Study on special surfaces in Walker 3-manifolds.
problem Characterizing totally umbilical surfaces in Walker 3-manifolds.
method Utilizes techniques from homogeneous Riemannian three-manifolds classification.
result Surfaces are either totally geodesic or ambient manifold is locally conformally flat.
First I will explain my motivation to introduce the δ-invariants for Riemannian manifolds. I will also recall the notions of ideal immersions and best ways of living. Then I will present a few of the many applications of δ-invariants to several areas in mathematics. Finally, I will present two optimal inequalities …
Graphs from van der Corput sequence embed into Chamanara surface.
problem Embedding graphs from van der Corput sequence into surfaces.
method Constructed 4-regular graphs from van der Corput sequence and Kronecker sequence, embedded into torus and Chamanara surface. result Graphs from van der Corput sequence embed into Chamanara surface with one edge removal.
Study finds multiple solutions for Van der Waals-Cahn-Hilliard equation on manifolds.
problem Finding multiple solutions for a specific equation on manifolds.
method Combines Lusternik-Schnirelman and Morse theory with a photography method.
result Establishes multiplicity of solutions using topological invariants.
The paper classifies Lagrangian submanifolds in complex space forms achieving a specific curvature inequality.
problem Classifying Lagrangian submanifolds in complex space forms that achieve a specific curvature inequality.
method Analyzing the equality case of a curvature inequality for Lagrangian submanifolds in complex space forms.
result Classification of δ(2,n−2)-ideal Lagrangian submanifolds in n-dimensional complex space forms. A cubic identity for the Infeld-van der Waerden field is found and its application to verifying an explicit formula for the spinor components of the metric connection is demonstrated.
Study finds multiple solutions to a complex equation with volume constraint.
problem Finding multiple solutions to a nonlinear elliptic equation with a specific potential.
method Analyzes a Van der Waals-Allen-Cahn-Hilliard equation with a linear volume constraint on a bounded Lipschitz domain.
result Estimates the number of solutions using topological and homological invariants.
Researchers define and evaluate quasi-local mass near axially symmetric null infinity.
problem Defining and evaluating quasi-local mass at null infinity.
method Using Bondi-van der Burg-Metzner coordinates, the researchers evaluate the Wang-Yau quasilocal mass on surfaces of unit size at null infinity of axi-symmetric spacetimes.
result Evaluation of quasi-local mass near axially symmetric null infinity.
Proves generic nondegeneracy for solutions under volume constraint in closed manifolds.
problem Proving nondegeneracy for solutions of the Van der Waals-Allen-Cahn-Hilliard equation.
method Adapting techniques from previous research to prove nondegeneracy.
result Generic nondegeneracy for solutions of the Van der Waals-Allen-Cahn-Hilliard equation under a volume constraint in closed manifolds.
This paper proves t-SNE can recover well-separated clusters, improving visualization and embedding quality.
problem The lack of mathematical foundations and inner workings of t-SNE.
method Proves t-SNE's ability to recover well-separated clusters, using early exaggeration phase and rigorous analysis.
result t-SNE in the early exaggeration phase can be rigorously analyzed and provides novel ways to set parameters.
New approach models brain dynamics using coupled van der Pol oscillators and LSTM.
problem Capturing nonlinear dynamics in brain calcium imaging data.
method Proposes a new approach combining van der Pol oscillators and LSTM for modeling brain activity.
result Shows improved accuracy and interpretability compared to LSTM and hybrid VDP-LSTM approach.
This paper surveys parallel submanifolds in Riemannian and pseudo-Riemannian manifolds.
problem Understanding parallel submanifolds in Riemannian and pseudo-Riemannian manifolds.
method Comprehensive survey of parallel submanifolds.
result Extrinsic invariants of parallel submanifolds do not vary from point to point.
Novel knot polynomials from Gaussian calculus show half vanish and determine Jones polynomials.
problem Understanding and characterizing knot polynomials from Gaussian calculus.
method Gaussian calculus of generating series for noncommutative algebras, connected sum of knots.
result Half of the polynomials vanish and three polynomials are explicitly given.
We give a complete classification of umbilical submanifolds of arbitrary dimension and codimension of $\Sf^n\times \R$, extending the classification of umbilical surfaces in $\Sf^2\times \R$ by Rabah-Souam and Toubiana as well as the local description of umbilical hypersurfaces in $\Sf^n\times \R$ by Van der Veken and …
We prove that any limit-interface corresponding to a locally uniformly bounded, locally energy-bounded sequence of stable critical points of the van der Waals--Cahn--Hilliard energy functionals with perturbation parameter tending to 0 is supported by an embedded smooth stable minimal hypersurface in low dimensions and …
Study MMD for critical transitions in fast-slow systems, showing it's a good binary classifier.
problem Detecting change points in multiscale systems with critical transitions.
method Link between dynamical theory of critical transitions and statistical MMD, leading-order approximation.
result MMD is a good binary classifier for detecting change points in critical transitions.
Deep learning model simulates noisy dynamical systems without distributional assumptions.
problem Simulating noisy dynamical systems with unknown distributional properties.
method DE-LSTM model using LSTM network for multi-label classification and penalized maximum log likelihood.
result DE-LSTM makes accurate predictions of probability distributions for noisy dynamical systems.
We give a complete description of all hypersurfaces of the product spaces $\Sf^n\times \R$ and $\Hy^n\times \R$ that have flat normal bundle when regarded as submanifolds with codimension two of the underlying flat spaces $\R^{n+2}\supset \Sf^n\times \R$ and $\Le^{n+2}\supset \Hy^n\times \R$. We prove that any such hyp…
Automatically learns summary features from time series data for likelihood-free inference.
problem Necessity of hand-tailored summary features for time series data in likelihood-free inference.
method Data-driven approach to automatically learn summary features.
result Learning summary features from data can outperform hand-crafted values in likelihood-free inference.
The present paper introduces a majority orienting model in which the dealers' behavior changes based on the influence of the price to show the oscillation of stock price in the stock market. We show the oscillation of the price for the model by applying the van der Pol equation which is a deterministic approximation of…
Responding to discussions on missing data models.
problem Clarifying causal and counterfactual views in missing data models.
method Summarizing and responding to comments from discussants.
result Summarized main contributions and addressed each discussion.
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…
We consider the concept of Stokes-Dirac structures in boundary control theory proposed by van der Schaft and Maschke. We introduce Poisson reduction in this context and show how Stokes-Dirac structures can be derived through symmetry reduction from a canonical Dirac structure on the unreduced phase space. In this way, …
Paper proposes robust methods to detect and treat outliers in multivariate loss reserving.
problem Distortion of traditional reserving techniques by outliers in past claims data.
method Two robust bivariate chain-ladder techniques: Adjusted Outlyingness and Bagdistance.
result Improved accuracy in estimating outstanding claim liabilities through robust methods.
Develops a Gaussian model to compute the Alexander polynomial of knots.
problem Computing the Alexander polynomial of knots.
method Uses perturbed Gaussian functions, Heisenberg algebra, and tensor-contraction formalism.
result Associates a Gaussian function to a knot whose partition function recovers the Alexander polynomial.
We prove an upper bound for the evaluation of all classical SU(2) spin networks conjectured by Garoufalidis and van der Veen. This implies one half of the analogue of the volume conjecture which they proposed for classical spin networks. We are also able to obtain the other half, namely, an exact determination of the s…
New derivation of knot invariants from universal invariant.
problem Deriving knot invariants from universal invariant.
method Using Hopf algebra D and a Mathematica implementation to compute ZD(K). result Derivation of large-color expansion from universal invariant.
Motivated by the problem of finding an explicit description of a developable narrow Moebius strip of minimal bending energy, which was first formulated by M. Sadowsky in 1930, we will develop the theory of elastic strips. Recently E.L. Starostin and G.H.M. van der Heijden found a numerical description for an elastic Mo…
This paper classifies flat submanifolds with a special type of curvature form.
problem Classifying flat submanifolds with a specific curvature property.
method Using Moebius geometry and curvature operators to classify submanifolds.
result Classification of umbilic-free isometric immersions with flat normal bundle and semi-parallel Moebius second fundamental form.
We use the formalism of Geometrothermodynamics to describe chemical reactions in the context of equilibrium thermodynamics. Any chemical reaction in a closed system is shown to be described by a geodesic in a 2−dimensional manifold that can be interpreted as the equilibrium space of the reaction. We first show this i…
Study satellite operations on knot invariant θ, proving additivity and distinguishing knots.
problem Behavior of knot invariant θ under satellite operations.
method Proved additivity, introduced computational tool, verified conjecture for 2977 knots.
result Distinguished knots using invariant θ and verified conjecture for 2977 primes.
Study character varieties of surfaces using cluster algebras and Poisson structures.
problem Character varieties of surfaces and their Poisson structures.
method Use Bonahon-Wong's trace map and cluster algebras associated with ideal triangulations.
result Recover Goldman Poisson algebra from cluster algebra structure and show automorphisms.
New knot invariants derived using quantum cluster algebras.
problem Deriving new knot invariants from quantum cluster algebras.
method Interpreting R-matrix of Uq(sl2) as cluster transformation, introducing auxiliary parameter ε. result Derives perturbed-Alexander invariants with higher-order terms in ε. The CAP slope is Bayes' theorem in cumulative coordinates, unlocking the weight of evidence, Somers' D, and Gini coefficient.
problem Calibration and weight of evidence
method Identifying the CAP slope as Bayes' theorem
result Revealing the CAP slope as Bayes' theorem
Study embeddability of 2-complexes in 4-space, proving Heawood family's excluded minors.
problem Whether a 2-dimensional CW complex embeds in R4. method Operations preserving embeddability, constructions of non-preserving transformations, study of 4-flat graphs.
result Prove 78 graphs of Heawood family are excluded minors for 4-flat graphs.
Paper proposes a new method for predicting DER adoption with hierarchical guarantees.
problem Accurately predicting DER adoption in electric grids with uncertainty and spatial disparity.
method Multivariate Hawkes process for modeling DER adoption dynamics and split conformal prediction algorithm for hierarchical validity.
result Empirical evaluation shows superior predictive accuracy and uncertainty calibration compared to existing methods.
Study of knot invariant growth for twisted knots.
problem Understanding the behavior of a knot invariant for families of knots.
method Analyzing the perturbed Alexander invariant for twisted knots.
result Coefficients of the invariant grow linearly as the number of twists increases.
The intention with this paper is to provide all the estimation concepts and techniques that are needed to implement a two-phases approach to the parametric estimation of probability of default (PD) curves. In the first phase of this approach, a raw PD curve is estimated based on parameters that reflect discriminatory p…
Paper introduces ℓ-DER for regression tasks using morphological operators and convex-concave procedure.
problem Developing a universal approximator for regression tasks.
method Introduces ℓ-DER model, trains it using a convex-concave procedure (CCP) to minimize least-squares. result Outperforms other hybrid morphological models and state-of-the-art approaches.
Researchers use manifold learning to analyze 4D-STEM data of graphene, revealing atomic structure details.
problem Challenges in processing and interpreting large 4D-STEM datasets, especially for light materials.
method Data-driven manifold learning approaches for visualization and exploration of 4D-STEM datasets.
result Extracted patterns relate to individual atom sites and sublattice structures, effectively discriminating single dopant anomalies.
New techniques prove bounded acyclicity results for semi-simplicial sets.
problem Homological stability of semi-simplicial sets and groups.
method Combinatorial techniques in simplicial bounded cohomology.
result Slope-1/2 stability results for certain groups. Paper introduces new metrics for evaluating model accuracy.
problem Improving model accuracy and calibration.
method Developed two second-order accuracy metrics with integral and numerical representations.
result Validates model calibration settings and reveals distortions.
Deep reinforcement learning solves complex differential equations.
problem Solving nonlinear differential equations.
method Rule-based deep reinforcement learning approach.
result Solver captures intrinsic nature of equations with high accuracy.
New approach uses machine learning to control DERs without centralized communication.
problem Optimal power flow requires extensive communication; new method uses local data.
method Data-driven approach to learn control policies for DERs to mimic centralized OPF solutions.
result Decentralized controllers closely match centralized OPF solution, providing near optimal performance.
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.
Researchers improve BART's theoretical foundation for better performance.
problem Improving theoretical understanding of BART's performance.
method Dive into branching process theory and random walks to derive optimal posterior convergence rate for BART.
result Obtained optimal rate of posterior convergence for BART.
We show that classical thermodynamics has a formulation in terms of Hamilton-Jacobi theory, analogous to mechanics. Even though the thermodynamic variables come in conjugate pairs such as pressure/volume or temperature/entropy, the phase space is odd-dimensional. For a system with n thermodynamic degrees of freedom it …
New method selects critical DER scenarios for distribution grid investment planning.
problem Determining critical DER adoption scenarios for risk assessment in distribution grids.
method Bayesian Optimization framework using Gaussian Process surrogates and Pareto-critical acquisition function.
result Statistical guarantee and significant speed-up over exhaustive search in selecting critical DER scenarios.