We study the structure of finite quandles in terms of subquandles. Every finite quandle decomposes in a natural way as a union of disjoint -complemented subquandles; this decomposition coincides with the usual orbit decomposition of . Conversely, the structure of a finite quandle with a given orbit decomposit…
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48 results for “finite structures”
Researchers find Frobenius manifold structures on orbits spaces of finite groups.
problem Understanding Frobenius manifold structures on orbits spaces of finite groups.
method Applying Dubrovin's method to various orbits spaces of linear representations of finite groups.
result Discoveries of non-trivial Frobenius manifold structures on orbits spaces.
Characterizes monodromies of projective structures on finite-type surfaces.
problem Understanding monodromies of projective structures on finite-type surfaces.
method Geometrical/topological study of local conical projective structures.
result Any representation can be represented as the holonomy of a branched projective structure.
No spin structures found in a hyperbolic 4D space.
problem Finding hyperbolic 4-manifolds without spin structures.
method Constructed a non-compact, orientable, hyperbolic 4-manifold.
result Demonstrated existence of a hyperbolic 4D space without spin structures.
Proves Vaisman solvmanifolds are finite quotients of Kodaira-Thurston manifolds.
problem Characterizing Vaisman solvmanifolds and their properties.
method Analyzing fundamental groups and quotient structures.
result Every Vaisman solvmanifold is a finite quotient of a Kodaira-Thurston manifold.
Generalizes pseudo-product structures with abnormal extremals.
problem Finiteness of symmetry algebras for non-degenerate pseudo-product structures.
method Modified universal prolongation of graded nilpotent Lie algebras and generalized finiteness criterion.
result Distributions with singularly transitive properties have finite-dimensional symmetries.
Finite groups can be automorphism groups of translation surfaces with poles.
problem Existence of finite automorphism groups on translation surfaces with poles.
method Analyzing translation surfaces with poles and extending results to branched projective structures.
result Finite groups can be automorphism groups of translation surfaces with poles.
This thesis introduces big mapping class groups and their structure.
problem Understanding mapping class groups of infinite-type surfaces.
method Systematic introduction and analysis of structure and topological generation.
result Key differences from finite-type mapping class groups.
Study on surfaces of genus g≥1 in 3D contact sub-Riemannian manifolds, proving finiteness or infiniteness of induced distance.
problem Determining the finiteness of the induced distance on surfaces of genus g≥1 in 3D contact sub-Riemannian manifolds.
method Analyzing the structural stability of the finiteness/not-finiteness of the induced distance on closed surfaces of genus g≥1.
result Closed surfaces of genus g≥1 can be embedded in such a way that the induced distance is either always finite or always infinite.
Every Feedforward Neural Network Definable in an o-Minimal Structure Has Finite Sample Complexitystat.ML
New findings show modern neural networks have finite sample complexity in o-minimal structures.
problem Understanding the learnability of modern neural networks in a broad context.
method Analyzing feedforward neural networks definable in o-minimal structures.
result Modern neural networks, including MLPs, CNNs, GNNs, and transformers, have finite sample complexity in the agnostic PAC setting.
We give elementary constructions of manifold with corner structures and associative gluing maps on compactifications of spaces of infinite, half infinite, and finite Morse flow lines.
The paper extends Nambu-Poisson structures to infinite dimensions.
problem Extending Nambu-Poisson structures to infinite dimensional settings.
method Adapting finite dimensional Nambu-Poisson structures to a convenient manifold setting.
result Classical results in finite dimensions can be extended to infinite dimensions for partial Nambu structures.
We present a sketch of the proof of the following theorems: (1) Every 3-manifold has only finitely many homotopy classes of 2-plane fields which carry tight contact structures. (2) Every closed atoroidal 3-manifold carries finitely many isotopy classes of tight contact structures.
Researchers address the generation of differential invariants for geometric structures.
problem Finite generation of differential algebra of relative differential invariants.
method Investigation of algebraic and differential properties, localization, weight analysis.
result Localization on a finite set of relative invariants makes the differential algebra finitely generated.
Paper detects gradual changes in cluster structure using MC fusion.
problem Detecting gradual changes in cluster structure over time.
method MC fusion for multiple mixture numbers, examining MC transition.
result Accurately captures cluster structure during transitional periods.
Finite 2-step nilpotent groups studied by Kim and Manturov.
problem Understanding the structure of groups defined by Kim and Manturov.
method Analyzing the groups for all .
result The groups are finite and 2-step nilpotent.
Finite resources limit false discovery rate control in structured hypothesis spaces.
problem Controlling false discovery rate in hypothesis testing with finite data and structured hypothesis spaces.
method Framework for exact FDR control and adaptive power maximization.
result Exact FDR control and adaptive power maximization.
Let be a finite abelian group. A dynamical system with transformation group is a triple , consisting of a unital locally convex algebra , the finite abelian group and a group homomorphism $α:Λ\rightarrow\Aut(A)$, which induces an action of on . In this paper we present a new, geometricall…
Unified framework for Arnold-type invariants via dual complexes and finite-difference structures.
problem Study of Arnold-type invariants of immersed curves and surfaces.
method Framework on dual complexes, locally normalized maps, finite-difference structures, and Shumakovitch-type identities.
result Unified evaluation of Arnold-type invariants and on dual skeleta.
Let V be a closed 3-manifold. In this paper we prove that the homotopy classes of plane fields on V that contain tight contact structures are in finite number and that, if V is atoroidal, the isotopy classes of tight contact structures are also in finite number.
The study classifies flat solvmanifolds and finds -structures on them.
problem Classifying and finding -structures on flat solvmanifolds.
method Classification of flat splittable solvmanifolds and search for compatible -structures.
result Examples of compact flat manifolds with specific -structures.
Extends Dirac structures to infinite dimensions, focusing on convenient Lie algebroids and manifolds.
problem Extending classical geometrical results from finite to infinite dimensions.
method Introduces partial Dirac structures on convenient Lie algebroids and manifolds, explores their properties and limits.
result Classical geometrical results can be extended to infinite dimensional contexts.
Currents on Lie groups form a Hopf algebra structure.
problem Understanding algebraic structure of currents on Lie groups.
method Defined Hopf algebra structure on currents using convolution and wedge product.
result Explicit formulas for Hopf algebra operations on currents are derived.
Finite-time Lyapunov exponents of deep neural networkscond-mat.dis-nn
Study of deep neural networks using finite-time Lyapunov exponents.
problem Understanding the geometric structures in input space formed by deep neural networks.
method Analogy with dynamical systems, computing finite-time Lyapunov exponents.
result Ridges of large positive exponents divide input space into regions associated with different classes.
The paper analyzes finite-time singularities in Spin(7)-structure flows using Shi-type estimates.
problem Analyzing finite-time singularities in Spin(7)-structure flows.
method Proves Shi-type derivative estimates and shows that Λ(x,t) must blow up at finite-time singularities.
result Establishes a general analytic framework for studying Spin(7)-structure flows.
Unified algebraic framework for virtual braid structures with strong structural consequences.
problem Unified algebraic framework for virtual braid structures with various types of crossings.
method Introducing the universal virtual braid group and proving its properties.
result Strong structural consequences including residual finiteness, linearity, and solvability of conjugacy problems.
On symplectic quandlesmath.QA
We study the structure of symplectic quandles, quandles which are also R-modules equipped with an antisymmetric bilinear form. We show that every finite dimensional symplectic quandle over a finite field F or arbitrary field F of characteristic other than 2 is a disjoint union of a trivial quandle and a connected quand…
Defines invariants for reflection groups and connects them to Frobenius structures.
problem Understanding invariants for reflection groups and their relation to Frobenius structures.
method Defines good basic invariants and shows their connection to Frobenius structures.
result Good basic invariants for reflection groups lead to Frobenius structure constants.
We consider four notions of maps between smooth C^r orbifolds O, P with O compact (without boundary). We show that one of these notions is natural and necessary in order to uniquely define the notion of orbibundle pullback. For the notion of complete orbifold map, we show that the corresponding set of C^r maps between …
In this paper, we extend Lotay-Wei's Shi-type estimate from Laplacian flow to more general flows of G structures including the modified Laplacian co-flow. Then we prove a version of -non-collapsing theorem. We will use both of them to study finite time singularities of general flows of G structures.
Using the categorical description of supergeometry we give an explicit construction of the diffeomorphism supergroup of a compact finite-dimensional supermanifold. The construction provides the diffeomorphism supergroup with the structure of a Frechet supermanifold. In addition, we derive results about the structure of…
Conditions of Stability for explicit finite difference scheme and some results of numerical analysis for a unified 2 factor model of structural and reduced form types for corporate bonds with fixed discrete coupon are provided. It seems to be difficult to get solution formula for PDE model which generalizes Agliardi's …
The existence of essential closed surfaces surfaces is proven for finite coverings of 3-manifolds that are triangulated by finitely many topological ideal tetrahedra and admit a regular, negatively curved, ideal structure.
Defines pre-Kähler structures and their properties.
problem Geometry of Levi degenerate CR hypersurfaces.
method Introduces pre-Kähler structures, holomorphic degeneration, and finite-nondegeneracy.
result Symmetry algebra is finite-dimensional if and only if structure is finitely nondegenerate.
We employ techniques of machine-learning, exemplified by support vector machines and neural classifiers, to initiate the study of whether AI can "learn" algebraic structures. Using finite groups and finite rings as a concrete playground, we find that questions such as identification of simple groups by "looking" at the…
The study proves how groups can be split with limited complexity.
problem Understanding the complexity of group splittings.
method Analyzing trees with finite stabilizers and their quotient structures.
result Deformation spaces of trees have maximal complexity.
A new layer learns abstract relations from graph structure using finite-state automata.
problem Learning abstract relations from graph structure for program analysis.
method Relaxing the problem into learning finite-state automata policies on a graph-based POMDP and training these policies using implicit differentiation.
result GFSA layer finds shortcuts in grid-world graphs and reproduces simple static analyses on Python programs.
The kth finite subset space of a topological space X is the space exp_k X of non-empty finite subsets of X of size at most k, topologised as a quotient of X^k. The construction is a homotopy functor and may be regarded as a union of configuration spaces of distinct unordered points in X. We show that the finite subset …
We characterize finite groups G generated by orthogonal transformations in a finite-dimensional Euclidean space V whose fixed point subspace has codimension one or two in terms of the corresponding quotient space V/G with its quotient piecewise linear structure.
Redundancy improves learning stability and generalization in structured systems.
problem Understanding redundancy in structured systems for learning and generalization.
method Developed a theoretical framework that redefines redundancy as a geometric principle unifying various measures.
result Redundancy balances structure and coupling, leading to optimal stability and generalization.
In this paper, using the structures of cone and bicone fields on vector bundles, the author introduces a ILB (inverse limit of Banach)- manifold structure on the space of Riemannian metrics on a noncompact manifold . In the last section, it is proven that, this way, on the open submanifold $\mathcal{M}_…
Finite-time blow-up in Yang-Mills flow for small energy initial connections.
problem Finite-time blow-up of Yang-Mills flow solutions.
method Analyzing the Yang-Mills flow on Riemannian and Kähler manifolds.
result Finite-time blow-up occurs for small energy initial connections.
In this survey, we study representations of finitely generated groups into Lie groups, focusing on the deformation spaces of convex real projective structures on closed manifolds and orbifolds, with an excursion on projective structures on surfaces. We survey the basics of the theory of character varieties, geometric s…
Characterizes CR manifolds in complex flag manifolds.
problem Closed real orbits in complex flag manifolds.
method Characterization through CR manifold structures and real forms.
result Closed orbits are finitely nondegenerate.
New examples of hyperbolic 3-manifolds with unique profinite structure.
problem Finding hyperbolic 3-manifolds with unique profinite structure.
method Examining fundamental groups of closed fibered hyperbolic 3-manifolds.
result First examples of closed fibered hyperbolic 3-manifolds with unique profinite structure.
Monodromy Equivalence for Lamé-type Equations I: Finite-gap Structures and Cone Spherical Metricsmath.CA
Study establishes monodromy equivalence for Lamé-type equations and constructs cone spherical metrics.
problem Investigating monodromy equivalence and finite-gap structures of Lamé-type equations.
method Analyzing finite-gap structures and constructing cone spherical metrics.
result Established monodromy equivalence between classical and generalized Lamé-type equations, derived finite-gap structures, and constructed cone spherical metrics.
This is the less official, English version of the proof of the fact that every closed atoroidal 3-manifold carries finitely many isotopy classes of tight contact structures.
Study of cluster and skein algebras for surfaces, showing their connection.
problem Understanding algebraic structures of curve algebras on surfaces.
method Generalization and explicit definition of maps between cluster and skein algebras.
result Explicit maps between cluster and skein algebras, showing their close relationship.