Proof shows volume equals integral points for certain manifolds.
problem Counting integral points on affine manifolds.
method Rational Ehrhart theory and Fourier analysis.
result Volume equals number of integral points for integral-integral affine manifolds.
Study oscillatory integrals with degenerate singular points in multivariable phase functions.
problem Analyzing oscillatory integrals with degenerate singular points in phase functions.
method Using asymptotic expansions and results from one variable, the study examines multivariable phase functions.
result Asymptotic expansions of oscillatory integrals for multivariable phase functions with degenerate singular points.
Defines a new distance for integral current spaces and proves convergence criteria.
problem Defining a new distance metric for integral current spaces.
method Defines a new distance function and proves convergence criteria.
result Establishes the compactness theorem for integral current spaces using a new distance function.
Integral points are potentially dense in character varieties of quasi-projective varieties.
problem Density of integral points in character varieties of quasi-projective varieties.
method Reduction to Riemann surfaces and use of Corlette-Simpson work.
result Integral points have Zariski-dense orbit under the mapping class group.
The Poisson equation on manifolds plays an fundamental role in many applications. Recently, we proposed a novel numerical method called the Point Integral method (PIM) to solve the Poisson equations on manifolds from point clouds. In this paper, we prove the convergence of the point integral method for solving the Pois…
A new model DKMPP integrates covariates and uses an integration-free method for spatio-temporal point processes.
problem Training intractable deep spatio-temporal point processes with multimodal covariates.
method DKMPP uses a deep kernel to model complex relationships and an integration-free score matching method.
result DKMPP and score-based estimators outperform baseline models in spatio-temporal point processes.
Enhances method for constructing integral manifolds of non-integrable Pfaffian systems.
problem Constructing maximal integral manifolds for non-integrable Pfaffian systems.
method Recurrent geometrical method developed by Élie Cartan and von Weber.
result Enhanced Jordan-Hölder integration procedure for constructing local maximal integral manifolds.
Integrability of mean curvature near degenerate points in Heisenberg group.
problem Integrability of sub-Riemannian mean curvature at degenerate characteristic points in the Heisenberg group.
method Introduction of mildly degenerate characteristic points and use of perimeter measure.
result The sub-Riemannian mean curvature is integrable in a neighborhood of these points.
A novel method for efficiently integrating spatiotemporal point processes.
problem Challenges in integrating spatiotemporal neural point processes, especially for flexible intensity functions.
method AutoSTPP (Automatic Integration for Spatiotemporal Neural Point Processes) extends a dual network approach to 3D STPP using ProdNet for decomposable parametrization of the integral network.
result AutoSTPP effectively sidesteps computational complexities and shows significant advantage in recovering complex intensity functions.
The Laplace-Beltrami operator (LBO) is a fundamental object associated to Riemannian manifolds, which encodes all intrinsic geometry of the manifolds and has many desirable properties. Recently, we proposed a novel numerical method, Point Integral method (PIM), to discretize the Laplace-Beltrami operator on point cloud…
The article constructs stochastic integration in Riemannian manifolds.
problem No specific problem stated; focuses on the construction of stochastic integration.
method Functional-analytic approach to stochastic integration in Riemannian manifolds.
result There are infinitely many stochastic integrals, and they are related by a simple formula.
The paper studies the smoothness of critical points of variational integrals on Hessian spaces.
problem The study focuses on the regularity of critical points of variational integrals defined on Hessian spaces.
method The approach involves solving a fourth order nonlinear equation and analyzing the Hessian of the critical points.
result Smooth critical points with bounded Hessian are shown to be smooth provided their Hessian has small BMO.
Multi-dimensional state-integrals of products of Faddeev's quantum dilogarithms arise frequently in Quantum Topology, quantum Teichmüller theory and complex Chern--Simons theory. Using the quasi-periodicity property of the quantum dilogarithm, we evaluate 1-dimensional state-integrals at rational points and express the…
New vector fields integrate first-order ODEs.
problem Integrating first-order ODEs.
method Relation between Riemannian manifolds and ODEs integration.
result Integration procedure for first-order ODEs.
Proposes a new method for better explaining neural network decisions.
problem Challenges in explaining neural network decisions due to base-point choice.
method Introduces tangentially aligned integrated gradients to maximize explanation tangential alignment.
result Optimal base-point maximizes explanation tangential alignment, leading to more accurate interpretations.
Integration of the form ∫a∞f(x)w(x)dx, where w(x) is either sin(ωx) or cos(ωx), is widely encountered in many engineering and scientific applications, such as those involving Fourier or Laplace transforms. Often such integrals are approximated by a numerical integration…
Generalized Huber's theorem for specific manifold curvature types.
problem Finite point conformal compactification on manifolds with certain curvature integrability.
method Generalization of Huber's theorem to higher dimensions with $L^ rac{n}{2}$ integrable Ricci curvatures.
result Validated finite point conformal compactification theorem for new class of manifolds.
Study local system points on surfaces using group descent.
problem Understanding integral points on moduli of local systems.
method Mapping class group descent and boundedness results for systoles.
result Established structure theorem for integral points.
Some years ago Moshé Flato pointed up that it could be interesting to develop the Nambu's idea to generalize Hamiltonian mechanic. An interesting new formalism in that direction was proposed by T. Takhtajan. His theory gave new perspectives concerning deformation quantization, and many authors have developed its mathem…
Bayesian Quadrature speeds up integration by selecting batches of points instead of single points.
problem Efficiently parallelizing Bayesian Quadrature for integration over non-negative integrands.
method Developed methods to select batches of points at each step, based on recent batch Bayesian Optimization.
result Significantly reduces computation time, especially for expensive integrands.
Study on curves on moduli spaces of local systems, proving structure and integral point determination.
problem Arithmetic of algebraic curves on moduli spaces of local systems.
method Proved structure theorem for morphisms and effectively determined integral points on curves.
result Effective determination of integral points on nondegenerate algebraic curves on moduli space.
Paper proposes a framework for probabilistic load forecasting by integrating point forecasts.
problem Short-term load forecasting for power systems energy management.
method Two-stage framework: first stage for point forecasting, second stage for probabilistic forecasting using feature integration.
result Numerical results show effectiveness of the proposed approach in hour-ahead load forecasting.
Detecting closed essential surfaces in 3-manifolds is challenging.
problem Detecting closed essential surfaces in 3-manifolds using ideal points of character varieties or algebraic non-integral representations.
method Using Chesebro's module-theoretic interpretation of Culler-Shalen theory, constructing examples of 3-manifolds with no algebraic non-integral representations.
result Construction of an infinite family of closed hyperbolic Haken 3-manifolds with no algebraic non-integral representations into PSL(2, C).
Sharp lower bound found for integral varifolds' mean curvature.
problem Finding a sharp lower bound for the mean curvature integral of integral varifolds.
method Developed a new approach using integral varifolds and mean curvature.
result A sharp lower bound on the mean curvature integral with critical power for integral varifolds.
New proof for 6D symplectic manifold with 4 fixed points.
problem Classifying the integral cohomology ring and total Chern class for 6D symplectic manifolds with 4 fixed points.
method New different argument using moment map values and weights of fixed points.
result Determined the sets of weights and global invariants for the manifold.
Paper studies invariant distributions of bi-Hamiltonian structures.
problem Integrability of invariant distributions in bi-Hamiltonian structures.
method Description and investigation of invariant distributions.
result All invariant distributions of non-degenerate bi-Hamiltonian structures are described.
Recently, a new embedding/compactness theorem for integral currents in a sequence of metric spaces has been established by the second author. We present a version of this result for locally integral currents in a sequence of pointed metric spaces. To this end we introduce another variant of the Ambrosio--Kirchheim theo…
The study bounds Hausdorff measure of flat singular points in area-minimizing currents.
problem Bounding Hausdorff measure of flat singular points in area-minimizing currents.
method Proving locally finite (m−2)-dimensional Hausdorff measure and Minkowski content bounds. result The set of flat singular points has locally finite (m−2)-dimensional Hausdorff measure. Functional integrals explain quantum mechanics and field theory.
problem Explaining quantum mechanics and field theory using functional integrals.
method Describes Feynman's path integral approach to quantum mechanics and field theory.
result Equivalence of path integral formalism to classical mechanics and quantum mechanics.
New method finds points for approximating distributions faster.
problem Approximating target probability distributions using finite points.
method Stationary MMD points computed via MMD gradient flows.
result Stationary MMD points converge faster than global minimizers.
The exponential map fails to be injective near critical points in sub-Riemannian geometry.
problem Injectivity failure of the exponential map at critical points in sub-Riemannian geometry.
method Analysis of the Hilbert invariant integral of the variational problem associated with the sub-Riemannian structure.
result Characterization of conjugate points in terms of metric structure.
Median-of-means sampling outperforms mean-of-means for large sample sizes in numerical integration.
problem Improving numerical integration accuracy in high dimensions.
method Median-of-means sampling compared to mean-of-means using RQMC methods.
result Median-of-means sampling is superior for large sample sizes, while mean-of-means is better for smaller sample sizes.
First-order ODEs linked to flat surfaces, leading to integrability.
problem Integrating first-order ODEs.
method Defined Riemannian metrics on variable spaces, studied surface properties, and established connections between Jacobi fields and Lie point symmetries.
result Flat associated surfaces lead to integrable first-order ODEs.
The paper introduces a new complex analytic invariant called the pointed harmonic volume and its relation to the Johnson homomorphism.
problem Exploring new complex analytic invariants related to the complex structure of Riemann surfaces.
method Defining and computing the pointed harmonic volume as a natural extension of Chen's iterated integrals.
result Established a relationship between the harmonic volume and the first extended Johnson homomorphism.
To give a criterion for the integrability of Banach-Lie triple systems, we follow the construction of the period group of a Lie algebra and define the period group of a Lie triple system as an analogous concept. We show that a Lie triple system is integrable if and only if its period group is discrete. Along the way, w…
Study analyzes Lévy process structure on manifolds with conjugate points.
problem Microlocal analysis of Lévy processes on manifolds with conjugate points.
method Microlocal analysis, pseudodifferential operators, Fourier integral operators.
result Generator can be expressed as sum of pseudodifferential and Fourier integral operators.
Proves unique continuation for area minimizing currents.
problem Ensuring area minimizing currents match minimal surfaces.
method Analyzes infinite order contact between currents and minimal surfaces.
result Currents and minimal surfaces coincide in a neighborhood.
Pairs (Hamiltonian system, Lagrangian distribution), called dynamical Lagrangian distributions, appear naturally in Differential Geometry, Calculus of Variations and Rational Mechanics. The basic differential invariants of a dynamical Lagrangian distribution w.r.t. the action of the group of symplectomorphisms of the a…
Study light ray transform on Lorentzian manifolds without conjugate points.
problem Recovering spacelike singularities from weighted light ray transforms.
method Fourier Integral Operator analysis and filtered back-projection.
result Recovery of spacelike singularities from weighted light ray transforms without conjugate points.
A new method calculates fractional moments using the moment-generating function.
problem Computing fractional moments from probability densities.
method Integral framework based on moment-generating function.
result Exact integral expressions for various types of moments.
Conservation laws vanishing along characteristic directions of a given system of PDEs are known as characteristic conservation laws, or characteristic integrals. In 2D, they play an important role in the theory of Darboux-integrable equations. In this paper we discuss characteristic integrals in 3D and demonstrate that…
Symplectic classification for a specific type of singularity in integrable systems.
problem Symplectic classification of integrable systems near singular points of type An. method Real-analytic symplectic normal forms and classification of Lagrangian foliations.
result All integrable systems are symplectically equivalent near singular points of this type.
Geodesic flows with diagonalisable integrals are orthogonal.
problem Understanding geodesic flows with specific integrals.
method Analyzing quadratic integrals for geodesic flows.
result Diagonalisable integrals imply orthogonal separation of variables.
Integrality of FJRW invariants for Lie algebras A_l, D_l, and E_6.
problem Integrality of FJRW invariants for Lie algebras.
method Analyzing Frobenius manifolds and Pochhammer symbols.
result FJRW invariants are integral and coincide with the coefficients of a generating function.
Study finds u-plane integral equals full correlator at strong coupling and matches Donaldson invariants.
problem Understanding u-plane integral contributions in N=2 gauge theories.
method Used mock modular forms and Appell-Lerch sums to efficiently determine u-plane correlators.
result u-plane correlators match Donaldson invariants and are entire functions of fugacities.
The paper examines rigidity of Einstein metrics using curvature functionals.
problem Characterizing rigidity of Einstein metrics.
method Critical points of quadratic curvature functionals and integral inequalities involving Weyl curvature, trace-less Ricci curvature, and Sobolev constant.
result Rigidity results for Einstein metrics on complete manifolds.
Harmonic maps from the plane to SO(3) are studied via integral iterations.
problem Minimal harmonic maps from the plane to SO(3) with polynomial cubic differentials.
method Fixed-point problem for integral operator, spectral networks, BPS state counts.
result Determining the asymptotic structure of g by a convex polygon Y(P) in RP2. Study on quantum invariants of twist knots using saddle point method.
problem Asymptotic expansion of Reshetikhin-Turaev invariants of twist knots.
method Saddle point method applied to integral q-surgery. result Asymptotic expansion formula for Reshetikhin-Turaev invariants.