Study the evolution of the Lorenz strange set using Conley index theory.
problem Understanding the evolution of the Lorenz strange set through parameter changes.
method Application of Conley index theory to analyze the global attractor and its Morse decompositions.
result Identification and analysis of bifurcations and the role of the strange set in these transformations.
New integral expression quantizes Arnold strangeness.
problem Quantifying Arnold strangeness of plane curves.
method Integrating curvatures multiplied by densities, reformulating Arnold strangeness using Shumakovitch's partition function.
result Quantized Arnold strangeness includes rotation number and higher invariant terms.
Paper defines a new invariant for surface immersions.
problem Detecting and classifying jumps in surface immersions.
method Defines an integer-valued function to classify jumps involving quadruple points and triple-line tangencies.
result Classifies quadruple point jumps into five geometrically distinct cases.
The paper defines and computes a knot complement invariant for simple links.
problem Defining and computing a knot complement invariant for simple links.
method Using the large color R-matrix to study the Gukov-Manolescu series.
result Presentation of strange identities for positive braid knots.
The paper proves congruences for Fishburn numbers at roots of unity.
problem Arithmetic properties of Fishburn numbers at roots of unity.
method Proves prime power congruences for generalized Fishburn numbers.
result Proves congruences for coefficients of specific series at roots of unity.
Proves resurgence properties for Habiro elements from radial limits of theta series.
problem Proving resurgence properties for Habiro elements.
method Using strange identities and Borel transform of formal power series.
result Proves Costin and Garoufalidis conjecture for two families of torus knots.
We give examples of local signatures, completely different from the usual ones, for general fibrations of genus 2 and genus 3.
New method reconstructs hidden dynamics from low-dimensional time series.
problem Reconstructing hidden dynamics from limited experimental data.
method Autoencoder trained with a novel latent-space loss function.
result Reconstructs strange attractors better than existing techniques.
Neural networks can model chaos efficiently by becoming geometrically chaotic.
problem Lack of theoretical understanding of how neural networks learn chaos.
method Employed a geometric perspective to show neural networks can model chaotic dynamics.
result Neural networks can reconstruct strange attractors and accurately predict local divergence rates.
We define the generalized connected sum for generic closed plane curves, generalizing the strange sum defined by Arnold, and completely describe how the Arnold invariants J± and St behave under the generalized connected sums.
Recently V. Arnold introduced Strangeness and J± invariants of generic immersions of an oriented circle to R2. Here these invariants are generalized to the case of generic immersions of an oriented circle to an arbitrary surface F. We explicitly describe all the invariants satisfying axioms, which naturall…
Transformer is reinterpreted as a numerical ODE solver, leading to improved architectures.
problem Understanding and improving the Transformer architecture.
method Interpreted Transformer as a numerical ODE solver for a multi-particle dynamic system, proposing new architectures.
result Macaron Net outperforms Transformer on supervised and unsupervised learning tasks.
We have carried out simulations of a financial model of the firm to analyse the validity of the concept of Trade on Equity in dynamics. The results exhibit the ability of the borrowing policy connected to a cautious dividend distribution to inject chaos into the profit motion. The 3D system built with the van der Pol's…
Constructing transitive nilpotent Lie algebras from dilations and analyzing their prolongations.
problem Understanding the derivations of Tanaka prolongations of transitive nilpotent Lie algebras.
method Constructing transitive nilpotent Lie algebras from dilations and analyzing their prolongations.
result Derivations of degree 0 are given by vector fields of degree 0, and the Tanaka prolongation recovers the whole algebra of polynomial vectors defined by the dilation.
Study of Fishburn numbers and their congruences for torus knots.
problem Arithmetic properties of Fishburn numbers and congruences for torus knots.
method Use of divisibility results and a new identity for Kontsevich-Zagier series.
result Prove prime power congruences for generalized Fishburn numbers.
We investigate the issue of model selection and the use of the nonconformity (strangeness) measure in batch learning. Using the nonconformity measure we propose a new training algorithm that helps avoid the need for Cross-Validation or Leave-One-Out model selection strategies. We provide a new generalisation error boun…
Quantum modularity proven for specific theta series.
problem Proving quantum modularity for partial theta series with periodic coefficients.
method Explicit proof using Kontsevich-Zagier series and colored Jones polynomials.
result Kontsevich-Zagier series is a weight 3/2 quantum modular form.
One computes the cohomology of the projective embedding of sl(m+1,R) acting on the differential operators on densities on R^m of various weights. This cohomology is non vanishing only for some special critical values of the weights. This allows us first to explain some strange feature pointed out by Gargoubi in his cla…
New risk models use chaotic attractors to predict extreme events.
problem Predicting Black Swan events in financial markets.
method Combining heavy-tailed priors with chaotic dynamics (Lorenz and Rossler systems).
result Models generate volatility clustering, fat tails, and extreme events.
It is shown that a lagrangian system whose Legendre transformation degenerates along a hypersurface behaves in a strange manner by jumping from time to time without any ''visible cause''. In such a jump the system changes instantaneously its coordinates as well as its momenta. The mathematical dscription of the phenome…
Let (M,g) be a compact manifold and let −Δφk=λkφk be the sequence of Laplacian eigenfunctions. We present a curious new phenomenon which, so far, we only managed to understand in a few highly specialized cases: the family of functions fN:M→R≥0 $$ f_N(x) = \sum_{k \leq N}{ \frac{…
Following \cite{citeSavelyevVirtualMorsetheoryonOmegaHam(Momega).}, we develop here a connection between Morse theory for the (positive) Hofer length functional L:ΩHam(M,ω)→R, with Gromov-Witten/Floer theory, for monotone symplectic manifolds (M,ω). This gives some immediate restrictio…
New method improves conformal prediction for machine learning models.
problem Improving the efficiency and informativeness of conformal prediction models.
method Introduces Penalized Inverse Probability (PIP) and Regularized PIP (RePIP) nonconformity score functions.
result PIP-based conformal classifiers strike a good balance between informativeness and efficiency.
Khovanov homology, an invariant of links in R3, is a graded homology theory that categorifies the Jones polynomial in the sense that the graded Euler characteristic of the homology is the Jones polynomial. Asaeda, Przytycki and Sikora generalized this construction by defining a double graded homology theory…
Study discrete analog of zeta-determinant maximization on triangulated surfaces.
problem Maximizing zeta-determinant for discrete Laplacian on triangulated surfaces.
method Analogous to Osgood, Phillips, and Sarnak's theorem, study stationary points of determinants for discrete cotan-Laplacian.
result Discrete metrics of constant discrete Gaussian curvature are stationary points of the determinant, suggesting minima.
SGD with large learning rates can converge to local maxima.
problem Understanding the behavior of SGD with large learning rates.
method Constructing worst-case optimization problems.
result SGD can converge to local maxima under certain conditions.
Wireless sensor networks usually comprise a large number of sensors monitoring changes in variables. These changes in variables represent changes in physical quantities. The changes can occur for various reasons; these reasons are highlighted in this work. Outliers are unusual measurements. Outliers are important; they…
The paper constructs chaotic solutions to the Euler equations on high-dimensional manifolds.
problem Chaos in fluid dynamics on high-dimensional manifolds.
method Constructs finite-dimensional families of non-steady solutions to the Euler equations.
result Existence of strange attractors and chaos in the phase space.
Bayesian Neural Networks improve uncertainty modeling in facial emotion recognition.
problem High aleatoric uncertainty and visual ambiguity in facial emotion recognition.
method Bayesian Neural Networks approximated using MC-Dropout, MC-DropConnect, or Ensemble methods.
result Bayesian Neural Networks produce more human-like output probabilities.
100 years ago exactly, in 1906, Hartogs published a celebrated extension phenomenon (birth of Several Complex Variables), whose global counterpart was stated in full generality later by Osgood (1929): holomorphic functions in a connected neighborhood V(bD) of a connected boundary bD contained in C^n (n >= 2) do extend …
Superized Kaehler manifolds with continuous parameters.
problem How to extend Lie algebra actions to Lie superalgebras.
method Continuous parameterization of Kaehler manifolds and hyper-Kaehler supermanifolds.
result Definitions of hyper-Kaehler supermanifolds with parameters.
A new warping-invariant distance improves nearest-neighbor classification efficiency.
problem dtw distance inconsistency and inefficiency in nearest-neighbor classification.
method Showed dtw is not warping-invariant, converted to twi distance.
result twi distance equivalent error rates to dtw, more efficient.
Study shows Lefschetz fibrations on Milnor fibers of certain singularities.
problem Understanding Lefschetz fibrations on Milnor fibers of specific singularities.
method Analyzes Milnor fibers of cusp and simple elliptic singularities to construct Lefschetz fibrations.
result Milnor fibers of cusp and simple elliptic singularities admit genus-one Lefschetz fibrations.
Proposes a new method combining Reservoir Computing and Normalizing Flow for predicting stochastic dynamical systems.
problem Predicting and capturing long-term behaviors of stochastic dynamical systems.
method Data-driven framework combining Reservoir Computing and Normalizing Flow, integrating error modeling and both approaches virtues.
result Successfully predicts the long-term evolution of stochastic dynamical systems and replicates dynamical behaviors.
Motivated by analogies with basic density theorems in analytic number theory, we introduce a notion (and variations) of the homological density of one space in another. We use Weil's number field/ function field analogy to predict coincidences for limiting homological densities of various sequences $\mathcal{Z}^{(d_1,\…
Study Figgie card game strategies using agent-based simulation.
problem Analyze strategies for Figgie card game and market behavior.
method Develop agent-based discrete-event market simulation to test strategies.
result Fundamentalist strategy is profit-maximizing in all tested combinations.
This paper generates natural-looking perturbations to fool classifiers.
problem Generating adversarial examples that mimic natural objects or signals.
method Employing generative adversarial networks and optimization algorithms.
result The approach can fool classification models in both image and audio domains.
Bayes rule replaces do-calculus for causal inference.
problem Representing and addressing causal problems with probability theory.
method Encoding causal graphical models in Probabilistic graphical models and using Bayesian statistics.
result Causal effects can be estimated entirely within the Bayesian paradigm.
In Part II of this paper, we concentrate our analysis on the price dynamical model with the moving average rules developed in Part I of this paper. By decomposing the excessive demand function, we reveal that it is the interplay between trend-following and contrarian actions that generates the price chaos, and give par…
The paper derives formulas for linear connections with totally anti-symmetric torsion in 3D generalized Berwald manifolds.
problem Understanding linear connections with specific torsion in generalized Berwald manifolds.
method Averaging of Finslerian quantities over indicatrix surfaces to express the torsion tensor.
result Explicit formulas for linear connections with totally anti-symmetric torsion in 3D generalized Berwald manifolds.
Quantum modularity proved for a knot manifold.
problem Proving quantum modularity for a specific closed hyperbolic 3-manifold.
method Using factorization of state integrals and proving quantum modularity for functions and q-series. result Quantum modularity for the closed manifold provides a unification of volume conjecture and Witten's asymptotic expansion conjecture.
In this thesis, we prove several results concerning field-theoretic invariants of knots and 3-manifolds. In Chapter 2, for any knot K in a closed, oriented 3-manifold M, we use SU(2) representation spaces and the Lagrangian field theory framework of Wehrheim and Woodward to define a new homological knot invariant…
Study geodesic X-ray transform and streaking artifacts on simple surfaces or spaces of constant curvature.
problem Streaking artifacts in CT images due to metal regions.
method Geodesic X-ray transform on nontrapping compact Riemannian manifolds with strictly convex boundaries.
result Streaking artifacts result from conormal singularities along common tangent geodesics.
In this paper we introduce a new algebraic device, which enables us to treat the quaternions as though they were a commutative field. This is of interest both for its own sake, and because it can be applied to develop an "algebraic geometry" of noncompact hypercomplex manifolds. The basic building blocks of the theory …
Machine learning (ML) has become a commodity in our every-day lives. We routinely ask ML empowered smartphones to suggest lovely food places or to guide us through a strange place. ML methods have also become standard tools in many fields of science and engineering. A plethora of ML applications transform human lives a…
The paper finds the extremal compatible linear connection on generalized Berwald manifolds.
problem Finding the extremal compatible linear connection on generalized Berwald manifolds.
method Minimizing the pointwise length of the torsion tensor, solving conditional extremum problems.
result An intrinsic algorithm to check the existence of compatible linear connections on Finsler manifolds.
Study shows Julia sets and gasket limit sets are quasiconformally different.
problem Quasiconformal non-equivalence of Julia sets and gasket limit sets.
method Proved quasiconformal non-equivalence of Julia sets and gasket limit sets.
result Julia sets and gasket limit sets are quasiconformally different.
New deep learning model for matching sets of items, preserving exchangeability.
problem Matching two different sets of items while preserving exchangeability.
method Exchangeable deep neural networks architecture and efficient training framework.
result Significant improvements in fashion set recommendation and group re-identification.