Survey of open problems in finite-dimensional integrable systems.
problem Open problems in finite-dimensional integrable systems.
method None specified; survey of existing open problems.
result Many open problems were identified from a conference.
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48 results for “integration problem”
Two EP frameworks ensure integrable beliefs in Bayesian estimation problems.
problem Non-integrable beliefs in EP can lead to infeasible solutions.
method Proposes two EP frameworks to keep messages non-integrable.
result Ensures integrable beliefs in EP, even with non-integrable messages.
Study variational problems for integral invariants of maps between pseudo-Riemannian manifolds.
problem Understanding variational properties of integral invariants defined from the second fundamental form.
method Derive first variational formulae for integral invariants of degree two, show Euler-Lagrange equation for Chern-Federer energy, and provide examples of submanifolds.
result The Euler-Lagrange equation of the Chern-Federer energy functional reduces to a second order PDE.
The paper solves the Integration Problem for principal connections.
problem Describing discrete connections associated with a principal connection.
method Using the Lie or derivative functor to induce connections on the principal bundle.
result For flat principal connections, the Integration Problem has a unique solution among flat discrete connections.
The paper approximates financial derivatives using neural networks and iterated integrals.
problem Approximating -integrable financial derivatives.
method Using iterated Stratonovich integrals and neural networks.
result Approximate solutions to the -hedging problem.
Survey of open problems linking integrable systems and Nijenhuis geometry.
problem Interplay between integrable systems and Nijenhuis geometry.
method Discussion of open problems and questions.
result Challenges and simple questions in the field.
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.
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.
Tensor networks improve integration accuracy for high-dimensional problems.
problem Integration of high-dimensional functions with exponential convergence.
method Regression-free tensor network representations for integration.
result Exponential convergence achieved for non-analytic integrands.
Solves integration problem for generalized complex manifolds.
problem Integration problem for generalized complex manifolds.
method Weakly holomorphic symplectic groupoid approach.
result Generalized complex manifolds are integrable if and only if their underlying real Poisson structure is integrable.
Study compares different integrals for optimal portfolio optimization with insider information.
problem Optimizing portfolios in a financial market with insider information.
method Anticipating stochastic calculus and various integrals (Russo-Vallois forward, Ayed-Kuo, Hitsuda-Skorokhod).
result The Hitsuda-Skorokhod and Ayed-Kuo integrals do not provide a financially meaningful investment strategy.
In this paper we present the solution to a longstanding problem of differential geometry: Lie's third theorem for Lie algebroids. We show that the integrability problem is controlled by two computable obstructions. As applications we derive, explain and improve the known integrability results, we establish integrabilit…
Unlike Legendrian submanifolds, the deformation problem of coisotropic submanifolds can be obstructed. Starting from this observation, we single out in the contact setting the special class of integral coisotropic submanifolds as the direct generalization of Legendrian submanifolds for what concerns deformation and mod…
Paper derives a Reilly type integral formula and applies it to inequalities and eigenvalue problems.
problem Developing a new integral formula and its applications in geometric inequalities and eigenvalue problems.
method Derives a Reilly type integral formula associated with the -Laplacian and applies it to inequalities and eigenvalue problems.
result Obtains Heintze-Karcher and Minkowski type inequalities, and eigenvalue relationships.
Paper constructs super integrable systems on color Lie algebra.
problem Super integrable systems on color Lie algebra.
method Using non-isospectral problems with matrices from color Lie algebra , constructing (1+1)- and (2+1)-dimensional systems.
result Super integrable systems and their Hamiltonian structures constructed on color Lie algebra .
In this lecture delivered at the Integrable and Quantum Field Theory at Peyresq sixth meeting, we review the Lychagin's Monge-Ampere operators theory and exhibit the link it establishes between the classical problem of local equivalence for non linear partial differential equations and the problem of integrability of s…
Study on geodesic flows on a two-torus for additional first integrals.
problem Existence of additional first integrals for geodesic flows on a two-torus.
method Analyzes polynomial and quadratic first integrals, relates to soliton equations.
result Nonexistence of an additional quadratic first integral for certain magnetic geodesic flows.
Solves inverse problem for Calderón using microlocal normal forms.
problem Recovering unknown coefficient from boundary measurements.
method Microlocal normal forms and propagation of singularities.
result Recovery of integrals of unknown coefficient over good bicharacteristic leaves.
To every Darboux integrable system there is an associated Lie group which is a fundamental invariant of the system and which we call the Vessiot group. This article shows that solving the Cauchy problem for a Darboux integrable partial differential equation can be reduced to solving an equation of Lie type for the …
The study shows how certain ODEs and integrals are regular under Borel summation.
problem Analyzing the regularity of solutions to ODEs and integration problems.
method Using geometric perspective on Laplace and Borel transforms, the study examines level 1 ODEs and exponential period integrals over Lefschetz thimbles.
result Solutions of certain ODEs and integration problems are Borel regular.
Novel duality theory for operator Frobenius algebras solves long-standing hydrodynamic integrable systems problem.
problem Long-standing Eisenhart-Stäckel problem for non-degenerate integrable systems.
method Introduce duality for operator Frobenius algebras and use mutual symmetry assumption.
result Construct new infinite-dimensional integrable systems of hydrodynamic type.
In this paper we propose {\it a region choice problem} for a knot projection. This problem is an integral extension of Shimizu's 'region crossing change unknotting operation.' We show that there exists a solution of the region choice problem for all knot projections.
This research improves deep neural networks for parameter identification and prediction in stochastic Volterra integral equations.
problem Parameter identification and prediction in Volterra integral equations driven by Gaussian noise.
method Improved deep neural networks framework that incorporates inter-output relationships into the loss function.
result The framework enhances parameter estimation accuracy and provides accurate solutions for modeling stochastic systems.
New integrable structures found with abnormal geodesics.
problem Integrable homogeneous sub-Riemannian structures with abnormal geodesics.
method Analysis of equivalence problem for sub-Riemannian Engel structures.
result First known family of examples of integrable homogeneous sub-Riemannian structures with strictly abnormal geodesics.
Dissertation tackles geodesic ray transform on Riemannian manifolds.
problem Determining functions from line integrals along geodesics.
method Establishes conditions for unique and stable determination of functions.
result New numerical model for computed tomography imaging created.
Local high-degree polynomial integrals of geodesic flows and the generalized hodograph methodmath.DS
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.
Problem of global integration of geometric structures arising in the theory of dynamical systems admitting the normal shift is considered. In the case when such integration is possible the problem of globalization for shift maps is studied.
Surveying inverse problems on manifolds with boundaries.
problem Inverse problems for geometric structures on manifolds with boundaries.
method Analyzes linear and non-linear geometric inverse problems.
result Recent results on geodesic X-ray transforms and Riemannian metrics.
A theorem proves integrability of Fréchet tangent distributions.
problem Integrability of Fréchet tangent distributions on manifolds.
method Introduced Condition W, applied variational approach, used differential forms.
result Existence and uniqueness of maximal foliations.
Develops a new quadrature method for Lebesgue integrals.
problem Finding optimal values and weights for non-Gaussian processes.
method Solves a generalized eigenvalue problem to find value-nodes and weights.
result Advantages in analyzing irregular and stochastic processes.
Improves sampling, rounding, and integration of logconcave functions.
problem Sampling, rounding, and integration of logconcave functions.
method Algorithmic diffusion approach.
result First complexity improvements in nearly two decades for general logconcave functions.
Gaussian process is a very promising novel technology that has been applied to both the regression problem and the classification problem. While for the regression problem it yields simple exact solutions, this is not the case for the classification problem, because we encounter intractable integrals. In this paper we …
By the classical Martingale Representation Theorem, replication of random vectors can be achieved via stochastic integrals or solutions of stochastic differential equations. We introduce a new approach to replication of random vectors via adapted differentiable processes generated by a controlled ordinary differential …
Lie algebroids can not always be integrated into Lie groupoids. We introduce a new object--``Weinstein groupoid'', which is a differentiable stack with groupoid-like axioms. With it, we have solved the integration problem of Lie algebroids. It turns out that every Weinstein groupoid has a Lie algebroid and every Lie al…
The paper solves a pricing problem for a multiple reset put option using integral equations.
problem Valuation of a multiple reset put option with reset rights.
method Formulated as a multiple optimal stopping problem, reduced to single optimal stopping problems, solved by induction and integral equations.
result Characterized optimal reset boundaries as solutions to nonlinear integral equations and derived reset premium representations.
Problems on region choices for knot and link diagrams solved using Alexander numbering.
problem Existence of solutions for region choice problems on knot and link diagrams.
method Alexander numbering for regions, alternative proofs, necessary and sufficient conditions.
result Existence of solutions for region choice problems on link diagrams.
Integral Transforms in a Physics-Informed (Quantum) Neural Network setting: Applications & Use-Casesquant-ph
This work integrates differentiation and integration in Physics-Informed Neural Networks.
problem Solving integro-differential equations and computing integral transforms.
method Augmenting Physics-Informed Neural Networks with automatic integration.
result Solving complex integral transforms and integro-differential equations.
Paper examines stability of Bayesian posterior measures using integral probability metrics.
problem Stability of Bayesian inference in large-scale inverse problems.
method New families of integral probability metrics for likelihood and prior perturbations.
result Constructs new stability results for Bayesian posterior measures.
Lie's Third Theorem, asserting that each finite-dimensional Lie algebra is the Lie algebra of a Lie group, fails in infinite dimensions. The modern account on this phenomenon is the integration problem for central extensions of infinite-dimensional Lie algebras, which in turn is phrased in terms of an integration proce…
On an integral equation for the free-boundary of stochastic, irreversible investment problemsq-fin.PM
In this paper, we derive a new handy integral equation for the free-boundary of infinite time horizon, continuous time, stochastic, irreversible investment problems with uncertainty modeled as a one-dimensional, regular diffusion . The new integral equation allows to explicitly find the free-boundary in s…
Study optimal trading strategies for mean-reverting spreads using integral equations.
problem Optimal timing for trading mean-reverting price spreads.
method Utilized local time-space calculus and nonlinear integral equations of Volterra-type.
result Derived optimal boundaries for trading strategies.
We study local normal forms for completely integrable systems on Poisson manifolds in the presence of additional symmetries. The symmetries that we consider are encoded in actions of compact Lie groups. The existence of Weinstein's splitting theorem for the integrable system is also studied giving some examples in whic…
Study optimal control of diffusion processes with infimum or supremum costs.
problem Optimizing control of a diffusion process with costs dependent on its infimum or supremum.
method Introduced novel integral operators to solve two-dimensional singular control problems.
result Explicit solutions for optimal dividend problem with time-dependent preferences.
This paper proves integrability of Birkhoff billiards inside convex cones.
problem Proving integrability of Birkhoff billiards in non-traditional shapes.
method Analyzing the billiard inside a convex cone, proving integrability using a first integral of degree two.
result The Birkhoff billiard inside a convex cone is integrable.
Integrates transitive Lie algebroids to Lie groupoids, explaining obstructions.
problem Integrating transitive Lie algebroids to Lie groupoids.
method Geometric explanation and explicit construction of integration, with obstructions considered.
result Obstructions explained and integration constructed when they vanish.
Study quantization on manifolds with embedded submanifolds using Fourier integral operators.
problem Constructing an algebra for relative elliptic problems on manifolds with embedded submanifolds.
method Using Fourier integral operators on Lie groupoids, the authors develop a calculus for elliptic problems.
result The constructed algebra is closed under composition and provides a noncommutative completion of the embedding.
Path Integral and Asian Optionsq-fin.PR
In this paper we analytically study the problem of pricing an arithmetically averaged Asian option in the path integral formalism. By a trick about the Dirac delta function, the measure of the path integral is defined by an effective action functional whose potential term is an exponential function. This path integral …
The paper solves numerical integration on graphs by optimizing vertex sampling and weights.
problem Finding efficient sampling and weights for graph functions.
method Rewriting integration as a geometric packing problem and constructing approximate solutions.
result Efficient numerical integration on graphs can be achieved through optimal packing of heat balls.