Knots and 4-manifolds linked via matrix kinking.
problem Understanding equivalence of symmetric matrices and their implications.
method Isotopy and kinking moves on Goeritz matrices.
result Every nonsingular symmetric integer matrix is kink-equivalent to positive or negative-definite matrices.
Study on manifolds with kinks and Gaussian kernel behavior.
problem Understanding the asymptotic behavior of graph Laplacian on manifolds with singularities.
method Introduced manifolds with kinks, derived asymptotic behavior of Graph Laplacian with Gaussian kernel, and validated results numerically.
result Asymptotic behavior of the Graph Laplacian is determined by the inward sector of the tangent space.
The paper resolves kinks on curves on surfaces with punctures.
problem Resolving kinks of curves on surfaces with punctures.
method Application of the Diamond Lemma.
result The resolution of kinks of a curve on a surface with punctures is uniquely determined up to homotopy.
In order to study a one-dimensional analogue of the spontaneous curvature model for two-component lipid bilayer membranes we consider planar curves that are made of a material with two phases. Each phase induces a preferred curvature to the curve, and these curvatures as well as phase boundaries may lead to the develop…
Two-layer neural networks must be robust, even with arbitrary weights.
problem Proving the robustness of two-layer neural networks with arbitrary weights.
method Developed a new function-space covering method to prove the robustness law, replacing parameter-space covering.
result Proved the conjectured law for two-layer networks with arbitrary real weights, biases, and affine skip connections.
We study necessary and sufficient conditions for the existence of Lorentzian and weak Lorentzian cobordisms between closed smooth manifolds of arbitrary dimension such that the structure group of the frame bundle of the cobordism is $\Spin(1, n)_0$. This extends a result of Gibbons-Hawking on $\Sl(2, \C)$-Lorentzian co…
Bayesian estimators for causal inference using hierarchical Gaussian Processes.
problem Estimating causal effects in sharp and fuzzy RD/RK designs.
method Hierarchical Gaussian Process models for regression and classification.
result Hierarchical GP models improve precision and coverage of RD/RK estimations.
The n-solvable filtration {Fn}n=0∞ of the smooth knot concordance group (denoted by C), due to Cochran-Orr-Teichner, has been instrumental in the study of knot concordance in recent years. Part of its significance is due to the fact that certain geometric characterizations of a knot …
We prove new improved endpoint, Lpc, pc=n−12(n+1), estimates (the "kink point") for eigenfunctions on manifolds of nonpositive curvature. We do this by using energy and dispersive estimates for the wave equation as well as new improved Lp, 2<p<pc, bounds of Blair and the author \cite{BSTop}, \…
The existence of K-instantons on a cylinder M^7 = R_tau x K/H over a homogeneous nearly K"ahler 6-manifold K/H requires a conformally parallel or a cocalibrated G_2-structure on M^7. The generalized anti-self-duality on M^7 implies a Chern-Simons flow on K/H which runs between instantons on the coset. For K-equivariant…
Dropout schedules can be optimized to significantly reduce model test loss.
problem Improving model performance in neural networks.
method Developed a mean-field theory of dropout at the edge of chaos, proposing front-loaded dropout schedules.
result Front-loaded dropout schedules reduce test loss by 18-35% over constant dropout.
Following a Geometrical Brownian Motion extension into an Irrational Fractional Brownian Motion model, we re-examine agent behaviour reacting to time dependent news on the log-returns thereby modifying a financial market evolution. We specifically discuss the role of financial news or economic information positive or n…
The simplest field theory description of the multivariate statistics of forward rate variations over time and maturities, involves a quadratic action containing a gradient squared rigidity term. However, this choice leads to a spurious kink (infinite curvature) of the normalized correlation function for coinciding matu…
Common models for two-phase lipid bilayer membranes are based on an energy that consists of an elastic term for each lipid phase and a line energy at interfaces. Although such an energy controls only the length of interfaces, the membrane surface is usually assumed to be at least C1 across phase boundaries. We consi…
This work shows how penalising bias terms in norm regularisation leads to sparse solutions.
problem Understanding the relation between parameter norm regularization and the sparsity of neural network solutions.
method Analyzes one hidden ReLU layer networks with unidimensional data, showing the norm required for function representation and the importance of the bias term's norm.
result Penalising the bias terms in regularisation leads to sparse solutions, enforcing the uniqueness and sparsity of the minimal norm interpolator.
Solitary waves are localized gravity waves that preserve their consistency and henceforth their visibility through properties of nonlinear hydrodynamics. Solitary waves have finite amplitude and spread with constant speed and constant shape. In this paper, we have used Lie group of transformation method to solve (3 + 1…
Given a braid presentation D of a hyperbolic knot, Hikami and Inoue consider a system of polynomial equations arising from a sequence of cluster mutations determined by D. They show that any solution gives rise to shape parameters and thus determines a boundary-parabolic PSL(2,C)-representation …
Optimizes leveraged staking strategies in decentralized finance.
problem Maximizing returns on staked assets in decentralized lending platforms.
method Developed a mathematical framework to optimize leveraged staking strategies, reducing the multi-market problem to convex allocation over market exposures.
result Rebalanced leveraged positions can achieve up to 6.2% APY, significantly higher than unleveraged staking.
Simulates financial market orders using anomalous diffusion models.
problem Anomalous diffusion in financial market order dynamics.
method Discrete Time Random Walk with Sibuya waiting times, non-uniform sampling, and cubic spline interpolation.
result Demonstrates price impact for different forcing functions and model parameters.
This article combines various methods of analysis to draw a comprehensive picture of penalty approximations to the value, hedge ratio, and optimal exercise strategy of American options. While convergence of the penalised solution for sufficiently smooth obstacles is well established in the literature, sharp rates of co…
This chapter covers different approaches to policy evaluation for assessing the causal effect of a treatment or intervention on an outcome of interest. As an introduction to causal inference, the discussion starts with the experimental evaluation of a randomized treatment. It then reviews evaluation methods based on se…
Study detects boundaries in unlabeled noisy images without labels.
problem Detecting boundaries in unlabeled noisy images without labels.
method Proposed a continuous hinge-type surrogate loss for boundary detection, combined with deep neural networks.
result Deep neural network achieves minimax-optimal boundary recovery rate under piecewise smooth boundary model.
Paper introduces differentiable sorting and ranking with O(nlogn) time complexity.
problem Non-differentiability of sorting and ranking operations in machine learning.
method Differentiable proxies constructed as projections onto the permutahedron and reduction to isotonic optimization.
result First differentiable sorting and ranking operators with O(nlogn) time and O(n) space complexity. We review our recent work on solitons in the Higgs phase. We use U(N_C) gauge theory with N_F Higgs scalar fields in the fundamental representation, which can be extended to possess eight supercharges. We propose the moduli matrix as a fundamental tool to exhaust all BPS solutions, and to characterize all possible modu…
This work improves interpretability and calibration of complex-valued neural networks using Newton-Puiseux analysis.
problem Insufficient interpretability and probability calibration of complex-valued neural networks.
method Newton-Puiseux framework to examine local decision geometry, fitting a polynomial surrogate and factorizing it using Newton-Puiseux expansions.
result Enhanced Expected Calibration Error in ECG and wireless modulation datasets compared to uncalibrated softmax and standard post-hoc baselines.
A neural network solves Black-Scholes PDE for option pricing with uncertainty quantification.
problem Solving the Black-Scholes equation for option pricing with uncertainty.
method Physics-informed neural network (PINN) that embeds BS operator and conditions, handles early exercise via relaxation, and uses anchored-ensemble fine-tuning for uncertainty quantification.
result The method achieves low errors and accurate predictions for European and American options, outperforming data-driven baselines.
Extends Tian theorem to Vaisman manifolds for approximations.
problem Approximating Vaisman metrics by immersions/embeddings.
method Study Vaisman metrics on compact manifolds.
result Extend Tian's theorem to Vaisman manifolds.
In 1972, K. Kenmotsu studied a class of almost contact Riemannian manifolds. Later, such a manifold was called a Kenmotsu manifold. This paper, we studied Kenmotsu manifolds with (2n+s)-dimensional s−contact metric manifold and this manifold, we have called generalized Kenmotsu manifolds. Necessary and sufficient c…
Study on a new type of manifolds that generalize almost C-manifolds.
problem Understanding weak nearly C-manifolds and their properties.
method Analyzing conditions for local Riemannian product structures and characterizing specific dimensions.
result Conditions for a weak nearly C-manifold to become locally a Riemannian product and characterization of specific dimensions.
Stabilized convex symplectic manifolds are equivalent to flexible Weinstein manifolds.
problem Understanding the equivalence between stabilized convex symplectic manifolds and flexible Weinstein manifolds.
method Analyzing the homotopy type and symplectic properties of the manifolds.
result Stabilized convex symplectic manifolds are symplectomorphic to flexible Weinstein manifolds.
New class of complex manifolds defined, properties studied.
problem Understanding properties of complex manifolds.
method Introduced and studied wHHR manifolds, proved metric equivalence.
result Bergman and Kobayashi metrics are biLipschitz equivalent for wHHR Stein manifolds.
A selfsimiar manifold is a Riemannian manifold (M,g) endowed with a homothetic vector field ξ. We characterize global selfsimilar manifolds and describe the structure of local selfsimilar manifolds. We prove that any selfsimilar manifold with a potential homothetic vector field is a conical Riemannian ma…
Smoothly embed 3-manifolds in 5-manifolds, simplifying topological to smooth.
problem Embedding 3-manifolds smoothly in 5-manifolds.
method Homotopy and small homotopy to achieve smooth embeddings.
result Locally flat embeddings are homotopic to smooth ones.
The study provides a structure theorem for a new class of noncompact 3-manifolds.
problem Understanding a new class of noncompact 3-manifolds.
method Proved a structure theorem for irreducible open graph manifolds.
result A canonical 'reduced' decomposition of irreducible open graph manifolds along embedded, incompressible 2-tori.
Condition for intersection of real flag manifolds in complex flag manifold.
problem Intersection conditions of real flag manifolds in a complex flag manifold.
method Condition given in terms of symmetric triad, antipodal intersection proven.
result Intersection of real flag manifolds is antipodal.
The paper studies extended quasi-Einstein manifolds with special geometric properties and solitons.
problem Exploring new types of manifolds in general relativity.
method Generalization of existing manifolds and construction of specific examples.
result Existence and properties of extended quasi-Einstein manifolds with solitons.
Conditions for flat manifolds as cusp cross-sections in arithmetic hyperbolic manifolds.
problem Determining when a flat manifold can be a cusp cross-section in arithmetic hyperbolic manifolds.
method Analyzing rational representations of holonomy groups and quasi-arithmetic manifolds.
result Conditions for a flat manifold to appear as a cusp cross-section in every commensurability class of arithmetic hyperbolic manifolds.
Classifies 3-manifolds from simplified (2,0)-trisections of 4-manifolds.
problem Classifying 3-manifolds from simplified (2,0)-trisections of 4-manifolds.
method Classifies vertical 3-manifolds as preimages of arcs on the plane for simplified (2,0)-trisection maps.
result Each 6-tuple of vertical 3-manifolds determines the source 4-manifold uniquely up to orientation reversing diffeomorphisms.
Study on 3D manifolds with specific tensor structures and their properties.
problem Characterizing 3D Riemannian manifolds with tensor structures.
method Investigation of locally conformal Riemannian product manifolds and their associated structures.
result Conditions for additional structures to be parallel and properties of almost Einstein and Einstein manifolds.
New manifold type PNDP-manifold defined with Einstein warped product structure.
problem Defining manifolds with non-standard dimensions.
method Einstein warped product manifold with special base and fiber structures.
result PNDP-manifolds are Einstein warped product manifolds with specific base and fiber properties.
Optimal inequalities for systole, inradius, and volume in hyperbolic 3-manifolds
problem Establishing optimal inequalities relating systole, inradius, and volume in hyperbolic 3-manifolds
method Using systole-volume inequalities for extremal manifolds
result Extremal manifolds for systole, inradius, and volume are identified
The paper explores F-manifolds and metrics, constructing canonical structures.
problem Understanding relationships between F-manifolds and metrics.
method Construction of canonical flat F-manifolds and homogeneous Riemannian F-manifolds.
result Construction of a canonical flat F-manifold associated to an arbitrary Riemannian F-manifold.
Defines s-manifolds and s-manifolds with corners for symplectic applications.
problem Creating a framework for singular manifolds with useful properties.
method Introducing categories of stratified manifolds and manifolds with corners.
result Fundamental classes and transverse fibre products in s-manifolds.
Embeds 3-manifolds in symplectic 4-manifolds with constraints.
problem Embedding 3-manifolds in symplectic 4-manifolds with topological and smooth properties.
method Topological and smooth embeddings, using homology cobordism and obstructions.
result 3-manifolds can be embedded in symplectic 4-manifolds with specific conditions.
We introduce a new general class of metric f-manifolds which we call (nearly) trans-S-manifolds and includes S- manifolds, C-manifolds, s-th Sasakian manifolds and generalized Kenmotsu manifold studied previously. We prove their main properties and we present many examples which justify their study.
The study provides homological characterizations for Q-manifolds and l2-manifolds.
problem Density of maps in characterizing Q-manifolds and l2-manifolds. method Investigates weakening the density of Zn-maps and Z-maps to homological maps. result Obtains homological characterizations for Q-manifolds and l2-manifolds. A locally conformally Kähler (LCK) manifold M is one which is covered by a Kähler manifold M~ with the deck transform group acting conformally on M~. If M admits a holomorphic flow, acting on M~ conformally, it is called a Vaisman manifold. Neither the class of LCK manifolds nor that of Vais…
Study weak quasi contact metric manifolds to generalize K-contact and Sasakian manifolds criteria.
problem Generalize K-contact and Sasakian manifolds criteria using weak quasi contact metric manifolds.
method Study weak quasi contact metric manifolds and generalize theorems for K-contact and Sasakian manifolds.
result Provide new criterions for K-contact and Sasakian manifolds in terms of curvature tensor and geometric objects.