Smooth 2-tori in R^4 can be approximated by polyhedral Lagrangian or isotropic tori.
problem Approximating smooth 2-tori in high-dimensional spaces.
method Polyhedral approximation using Lagrangian and isotropic tori.
result Smooth 2-tori can be approximated by polyhedral Lagrangian or isotropic tori in C0 or C1 sense.
Proves existence and uniqueness of solutions to a quaternionic Monge-Ampère equation.
problem Solving the quaternionic Monge-Ampère equation for (n−1)-quaternionic plurisubharmonic functions on a hyperKähler manifold. method Proves existence and uniqueness of solutions using a Cherrier-type inequality and C1 and C2 estimates. result Obtains smooth solutions to the quaternionic Monge-Ampère equation.
New L0 norm added to TDA for market analysis.
problem Improving TDA tools for market prediction.
method Defined and applied L0 norm in TDA for four markets.
result Enhanced TDA tools for market analysis.
Paper proves Hamilton-Tian conjecture using partial C0-estimate.
problem Existence of Kähler-Einstein metrics on Fano manifolds.
method Using Liu-Székelyhidi's partial C0-estimate for polarized Kähler metrics with Ricci bounded below.
result Proves Hamilton-Tian conjecture for Kähler-Ricci flow.
We are concerned with the saddle solutions of the Allen-Cahn equation constructed by Cabré and Terra \cite{C,C2} in R2m. These solutions vanish precisely on the Simons cone. The existence and uniqueness of saddle solution are shown in \cite{C,C2,C1}. Regarding the stab…
Differentiable sorting and rank normalization are incompatible, with specific conditions for admissibility.
problem Incompatibility between differentiable sorting and rank normalization.
method Formalized admissibility through monotone invariance, batch independence, and rank-space stability conditions.
result Different gap-sensitive and batchwise relaxations of rank normalization violate the conditions for admissibility.
We study various aspects related to boundary regularity of complete properly embedded Willmore surfaces in H3, particularly those related to assumptions on boundedness or smallness of a certain weighted version of the Willmore energy. We prove, in particular, that small energy controls C1 boundary regularity. We examin…
The present paper attempts to show an alternative approach with regards to rational Pythagorean-hodograph (PH) curves and especially more natural approach for rational PH helices (i.e. rational helices). It exploits geometric features of rational helices to obtain a simpler construction of these curves and apply this t…
Proves local isometric embedding of low-differentiability metrics in 3D space.
problem Isometric embedding of metrics of low differentiability in Euclidean 3-space.
method Simplified notation, geodesic and level parameters, solutions of initial value problems for first order non-linear PDEs, classical linear algebraic systems.
result Local isometric embedding exists for metrics of C1 differentiability.
Proves some flat spacetimes can't be extended smoothly.
problem Proving smooth extension of certain FLRW spacetimes.
method Utilizing Sbierski's C0-inextendibility techniques. result Proves C0-inextendibility of spatially flat FLRW spacetimes. We define disentanglement in generative models and prove it's related to identifiable factors.
problem Understanding disentanglement in generative models like VAEs and GANs.
method Characterized disentanglement in smooth generative pushforward models using the SVD of the Jacobian.
result Disentanglement is identifiable under certain conditions on the generator, promoting separable factors.
Theorem proves integrability for piecewise-smooth distributions.
problem Integrability of piecewise-smooth distributions.
method Generalizations of Frobenius integrability theorem.
result Sufficient criteria for complete integrability with bi-Lipschitz coordinates.
Motivated by the computations done in \cite{C1}, where I introduced and discussed what I called the groupoid of generalized gauge transformations, viewed as a groupoid over the objects of the category BunG,M of principal G-bundles over a given manifold M, I develop in this paper the same ideas for the…
New rigidity result for metrics with positive scalar curvature and specific decay.
problem Understanding metrics with positive scalar curvature and C0 decay. method Analyzing metrics with non-negative scalar curvature and C0 decay properties. result Metrics with specific decay properties must be flat.
Revises Schwarzschild manifold rigidity proof for spin manifolds.
problem Rigidity of Schwarzschild manifold for spin manifolds.
method Spinorial proof approach.
result Generalizes and includes classical and recent black hole uniqueness theorems.
B.-Y. Chen initiated the study of warped product submanifolds in his fundamental seminal papers \cite{C1,C2,C2.1}. In this paper, we study contact CR-warped product submanifolds of cosymplectic space forms and prove an optimal inequality by using Gauss and Codazzi equations. In addition, we obtain two geometric inequal…
Paper proves smoothness of solutions to a complex geometric problem.
problem Smoothness of solutions to the degenerate Lp Dual Minkowski problem. method Inspired by Guan and Li's approach for the Aleksandrov problem, the authors derive C1,1 estimates. result Proves solutions are C1,1 regular. Generalizes Hecke algebra for double torus, linking to skein algebra.
problem Understanding algebraic structures on double torus.
method Introducing Heegaard dual operators and Dehn twists.
result Established relationship between Hecke algebra and skein algebra.
We consider the problem of detecting a tight community in a sparse random network. This is formalized as testing for the existence of a dense random subgraph in a random graph. Under the null hypothesis, the graph is a realization of an Erdös-Rényi graph on N vertices and with connection probability p0; under the …
A spacetime can be embedded in an enveloping space with all its extensions.
problem Existence and uniqueness of C0-maximal extensions in globally hyperbolic conformally flat spacetimes.
method Proving conformal embedding into an enveloping space containing all extensions.
result Existence and uniqueness of C0-maximal extensions proven.
In this paper we study the stochastic evolution equation (1.1) in martingale-type 2 Banach spaces (with the linear part of the drift being only a generator of a C0-semigroup). We prove the existence and the uniqueness of solutions to this equation. We apply the abstract results to the Heath-Jarrow-Morton-Musiela (HJMM)…
Let (X,L) be a polarized Kähler manifold that admits an extremal Kähler metric in c1(L). We show that on a nearby polarized deformation that preserves the symmetry induced by the extremal vector field of (X,L), the modified K-energy is bounded from below. This generalizes a result of Chen, Székelyhidi and Tosatti to ex…
A remarkable and elementary fact that a locally compact set F of Euclidean space is a smooth manifold if and only if the lower and upper paratangent cones to F coincide at every point, is proved. The celebrated von Neumann's result (1929) that a locally compact subgroup of the general linear group is a smooth manifold,…
Study shows a mass quantity for C0 metrics that agrees with ADM mass.
problem Understanding ADM mass for C0 metrics and its behavior under Ricci-DeTurck flow. method Developed a C0 mass quantity and analyzed its behavior under Ricci-DeTurck flow. result The C0 mass at infinity is independent of coordinate charts and has controlled distortion under Ricci-DeTurck flow. Simplified proof and new C0 estimate for Kähler-Einstein metrics.
problem Existence of Kähler-Einstein metrics on Calabi-Yau manifolds.
method Alternative C0 a priori estimate for the Monge-Ampère equation. result Established a new uniform bound for the solution of the Monge-Ampère equation.
The paper proves the existence of infinitely many nodal solutions to a Paneitz-type equation.
problem Proving the existence of nodal solutions to a specific type of partial differential equation.
method First, a C0−estimate for positive f-invariant solutions is proven. Then, the existence of mountain pass solutions with arbitrarily large energy is established. result The existence of infinitely many nodal solutions to the equation Δ2u−αΔu+βu=uq is proven. We propose a simple change to existing neural network structures for better defending against gradient-based adversarial attacks. Instead of using popular activation functions (such as ReLU), we advocate the use of k-Winners-Take-All (k-WTA) activation, a C0 discontinuous function that purposely invalidates the neural …
Box Thirding identifies the best arm efficiently under limited samples.
problem Efficiently identifying the best arm with limited sampling.
method Iterative ternary comparison of arms, discarding the weakest and exploring the best.
result Achieves comparable performance to Successive Halving with less predefined parameters.
Study shows Hamiltonian diffeomorphisms form a connected component in C0-topology for most symplectic rational surfaces.
problem Understanding the C0-topology of symplectic diffeomorphisms on rational surfaces. method Combining techniques from symplectic mapping class groups and C0-symplectic topology, establishing C0-distance estimates. result Hamiltonian diffeomorphisms form a connected component in C0-topology for all but a few exceptions on rational surfaces. The paper examines robustness of topological entropy in geodesic flows.
problem Entropy robustness in geodesic flows under C0 perturbations. method Study of topological entropy on Riemannian metrics with C0 topology. result Metrics with contractible closed geodesics have robust entropy.
Study shows nonextendibility of warped spacelike singularities in specific spacetimes.
problem Nonextendibility of warped spacelike singularities in specific spacetimes.
method Establishes a local obstruction through integrability conditions and radial compression.
result Imply C0-inextendibility for the one-horizon Birmingham-Kottler family. We present the recent advances along with an error analysis of the IBM speaker recognition system for conversational speech. Some of the key advancements that contribute to our system include: a nearest-neighbor discriminant analysis (NDA) approach (as opposed to LDA) for intersession variability compensation in the i-…
Develops analysis of weak immersions with bounded second fundamental forms in critical Sobolev space.
problem Analyzing weak immersions with bounded second fundamental forms in a critical Sobolev space.
method Develops analysis of Lipschitz immersions with bounded second fundamental forms in W2n−1,2 space. result Proves existence of C1 differential structure from weak immersions with bounded second fundamental forms. The paper studies fundamental groups of compact Kahler varieties with nef anti-canonical bundles.
problem Understanding the fundamental groups of compact Kahler varieties with specific properties.
method Application of geometric analytic theory of Kahler spaces and study of Albanese maps.
result The fundamental group of compact Kahler varieties with nef anti-canonical bundle is almost Abelian.
The study provides conditions for approximating Riemannian manifolds with polyhedral metrics.
problem Approximating Riemannian manifolds with polyhedral metrics.
method Conditions on curvature tensors for Lipschitz and local polyhedral approximations.
result Conditions are sufficient for local polyhedral approximations, conjectured to be sufficient for global approximations.
Geometric Gaussian approximations capture any distribution.
problem Approximating complex probability distributions.
method Geometric Gaussian approximations through diffeomorphisms or exponential maps.
result Geometric Gaussian approximations are universal, capturing any distribution.
Method approximates Riemannian barycenter on manifolds.
problem Computing the exact Riemannian barycenter is computationally expensive.
method Uses under- and over-approximations of Riemannian distance to compute an approximate barycenter.
result Approximation method is more efficient than exact methods and steepest descent.
We consider in this paper the optimal approximations of convex univariate functions with feed-forward Relu neural networks. We are interested in the following question: what is the minimal approximation error given the number of approximating linear pieces? We establish the necessary and sufficient conditions and uniqu…
Efficiently reduces tensor ranks using mean-field approximation.
problem Low-rank approximation of non-negative tensors.
method Mean-field approximation of tensor rank reduction.
result Our algorithm achieves faster and competitive tensor rank reduction.
Study approximates unknown function levels with queries.
problem Approximating unknown function levels through sequential queries.
method Introduce Bisect and Approximate algorithms to reduce to local function approximation.
result Rate-optimal sample complexity guarantees for H{ö}lder functions.
We study sparse approximate solutions to convex optimization problems. It is known that in many engineering applications researchers are interested in an approximate solution of an optimization problem as a linear combination of elements from a given system of elements. There is an increasing interest in building such …
Softmax attention approximates complex functions and subsumes many known universal approximators.
problem Universal approximation of continuous sequence-to-sequence functions.
method Interpolation-based analysis of attention's internal mechanism, showing its ability to approximate ReLU functions.
result Softmax attention is a universal approximator for continuous sequence-to-sequence functions.
Improved matrix approximation using randomized algorithms.
problem Finding better approximations of given matrices.
method Randomized algorithms to compute (HT) as an improved approximation. result Computed (HT) provides a better approximation than given F∗. Deviation inequalities for stochastic approximation methods.
problem Establishing bounds on the deviation of stochastic approximation methods.
method Martingale approximation method for separately Lipschitz functions.
result Established various deviation inequalities for stochastic approximation by averaging and minimization.
Neural approximate computing gains enormous energy-efficiency at the cost of tolerable quality-loss. A neural approximator can map the input data to output while a classifier determines whether the input data are safe to approximate with quality guarantee. However, existing works cannot maximize the invocation of the a…
Approximate symmetries of geodesic equations on 2-spheres are studied. These are the symmetries of the perturbed geodesic equations which represent approximate path of a particle rather than exact path. After giving the exact symmetries of the geodesic equations, two different approaches to study the approximate symmet…
We are concerned with an approximation problem for a symmetric positive semidefinite matrix due to motivation from a class of nonlinear machine learning methods. We discuss an approximation approach that we call {matrix ridge approximation}. In particular, we define the matrix ridge approximation as an incomplete matri…
Transformers use ReLUs to approximate softmax efficiently.
problem Analyzing resource usage in softmax transformer models.
method Translating ReLU approximation results to softmax attention mechanisms.
result Economic resource bounds for softmax attention mechanisms.