A new method for uncertainty estimation in neural networks using Gaussian-softmax integration.
problem Quantifying uncertainty in neural network predictions.
method Proposes a single-model approach integrating Gaussian distribution with softmax outputs, using mean-field approximation.
result Competitive performance on uncertainty estimation tasks and outperforms many methods on out-of-distribution detection.
Investigates statistical properties of perturb-softmax and perturb-argmax distributions.
problem Underexplored statistical properties of Gumbel-Softmax and Gumbel-Argmax distributions.
method Investigates convexity and differentiability to determine completeness and minimality of these distributions.
result Identifies parameters that admit complete and minimal representation of probability distributions.
Topic sparsity refers to the observation that individual documents usually focus on several salient topics instead of covering a wide variety of topics, and a real topic adopts a narrow range of terms instead of a wide coverage of the vocabulary. Understanding this topic sparsity is especially important for analyzing u…
The study examines stability of Hamiltonian Poisson integrators on both integrable and non-integrable systems.
problem Investigating stability properties of Hamiltonian Poisson integrators.
method Examples of Lotka-Volterra dynamics and numerical investigations of a non-integrable system are used.
result The existence of a modified Hamiltonian is crucial for the stability of Hamiltonian Poisson integrators.
Geometrically interprets integrability of geodesic flow using web theory.
problem Integrability of geodesic flow by quadratic integrals.
method Geometric interpretation through web theory and integrable billiards construction.
result Constructs integrable billiards on surfaces with quadratic geodesic integrals.
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.
Integrates rough geometric forms on manifolds.
problem Integrating rough forms on complex manifolds.
method Combines Whitney's geometric integration and sewing approaches.
result Introduced distributional k-forms for integration.
The paper defines and analyzes set-valued stochastic integrals for Lévy processes.
problem Defining and analyzing set-valued stochastic integrals for Lévy processes.
method Extending classical definitions to convoluted integrals with square-integrable kernels, and proving properties of set-valued convoluted stochastic integrals.
result Set-valued convoluted stochastic integrals can be explosive and take extended vector values.
New integration theory on topological spaces, including fractals.
problem Developing a universal integration theory for arbitrary topological spaces.
method Introducing a new integration framework using unital magma valued functions and measures.
result Integration, differentiation, and orientation defined for arbitrary topological spaces.
We discuss a recurrent geometrical method, due to Élie Cartan and von Weber ([1],[11]) enabling us to determine, step by step, the maximal integral manifolds of a not necessarily integrable nor regular Pfaffian system. The dimensions of such integral manifolds can, of course, vary from point to point but more so can va…
We define a non-absolutely convergent integration on integral currents of dimension 1 in Euclidean space. This integral is closely related to the Henstock-Kurzweil and Pfeffer Integrals. Using it, we prove a generalized Fundamental Theorem of Calculus on these currents. A detailed presentation of Henstock-Kurzweil Inte…
The paper defines and proves the existence of decompositions of integral varifolds.
problem Existence of integral varifold decompositions.
method Introducing and proving the existence of decompositions of integral varifolds into countably many integral varifolds.
result Existence of decompositions of integral varifolds whose first variation is representable by integration.
Counterexample shows Ito integrand needn't be locally square integrable.
problem Ito integrand's square integrability condition is not always met.
method Provided a counterexample to Ito's Lemma's integrability condition.
result Ito integrand needn't be locally square integrable.
Study integrable geodesic flows on 2-surfaces with high-degree polynomial first integrals.
problem Integrable geodesic flows on 2-surfaces with high-degree polynomial first integrals.
method Semi-Hamiltonian systems of PDEs and generalized hodograph method.
result Construction of many local explicit and implicit integrable examples with polynomial first integrals of degrees 3, 4, 5.
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.
We introduce renormalized integrals which generalize conventional measure theoretic integrals. One approximates the integration domain by measure spaces and defines the integral as the limit of integrals over the approximating spaces. This concept is implicitly present in many mathematical contexts such as Cauchy's pri…
TQ separates sampling and integration for high-dimensional integrals.
problem High-dimensional integration challenges in science.
method Tree Quadrature (TQ) constructs a surrogate model using regression trees.
result TQ outperforms existing methods in up to 15 dimensions.
Method finds differential equations for integrable billiard tables.
problem Finding differential equations for integrable billiard tables.
method Introducing a method to find differential equations for functions defining tables.
result Illustrated method in three billiard systems.
Constructs Lie groupoid integrating elliptic tangent bundles and Poisson structures.
problem Existence and integration of elliptic tangent bundles and Poisson structures.
method Explicit construction of Lie groupoids and local models for symplectic integration.
result Necessary and sufficient topological condition for integration of elliptic tangent bundles.
Investigates integrable systems with linear periodic integral for e(3) Lie algebra.
problem Analyzes singularities and topological properties of integrable systems.
method Examines singularities of Liouville foliation, bifurcation diagram, transformations of Liouville tori, and isoenergy surfaces.
result Discovers topological properties of integrable systems with linear periodic integral.
We use neural networks as control variates with geometric integration techniques.
problem Analytic integration of neural network approximations for variance reduction.
method Integration domain subdivision using computational geometry for MLPs with continuous piecewise linear activation functions.
result Neural networks can be used as control variates with geometric integration methods.
Integral foliated simplicial volume is a version of simplicial volume combining the rigidity of integral coefficients with the flexibility of measure spaces. In this article, using the language of measure equivalence of groups we prove a proportionality principle for integral foliated simplicial volume for aspherical m…
This paper is an exposition of heuristics related to Witten's functional integral, relating it to Vassiliev invariants and to the Kontsevich integrals that can be used to produce Vassiliev invariants of knots and links.In particular, we give a simplified version of the appearance of the Kontsevich integrals in the pert…
Surveying integrability of Lie algebroids and structures.
problem Integrability of Lie algebroids and structures.
method Survey and recent results on integrability.
result Recent findings on local and global integrability.
This paper provides an existence-and-uniqueness theorem characterizing the stochastic integral with respect to a Wiener process. The integral is represented as a mapping from the space of measurable and adapted pathwise locally integrable processes to the space of continuous adapted processes. It is characterized in te…
We construct an infinite-dimensional symplectic 2-groupoid as the integration of an exact Courant algebroid. We show that every integrable Dirac structure integrates to a "Lagrangian" sub-2-groupoid of this symplectic 2-groupoid. As a corollary, we recover a result of Bursztyn-Crainic-Weinstein-Zhu that every integrabl…
Integrable LCK manifolds characterized as Kähler Lie algebras.
problem Characterizing LCK manifolds with integrable anti-Lee forms.
method Examining LCK manifolds with integrable anti-Lee forms and applying to Lie algebras.
result Unimodular integrable LCK Lie algebras are Kähler Lie algebras with specific derivations.
The linking integral is an invariant of the link-type of two manifolds immersed in a Euclidean space. It is shown that the ordinary Gauss integral in three dimensions may be simplified to a winding number integral in two dimensions. This result is then generalized to show that in certain circumstances the linking integ…
New integrable deformations for topological hierarchies from Frobenius manifolds.
problem Integrable deformations of topological hierarchies from Frobenius manifolds.
method Construction of integrable deformations with polynomial tau-structures.
result Conjecture of universal object for Riemann--Hopf hierarchy.
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.
Proves a theorem for normal distributions on manifolds with boundary.
problem Normal distributions on manifolds with boundary require a new approach to integration.
method Introduces neat integral manifolds with boundary and conditions for integrability.
result Conditions for integrability expressed in terms of adapted collars and integrability on interior and boundary.
The paper proves conditions for Darboux integrability in diagonal hydrodynamic systems.
problem Conditions for Darboux integrability in diagonal hydrodynamic systems.
method Proof of conditions using Laplace transformation sequences and geometric interpretations.
result Diagonal systems of hydrodynamic type are Darboux integrable if and only if the corresponding systems for commuting flows are Darboux integrable.
A new method integrates forms on Riemann surfaces, leading to modular forms.
problem Integrating differential forms with poles on Riemann surfaces.
method Simple procedure to integrate differential forms with arbitrary holomorphic poles, establishing an analytic theory for integrals over configuration spaces.
result Regularized graph integrals on elliptic curves are almost-holomorphic modular forms.
Researchers prove integrability of magnetic systems on spheres up to dimension 6.
problem Integrability of magnetic systems on spheres restricted to their surface.
method Proved complete integrability for n ≤ 6, noncommutative integrability for n ≥ 7, conjectured integrability for all n.
result Complete integrability of magnetic flows on spheres for n ≤ 6, noncommutative integrability for n ≥ 7.
A new adaptive binarization technique using fuzzy integrals improves image quality.
problem Improving image thresholding quality.
method FLAT (Fuzzy Local Adaptive Thresholding) based on fuzzy integrals.
result The proposed FLAT method produces better image quality than traditional algorithms and neural networks.
Starting from a homogeneous polynomial in momenta of arbitrary order we extract multi-component hydrodynamic-type systems which describe 2-dimensional geodesic flows admitting the initial polynomial as integral. All these hydrodynamic-type systems are semi-Hamiltonian, thus implying that they are integrable according t…
A new geometric definition of integration for differential forms.
problem Standard integration definitions are coordinate-dependent and not suitable for certain contexts.
method Uses triangulations and cochains on the pair groupoid to define integration.
result Natural definition in Lie algebroids, stochastic integration, and quantum field theory.
We prove that a holomorphic Lie algebroid is integrable if, and only if, its underlying real Lie algebroid is integrable. Thus the integrability criteria of Crainic-Fernandes do also apply in the holomorphic context without any modification. As a consequence we give another proof of the following theorem: a holomorphic…
Integral filling volume of mapping tori grows sublinearly with complexity.
problem Characterizing mapping classes with vanishing integral filling volume.
method Analyzing Dehn twists and mapping tori, using simplicial volume and complexity.
result Integral simplicial volume of mapping tori grows sublinearly with respect to the monodromy power.
New integral theorems improve density function estimations.
problem Improving density function estimations.
method Integrals based on cyclic functions and Riemann sums, Fourier integral theorem, Monte Carlo methods, variational approach, Cauchy residue theorem.
result Optimal cyclic functions minimize square integrals, improving density estimations.
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…
Joachimsthal integrals characterize conics in various geometries.
problem Characterizing conics in different geometries using Joachimsthal integrals.
method Extending Joachimsthal integrals to spherical and hyperbolic geometries and connecting them to the Poritsky property.
result Existence of Joachimsthal integrals characterizes conics in various geometries.
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.
We introduce a "minimal" Kontsevich integral that generates the original Kontsevich integral while at the same time producing ribbons whose boundaries are the braids on which the minimal Kontsevich integral is evaluated. We generalize the definition of the Kontsevich integral to that of graphs in R^3 and study the beha…
New integrators for mechanical systems on Lie groups simplify based on group properties.
problem Designing numerical integrators for mechanical systems on Lie groups.
method Leverage retraction maps and Lie group properties to design structure-preserving integrators.
result Simplified design of integrators for Euler-Poincare and Lie-Poisson equations.
Geodesic flows with specific integrals are linked to special 4-webs.
problem Characterizing geodesic flows with commuting quadratic integrals.
method Characterization through geodesic 4-webs and geometric hypothesis.
result Metrics of Stäckel type for geodesic flows with specific integrals.
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.
Basic elements of integral calculus over algebras of iterated differential forms, are presented. In particular, defining complexes for modules of integral forms are described and the corresponding berezinians and complexes of integral forms are computed. Various applications and the integral calculus over the algebra $…