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…
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48 results for “state integrals”
Proves summability of state integrals for specific hyperbolic knots.
problem Summability of perturbative series for hyperbolic knots.
method Algorithm to compute Borel-Laplace resummation as state integrals.
result Complete description of resurgent structure and explicit computations of Stokes constants.
It is well-known to the experts that multi-dimensional state integrals of products of Faddeev's quantum dilogarithm which arise in Quantum Topology can be written as finite sums of products of basic hypergeometric series in q=e^{2πiτ} and \tilde{q}=e^{-2πi/τ}. We illustrate this fact by giving a detailed proof for a fa…
The paper reinterprets a quantum invariant using state integrals and contour integrals.
problem Quantum invariants of knots and their asymptotic behavior.
method Expressing the invariant as a sum over contour integrals in hyperbolic structures.
result Establishes a new integral representation for quantum invariants.
We investigate the properties of a modulus of a foliation on a Riemannian manifold. We give necessary and sufficient conditions for the existence of an extremal function and state some of its properties. We obtain the integral formula which, in a sense, combines the integral over the manifold with integral over the lea…
Anomalies in (2+1)D fermionic topological phases and (3+1)D path integral state sums for fermionic SPTscond-mat.str-el
Constructs a path integral for fermionic SPTs, solving anomalies in 2+1D topological orders.
problem Anomalies in (2+1)D fermionic topological phases and their computation.
method Combining (2+1)D fermionic topological order with symmetry fractionalization data to construct a (3+1)D path integral.
result Reproduces the anomaly indicator for time-reversal symmetric topological superconductors.
This paper proposes a new method to learn integration schemes for complex ODEs.
problem Learning efficient integration schemes for non-linear ODEs and their identification.
method A novel framework to learn integration schemes that minimize an integration-related cost function.
result The proposed learning-based approach provides integration schemes close to analytical solutions.
Study differential and integral calculus on noncommutative C*-algebras.
problem Develop calculus on noncommutative spaces.
method Formal smooth structure on nonpure states of C*-algebras.
result Prove Stokes' theorem in both commutative and noncommutative settings.
Gradient matching with Gaussian processes is a promising tool for learning parameters of ordinary differential equations (ODE's). The essence of gradient matching is to model the prior over state variables as a Gaussian process which implies that the joint distribution given the ODE's and GP kernels is also Gaussian di…
This paper deals with the evaluation of double line integrals of the squared exponential covariance function. We propose a new approach in which the double integral is reduced to a single integral using the error function. This single integral is then computed with efficiently implemented numerical techniques. The perf…
In this note we define a distance between two pointed locally integral current spaces. We prove that a sequence of pointed locally integral current spaces converges with respect to this distance if and only if it converges in the sense of Lang-Wenger. This enables us to state the compactness theorem by Lang-Wenger for …
Proves magnetic geodesic flow on sphere is integrable with constant 2-form.
problem Proving integrability of magnetic geodesic flow on sphere.
method Analyzes magnetic geodesic flow on sphere with constant 2-form.
result Proves Liouville integrability of magnetic geodesic flow on sphere.
New modifiers improve noisy RNN replay in hippocampal networks.
problem Improving noisy RNN replay in hippocampal networks.
method Three approaches: hidden state leakage, adaptation, and momentum.
result Hidden state leakage, adaptation, and momentum improve noisy RNN replay.
Itô maps for any-step SDEsstat.ML
Itô maps provide a method for any-step SDE integration.
problem Stochastic dynamics
method Itô map formulation
result Empirical results on synthetic and image-generation benchmarks
Geometric approach to Dirac operator evolution on spacetimes.
problem Constructing the Cauchy evolution operator for Lorentzian Dirac operators.
method Realizing the operator as a sum of oscillatory integrals, relating to Feynman propagator.
result Relating Cauchy evolution operators to Feynman propagators and constructing Hadamard states.
Holomorphic quantum modular forms linked to knot volumes.
problem Understanding algebraic properties of quantum modular forms.
method Analyzing descendant state integrals for specific knots.
result Illustrated algebraic properties for the (-2,3,7)-pretzel knot.
Representation Learning for Integrating Multi-domain Outcomes to Optimize Individualized Treatmentscs.LG
Framework integrates mental disorder measurements for personalized treatment.
problem Optimizing treatment for mental disorders with latent mental states and heterogeneity.
method Measurement theory and multi-layer neural network for complex treatment effects.
result Learned treatment policies outperform alternatives on heterogeneous treatment effects.
By using the Weil-Gel'fand-Zak transform of Faddeev's quantum dilogarithm, we propose a new state-integral model for the Teichmüller TQFT, where the circle valued state variables live on the edges of oriented leveled shaped triangulations.
In this paper we investigate compatible overdetermined systems of PDEs on the plane with one common characteristic. Lie's theorem states that its integration is equivalent to a system of ODEs, and we relate this to the geometry of rank 2 distributions. We find a criterion for integration in quadratures and in closed fo…
The generator gradient estimator is an adjoint state method for stochastic differential equationsmath.OC
New estimator for SDEs is shown to be an adjoint state method.
problem Estimating gradients for overparameterized SDEs efficiently.
method Demonstrates generator gradient estimator as an adjoint state method.
result Generator gradient estimator is an adjoint state method for SDEs.
We use the 3d-3d correspondence together with the DGG construction of theories labelled by 3-manifolds M to define a non-perturbative state-integral model for SL(n,C) Chern-Simons theory at any level k, based on ideal triangulations. The resulting partition functions generalize a widely studied k=1 state-integ…
We first analyze the integrated density of states (IDS) of periodic Schrödinger operators on an amenable covering manifold. A criterion for the continuity of the IDS at a prescribed energy is given along with examples of operators with both continuous and discontinuous IDS'. Subsequently, alloy-type perturbations of th…
Study wave functions in complex Chern-Simons theory, finding integrality and rational points.
problem Understanding wave functions in complex Chern-Simons theory.
method Conjecture and prove integrality structure, develop techniques to determine wave functions at rational points.
result Wave functions have integrality structure and can be determined at rational points.
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.
Quantum dilogarithms help define invariants of 3-manifolds.
problem Defining invariants of 3-manifolds using quantum dilogarithms.
method Associate quantum dilogarithms to local fields, construct TQFTs, and use partition functions.
result Quantum dilogarithms yield invariants related to -polynomial curves.
Hypergeometric integrals associated with hypersphere arrangements and Cayley-Menger determinantsmath.DG
The n-dimensional hypergeometric integrals associated with a hypersphere arrangement are formulated by the pairing of n-dimensional twisted cohomology and its dual. Under the condition of general position there are stated some results which concern an explicit representation of the standard form by a special (NBC) basi…
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.
Assessing world-wide financial integration constitutes a recurrent challenge in macroeconometrics, often addressed by visual inspections searching for data patterns. Econophysics literature enables us to build complementary, data-driven measures of financial integration using graphs. The present contribution investigat…
Proposes an INLA-based method for state and parameter estimation in nonlinear systems.
problem Difficulty in learning parameters accurately in nonlinear dynamical systems.
method Iterated INLA for state and parameter estimation in nonlinear dynamical systems.
result Outperforms existing methods on data assimilation tasks.
TSAW improves MCMC integral estimation with faster convergence.
problem Estimating integrals using MCMC with standard random walks is slow.
method Introduces TSAW to penalize overuse in finite-state adaptive sampling.
result TSAW-based estimators converge faster, achieving error.
On the Representation of General Interest Rate Models as Square Integrable Wiener Functionalsq-fin.GN
In the setting proposed by Hughston & Rafailidis (2005) we consider general interest rate models in the case of a Brownian market information filtration . Let be a square-integrable -measurable random variable, and assume the non-degeneracy condition that for all $t<\in…
Scroll structures on solutions of 4D integrable equations are involutive and governed by a dispersionless hierarchy.
problem Characterizing the geometry of solutions to 4D integrable equations.
method Defining rational normal scrolls and showing their involutivity.
result Involutive scroll structures are governed by a dispersionless integrable hierarchy.
Restricts quantum representations of mapping class groups to integral coefficients.
problem Integrality of non-semisimple quantum representations of mapping class groups.
method Exhibits explicit bases of states spaces that span -lattices invariant under mapping class groups.
result Restricts quantum representations to integral coefficients from to .
There is renewed interest in formulating integration as an inference problem, motivated by obtaining a full distribution over numerical error that can be propagated through subsequent computation. Current methods, such as Bayesian Quadrature, demonstrate impressive empirical performance but lack theoretical analysis. A…
Study efficient reinforcement learning for partially observed systems with linear structure.
problem Efficient reinforcement learning for partially observed Markov decision processes with linear structure.
method Proposes OP-TENET algorithm using a Bellman operator with finite memory, adversarial integral equation, and optimistic exploration.
result Achieves ε-optimal policy within O(1/ε^2) episodes with polynomial sample complexity in intrinsic dimension.
We replace the usual Convex Integration formula by a Corrugation Process and introduce the notion of Kuiper differential relations. This notion provides a natural framework for the construction of solutions with self-similarity properties. We consider the case of the totally real relation, we prove that it is Kuiper an…
New method for sampling from multivariate distributions using optimal control and quantum mechanics.
problem Sampling from continuous multivariate probability distributions efficiently and accurately.
method Harmonic Path Integral Diffusion (H-PID) framework, formulated as a Stochastic Optimal Control problem.
result Efficient sampling algorithms without neural networks, revealing dynamic phase transitions.
Estimates crypto risk premia using hidden factors and finds significant integration with traditional markets.
problem Estimating risk premia in cryptocurrency returns.
method Giglio-Xiu (2021) three-pass approach, controlling for latent factors and non-tradable state variables.
result Latent factors significantly impact crypto returns, highlighting the importance of controlling for unobserved risks.
Bayesian control prepares GHZ states in Rydberg atoms.
problem Preparing non-classical states robustly for quantum sensors.
method Bayesian optimal control of trapped Rydberg atoms.
result Control pulses drive atoms into GHZ states with scalable preparation times.
Entanglement and productsmath.DG
Quantum states are not entangled if submanifold is a product.
problem Understanding entanglement in quantum states associated with product submanifolds.
method Analyzing quantum states on submanifolds of product Kähler manifolds in the semiclassical limit.
result States are not entangled when submanifold is a product.
A path integral on a link complement of a three-sphere fixes a vector (the "link state") in Chern-Simons theory. The link state can be written in a certain basis with the colored link invariants as its coefficients. We use symmetric webs to systematically compute the colored link invariants, by which we can write down …
Proposes integrating global and local entropy for more reliable LLMs.
problem Uncertainty in large language models (LLMs) leads to unreliable predictions.
method Measures global uncertainty from hidden-state matrices and local uncertainty from tokens, combining them via a multiplicative gate.
result Global-Local Uncertainty (GLU) outperforms unsupervised baselines across multiple models and benchmarks.
A multiplicatively closed, horizontal foliation on a Lie groupoid may be viewed as a "pseudoaction" on the base manifold . A pseudoaction generates a pseudogroup of transformations of in the same way an ordinary Lie group action generates a transformation group. Infinitesimalizing a pseudoaction, one obtains the…
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.
The aim of this article is to design a moment transformation for Student- t distributed random variables, which is able to account for the error in the numerically computed mean. We employ Student-t process quadrature, an instance of Bayesian quadrature, which allows us to treat the integral itself as a random variable…
We analyze relations between BPS degeneracies related to Labastida-Marino-Ooguri-Vafa (LMOV) invariants, and algebraic curves associated to knots. We introduce a new class of such curves that we call extremal A-polynomials, discuss their special properties, and determine exact and asymptotic formulas for the correspond…
The various scalar curvatures on an almost Hermitian manifold are studied, in particular with respect to conformal variations. We show several integrability theorems, which state that two of these can only agree in the Kähler case. Our main question is the existence of almost Kähler metrics with conformally constant Ch…
In this survey, we report on the state of the art of some of the fundamental problems in the Lie theory of Lie groups modeled on locally convex spaces, such as integrability of Lie algebras, integrability of Lie subalgebras to Lie subgroups, and integrability of Lie algebra extensions to Lie group extensions. We furthe…