The heights of Alexandroff square transformation groups are computed and proven.
problem Computing possible heights of Alexandroff square transformation groups.
method Analyzing the heights of transformation groups for Alexandroff square, unit square with lexicographic order, and unit square with Euclidean topology.
result Proven heights for transformation groups of Alexandroff square, unit square with lexicographic order, and unit square with Euclidean topology.
We investigate the classical Alexandroff-Borsuk problem in the category of non-triangulable manifolds: Given an n-dimensional compact non-triangulable manifold Mn and ε>0, does there exist an ε-map of Mn onto an n-dimensional finite polyhedron which induces a homotopy equivalence?
Generalizes Alexandroff's Vn-continua to cohomological dimensions.
problem Extending Alexandroff's concept of Vn-continua to cohomological dimensions. method Proves that strongly locally homogeneous generalized continua with cohomological dimension n are generalized Vn-spaces. result Every strongly locally homogeneous continuum of covering dimension n is a Vn-continuum in the sense of Alexandroff. Universal spaces for finite topological spaces simplify shape descriptions.
problem Describing shape properties of compact metric spaces.
method Inverse limits of finite spaces and Alexandroff extensions.
result Universal spaces simplify shape descriptions of compact metric spaces.
Stability results for geometric equations in warped product spaces.
problem Geometric partial differential equations in warped product spaces.
method Stability theorem development for level sets of functions.
result Quantitative stability theorems for Serrin's problem and Alexandroff's theorem.
The article defines hyperconnected relator spaces and their properties.
problem Understanding the nearness of path-connected sub-complexes in CW spaces.
method Introduces hyperconnectedness and applies it to CW complexes and continuous functions.
result Existence of continuous functions that are paths in hyperconnected relator spaces.
Unified treatment of stability problems in geometry and analysis.
problem Spherical closeness of hypersurfaces under geometric constraints.
method Estimate relating distance to geodesic spheres with norms of traceless Hessian operator.
result Unified treatment of stability problems in geometry and analysis.
We prove that for every n>2, the Banach-Mazur compactum Q(n) is the compactification of a Hilbert cube manifold by the Euclidean point. For n=2 this result was proved earlier.
The paper proves Hölder continuity for solutions of degenerate parabolic equations in any dimension.
problem Proving Hölder continuity for solutions of degenerate parabolic equations in arbitrary dimensions.
method Establishing Alexandroff-Bakelman-Pucci estimate, Harnack inequality, Hölder regularity, and Schauder estimates for a class of degenerate parabolic equations.
result The paper proves Hölder continuity for solutions of degenerate parabolic equations in all dimensions.
We prove the following result announced in Todorov and Valov: Any homogeneous, metric ANR-continuum is a VGn-continuum provided dimGX=n≥1 and Hˇn(X;G)=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 specify a result of Yokoi \cite{yo} by proving that if G is an abelian group and X is a homogeneous metric ANR compactum with dimGX=n and Hˇn(X;G)=0, then X is an (n,G)-bubble. This implies that any such space X has the following properties: Hˇn−1(A;G)=0 for every closed…
This paper defines ribbons and ribbon complexes in CW spaces and analyzes their topological properties.
problem Characterizing and analyzing topological structures in CW spaces.
method Introducing planar ribbons, ribbon complexes, and ribbon nerves in Alexandroff-Hopf-Whitehead CW spaces, and studying their topological properties.
result Characterization of ribbons and ribbon nerves by Betti numbers and homotopy types.
This paper uses Ghrist barcodes to track persistent shapes in video frames.
problem Detecting and tracking persistent shapes in video frames.
method Introduces Ghrist barcodes for persistent Betti numbers derived from vortex nerve complexes in triangulated video frames.
result Persistent Betti numbers of vortex nerves are k+2 for k edges. Detects outliers in VAE latent space by identifying vacant holes.
problem Outliers detection in VAE latent space.
method Compactness enforced via Alexandroff extension and fixed Lipschitz continuity.
result Anomalous inputs land on latent holes, enabling successful identification.
The homological dimension dG of metric compacta was introduced by Alexandroff. In this paper we provide some general properties of dG, mainly with an eye towards describing the dimensional full-valuedness of compact metric spaces. As a corollary of the established properties of dG, we prove that any two-dimens…
Sharp ABP estimate on metric spaces via optimal transport.
problem Sharp ABP estimate on metric measure spaces.
method Optimal transport theory.
result Established a sharp ABP estimate on metric measure spaces.
A classical theorem of Alexandroff states that every n-dimensional compactum X contains an n-dimensional Cantor manifold. This theorem has a number of generalizations obtained by various authors. We consider extension-dimensional and infinite dimensional analogs of strong Cantor manifolds, Mazurkiewicz manifolds,…
Let (M^n_i,g_i,p_i) be a sequence of smooth pointed complete n-dimensional Riemannian Manifolds with uniform bounds on the sectional curvatures and let (X,d,p) be a metric space such that (M^n_i,g_i,p_i) -> (X,d,p) in the Gromov-Hausdorff sense. Let O \subseteq X be the set of points x \in X such that there exists a ne…
The paper introduces vortex nerve complexes and new Betti numbers in CW spaces.
problem Understanding the structure and properties of CW complexes and their nerves.
method Introducing vortex nerve complexes and defining new Betti numbers for CW complexes.
result New Betti numbers (vortex Bvtex, vortex nerve BvNrv, shape Bsh) are introduced and studied. We introduce and investigate the notion of (strong) KGn-manifolds, where G is an abelian group. One of the result related to that notion (Theorem 3.4) implies the following partial answer to the Bing-Borsuk problem \cite{bb}, whether any partition of a homogeneous metric ANR-space X of dimension n is cyclic…
A distance-squared function is one of the most significant functions in the application of singularity theory to differential geometry. Moreover, distance-squared mappings are naturally extended mappings of distance-squared functions, wherein each component is a distance-squared function. In this paper, compositions of…
We study square-tiled tori, that is, tori obtained from a finite collection of unit squares by parallel side identifications. Square-tiled tori can be parametrized in a natural way that allows to count the number of square-tiled tori tiled by a given number of square tiles. There is a natural $\mathrm{SL}(2,\mathbf{Z})…
Study finds Hilbert square of real surfaces can be maximal even when the surface has disconnected real locus.
problem Exploring conditions for maximality of Hilbert square of real surfaces.
method Analyzing Hilbert square of maximal real surfaces and examining specific examples.
result Hilbert square can be maximal even for surfaces with disconnected real locus.
Directly simulates squared Bessel processes efficiently.
problem Simulating squared Bessel processes accurately and efficiently.
method Two-dimensional Chebyshev expansion for non-central chi-square distribution inverse.
result Accurate and efficient simulation for various degrees of freedom.
The paper studies counting problems on square-tiled surfaces.
problem Understanding the frequency of properties in square-tiled surfaces.
method Examining properties of the square torus and their implications in translation surfaces.
result Implications between properties and their frequency in translation surfaces.
CD converges linearly for MCP/SCAD penalized least squares.
problem Recovering sparse signals from data.
method Coordinate descent for MCP/SCAD penalized least squares.
result CD converges linearly to solutions of MCP/SCAD penalized least squares.
Study on Nijenhuis tensor forms and vanishing properties.
problem Understanding the properties of Nijenhuis tensor.
method Provided strong and weak forms of the square of Nijenhuis tensor, and identified vanishing results.
result Identified new vanishing results for the square of Nijenhuis tensor.
Study differential properties of matrix square roots in specific cases.
problem Understanding matrix square roots in semi-simple, symmetric, and orthogonal cases.
method Analysis of differential and metric structures of real square roots of matrices under specific conditions.
result Differential properties of matrix square roots in semi-simple, symmetric, and orthogonal cases.
Square-like quadrilaterals inscribed in space curves proven for finite total curvature.
problem Square-peg problem in space curves.
method Local curvature analysis and limiting argument on approximating curves.
result Every embedded curve with finite total curvature has an inscribed square-like quadrilateral.
Shear moves connect square-tiled surfaces in quadratic differentials.
problem Connecting square-tiled surfaces via specific moves.
method Shear moves corresponding to diagonal flips preserving square-tiled properties.
result Connected components of reconfiguration problem are in bijection with moduli space of quadratic differentials.
A square complex is a 2-complex formed by gluing squares together. This article is concerned with the fundamental group Γ of certain square complexes of nonpositive curvature, related to quaternion algebras. The abelian subgroup structure of Γ is studied in some detail.
Square can fit inside curves close to smooth ones.
problem Finding inscribed squares in nearly smooth curves.
method Using curvature and a map to relate curves, proving existence of inscribed squares.
result Curves close to smooth ones contain inscribed squares.
A distance-squared function is one of the most significant functions in the application of singularity theory to differential geometry. In this paper, we define naturally extended mappings of distance-squared functions, wherein each component is a distance-squared function. We investigate the properties of these mappin…
The study finds arithmetic groups often in square-tiled surface monodromies.
problem Understanding arithmetic properties of square-tiled surfaces.
method Analyzing variations of Hodge structures and Kontsevich-Zorich monodromies.
result Arithmetic groups are frequent in low genus square-tiled surfaces.
New expressions for Nijenhuis tensor squares found.
problem Understanding Nijenhuis tensor squares.
method Expressed dual forms in terms of exterior derivative derivatives.
result New vanishing results for Nijenhuis tensor squares.
In the context of Synthetic Differential Geometry, we describe the square volume of a ``second-infinitesimal simplex'', in terms of square-distance between its vertices. The square-volume function thus described is symmetric in the vertices. The square-volume gives rise to a characterization of the volume form in the t…
A new method simulates square-root processes efficiently.
problem Simulating square-root processes accurately and efficiently.
method Simulate the integrated square-root process instead of the square-root process itself.
result High precision with low number of time steps, and exact limiting Inverse Gaussian distributions.
Square metrics is an important class of Finsler metrics. Recently, we introduced a special class of non-regular Finsler metrics called singular square metrics. The main purpose of this paper is to provide a necessary and sufficient condition for singular square metrics to be of constant Ricci or flag curvature when dim…
New method corrects least-squares temporal difference for better lambda-return estimation.
problem Improving lambda-return estimation in reinforcement learning.
method Uncorrected least-squares temporal difference with a correction method.
result Enhanced accuracy in temporal difference learning.
Cross validation residuals are well known for the ordinary least squares model. Here leave-M-out cross validation is extended to generalised least squares. The relationship between cross validation residuals and Cook's distance is demonstrated, in terms of an approximation to the difference in the generalised residual …
The study calculates the Smith-Thom deficiency of Hilbert squares and provides conditions for maximality.
problem Calculating the Smith-Thom deficiency of Hilbert squares and conditions for maximality.
method Using Mayer-Vietoris mapping and rank calculations.
result Established necessary and sufficient conditions for maximality of Hilbert squares in projective complete intersections.
Square percolation determines threshold for group divergence in random graphs.
problem Threshold for quadratic divergence in random right-angled Coxeter groups.
method Square-graph analysis of random graphs to determine connectivity and divergence.
result Threshold probability for quadratic divergence is \( p_c(n) = \sqrt{\sqrt{6}-2}/\sqrt{n} \).
Defines a new Steenrod square for virtual links, linking to Khovanov-Lipshitz-Sarkar stable homotopy type.
problem Studying Steenrod squares for virtual links.
method Defines a second Steenrod square for virtual links.
result First meaningful nontrivial example of the second Steenrod square on Khovanov homology.
Formulae count square-tiled surfaces in genus two.
problem Counting square-tiled surfaces in genus two.
method Parametrized classification into four diagrams, provided formulae for enumeration.
result Formulae for enumeration of square-tiled surfaces in four diagrams, completing the count for genus two.
The Lorentzian length, which is one of the most significant functions in Lorentzian geometry, is a complex-valued function. Its square gives a real-valued non-degenerate quadratic function. In this paper, we define naturally extended mappings of Lorentzian distance-squared functions, wherein each component is a Lorentz…
New model for STSs with restricted horizontal gluings, focusing on maximal horizontal cylinders.
problem Modeling STSs with specific horizontal restrictions.
method Modified model with conjugacy classes of permutations to restrict horizontal gluings.
result Asymptotic analysis of components, genus distribution, and saddle connections.
Study of group orderability using tensor and exterior squares.
problem Circular orderability of groups and torsion elements.
method Analysis of tensor and exterior squares of groups.
result Circularly orderable groups have left-orderable tensor and exterior squares under certain conditions.
We consider a special class of Finsler metrics --- square metrics which are defined by a Riemannian metric and a 1-form on a manifold. We show that an analogue of the Beltrami Theorem in Riemannian geometry is still true for square metrics in dimension n≥3, namely, an n(≥3)-dimensional square metric is locall…