The paper sets lower bounds for Laplacian eigenvalues and clamped plate problem eigenvalues.
problem Eigenvalues of the Laplace operator and clamped plate problem.
method Sharp lower bounds for Laplacian eigenvalues and clamped plate problem eigenvalues.
result Sharp lower bounds for Laplacian eigenvalues and clamped plate problem eigenvalues.
The paper estimates the gap between eigenvalues of a clamped plate problem.
problem Estimating the gap between eigenvalues of a clamped plate problem.
method Using the asymptotic formula of Agmon and Pleijel, the paper gives an estimate for the gap between eigenvalues.
result The gap between eigenvalues is bounded by a term with a lower order $k^{ rac1n}$.
In this paper, we study eigenvalues of a clamped plate problem. We obtain a lower bound for eigenvalues, which gives an important improvement of results due to Levine and Protter.
In this paper, we study estimates for eigenvalues of the clamped plate problem. A sharp upper bound for eigenvalues is given and the lower bound for eigenvalues in [10] is improved.
For a bounded domain Ω in a complete Riemannian manifold Mn, we study estimates for lower order eigenvalues of a clamped plate problem. We obtain universal inequalities for lower order eigenvalues. We would like to remark that our results are sharp.
Affirm Lord Rayleigh's conjecture on curved spaces for clamped plates.
problem Lord Rayleigh's conjecture for vibrating clamped plates on curved spaces.
method Nodal-decomposition argument, Lévy-Gromov isoperimetric inequality, Gaussian hypergeometric functions, sharp spectral gap estimates.
result Positive curvature enhances genuine differences between low- and high-dimensional settings.
Extends plate problems to differential forms on manifolds.
problem Eigenvalue problems for buckling and clamped plates on differential forms.
method Characterizes smallest eigenvalues, proves spectra equivalence, obtains estimates.
result Spectra of plate problems on forms coincide with functions in bounded domains.
Study fundamental tones of clamped plates on curved spaces.
problem Prove fundamental tone bounds and isoperimetric inequalities on curved spaces.
method Prove spectral gap estimates and isoperimetric inequalities.
result Fundamental tone bounds and isoperimetric inequalities for clamped plates on nonpositively curved spaces.
The paper studies eigenvalue inequalities for a clamped plate problem involving a generalized elliptic differential operator.
problem Eigenvalue inequalities for a clamped plate problem involving a generalized elliptic differential operator.
method Extending the LII operator to Lν, establishing a general formula for eigenvalues, and applying it to estimate eigenvalues on Riemannian manifolds. result Established eigenvalue inequalities for the Lν2 operator on translating solitons and other geometric settings. New isoperimetric inequality for clamped plates in RCD(0,N) spaces, sharp and stable.
problem Fine properties of the principal frequency of clamped plates in RCD(0,N) spaces.
method Analyzing the RCD(0,N) spaces and applying isoperimetric inequalities.
result Sharp isoperimetric inequality for the principal frequency of clamped plates in RCD(0,N) spaces.
Proves Payne conjecture for buckling and membrane eigenvalues.
problem Proving Payne conjecture for buckling and membrane eigenvalues.
method Analytical proof for buckling and membrane eigenvalues.
result Proves Payne conjecture for n-dimensional case (n≥2). Paper studies eigenvalues of a specific operator on Riemannian manifolds.
problem Eigenvalues of a specific operator on Riemannian manifolds.
method Established a general formula for eigenvalues and derived estimates.
result Obtained universal inequalities for the eigenvalues on translating solitons.
Sharp spectral gap estimates for higher-order operators on hyperbolic spaces.
problem Estimating spectral gaps for higher-order operators on Cartan-Hadamard manifolds.
method Symmetrization-free proofs based on general functional inequalities.
result Solves a sharp asymptotic problem from Cheng and Yang and answers a question from Kristály.
In this paper, we establish sharp inequalities for four kinds of classical eigenvalues on a bounded domain of a Riemannian manifold. We also establish asymptotic formulas for the eigenvalues of the buckling and clamped plate problems. In addition, we give a negative answer to the Payne conjecture for the one-dimensiona…
In this paper, we establish universal inequalities for eigenvalues of the clamped plate problem on compact submanifolds of Euclidean spaces, of spheres and of real, complex and quaternionic projective spaces. We also prove similar results for the biharmonic operator on domains of Riemannian manifolds admitting spherica…
Solves the Poisson problem for elastic plates with specific boundary conditions.
problem Finding an immersed surface minimizing Germain's elastic energy.
method Minimizes total curvature energy E(Σ) variationally. result The minimum is an immersed disk with branch points, extending to a C0,α Gauss map. Study proves inequalities for eigenvalues of fourth-order elliptic operators on Riemannian manifolds.
problem Eigenvalue inequalities for fourth-order elliptic operators on Riemannian manifolds.
method Proves inequalities using Payne-Pólya-Weinberger-Yang type for eigenvalues of fourth-order elliptic operators in divergence form on complete Riemannian manifolds.
result Generalizes eigenvalue inequalities for the clamped plate problem to complete Riemannian manifolds.
The paper explores inequalities between eigenvalues on Riemannian manifolds.
problem Investigating relationships between eigenvalues on Riemannian manifolds.
method Constructing gradient estimates for a first eigenfunction to derive inequalities.
result Obtained some relationships between weighted p-Laplacian first eigenvalues. In this paper, we obtain a new abstract formula relating eigenvalues of a self-adjoint operator to two families of symmetric and skew-symmetric operators and their commutators. This formula generalizes earlier ones obtained by Harrell, Stubbe, Hook, Ashbaugh, Hermi, Levitin and Parnovski. We also show how one can use t…
The paper connects eigenvalue problems for various operators and establishes inequalities and asymptotic formulas for heat traces.
problem Eigenvalue problems and heat trace asymptotics for different operators.
method Establishes connections and inequalities for eigenvalues and heat traces.
result Eigenvalue inequalities and three-term asymptotic formulas for heat traces of various operators.
Improved lower bounds for poly-Laplacian eigenvalues in arbitrary dimensions.
problem Lower bounds for higher eigenvalues of the poly-Laplacian operator.
method Sharp inequalities and eigenvalue bounds in low and arbitrary dimensions.
result Improved lower bounds for eigenvalues of the poly-Laplacian in arbitrary dimensions.
Deep RNN predicts vehicle license plate auction prices with high accuracy.
problem Predicting the auction price of vehicle license plates with desirable numbers.
method Constructed a deep recurrent neural network (RNN) to predict prices based on license plate characters.
result Deep RNN predictions explain over 80 percent of price variations, significantly outperforming previous models.
Game of plates and olives counts Morse functions on a sphere.
problem Counting Morse functions on a sphere.
method Analyzing the game's rules and mapping to Morse functions.
result Confirming the speculation that log M_n ~ n log n.
CLAMP uses neural manifold packing to improve self-supervised learning.
problem Improving self-supervised learning for vision tasks.
method CLAMP recasts representation learning as a manifold packing problem, introducing a loss function inspired by particle systems.
result CLAMP achieves competitive performance with state-of-the-art models and separates neural manifolds effectively.
Study examines forces between partially immersed plates and singular configurations.
problem Forces between partially immersed parallel plates and singular configurations.
method Analyzes forces and singular configurations of partially immersed parallel plates in an infinite liquid bath.
result New estimates on meniscus height details presented.
Tensor variable elimination for plated factor graphs enables exact inference in models with repeated structure.
problem Efficient inference in models with repeated structure.
method Generalized variable elimination to tensor variable elimination on plated factor graphs.
result Tractable inference for a class of plated factor graphs.
Euler's elastica with monotone curvature is uniquely minimal.
problem Global minimality of planar elastica with monotone curvature.
method Proof of global minimality using clamped boundary conditions and length penalization.
result Every planar elastica with non-constant monotone curvature is uniquely minimal.
We examine the effect of clamping variables for approximate inference in undirected graphical models with pairwise relationships and discrete variables. For any number of variable labels, we demonstrate that clamping and summing approximate sub-partition functions can lead only to a decrease in the partition function e…
Estimates for plate eigenvalues with nonzero Poisson's ratio.
problem Estimating eigenvalues of a free plate with nonzero Poisson's ratio.
method Using Fourier transform to derive estimates.
result Kroger-type estimates for sums of eigenvalues.
Study develops efficient algorithm for probabilistic penetration response of composite plates.
problem Probabilistic modeling of discrete structural response, focusing on binary events like buckling.
method Adaptive domain-based decomposition, sparse grid sampling, assumption of monotonic behavior.
result Efficient computational framework for probabilistic penetration response of composite plates.
Bayesian method improves hit identification in compound screening.
problem Identifying candidate hits from thousands to millions of compounds.
method Bayesian nonparametric modeling for cross-plate correlation and statistical strength.
result Significant improvements in hit identification sensitivity and specificity.
PAVI speeds up Bayesian inference for large datasets.
problem Challenges in Bayesian inference for large population studies.
method Plate-amortized Variational Inference (PAVI) that shares parameterization across i.i.d. variables.
result Significant speedup in training variational distributions, orders of magnitude faster.
Study curves evolving by gradient flow of elastic energy, proving existence, smoothing, and convergence.
problem Evolution of curves with fixed length and clamped boundary conditions.
method Negative L2-gradient flow of elastic energy, existence, parabolic smoothing, constrained Lojasiewicz-Simon gradient inequality. result Convergence to a critical point as time tends to infinity.
Recent work by Jaffe and Scardicchio has expressed the optical approximation to the Casimir effect as a sum over geometric quantities. The first two authors have developed a technique which uses the complex geometry of the space of oriented affine lines in R3 to describe reflection of rays off a surface. Thi…
This research optimizes plate structures to reduce vibrations in vehicles and aircraft.
problem Minimizing structural vibrations in engineering systems for improved passenger comfort.
method Guided flow matching design optimization integrating generative flow matching and surrogate model.
result Generated plate designs with reduced vibrations compared to random search and other methods.
Flat systems of up to 2 dimensions have flat subsystems.
problem Characterizing flat subsystems in flat systems of differential dimension 2.
method Analyzing subsystems of a flat system of differential dimension at most 2.
result Flat subsystems of a flat system of differential dimension at most 2 exist and can have independent time-uniform outputs.
Paper tackles vehicle make & model classification with improved accuracy.
problem High classification accuracy and reduced annotation time for vehicle images.
method Created a fine-grained database and proposed a pipeline combining SSD and CNN models.
result Approximately 4% better classification accuracy compared to conventional CNN model.
The study introduces a new stickiness parameter for stock prices using a non-linear model.
problem Understanding how closely individual stocks follow a stock index's price movements.
method Developed a non-linear pricing model inspired by tectonic plate movements to measure stickiness.
result Defined a stickiness parameter for stock price returns using a novel model.
Jacobian conditioning predicts GAN quality, and clamping improves GAN performance.
problem Predicting and improving GAN performance through Jacobian conditioning.
method Examined Jacobian singular values in GANs, proposed Jacobian Clamping as a regularization technique.
result Jacobian Clamping improves GAN metrics like Inception Score and FID.
Paper proves rigidity estimates for hyperbolic shells and applies them to \(Γ\)-limit theory.
problem Rigidity of hyperbolic shells and their \(Γ\)-limit behavior.
method Nonlinear rigidity estimates for \(H^1\) deformations and hyperbolic shells with clamped lateral boundary.
result Derives the optimal exponent \(h^{-4/3}\) for hyperbolic shells.
We derive a dimensionally-reduced limit theory for an n-dimensional nonlinear elastic body that is slender along k dimensions. The starting point is to view an elastic body as an n-dimensional Riemannian manifold together with a not necessarily isometric W1,2-immersion in n-dimensional Euclidean space. The…
We study the conformal plate buckling equation (Laplace--Beltrami)^2 u =1, where the L-B operator is for the metric g = e^{2u}g_0, with g0 the standard Euclidean metric on R^2. This conformal elliptic PDE of fourth order is equivalent to the nonlinear system of elliptic PDEs of second order, Delta u +K_g e^(2u)=0, D…
Physics-informed neural network identifies and characterizes surface cracks in metals.
problem Identifying and characterizing surface-breaking cracks in metals using ultrasound.
method Physics-informed neural network (PINN) trained with ultrasonic surface wave data and adaptive activation functions.
result PINN accurately estimates the speed of sound and identifies crack locations in metals.
Develops control and observer methods for complex systems.
problem Controlling and observing infinite-dimensional systems with boundary actuation.
method Energy-Casimir method and port-Hamiltonian system representation.
result Control law and observer designed for Kirchhoff-Love plate example.
The paper bridges stochastic control and deep hedging for European call options with transaction costs.
problem Hedging and pricing European call options with proportional transaction costs.
method Complementary perspectives: stochastic control and deep hedging. Two architectures proposed: NTBN-Delta and WW-NTBN.
result WW-NTBN converges faster, matches no-transaction bands more closely, and generalizes well across transaction cost regimes.
New algorithm uses random matrices for neural network training without synaptic weight symmetries.
problem Training neural networks efficiently and without synaptic weight symmetries.
method Contrastive Hebbian learning with random feedback weights.
result Random contrastive Hebbian learning achieves better computational models for learning.
PAVI speeds up VI for large-scale studies by sharing parameterization across i.i.d. variables.
problem Challenges in Bayesian inference for large population studies with many latent parameters.
method Designing plate-amortized variational inference (PAVI) to share parameterization across i.i.d. variables.
result Significant speedup in training large-scale hierarchical variational distributions.
Layered graphical models improve discriminative learning efficiency.
problem Improving discriminative learning efficiency in graphical models.
method Designing layered graphical models (LGMs) in analogy to neural networks, using tensorized truncated variational inference and backpropagation.
result LGMs achieve competitive results in image classification, comparable to neural networks.