Heaps are para-associative ternary operations bijectively exemplified by groups via the operation . They are also ternary self-distributive, and have a diagrammatic interpretation in terms of framed links. Motivated by these properties, we define para-associative and heap cohomology theories…
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48 results for “Heap theory”
Heap theory applied to framed links yields new invariants.
problem Developing invariants for framed links using heap theory.
method Introducing fundamental heap, defining cocycle invariant using ternary cohomology.
result Found cocycles and computed invariants for specific link families.
Introduces fundamental heaps for surfaces, linking them to cocycle invariants.
problem Isotopy invariants of surface embeddings in 3-space.
method Definition of fundamental heaps using surface ribbons and heap operations.
result Fundamental heaps have a free part whose rank matches the number of connected components.
New braided Frobenius algebras created from specific Hopf algebras.
problem Creating new algebraic structures from Hopf algebras.
method Heap operation and Yang-Baxter operator on tensor product.
result Heap operation induces a braiding compatible with Frobenius operations.
In this paper, we determine the abelianization of the level d mapping class group for d=2 and odd d. We also extend the homomorphism of the Torelli group defined by Heap to a homomorphism of the level 2 mapping class group.
The paper constructs Yang-Baxter solutions using categorical augmented racks.
problem Solutions to the Yang-Baxter equation in knot theory.
method Interpreting augmented racks in tensor categories and constructing solutions using quantum heaps and Hopf algebra modules.
result Explicit constructions and infinite families of Yang-Baxter solutions are provided.
Bayesian framework for semiparametric regression of discrete data.
problem Complex distributional features of discrete data.
method Semiparametric modeling with nonparametric marginal and latent linear regression.
result Posterior consistency and analytical/posterior predictive distributions.
Expanding on prime knots with 6 or less mosaic tiles, this paper analyzes those with 7 tiles.
problem Determining the tile number and space-efficiency for prime knots with mosaic number 7.
method Extending the methods of Heap and Knowles (2017) to include prime knots with mosaic number 7.
result Identifying the possible tile numbers and space-efficient layouts for all prime knots with mosaic number 7.
Analyzes how BPE tokenisation affects corpus statistics and model entropy in transformer models.
problem Understanding how natural language properties relate to tokenisation schemes in transformer models.
method Analyzes Shannon entropy of corpora under Zipfian distribution, investigates BPE transformations, trains language models, and uses attention diagnostics.
result Transformer models trained on BPE-tokenised corpora increasingly agree with Zipfian predictions as BPE depth increases, indicating reduced local token dependencies.
Novel ternary structures reveal new interpretations of linear connections.
problem Examining the ternary structure of Lie algebroid connections.
method Study of endomorphisms and explicit presentation of the endomorphism truss.
result Explicitly presented endomorphism truss of linear connections.
Kanerva++: extending The Kanerva Machine with differentiable, locally block allocated latent memorycs.NE
New memory allocation scheme improves image generation performance.
problem Improving episodic and semantic memory representation in neural networks.
method Developed a hierarchical latent variable model with differentiable, locally block allocated latent memory.
result Improved conditional likelihood values on various datasets.
New tile types for knots and links reduce complexity.
problem Determining the minimum number of tiles needed for knot representations.
method Introduced new tile types and analyzed their impact on knot complexity.
result Corner tile number lies between tile number and 3 times tile number.
A novel multilayer network approach for text analysis.
problem Clustering documents and finding topics in large collections with metadata and hyperlinks.
method Multilayer Networks and Stochastic Block Models applied to multiple data types.
result Taking into account multiple types of information improves topic and document clustering.
Online reviews provided by consumers are a valuable asset for e-Commerce platforms, influencing potential consumers in making purchasing decisions. However, these reviews are of varying quality, with the useful ones buried deep within a heap of non-informative reviews. In this work, we attempt to automatically identify…
Semiparametric STAR model improves mental health data analysis.
problem Overdispersed, zero-inflated, bounded count data in self-reported mental health surveys.
method STAR transformation and rounding of latent Gaussian model, nonparametric transformation estimation, EM algorithm for maximum likelihood.
result Substantial improvements in goodness-of-fit compared to existing models.
Large-scale industrial recommender systems are usually confronted with computational problems due to the enormous corpus size. To retrieve and recommend the most relevant items to users under response time limits, resorting to an efficient index structure is an effective and practical solution. The previous work Tree-b…
Paper proves corner connection tiles can represent knots with fewer tiles.
problem Finding the minimum number of tiles for knot representation.
method Developed corner connection tiles and proved their efficiency.
result Corner connection tiles can represent knots with fewer tiles than traditional tiles.
The paper connects function theory, dynamics, and ergodic theory via Thurston's theory.
problem Function theory on Teichmüller space and dynamics of mapping class groups.
method Utilizes Thurston's theory and Sullivan's theory on discrete subgroups of hyperbolic space.
result Establishes connections between function theory, dynamics, and ergodic theory.
Survey of Floer theories and their connections.
problem None explicitly stated; focuses on surveying theories.
method None explicitly stated; focuses on surveying theories.
result None explicitly stated; focuses on surveying theories.
Lectures on topological field theories and differential cohomology.
problem Exploring topological field theories and their connections to differential cohomology.
method Introduction to topological field theory and generalized Abelian gauge theories.
result Explains the relationship between topological field theories and differential cohomology.
The paper defines strong emergence in field theories and proves it exists between certain theories.
problem Defining and proving the existence of strong emergence phenomena between field theories.
method Formal definition and sufficient conditions for emergence, proving existence in Euclidean background.
result Strong emergence exists between certain parameterized Lagrangian field theories.
This is the first paper in a series introducing a generalized Fredholm theory in a new class of smooth spaces called polyfolds. The theory will be illustrated in upcoming papers by applications to Floer Theory, Gromov-Witten Theory and Symplectic Field Theory.
Unified Higgs bundle vacua from M-theory on Spin(7) spaces.
problem Unifying Higgs bundle vacua from different string compactifications.
method Developed formalism for M-theory on local Spin(7) spaces and constructed explicit solutions.
result Unified 3D effective field theory from 4D M- and F-theory vacua.
Researchers find new -conifolds in -theory with potential field theory duals.
problem Exploring the field theory interpretation of -theory -conifolds.
method Constructing -holonomy orbifolds from circle bundles over Calabi-Yau cones.
result Many UV perturbative gauge theories have an infrared dual described by smooth -holonomy backgrounds in -theory.
We survey three different ways in which K-theory in all its forms enters quantum field theory. In Part 1 we give a general argument which relates topological field theory in codimension two with twisted K-theory, and we illustrate with some finite models. Part 2 is a review of pfaffians of Dirac operators, anomalies, a…
Distributivity in algebraic structures appeared in many contexts such as in quasigroup theory, semigroup theory and algebraic knot theory. In this paper we give a survey of distributivity in quasigroup theory and in quandle theory.
New theory captures framing anomaly in gauge theory.
problem Capturing framing anomaly in gauge theory.
method Constructs a relative Crane-Yetter theory from non-semisimple data.
result Establishes invertibility property for the theory.
Survey on algebraic K- and L-theory conjecture.
problem Algebraic K- and L-theory of groups rings.
method Not specified in the abstract, likely involves algebraic and geometric approaches.
result Applications to algebra, geometry, group theory, and topology.
Study pin manifolds using Clifford linear Dirac operator and KO-theory.
problem Index theory on Pin manifolds.
method Clifford linear Dirac operator and differential KO-theory.
result Systematic treatment of index theory on Pin manifolds.
Main mathematical applications of Frobenius manifolds are in the theory of Gromov - Witten invariants, in singularity theory, in differential geometry of the orbit spaces of reflection groups and of their extensions, in the hamiltonian theory of integrable hierarchies. The theory of Frobenius manifolds establishes rema…
In this paper, we construct a new homology theory for semi-groups satisfying the self distributivity axiom or the idempotency axiom. Next, we consider the geometric realization corresponding to the homology theory. We continue with the comparison of this homology theory with one term and two term (rack) homology theori…
Quantum field theory uses Lorentzian bordisms to describe time evolution.
problem Describing the time evolution of quantum field theories.
method Defines a functorial field theory on Lorentzian bordism pseudo-category.
result Lorentzian bordisms naturally arise in algebraic quantum field theory.
This thesis proposes a global geometric formulation of Extended Field Theories.
problem Global understanding of Extended Field Theories remains an open problem.
method Introducing an atlas for the principal infinity-bundle, unifying metric and higher gauge field.
result Global abelian T-duality and Poisson-Lie T-duality are automatically recovered.
This is the revised version of the second paper in a series introducing a generalized Fredholm theory in a new class of smooth spaces called polyfolds. The theory will be illustrated in upcoming papers by applications to Floer Theory, Gromov-Witten Theory and Symplectic Field Theory. Some proofs have been improved and …
The paper quantizes hybrid topological-holomorphic field theories on .
problem Quantizing hybrid topological-holomorphic field theories rigorously.
method Constructing perturbative, one-loop quantizations on .
result The one-loop obstruction to quantization vanishes when .
3D dual field theories for Virasoro minimal models constructed using Seifert fiber spaces.
problem Constructing 3D dual field theories for Virasoro minimal models.
method 3D-3D correspondence and Seifert fiber spaces.
result 3D dual field theories constructed for Virasoro minimal models.
We propose a new partially topological theory in three dimensions which couples Chern-Simons theory to matter. The 3-manifolds needed for this construction admit transverse holomorphic foliation (THF). The theory depends only on the choice of such a structure, but not on a choice of metric and in this sense, it is topo…
We show that Chern-Simons gauge theory with appropriate cutoffs is equivalent, term by term in perturbation theory, to a Fermionic theory with a nonlocal interaction term. When an additional cutoff is placed on the Fermi fields, this Fermionic theory gives rise to a convergent perturbation expansion. This leads us to c…
From Euclidean field theory to hyperkähler Floer theory via regularized polysymplectic geometrymath.SG
Abstract: Linking field theory to Floer theory via regularization.
problem Finding periodic solutions of Hamilton's equation.
method Regularization scheme for polysymplectic formalism linking Euclidean field theory to hyperkähler Floer theory.
result Proved a cuplength estimate.
Unified treatment of gauge theories and Yang-Mills theory duality.
problem Unified treatment of gauge theories and Yang-Mills theory duality.
method Cohomological localization techniques and Atiyah-Singer index theorem.
result Unified framework and simplified derivations of localization formulas.
New theory connects string theory to swampland distance conjecture.
problem Connecting string theory to swampland distance conjecture.
method Deformations of the heterotic superpotential, treating separately for large fluxes or large distances, integrating out fields to obtain a new field theory.
result New holomorphic theory defined, connects to swampland distance conjecture.
Identifies all perturbative vacua in bosonic string theory.
problem Identifying all perturbative vacua in bosonic string theory.
method Completely identified perturbative vacua through string fluctuations.
result Derivation of path-integrals up to any order from fluctuations.
In this short note we show how Dubrovin's integrable hierarchies, defined using the Gromov-Witten theory of a closed symplectic manifold, generalizes to Hamiltonian Floer theory. In particular, we show how the required generalization of the PSS isomorphism, relating Gromov-Witten theory and Hamiltonian Floer theory, ca…
Semisimple 4D field theories can't distinguish smooth 4-manifolds.
problem Detecting exotic smooth structures in 4-manifolds.
method Proving field theories lead to stable invariants, distinguishing only homeomorphic and homotopy equivalent manifolds.
result Semisimple 4D field theories can't distinguish homotopy equivalent 4-manifolds.
Paper introduces a new geometric homology theory and applies it to Gromov-Witten theory.
problem Developing a new homology theory for orbifolds with corners.
method Using stratification and triangulation theories of Lie groupoids and their orbit spaces, extending to Lie groupoids with corners.
result Proposes and proves the geometric homology theory (GHT), a flexible generalization of singular homology.
New geometric approach realizes 5D bulk theories with 4D edge modes.
problem Realizing novel higher-dimensional junctions of theories coupled to localized edge modes.
method M-theory on singular, asymptotically conical G2-holonomy orbifolds.
result Geometric approach shows how bulk generalized symmetries are inherited in the boundary system.
Paper reinterprets marginal productivity theory using vectorial products, challenging traditional ethical interpretations.
problem Challenges traditional ethical interpretations of marginal productivity theory.
method Formulates marginal productivity theory using vectorial marginal products, contrasting with traditional scalar approach.
result Vectorial marginal products conflict with traditional distributive shares picture of property.
Category theory generalizes finite type invariants using diagrams systems.
problem Generalizing finite type invariants using category theory.
method Relating generating sets for generalized finite type theories with diagrams systems.
result Demonstrates the correspondence between finite type theories and diagrams systems.