New boundary conditions for knot polynomials solved via elliptic KW equations.
problem Computing Jones polynomial coefficients from knot solutions.
method Formulated generalized Nahm pole boundary conditions for KW equations, proved ellipticity, analyzed polyhomogeneity.
result KW equations with generalized Nahm pole conditions are elliptic and solutions are polyhomogeneous.
Enhances KWS in vehicles with multi-source fusion.
problem Improving precision and recall rates in vehicle keyword spotting.
method Integrates vehicle information into a DNN for speech classification and selects optimal sensitivity parameters.
result Significantly improved performance metrics (precision, recall, MSE) compared to baseline.
This expository article introduces the Kapustin-Witten equations to mathematicians. We discuss the connections between the Complex Yang-Mills equations and the Kapustin-Witten equations. In addition, we show the relation between the Kapustin-Witten equations, the moment map condition and the gradient Chern-Simons flow.…
New KWS neural networks improve accuracy and power efficiency.
problem Maximizing accuracy and power efficiency in KWS systems.
method Hardware Aware Training (HAT) for LMU-based neural networks.
result Achieved state-of-the-art accuracy and low parameter counts.
The study classifies manifolds based on their geometric properties and invariants.
problem Classifying manifolds based on their geometric and topological properties.
method Analyzing metrics through isometric embeddings and deformations, considering scalar curvature, Ricci tensor, and Einstein metrics.
result The KW type classification of manifolds and sigma invariant calculations.
User-specific KWS system learns new keywords on-device.
problem Out-of-vocabulary problem in traditional KWS systems.
method Query-by-example enrollment and testing, using phonetic posteriors and FST.
result Promising performance on two keywords, preserving simplicity.
Paper optimizes KWS models using NAS and quantization for limited resources.
problem Developing efficient keyword spotting models in resource-constrained environments.
method Neural Architecture Search (NAS) for model structure optimization and quantization of weights and activations.
result Achieved high accuracy (95.55%) with minimal parameters and operations using NAS and quantization.
We consider the problem of designing an allocation rule or an "online learning algorithm" for a class of bandit problems in which the set of control actions available at each time s is a convex, compact subset of Rd. Upon choosing an action x at time s, the algorithm obtains a noisy value of the unkno…
On a smooth line bundle L over a compact Kähler Riemann surface Σ, we study the family of vortex equations with a parameter s. For each s∈[1,∞], we invoke techniques in \cite{Br} by turning the s-vortex equation into an s-dependent elliptic partial differential equation, studied in \cite{kw}, provi…
Sparse algorithms reduce model size for faster keyword spotting.
problem Large models are hard to deploy on resource-limited devices.
method Apply sparse algorithms to reduce parameter count in DNN KWS models.
result Sparse models perform better with minimal parameter loss.
Enhanced DFO using adaptive batch-based FD estimates.
problem Derivative-free optimization with imprecise gradient estimates.
method Adaptive batch-based finite difference estimation and dynamic sampling strategy.
result Algorithm achieves convergence rate similar to KW and SPSA methods.
Improved ASR-free keyword spotting in under-resourced languages.
problem Dynamic time warping for keyword spotting in languages with limited resources.
method Multilingual bottleneck extractor and correspondence autoencoder integration.
result More than 11% absolute improvement in ROC AUC over MFCCs.
Study numerical invariants under retraction maps between topological spaces.
problem Understand behavior of invariants like cohomological dimensions under retractions.
method Introduced a notion of retraction and studied several numerical invariants.
result Proved inequalities between invariants hold under retractions.
Study numerical invariants for groups, computing for cyclic groups and surfaces.
problem Numerical invariants for groups and their computation.
method Computational and theoretical analysis of groups, including finite cyclic groups and nonorientable surfaces.
result Formula for the numerical invariant of free products of groups.
New complexity defined for groups, inspired by topological spaces.
problem Complexity of finitely presented groups.
method Inspired by Karoubi-Weibel work, introduces combinatorial complexity.
result New complexity (covering type) defined and properties considered.
We propose a max-pooling based loss function for training Long Short-Term Memory (LSTM) networks for small-footprint keyword spotting (KWS), with low CPU, memory, and latency requirements. The max-pooling loss training can be further guided by initializing with a cross-entropy loss trained network. A posterior smoothin…
Data augmentation improves keyword spotting accuracy in noisy conditions.
problem Maintaining low false reject rates in far-field KWS with playback interference.
method Artificially corrupted training data with mixed music and TV audio.
result 30-45% reduction in false reject rates under audio playback.
On simply connected five manifolds Sasakian-Einstein metrics coincide with Riemannian metrics admitting real Killing spinors which are of great interest as models of near horizon geometry for three-brane solutions in superstring theory [KW]. We expand on the recent work of Demailly and Kollár [DK] and Johnson and Kollá…
A framework reduces bias in sampling from posterior distributions.
problem Reducing bias in sampling from posterior distributions.
method A black-box debiasing scheme generating weighted samples.
result Improves accuracy of posterior sampling without increasing variance.
Equation embeddings learn from surrounding words to represent math equations.
problem Math equations are hard to analyze due to their uniqueness.
method Equation embeddings use surrounding words to find good representations of equations.
result Equation embeddings provide better models than existing word embedding approaches.
Paper finds new equations for pseudospherical surfaces with isometric immersions.
problem Identifying equations with isometric immersions for pseudospherical surfaces.
method Provided families of second order non-linear PDEs with local isometric immersions in E^3.
result Found equations with principal curvatures depending on finite-order jets of solutions.
Study Galois groupoids of discret Painlevé equations.
problem Computing Galois groupoids for discret Painlevé equations.
method Using semi-continuity theorem for Galois groupoid in confluence of difference to differential equations.
result Computed Galois groupoids for discret Painlevé equations.
Proves solvability of general inverse σ_k equations with constant coefficients.
problem Solvability of general inverse σ_k equations with constant coefficients.
method Proves existence of unique solution if a C-subsolution exists.
result Confirms analytical conjecture for deformed Hermitian--Yang--Mills equation.
Paper establishes estimates for nonlinear equations on compact manifolds.
problem Estimating solutions to fully nonlinear equations with gradient terms on compact almost Hermitian manifolds.
method Establishes second order estimates and proves existence of solutions for specific equations.
result Proves existence of solutions for various equations, including Monge-Ampère and Hessian equations.
Proves C^2,alpha estimates for elliptic equations on hyperkähler manifolds.
problem Elliptic equations on hypercomplex manifolds.
method Proves C^2,alpha estimates under suitable assumptions.
result Solutions to specific elliptic equations on hyperkähler manifolds satisfy C^2,alpha estimates.
The paper generalizes Monge-Ampère equations and their solutions in differential geometry.
problem Understanding the structure of Monge-Ampère equations and their solutions.
method Generalizing Monge-Ampère equations to higher-order systems and proving their solutions correspond to integral manifolds of exterior differential systems.
result The Korteweg-de Vries (KdV) equation and Cauchy-Riemann equations are examples of generalized Monge-Ampère equations.
We study four distinct second-order nonlinear equations of Rabelo which describe pseudospherical surfaces. By transforming these equations to the constant-characteristic form we relate them to some well-studied integrable equations. Two of the Rabelo equations are found to be related to the sine-Gordon equation. The ot…
Introduces a new PDE involving differential forms for Kähler geometry.
problem Solving a unified PDE for various important equations in Kähler geometry.
method Introduces a fully nonlinear PDE with differential form Λ and proves solvability conditions.
result Generalizes previous works and proves a conjecture for the dHYM equation.
Sharp sub-Gaussian bounds for subsolutions of Trudinger's equation on Riemannian manifolds.
problem Bounding weak subsolutions of Trudinger's equation on Riemannian manifolds.
method Proving sub-Gaussian upper bounds for weak subsolutions.
result The upper bounds are sharp for specific classes of manifolds, including \(\mathbb{R}^{n}\).
In this paper we perform a blow-up and quantization analysis of the following nonlocal Liouville-type equation \begin{equation}(-Δ)^\frac12 u= κe^u-1~\mbox{in S1,} \end{equation} where (−Δ)21 stands for the fractional Laplacian and κ is a bounded function. We interpret the above equation as the prescri…
The paper studies curvature equations and their solvability.
problem Solving curvature type equations and their Dirichlet problems.
method General class of fully nonlinear curvature equations, Christoffel-Minkowski problem, degenerate equations.
result Solvability of curvature type equations and Dirichlet problems.
The paper derives gradient estimates for porous medium and fast diffusion equations on metric measure spaces.
problem Gradient estimates for porous medium and fast diffusion equations on metric measure spaces.
method Derives Li-Yau and Souplet-Zhang type gradient estimates for the given equations.
result Gradient estimates for the equations on complete noncompact metric measure spaces with compact boundary.
New periodic solutions found for a specific equation.
problem Finding solutions to a specific partial differential equation.
method Using Bäcklund transformation for the sine-Gordon equation to generate solutions.
result New periodic exact solutions of the constant astigmatism equation.
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.
Paper solves Hessian equations on Kähler manifolds.
problem Solving Hessian equations on Kähler manifolds.
method Combines elementary symmetric functions; provides sufficient and necessary condition.
result Generalizes results for Hessian and Hessian quotient equations.
The paper introduces new equations in Kähler geometry and proves their solutions and convexity.
problem Solving equations in Kähler geometry and understanding their geometric implications.
method Using moment map pictures to motivate and prove solutions for the equations.
result The Mabuchi functional for certain equations is shown to be convex.
Probabilistic grammars improve equation discovery from data.
problem Discovering scientific laws from data using equations.
method Proposed probabilistic context-free grammars to encode soft constraints and a Monte-Carlo algorithm.
result Probabilistic grammars lead to more efficient equation discovery.
We describe a method to reduce partial differential equations of Monge-Ampère type in 4 variables to complex partial differential equations in 2 variables. To illustrate this method, we construct explicit holomorphic solutions of the special lagrangian equation, the real Monge-Ampère equations and the Plebanski equatio…
The paper generalizes the Bott-Virasoro group and derives new Euler equations.
problem Understanding the generalized Bott-Virasoro group and its dynamics.
method Generalizing the Bott-Virasoro group using connection cochain and deriving Euler equations.
result New Euler equations derived from the generalized Bott-Virasoro group.
The paper proves constant rank theorems for special Lagrangian equations.
problem Understanding saddle solutions and Liouville type results for special Lagrangian equations.
method Argument based on saddle solutions and Liouville type results for the special Lagrangian equation.
result Obtained constant rank theorems for saddle solutions to the special Lagrangian equation and the quadratic Hessian equation.
Study finds conservation laws for a specific class of parabolic equations.
problem Existence and structure of conservation laws for evolutionary scalar second-order differential equations.
method Calculation of linearized characteristic cohomology to find conservation laws, showing dependence on second derivatives.
result Only Monge-Ampère type equations have non-trivial conservation laws.
Study solves HJB equations for time-inconsistent control problems.
problem Time-inconsistent deterministic linear quadratic control problems.
method Characterized solutions using Riccati equations with integral terms, proving uniqueness.
result Uniqueness of solutions to equilibrium HJB equations proved.
Sharp rates and symmetry for higher order conformally invariant equations near singularities.
problem Understanding solutions near isolated singularities for higher order conformally invariant equations.
method Blow-up analysis for local integral equations, Fowler solutions, Harnack inequality.
result Sharp blow-up rates and asymptotic radial symmetry of solutions near singularities.
In this thesis, we consider the suitability of using the charged cold fluid model in the description of ultra-relativistic beams. The method that we have used is the following. Firstly, the necessary notions of kinetic theory and differential geometry of second order differential equations are explained. Then an averag…
We introduce a class of overdetermined systems of partial differential equations of finite type on (pseudo)-Riemannian manifolds that we call the generalised Ricci soliton equations. These equations depend on three real parameters. For special values of the parameters they specialise to various important classes of equ…
Combines geometric hydrodynamics with magnetic systems to derive new equations and prove well-posedness.
problem Deriving new equations for magnetic systems and proving their well-posedness.
method Introducing the magnetic Euler-Arnold equation and proving well-posedness for specific equations.
result Local and global well-posedness results for the magnetic Euler-Arnold equation associated with the global quasi-geostrophic equations.
Studies projective geometry and partial differential equations prolongation.
problem Understanding the prolongation of overdetermined geometric partial differential equations.
method Introduction to differential geometry and tractor calculus, study of prolongation of equations.
result Recovery of projective tractor and cotractor connections via partial differential equations prolongation.
We determine the Lie point symmetries of the Fokker-Planck equation and provide examples of solutions of this equation. The Fokker-Planck equation admits a conserved form, hence there is an auxiliary system associated to this equation and whose point symmetries give rise to potential symmetries of the Fokker-Planck equ…