smooth knots can be embedded into a specific Menger continuum.

The fundamental group of the Menger universal curve is uncountable and not free, although all of its finitely generated subgroups are free. It contains an isomorphic copy of the fundamental group of every one-dimensional separable metric space and an isomorphic copy of the fundamental group of every planar Peano contin…

We show that every inner metric space X is the metric quotient of a complete R-tree via a free isometric action, which we call the covering R-tree of X. The quotient mapping is a weak submetry (hence, open) and light. In the case of compact 1-dimensional geodesic space X, the free isometric action is via a subgroup of …

Constructs infinitely many non-equivalent wild knots in Menger sponge.

The boundary of certain hyperbolic groups is like a Menger curve.

Gradient flow preserves speed for integral Menger curvature curves.

Characterizes Coxeter groups with specific boundary shapes.

Knots can be embedded into fractals like the Menger Sponge and Sierpinski Tetrahedron.

We prove that the boundary of a right-angled hyperbolic building is a universal Menger space. Corollary: the 3-dimensional universal Menger space is the boundary of some Gromov-hyperbolic group.

Karl Menger's 1934 paper on the St. Petersburg paradox contains mathematical errors that invalidate his conclusion that unbounded utility functions, specifically Bernoulli's logarithmic utility, fail to resolve modified versions of the St. Petersburg paradox.

We show that the n-homotopy category of connected (n+1)-dimensional Menger manifolds is isomorphic to the homotopy category of connected Hilbert cube manifolds whose k-dimensional homotopy groups are trivial for each k > n.

A generic finite presentation defines a word hyperbolic group whose boundary is homeomorphic to the Menger curve. In this article, we produce the first known examples of non-hyperbolic $CAT(0)$ groups whose visual boundary is homeomorphic to the Menger curve. The examples in question are the Coxeter groups whose nerve …

Researchers found a new hyperbolic 3-orbifold using a Menger curve.

Triangulates surfaces with bounded energy using diffeomorphisms.

The n-dimensional hypergeometric integrals associated with a hypersphere arrangement are formulated by the pairing of n-dimensional twisted cohomology and its dual. Under the condition of general position there are stated some results which concern an explicit representation of the standard form by a special (NBC) basi…

Two groups with specific limit sets in hyperbolic spaces are identified.

In each Menger manifold $M$ we construct: (i) a closed nowhere dense subset $M_0$ which is homeomorphic to $M$ and is universal nowhere dense in the sense that for each nowhere dense set $A\subset M$ there is a homeomorphism $h$ of $M$ such that $h(A)\subset M_0$; (ii) a meager $F_σ$-set $Σ_0\subset M$ which is univers…

We investigate knot-theoretic properties of geometrically defined curvature energies such as integral Menger curvature. Elementary radii-functions, such as the circumradius of three points, generate a family of knot energies guaranteeing self-avoidance and a varying degree of higher regularity of finite energy curves. …

We prove some infinitesimal analogs of classical results of Menger, Schoenberg and Blumenthal giving the existence conditions for isometric embeddings of metric spaces in the finite-dimensional Euclidean spaces.

We generalize the notion of integral Menger curvature introduced by Gonzalez and Maddocks by decoupling the powers in the integrand. This leads to a new two-parameter family of knot energies $intM^{p,q}$. We classify finite-energy curves in terms of Sobolev-Slobodeckij spaces. Moreover, restricting to the range of para…

We extend the edge version of the classical Menger's Theorem for undirected graphs to $n$-dimensional simplicial complexes with chains over the field $\mathbb{F}_2$. The classical Menger's Theorem states that two different vertices in an undirected graph can be connected by $k$ pairwise edge-disjoint paths if, and only…

We construct and embedding of a Nöbeling space $N^n_{n-2}$ of codimension $2$ into a Menger space $M^n_{n-2}$ of codimension $2$. This solves an open problem stated by R.~Engelking in 1978 in codimension~$2$.

We prove that random groups in the Gromov density model, at any density, satisfy property (FA), i.e. they do not act non-trivially on trees. This implies that their Gromov boundaries, defined at density less than 1/2, are Menger curves.

We prove the following result announced in Todorov and Valov: Any homogeneous, metric $ANR$-continuum is a $V^n_G$-continuum provided $\dim_GX=n\geq 1$ and $\check{H}^n(X;G)\neq 0$, where $G$ is a principal ideal domain. This implies that any homogeneous $n$-dimensional metric $ANR$-continuum with $\check{H}^n(X;G)\neq…

We introduce the continuum self-similar tree (CSST) and characterize it topologically. We apply this to answer a question of Curien about the topology of the continuum random tree (CRT). We also give a topological characterization of other trees with branch points of finite or infinite valences.

We develop a theory of Nobeling manifolds similar to the theory of Hilbert space manifolds. We show that it reflects the theory of Menger manifolds developed by M. Bestvina and is its counterpart in the realm of complete spaces. In particular, the Nobeling manifold characterization conjecture is proven.

In this thesis, we consider the knot energy "integral Menger curvature" which is the triple integral over the inverse of the classic circumradius of three distinct points on the given knot to the power $p\in [2,\infty)$. We prove the existence of the first variation for a subset of a certain fractional Sobolev space if…

Continuum Dropout improves neural differential equations by preventing overfitting.

Dimension reduction of multivariate data supervised by auxiliary information is considered. A series of basis for dimension reduction is obtained as minimizers of a novel criterion. The proposed method is akin to continuum regression, and the resulting basis is called continuum directions. With a presence of binary sup…

Derives continuum model from discrete $\varepsilon$-graphs with connectivity functional.

We present two identities (contiguity relation and variation formula) concerning the volume of a spherically faced simplex in the Euclidean space. These identities are described in terms of Cayley-Menger determinants and their differentials involved with hypersphere arrangements. They are derived as a limit of fundamen…

The present chapter gives an overview on results for discrete knot energies. These discrete energies are designed to make swift numerical computations and thus open the field to computational methods. Additionally, they provide an independent, geometrically pleasing and consistent discrete model that behaves similarly …

An important question that discrete approaches to quantum gravity must address is how continuum features of spacetime can be recovered from the discrete substructure. Here, we examine this question within the causal set approach to quantum gravity, where the substructure replacing the spacetime continuum is a locally f…

Generalizes Alexandroff's $V^n$-continua to cohomological dimensions.

Proves continuum limits of Lipschitz learning using Γ-convergence.

This paper studies continuum-armed bandits under Besov smoothness conditions and derives minimax rates.

A mesh-free method solves continuum-marginal optimal transport problems.

Optimal reinsurance contracts designed for a continuum of risk types.

We construct a Fredholm module on self-similar sets such as the Cantor dust, the Sierpinski carpet and the Menger sponge. Our construction is a higher dimensional analogue of Connes' combinatorial construction of the Fredholm module on the Cantor set. We also calculate the Dixmier trace of two operators induced by the …

We characterize those planar Peano continua that are homotopy equivalent to 1-dimensional sets. While many planar Peano continua are not homotopically 1-dimensional, we prove that each has fundamental group that embeds in the fundamental group of a 1-dimensional planar Peano continuum. We leave open the following quest…

This work proves the continuum limit of t-SNE for data visualization.

Gradient flows on graphons converge to curves on graphon space.

Many statistical learning problems can be posed as minimization of a sum of two convex functions, one typically a composition of non-smooth and linear functions. Examples include regression under structured sparsity assumptions. Popular algorithms for solving such problems, e.g., ADMM, often involve non-trivial optimiz…

Continuum transformers learn operators in context via gradient descent.

Given a trivalent graph in the 3-dimensional Euclidean space, we call it a discrete surface because it has a tangent space at each vertex determined by its neighbor vertices. To abstract a continuum object hidden in the discrete surface, we introduce a subdivision method by applying the Goldberg-Coxeter subdivision and…

We show how to associate an R-tree to the set of cut points of a continuum. If X is a continuum without cut points we show how to associate an R-tree to the set of cut pairs of X.

Continuum-wise hyperbolicity is exactly the pseudo-Anosov dynamics with spine singularities.

This paper establishes the consistency of spectral approaches to data clustering. We consider clustering of point clouds obtained as samples of a ground-truth measure. A graph representing the point cloud is obtained by assigning weights to edges based on the distance between the points they connect. We investigate the…