Loops linked to p-adic manifolds studied.
problem Analyzing loops in p-adic differential geometry.
method Construction of analytic loops associated with p-adic differential manifolds.
result New insights into loops of Lie type.
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48 results for “analytic loops”
Localization reveals geometric and analytic properties of the Witten genus.
problem Understanding the Witten genus and its modularity properties.
method Equivariant localization techniques and geometric interpretations.
result Identifies the elliptic Bismut-Chern character as a candidate target.
Action of loop groups on Cuntz algebras constructs geometric twists.
problem Defining actions of loop groups on Cuntz algebras.
method Using representations of Cuntz algebras and analytic loop groups to construct actions and bundles.
result Explicit construction of geometric twists in higher -theory.
Researchers compute knot invariants in Seifert manifolds using Wilson loops.
problem Calculating BPS invariants for knots in Seifert manifolds.
method Analytically continued Chern-Simons level and representation from Wilson loop operator.
result Homological blocks for knots in Seifert manifolds and Seifert integer homology spheres.
Study topological G₂ and Spin(7) strings at 1-loop using double complexes.
problem Calculate topological string partition functions at 1-loop.
method Define double complexes for supersymmetric backgrounds using generalised geometry, compute partition functions as alternating products of determinants of Laplacians.
result Reproduce known results for G₂ string and predict for Spin(7) string.
Study of loops in sums of Laplace eigenfunctions on surfaces.
problem Uniform bound for the number of nested loops in sums of Laplace eigenfunctions.
method Real-analytic category analysis and biharmonic function construction.
result Uniform bound for the number of rooted double nests in terms of surface, root, and spectral cutoff.
On the unit sphere in a real Hilbert space , we derive a binary operation such that is a power-associative Kikkawa left loop with two-sided identity , i.e., it has the left inverse, automorphic inverse, and properties. The operation is co…
The paper proves an index theorem for loop spaces of compact manifolds.
problem Defining an index theorem for loop spaces of compact manifolds.
method Formulated and proved an equivariant index theorem for non-compact manifolds with -actions, using a ring of formal power series.
result Found an appropriate form of the index theorem for loop spaces.
Develops Poisson structures on weak Sobolev loop spaces for integrable systems.
problem Analyzing integrable systems on low regularity loop spaces.
method Extending Mokhov's constructions to weak Sobolev spaces, constructing presymplectic and Poisson structures.
result Valid Poisson structures and deformations for weak Sobolev loops, extending Hamiltonian formalisms.
Toward an efficient hybrid method for pricing barrier options on assets with stochastic volatilityq-fin.CP
Efficient hybrid method for pricing barrier options with stochastic volatility.
problem Valuation of barrier options on assets with stochastic volatility.
method Combining Monte Carlo simulation and semi-analytical heat potential method.
result Our method provides better accuracy and is orders of magnitude faster than existing methods.
The classical Björling problem is to find the minimal surface containing a given real analytic curve with tangent planes prescribed along the curve. We consider the generalization of this problem to non-minimal constant mean curvature (CMC) surfaces, and show that it can be solved via the loop group formulation for suc…
Abstract: Extends Drinfeld correspondence to infinite-dimensional Lie groups.
problem Establishing Drinfeld correspondence in infinite dimensions.
method Extending Drinfeld correspondence to Poisson Lie groups and Lie bialgebras in infinite-dimensional settings.
result Extended Drinfeld correspondence to regular Lie groups modeled on nuclear Fréchet and Silva spaces.
Survey of mathematical developments in gauge theory using thin homotopy.
problem Formalizing gauge fields as group homomorphisms on smooth connections.
method Introducing group structures on spaces of loops via thin homotopy equivalence.
result Clarified difference between thin and retrace equivalence for loops.
We study two aspects of the loop group formulation for isometric immersions with flat normal bundle of space forms. The first aspect is to examine the loop group maps along different ranges of the loop parameter. This leads to various equivalences between global isometric immersion problems among different space forms …
The monodromy action in the homology of level sets of Morse functions on stratified singular analytic varieties is studied. The local variation operators in both the standard and the intersection homology groups defined by the loops around the critical values of such functions are reduced to similar operators in the ho…
First time projective elliptic genera constructed for oriented manifolds.
problem Defining and proving properties of projective elliptic genera.
method Topological construction and analytic interpretation via fractional index theorem.
result Modularity properties of projective elliptic genera proven.
Paper achieves sample complexity for actor-critic methods with minimal assumptions.
problem Achieving sample complexity for actor-critic methods under minimal assumptions.
method Single-loop, single-timescale implementation; coupled Lyapunov drift framework.
result First sample complexity guarantee for finding an -optimal policy.
Automated feature recommendation for signal analytics with interpretability.
problem Signal data analytics problems requiring feature interpretation.
method Wide Learning architecture for feature recommendation with interpretability.
result Effective feature recommendation and interpretation techniques for signal analytics.
This paper introduces TDA and TSI for better business analytics.
problem Nonlinear, multi-scale business datasets under-represented by traditional tools.
method Topological Data Analysis (TDA) and Topological Stability Index (TSI).
result TSI reveals structural variability in business data.
Study of operators on loop spaces using Fermionic calculus and stochastic methods.
problem Analyzing operators on loop spaces arising from self-adjoint and closed operators.
method Fermionic calculus and stochastic methods to derive regularity and stochastic representations.
result Derivation of a stochastic refinement of the Duistermaat-Heckman localization formula.
Area-Constrained Planar Elasticacond-mat.soft
We determine the equilibria of a rigid loop in the plane, subject to the constraints of fixed length and fixed enclosed area. Rigidity is characterized by an energy functional quadratic in the curvature of the loop. We find that the area constraint gives rise to equilibria with remarkable geometrical properties: not on…
We define a large class of integrable nonlinear PDE's, \emph{-symmetric AKS systems}, whose solutions evolve on finite dimensional subalgebras of loop algebras, and linearize on an associated algebraic curve. We prove that periodicity of the associated algebraic data implies a type of quasiperiodicity for the soluti…
A Human-in-the-Loop Bayesian Optimization Framework for Constraint-Aware Bioprocess Developmentcs.LG
A Human-in-the-Loop Bayesian Optimization framework for constraint-aware bioprocess development.
problem Bioprocess development
method Pareto Front Guided Sampling (PFGS) with Bayesian Optimization (BO)
result Systematic identification of high-performing, feasibility-compliant, and perturbation-resilient operating conditions.
Study efficient pricing for barrier options in stochastic-volatility models with leverage correction.
problem Barrier options are sensitive to volatility dynamics, especially leverage, making accurate pricing difficult.
method Developed a class of continuous-path stochastic-clock volatility models and a systematic small-ρ expansion to incorporate leverage.
result Transform-only pricing formulas for barrier derivatives are fast and numerically stable, even for negative leverage.
Constructs an analytic index for infinite dimensional manifolds with -action.
problem Analyzing infinite dimensional manifolds with -action.
method Defines an analytic -equivariant index using a Hilbert space, Dirac operator, and crossed product.
result Justifies the analytic index in terms of noncommutative geometry.
Algorithm creates polynomials for knotted surfaces, with bounds on degree.
problem Creating polynomials for knotted surfaces with constraints.
method Algorithm constructs polynomials based on loop braids and surface braids.
result Upper bounds on the degree of polynomials for knotted surfaces.
Explicit BCH series radii found for special Banach-Malcev shift algebras.
problem Finding convergence radii for BCH series in specific algebraic structures.
method Established explicit convergence radii using continuity estimates and algebraic properties.
result Explicit formula for convergence radii derived and validated for various shift algebras.
To a knot in 3-space, one can associate a sequence of Laurent polynomials, whose th term is the th colored Jones polynomial. The Volume Conjecture for small angles states that the value of the -th colored Jones polynomial at $e^{\a/n}$ is a sequence of complex numbers that grows subexponentially, for a fixed s…
Projective loops generate rational loop groups without needing nilpotent loops.
problem Generating rational loop groups with noncompact reality conditions.
method Used projective loops to prove rational loop groups can be generated.
result Projective loops alone are sufficient to generate rational loop groups.
Study loop ensembles on graphs, linking group theory and topology.
problem Understanding loop homotopy classes and homologies on graphs.
method Determined distributions of loop homotopy classes and homologies using the lower central series of the fundamental group.
result Distributions of loop homotopy classes and homologies defined by the lower central series of the fundamental group.
Study smooth loops and loop bundles, relating to -structures.
problem Properties of smooth loops and their applications.
method Analyze smooth loops, introduce loop bundles, define torsion and curvature.
result Showed how loop bundles relate to -structures.
Study asymptotic expansion of graph Laplacian on discretized surfaces, relating spanning trees and cycle-rooted forests.
problem Asymptotic expansion of graph Laplacian on discretized surfaces.
method Relate spanning trees and cycle-rooted spanning forests to zeta-regularized determinants.
result Explicit formula for limit of cycle-rooted spanning forest probability and topological observables.
Skew loops in flat torimath.DG
We produce skew loops -- loops having no pair of parallel tangent lines -- homotopic to any loop in a flat torus or other quotient of R^n. The interesting case here is n=3. More subtly for any n, we characterize the homotopy classes that will contain a skew loop having a specified loop in the unit sphere as tangent ind…
The paper proves T-duality and Hori formulae for winding loop spaces.
problem Realizing T-duality and Hori formulae for loop spaces.
method Proving T-duality and Hori formulae for winding q-loop spaces.
result T-duality and Hori formulae for winding q-loop spaces are proven.
Paper establishes loop space T-duality formulae and refines earlier work.
problem Establishing T-duality on loop spaces with background flux.
method Developed loop Hori map and twisted Bismut-Chern character.
result Loop Hori map induces quasi-isomorphism on loop spaces.
The 2-loop polynomial is a polynomial presenting the 2-loop part of the Kontsevich invariant of knots. We show a cabling formula for the 2-loop polynomial of knots. In particular, we calculate the 2-loop polynomial for torus knots.
Kähler manifold loop space inherits Kähler structure and is complete.
problem Characterizing the geometric properties of loop spaces of Kähler manifolds.
method Proving the Kähler structure and completeness of the loop space.
result The loop space of a Kähler manifold is complete and inherits a Kähler structure.
Regshock visualizes financial risks to help regulators manage systemic shocks.
problem Managing systemic risks in financial networks.
method Risk-island visualization algorithm and regshock visual exploration approach.
result Demonstrated improved risk management and control capabilities.
Bayesian optimization with Gaussian processes speeds up searches for stationary points.
problem Accelerating searches for stationary points on potential energy surfaces.
method Unified Bayesian optimization view using Gaussian process regression with derivative observations, inverse-distance kernels, and active learning.
result Surrogates can reduce the number of expensive electronic structure evaluations by an order of magnitude.
New structure found in loops on quasi-surfaces.
problem Understanding operations on loops in quasi-surfaces.
method Defined a quasi-Lie bialgebra structure.
result Natural operations on loops form a quasi-Lie bialgebra.
Using the relations between the theory of differentiable Bol loops and the theory of affine symmetric spaces we classify all connected differentiable Bol loops having an at most -dimensional semi-simple Lie group as the group topologically generated by their left translations. We show that all these Bol loops are is…
A skew loop is a closed curve without parallel tangent lines. We prove: The only complete surfaces in euclidean 3-space with a point of positive curvature and no skew loops are the quadrics. In particular, ellipsoids are the only closed surfaces without skew loops. We also prove results about skew loops on cylinders an…
Introduces string structures linking to loop spaces.
problem Understanding geometric string structures.
method Explains connections to loop spaces.
result String structures linked to loop spaces.
Study shows looping a 6-manifold over a 4-manifold results in a product of loops on spheres.
problem Understanding the homotopy properties of 6-manifolds over 4-manifolds.
method Analyzes the homotopy equivalence and rational homotopy of the total space of a sphere bundle over a 4-manifold.
result Looping a 6-manifold over a 4-manifold is homotopy equivalent to a product of loops on spheres.
Integral invariants for framed 3-manifolds associated to trivalent graphs possibly with self-loopsmath.GT
New theory defines invariants for framed 3-manifolds using trivalent graphs with self-loops.
problem Defining invariants for framed 3-manifolds with semi-simple Lie groups.
method Constructing graph complexes and cocycles, including self-loops, to define invariants.
result Higher-loop invariants can be defined by graph cocycles with or without self-loops.
Geodesic loops on tetrahedra are studied in spherical and hyperbolic spaces.
problem Characterize geodesic loops on tetrahedra in different types of spaces.
method Analytical proofs for spherical and hyperbolic spaces.
result Existence and properties of geodesic loops on tetrahedra in various spaces.
Method calculates loop intersections on surfaces.
problem Computing intersections of loops on surfaces.
method Nielsen fixed point theory and Gröbner-Shirshov basis.
result Simple method to compute intersection numbers.
Training-free looped transformers improve model performance without additional training.
problem Improving model performance without additional training or fine-tuning.
method A lightweight inference-time wrapper loops a contiguous mid-stack block of layers of a frozen checkpoint without additional fine-tuning.
result Our method improves model performance across various model families.