Final revision. To appear in the Journal of Differential Geometry. This paper studies knots that are transversal to the standard contact structure in , bringing techniques from topological knot theory to bear on their transversal classification. We say that a transversal knot type $\cTK$ is {\it transversally…
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48 results for “Lück's theorem”
We construct a cubical CW-complex CK(M^3) whose rational cohomology algebra contains Vassiliev invariants of knots in the 3-manifold M^3. We construct \bar{CK}(R^3) by attaching cells to CK(R^3) for every degenerate 1-singular and 2-singular knot, and we show that π_1(\bar{CK}(R^3))=1 and π_2(\bar{CK}(R^3))=Z. We give …
CK simplifies ML model deployment and reproducibility with open APIs and DevOps.
problem Making ML models reproducible and deployable across different environments.
method Decompose complex systems into reusable sub-components with unified APIs and DevOps principles.
result Automatically co-design and optimize ML models for speed, accuracy, energy, and size.
Eigen-Spike Emergence and Quadratic Equivalents for Conjugate Kernels on Nonlinearly Separable Datastat.ML
Study of eigenvalues in nonlinear kernels for classification of separable data.
problem Understanding the applicability of linear equivalents in nonlinearly separable data classification.
method Analysis of conjugate kernels and their quadratic equivalents for a canonical nonlinearly separable dataset (XOR problem).
result Identification of regimes where nonlinear kernels deviate from linear equivalents, leading to label-aligned eigenspaces.
Study eigenvalue distributions of neural kernels for linear-width networks.
problem Eigenvalue distributions of neural kernels in linear-width networks.
method Asymptotic analysis of Conjugate Kernel and Neural Tangent Kernel under random initialization and approximate orthogonality.
result Eigenvalue distributions converge to deterministic limits, described by recursive fixed-point equations.
Study eigenvalues and eigenvectors in neural networks, focusing on signal propagation.
problem Characterize signal eigenvalues and eigenvectors in neural networks.
method Characterizes signal eigenvalues and eigenvectors for a nonlinear spiked covariance model.
result Provides precise quantitative characterizations of signal eigenvalues and eigenvectors in neural networks.
DEQs and explicit networks are nearly equivalent for Gaussian mixtures.
problem Understanding the equivalence between DEQs and explicit neural networks.
method Random matrix theory and analysis of kernel matrices.
result A shallow explicit network can mimic the kernel of a DEQ.
Researchers derived Kauffman bracket polynomial for Celtic link shadows using two methods.
problem Calculating the Kauffman bracket polynomial for Celtic link shadows.
method Two complementary approaches: recursive relation and 4-tangle algebra.
result Derived Kauffman bracket polynomial for shadows.
We consider canonical metrics on Fano manifolds. First we introduce a norm-type functional on Fano manifolds, which has Kahler-Einstein or Kahler-Ricci soliton as its critical point and the Kahler-Ricci flow can be viewed as its (reduced) gradient flow. We then obtain a natural lower bound of this functional. As an app…
Analyzes neural networks using spectral perspectives.
problem Understanding neural network initialization and training.
method Examines the Conjugate Kernel and Neural Tangent Kernel spectra.
result Lends insights into neural network initialization and training properties.
The harmonic oscillator on Riemannian and Lorentzian configuration spaces of constant curvaturemath-ph
The harmonic oscillator as a distinguished dynamical system can be defined not only on the Euclidean plane but also on the sphere and on the hyperbolic plane, and more generally on any configuration space with constant curvature and with a metric of any signature, either Riemannian (definite positive) or Lorentzian (in…
The twisting technique creates infinite links.
problem Creating infinite links from given ones.
method Applying the twisting technique to adequate, homogeneous, or alternative links.
result These three classes of links are infinite.
Proper free-boundary minimal hypersurfaces with a rotational symmetry in the Schwarzschild spacemath.DG
Study finds new minimal surfaces in Schwarzschild space.
problem Existence of non-totally geodesic minimal surfaces in Schwarzschild space.
method Family of properly embedded free boundary minimal hypersurfaces of revolution.
result Existence of new minimal surfaces with circular boundaries in Schwarzschild space.
We study spectral behavior of the complex Laplacian on forms with values in the tensor power of a holomorphic line bundle over a smoothly bounded domain with degenerated boundary in a complex manifold. In particular, we prove that in the two dimensional case, a pseudoconvex domain is of finite type if a…
Positive solutions for singularly perturbed nonlinear elliptic problem on manifolds via Morse theorymath.AP
Given (M, g0) we consider the problem -ε^2Delta_{g0+h}u + u = (u+)^{p-1} with (ε, h) \in (0, ε0) \times Bρ. Here Bρ is a ball centered at 0 with radius ρ in the Banach space of all Ck symmetric covariant 2-tensors on M. Using the Poincaré polynomial of M, we give an estimate on the number of nonconstant solutions with …
A hypercomplex manifold M is a manifold with a triple I,J,K of complex structure operators satisfying quaternionic relations. For each quaternion L=aI +bJ+cK, L^2=-1, L is also a complex structure operator on M, called an induced complex structure. We are studying compact complex subvarieties of (M,L), when L is a gene…
Study quasi-Einstein metrics on real hypersurfaces of complex space forms.
problem No Einstein metrics in non-flat complex space forms, explore quasi-Einstein condition.
method Analyze real hypersurfaces in complex Euclidean space, classify quasi-Einstein metrics.
result Classify quasi-Einstein metrics on real hypersurfaces of complex Euclidean space.
We develop an algebraic representation for (1,1)-knots using the mapping class group of the twice punctured torus MCG(T,2). We prove that every (1,1)-knot in a lens space L(p,q) can be represented by the composition of an element of a certain rank two free subgroup of MCG(T,2) with a standard element only depending on …
The paper studies neural networks with wide layers and finds a deformed semicircle law.
problem Investigating spectral distributions of neural networks in the ultra-wide regime.
method Analyzes empirical kernel matrices, proves deformed semicircle law, provides nonlinear Hanson-Wright inequality.
result Emergence of a deformed semicircle law in the ultra-wide neural network regime.
Twistor space of hypercomplex manifolds is never Moishezon.
problem Characterize the twistor space of compact hypercomplex manifolds.
method Analyzing the twistor family and its total space.
result The twistor space of a compact hypercomplex manifold is never Moishezon.
This paper analyzes the Bochner formula for Riemannian flows and derives eigenvalue estimates.
problem Analyzing the Bochner formula for Riemannian flows and deriving eigenvalue estimates.
method The approach involves studying the curvature term in the Bochner-Weitzenb{ö}ck formula of the basic Laplacian on M, splitting it into two parts, and establishing eigenvalue estimates.
result Established an eigenvalue estimate of the basic Laplacian on basic forms, and discussed the limiting case of the estimate.
Paper analyzes learning dynamics in quasi-periodic environments, showing consistent solutions.
problem Challenges in stochastic gradient learning for complex environments.
method Uses energy balance equations derived from Caldirola-Kanai Hamiltonian to model learning.
result In quasi-periodic environments, learning yields consistent solutions for similar patterns.
Paper shows how a Lie superalgebra can be realized using matrices.
problem Realizing the Lie superalgebra of contact projective vector fields.
method Using embedding techniques from projectively equivariant quantizations, the paper constructs a matrix realization.
result The Lie superalgebra is realized as the intersection of and .
Proposes a linear model for facial action recognition without requiring large datasets.
problem Limited annotated data for facial expression and action units.
method Exploits low-rank property across frames and group sparsity to subtract neutral faces and recognize actions.
result One-shot automatic method on raw face videos performs competitively and better than previous methods.
New insights into query complexity for Nash equilibrium learning.
problem Characterizing the number of queries needed to learn approximate Nash equilibria in matrix games.
method Introduced a new technique to prove lower bounds on query complexity, improving previous techniques.
result Lower bounds of order for any , where is a constant.
Study estimates squared error in high-dimensional binary regression, revealing phase transitions and structural properties.
problem Estimating squared error in high-dimensional regression with binary coefficients.
method Novel conditional second moment method to approximate optimal squared error.
result Establishes a phase transition point \( n^* = 2k \log p / \log (2k/\sigma^2 + 1) \) for binary regression, revealing structural properties and information-theoretic threshold.
New algorithm reduces online learning regret in uninformed Markov games.
problem Achieving no external regret in uninformed Markov games is impossible.
method Empirical Nash-value regret, parameter-free algorithm, adaptive restart.
result Achieves regret bound.
Theorem analogues proven using Artin's approximation theorem.
problem Proving analogues of Moser's Theorem.
method Using Artin's approximation theorem.
result Few analogues of Moser's Theorem proven.
AI generates theorems and proofs for training theorem provers.
problem Limited human-written theorems and proofs for supervised learning.
method Proposes a neural generator to automatically synthesize theorems and proofs.
result Synthetic data improves automated theorem proving in Metamath.
Global inverse function theorem proved easily using Riemannian geometry.
problem Global inverse function theorem in Riemannian geometry.
method Hopf--Rinow theorem in Riemannian geometry.
result Hadamard's global inverse function theorem is proven easily.
A new comparison theorem for geometric spaces.
problem Geometric space comparison theorems.
method Relative form of Toponogov comparison theorem.
result New geometric space comparison theorem established.
The paper proves a new theorem in Riemannian geometry and offers a new proof for Toponogov's theorem in Alexandrov geometry.
problem Proving new theorems in Riemannian and Alexandrov geometries.
method Inspired by the proof of the Schur-Toponogov theorem, a new proof of Toponogov's theorem is provided.
result A new theorem in Riemannian geometry and a new proof of Toponogov's theorem in Alexandrov geometry.
Revises a theorem by Thurston, finding a counter-example and a weaker version.
problem The bounded image theorem in Haken manifolds.
method Providing a counter-example and a weaker version of the second statement of Thurston's theorem.
result A counter-example and a weaker version of the second statement of Thurston's theorem are presented.
The paper proves three circles theorems and Liouville type theorems for subharmonic and holomorphic functions.
problem Establishing theorems for subharmonic and holomorphic functions on specific geometric structures.
method Using subharmonic and holomorphic functions on Riemannian manifolds and gradient shrinking Ricci solitons.
result Proves Liouville type theorems as applications of the established theorems.
Paper develops formulas and theorems in Hermitian geometry.
problem None explicitly stated in the abstract.
method Develops second variational formulas and index forms in Hermitian geometry.
result Establishes results analogous to classical theorems in Riemannian geometry.
Fixed-point theorems for set-valued maps using homological methods.
problem Finding fixed points for specific types of set-valued maps.
method Homological selection theorems applied to finite-dimensional spaces.
result Established fixed-point theorems for usco homologically UV^n set-valued maps.
Proves Markov theorem for trivalent braids using L-move approach.
problem Proving Markov theorem for trivalent braids.
method Follows L-move approach to prove Markov theorem.
result Proves one-move Markov-type theorem and algebraic Markov-type theorem for trivalent braids.
Proofs for Moon's theorem and its generalization.
problem Proving Moon's theorem and its generalization.
method Proofs based on key lemmas.
result Generalization of the four-vertex theorem.
New measure proves Poncelet-type theorems.
problem Proving Poncelet-type theorems.
method Introducing a new invariant measure on the circle.
result Simple proof of Emch closing theorem.
New theorem for doodles on sphere, similar to Markov's.
problem Understanding doodles on a sphere.
method Description of twins with equivalent closures.
result Analogous to Markov's theorem for doodles.
Investigates proving geometric theorems over complex and real numbers using tilings.
problem Proving incidence theorems over C and R using the master theorem.
method Formalizes tiling proofs and introduces a hierarchy of theorems based on topological spaces.
result Identifies which theorems can or cannot be proved over C and R.
Analyzes Saito vanishing theorem using methods.
problem Proving the Saito vanishing theorem.
method Uses -methods to prove the theorem.
result Analytic proof of the Saito vanishing theorem.
Extends calculus theorem to higher dimensions.
problem Calculus theorem limitations in higher dimensions.
method Type Stokes' theorem for type -chains.
result Extends fundamental theorem of calculus.
Extends symplectic reduction and theorem to Lie algebroids.
problem Symplectic reduction and theorem for Lie algebroids.
method Extends Marsden-Weinstein reduction and Darboux-Moser-Weinstein theorems.
result Obtained coisotropic embedding theorem for symplectic Lie algebroids.
Paper generalizes complex Brunn-Minkowski theory and proves new extension theorems.
problem Complex Brunn-Minkowski theory and extension theorems.
method Hilbert bundle approach to complex Brunn-Minkowski theory.
result Generalizes Guan's sharp strong openness theorem and sharp Ohsawa-Takegoshi extension theorem.
Proves Thurston's bounded image theorem for Haken manifolds.
problem Proving Thurston's bounded image theorem for Haken manifolds.
method Using recent developments in Kleinian group theory.
result A proof of Thurston's original bounded image theorem.
Method upgrades limit theorems to mixing limit theorems for dynamical systems.
problem Improving limit theorems for dynamical systems.
method General method for upgrading limit theorems to mixing limit theorems.
result Mixing limit theorems for specific subbundles of the Kontsevich-Zorich cocycle.
Formulates Index III lemma and Rauch III theorem with applications.
problem Develops new mathematical theorems based on existing ones.
method Formulation of Index III lemma and Rauch III theorem based on Index I, II lemmas and Rauch I, II theorems.
result Presented Rauch's type theorem and volume comparison result as applications.