Proposes a new framework for uncertainty evaluation in ML classification models.
problem Uncertainty evaluation for ML classification models not addressed by existing metrological guidelines.
method Develops a metrological framework based on probability mass functions and summary statistics.
result Extends the GUM to uncertainty for nominal properties, applicable to ML classification models.
Lecture notes on Teichmüller spaces with boundary examples.
problem Understanding higher Teichmüller spaces of Riemann surfaces with boundaries.
method Explicit and computationally tractable examples, chewing-gum move, snake calculus.
result The chewing-gum move is the inverse of amalgamation in bordered cusped Teichmüller space.
GUM tackles MARL by avoiding overestimation through state-marginal restriction.
problem Overestimation of values in large joint state-action spaces.
method Greedy UnMixing through state-marginal restriction and unmixing.
result Superior performance compared to existing Q-learning and general MARL algorithms.
This paper compares HMC and RNN expressivity using SRT.
problem Comparing expressivity of HMC and RNN models.
method Embed HMC and RNN in a GUM, use SRT to compare structured covariance series.
result Conditions for realizing covariance series by GUM, HMC, or RNN.
Proves HNN extensions of nilpotent groups are left-orderable, constructs non-left-orderable examples.
problem Characterizing left-orderability in HNN extensions of groups.
method Analyzes HNN extensions of torsion-free nilpotent groups and left-orderable groups.
result Constructs examples of non-left-orderable HNN extensions of left-orderable groups.
Examines differential smoothness in a specific skew PBW extension family.
problem Differential smoothness in skew PBW extensions.
method Investigates a specific family of skew PBW extensions.
result Results on differential smoothness of the family.
New insights into identifying mixtures of product distributions using Hadamard extensions.
problem Identifying mixtures of product distributions on binary variables.
method Analysis of Hadamard extensions of matrix products.
result Conditions for full column rank of Hadamard extensions.
A spacetime can be embedded in an enveloping space with all its extensions.
problem Existence and uniqueness of C0-maximal extensions in globally hyperbolic conformally flat spacetimes.
method Proving conformal embedding into an enveloping space containing all extensions.
result Existence and uniqueness of C0-maximal extensions proven.
We give a new variant of L2-extension theorem for the jets of holomorphic sections and discuss the relation between the extension problem of singular Hermitian metrics with semipositive curvature.
We generalize the prequantization central extension of a group of diffeomorphisms preserving a closed 2-form ω(ω-invariant diffeomorphisms) to an abelian extension of a group of diffeomorphisms preserving a closed vector valued 2-form ω, up to a linear isomorphism (ω-equivariant diffeomorphisms). Every abelian extensio…
The purpose of this paper is to show how central extensions of (possibly infinite-dimensional) Lie algebras integrate to central extensions of étale Lie 2-groups. In finite dimensions, central extensions of Lie algebras integrate to central extensions of Lie groups, a fact which is due to the vanishing of π_2 for each …
We construct a Kruskal-Szekeres-type analytic extension of the Emparan-Reall black ring, and investigate its geometry. We prove that the extension is maximal, globally hyperbolic, and unique within a natural class of extensions. The key to those results is the proof that causal geodesics are either complete, or approac…
Analytic linearization and holomorphic extensions for proper groupoids.
problem Analytic linearization and holomorphic extensions of proper groupoids.
method Establish analytic linearization around invariant submanifolds and apply to holomorphic extensions.
result Proper groupoids admit holomorphic extensions.
We study the properties of Modified Riemann extensions evolving under Ricci flow. We obtain the necessary and sufficient condition for modified Riemann extension under Ricci flow to stay as modified Riemann extension. We also discuss the properties of the curvature tensors under Ricci flow.
Let X=X∪Z be a data set in RD, where X is the training set and Z is the test one. Many unsupervised learning algorithms based on kernel methods have been developed to provide dimensionality reduction (DR) embedding for a given training set $Φ: \mathbf{X} \to \mat…
The paper examines differential smoothness in skew PBW extensions over polynomial rings.
problem Differential smoothness in skew PBW extensions over polynomial rings.
method Investigation of skew PBW extensions over commutative polynomial rings.
result Results on differential smoothness for skew PBW extensions over polynomial rings.
Simple construction of Lie 2-groups from loop group extensions.
problem Constructing Lie 2-groups from loop group extensions.
method Using conjugation action of loop group on its central extension.
result Simple construction of string 2-group as a strict Fréchet Lie 2-group.
Proves Girth Alternative for some HNN extensions, finds counterexamples.
problem Girth Alternative for HNN extensions of finitely generated groups.
method Proves Girth Alternative for a specific class of HNN extensions.
result Girth Alternative holds for some HNN extensions but fails in general.
We determine the universal central extension of the Lie algebra of hamiltonian vector fields, thereby classifying its central extensions. Furthermore, we classify the central extensions of the Lie algebra of symplectic vector fields, of the Poisson Lie algebra, and of its compactly supported version.
Kan extensions help in data science extrapolation and learning.
problem Generalizing functions over larger sets in data science.
method Kan extensions in category theory applied to data science problems.
result Kan extensions can be used to derive classification and clustering algorithms.
Optimal L2 extension theorem for holomorphic vector bundles with singular metrics.
problem Establishing conditions for optimal L2 extension in complex geometry.
method Analyzing singular Nakano positivity and applying L2 extension theorem.
result Necessary condition for equality in optimal L2 extension theorem.
The paper solves conditions for non-singular extensions of fold maps.
problem Conditions for the existence of non-singular extensions of horizontal stable fold maps.
method Defined a combinatorial object called a pairing map to prove equivalence of existence.
result Existence of non-singular extensions is equivalent to the existence of a pairing map.
Criterion for lifting smooth contact maps between Carnot groups to central extensions.
problem Existence of smooth contact map lifts between Carnot groups and their central extensions.
method Criterion using pullbacks and Lie algebra cohomology classes.
result Necessary and sufficient conditions for lifting are formulated.
Existence proved for static vacuum extensions near Schwarzschild spheres.
problem Proving existence of static vacuum extensions near Schwarzschild spheres.
method Existence and local uniqueness of static vacuum extensions for Bartnik data on a sphere near a Schwarzschild sphere.
result Existence of static vacuum extensions near Schwarzschild spheres.
We present a geometric construction of central extensions of covering groups of the group of volume preserving diffeomorphisms, integrating central extensions of the Lie algebra of divergence free vector fields defined by Lichnerowicz cocycles. Certain covering spaces of non-linear Grassmannians can be realized as preq…
Harmonic extension of Weil-Petersson circle homeomorphisms
problem Harmonic maps from the Weil-Petersson disk to the hyperbolic disk
method Anti-holomorphic L2-energy result Harmonic extension of Weil-Petersson circle homeomorphisms minimizes the L2-energy The Jacobian of Douady-Earle extension equals 1 only for isometries.
problem Investigating the Jacobian of Douady-Earle extension maps.
method Analyzing the Jacobian of the Douady-Earle extension map and constructing sequences of hyperbolic surfaces.
result The Jacobian of the Douady-Earle extension map is 1 only when the map is an isometry, and it can grow arbitrarily large for certain sequences of surfaces.
The paper explores uniqueness and non-uniqueness of spacetime extensions in general relativity.
problem Investigating the uniqueness and non-uniqueness of spacetime extensions in general relativity.
method Analyzes the extension of globally hyperbolic Lorentzian manifolds with a focus on low regularities.
result Local uniqueness of anchored extensions for certain regularity classes of extensions.
An Artin HNN-extension is an HNN-extension of an Artin group in which the stable letter conjugates a pair of suitably chosen subsets of the standard generating set. We show that some finite index subgroup of an Artin HNN-extension embeds in an Artin group. We also obtain an analogous result for Coxeter groups.
This paper characterizes extensions of augmented racks and constructs invariants for surfaces.
problem Characterizing extensions of augmented racks and constructing invariants for surfaces.
method Characterization of rack extensions through fibrant and additive cohomology, construction of invariants using cocycles.
result Characterization of extensions of augmented racks and construction of surface invariants.
New extension theorem for projective manifolds.
problem Extension of twisted canonical forms on hypersurfaces.
method Established a new extension result for canonical forms on hypersurfaces with simple normal crossings.
result Obtained sharp bounds for the extension.
In this note, we generalize a theorem of Juan Souto on rank and Nielsen equivalence in the fundamental group of a hyperbolic fibered 3-manifold to a large class of hyperbolic group extensions. This includes all hyperbolic extensions of surfaces groups as well as hyperbolic extensions of free groups by convex cocompact …
Study optimal holomorphic extensions on complex manifolds with transitivity property.
problem Optimal holomorphic extensions on complex manifolds with transitivity property.
method Use Toeplitz operators and transitivity property for optimal holomorphic extensions.
result Transitivity property of optimal holomorphic extensions with small defect.
In this paper we study the problem of extension of holomorphic sections of line bundles/vector bundles from reduced unions of strata of divisors. An extension theorem of Ohsawa--Takegoshi type is proved. As consequences we deduce several qualitative results on extension from snc divisors and generic global generation o…
Study the geometry of graph product extension graphs.
problem Properties of graph products.
method Introduce and study the extension graph of graph products of groups.
result Extension graph is isomorphic to crossing graph of a quasi-median graph and exhibits asymptotic dimension similar to quasi-trees.
The paper classifies extensions of Yang-Mills-type theories, proving maximality and universality are dense properties.
problem Classifying extensions of Yang-Mills-type theories.
method Categorical characterization and dense properties analysis.
result Maximality and universality are dense properties in the one-point compactification of extension classes.
Study on unique spacetime extensions in 1+1 dimensions with applications to weak null singularities.
problem Understanding unique spacetime extensions across null boundaries in 1+1 dimensions.
method Analyzing the C0- and C1-structures of continuous spacetime extensions. result Extensions can have the same C0-structure but different C1-structures. The paper explores extensions of local moves on string links and their relation to ribbon surfaces.
problem Finding a unique extension of classical local moves on string links.
method Relating local moves to surgeries on ribbon surfaces in R^4.
result There can be at most one unique welded extension of a classical move that is a ribbon residue.
The paper studies twisted Alexander polynomials for knot groups in various extensions.
problem Understanding twisted Alexander polynomials in knot groups for different extensions.
method Developed mod p formula for twisted Alexander polynomials and studied central extensions.
result Established formulas for twisted Alexander polynomials in knot groups for various extensions.
Let Omega^3(SU(n)) be the Lie group of based mappings from S^3 to SU(n). We construct a Lie group extension of Omega^3(SU(n)) for n>2 by the abelian group of the affine dual space of SU(n)-connections on S^3. In this article we give several improvement of J. Mickelsson's results in 1987, especially we give a precise de…
Hyperbolicity proven for a specific type of group extension.
problem Proving hyperbolicity of a specific group extension.
method Analyzing ascending HNN extension of groups with a free factor system and an injective endomorphism.
result Ascending HNN extension of a group is hyperbolic relative to a collection of maximal parabolic subgroups.
The study examines extensions of Lie algebras with specific geometric structures.
problem Conditions for preserving geometric structures in Lie algebra extensions.
method Analyzes extensions of Sasakian and Frobenius-Kähler Lie algebras.
result Conditions for maintaining Sasakian or Frobenius-Kähler structures after extensions.
Let D be a 2-dimensional closed unit disk and Symp(D,0)rel the group of symplectomorphisms preserving the origin and the boundary ∂D pointwise. We consider the R-valued flux homomorphism on Symp(D,0)rel and define the central R-extension called the $\mathb…
We study the extension of a Lie algebroid by a representation up to homotopy, including semidirect products of a Lie algebroid with such representations. The extension results in a higher Lie algebroid. We give exact Courant algebroids and string Lie 2-algebras as examples of such extensions. We then apply this to obta…
We show that every paracomplex space form is locally isometric to a modified Riemannian extension and give necessary and sufficient conditions so that a modified Riemannian extension is Einstein. We exhibit Riemannian extension Osserman manifolds of signature (3,3) whose Jacobi operators have non-trivial Jordan normal …
The paper explores relationships between quandle cohomology, extensions, and automorphisms.
problem Understanding the structure of quandle extensions and automorphisms.
method Establishes a four-term exact sequence and proves relationships involving cohomology, extensions, and automorphisms of quandles.
result Derives new relationships between quandle cohomology, extensions, and automorphisms.
The central extension of the Thompson group T that arises in the quantized Teichmüller theory is 12 times the Euler class. This extension is obtained by taking a (partial) abelianization of the so-called braided Ptolemy-Thompson group introduced and studied in \cite{FK2}. We describe then the cyclic central extension…
We prove that a transfinite extension of asymptotic dimension asind is trivial. We introduce a transfinite extension of asymptotic dimension asdim and give an example of metric proper space which has transfinite infinite dimension.