arXiv research

Rational loops played a central role in Uhlenbeck's construction of harmonic maps into U(n) (chiral model in physics), and they are generated by simple elements with one pole and one zero constructed from Hermitian projections. It has been believed for long time that nilpotent loops should be added to generate rational…

Training-free looped transformers improve model performance without additional training.

New loops found in universe's timeline, challenging traditional time direction.

The space $Ω(G)$ of all based loops in a compact semisimple simply connected Lie group $G$ has an action of the maximal torus $T\subset G$ (by pointwise conjugation) and of the circle $S^1$ (by rotation of loops). Let $μ: Ω(G)\to (\t\times i\mathbb{R})^*$ be a moment map of the resulting $T\times S^1$ action. We show t…

New methods detect nontrivial loops of spheres in 4-manifolds.

Polynomial-time algorithm finds short non-orientable loops intersecting graph edges up to 30 times.

Criteria for loop separability on surfaces using Goldman bracket.

Paper tackles stochastic control with mean and higher-order moments, finding Nash equilibria.

Investigates time-inconsistent portfolio selection under MMV preferences.

In this paper, we study a time-inconsistent consumption-investment problem with random endowments in a possibly incomplete market under general discount functions. We provide a necessary condition and a verification theorem for an open-loop equilibrium consumption-investment pair in terms of a coupled forward-backward …

This paper studies the question of when a loop $φ$ in the group Symp$(M,ω)$ of symplectomorphisms of a symplectic manifold $(M,ω)$ is isotopic to a loop that is generated by a time-dependent Hamiltonian function. (Loops with this property are said to be Hamiltonian.) Our main result is that Hamiltonian loops are rigid …

Program verification offers a framework for ensuring program correctness and therefore systematically eliminating different classes of bugs. Inferring loop invariants is one of the main challenges behind automated verification of real-world programs which often contain many loops. In this paper, we present Continuous L…

Novel algorithm reduces computational burden in IRL with finite-time guarantees.

This paper addresses the problem of learning the optimal control policy for a nonlinear stochastic dynamical system with continuous state space, continuous action space and unknown dynamics. This class of problems are typically addressed in stochastic adaptive control and reinforcement learning literature using model-b…

We study the small-time fluctuations for diffusion processes which are conditioned by their initial and final positions, under the assumptions that the diffusivity has a sub-Riemannian structure and that the drift vector field lies in the span of the sub-Riemannian structure. In the case where the endpoints agree and t…

We consider the problem of online planning in a Markov Decision Process when given only access to a generative model, restricted to open-loop policies - i.e. sequences of actions - and under budget constraint. In this setting, the Open-Loop Optimistic Planning (OLOP) algorithm enjoys good theoretical guarantees but is …

New algorithm finds more arbitrage opportunities in DEXs.

We derive the Do and Norbury recursion formula for the one-loop mean of an irregular spectral curve from a variant of replica method by Brezín and Hikami. We express this recursion in special times in which all terms $W_1^{(g)}$ of the genus expansion of the one-loop mean are polynomials. We find a generalization of th…

We characterize constant mean curvature surfaces in the three-dimensional Heisenberg group by a family of flat connections on the trivial bundle $\D \times \GL$ over a simply connected domain $\mathbb{D}$ in the complex plane. In particular for minimal surfaces, we give an immersion formula, the so-called Sym-formula, …

We review the caloron correspondence between $G$-bundles on $M \times S^1$ and $ΩG$-bundles on $M$, where $ΩG$ is the space of smooth loops in the compact Lie group $G$. We use the caloron correspondence to define characteristic classes for $ΩG$-bundles, called string classes, by transgression of characteristic classes…

Study smooth loops and loop bundles, relating to $G_{2}$-structures.

We consider various generalisations of the string class of a loop group bundle. The string class is the obstruction to lifting a bundle whose structure group is the loop group $LG$ to one whose structure group is the Kac-Moody central extension of the loop group. We develop a notion of higher string classes for bundles…

The paper quantizes hybrid topological-holomorphic field theories on $\mathbb{R}^m imes \mathbb{C}^n$.

In classical knot theory, Markov's theorem gives a way of describing all braids with isotopic closures as links in $\mathbb{R}^3$. We present a version of Markov's theorem for extended loop braids with closure in $B^3 \times S^1$, as a first step towards a Markov's theorem for extended loop braids and ribbon torus-link…

Study homology products on loop spaces with orientation reversal.

Our purpose is to explore, in the context of loop ensembles on finite graphs, the relations between combinatorial group theory, loops topology, loop measures, and signatures of discrete paths. We determine the distributions of the loop homotopy class, and of the first and second homologies, defined by the lower central…

New operations transform braids into P-fibered braids.

We produce skew loops -- loops having no pair of parallel tangent lines -- homotopic to any loop in a flat torus or other quotient of R^n. The interesting case here is n=3. More subtly for any n, we characterize the homotopy classes that will contain a skew loop having a specified loop in the unit sphere as tangent ind…

The paper proves T-duality and Hori formulae for winding loop spaces.

Paper establishes loop space T-duality formulae and refines earlier work.

The 2-loop polynomial is a polynomial presenting the 2-loop part of the Kontsevich invariant of knots. We show a cabling formula for the 2-loop polynomial of knots. In particular, we calculate the 2-loop polynomial for torus knots.

Kähler manifold loop space inherits Kähler structure and is complete.

Study time-inconsistent portfolio optimization for competitive agents with relative performance criteria.

Markov Chain Monte Carlo (MCMC) and Belief Propagation (BP) are the most popular algorithms for computational inference in Graphical Models (GM). In principle, MCMC is an exact probabilistic method which, however, often suffers from exponentially slow mixing. In contrast, BP is a deterministic method, which is typicall…

New example shows noncontractible fiberings for a 3-manifold.

Using the relations between the theory of differentiable Bol loops and the theory of affine symmetric spaces we classify all connected differentiable Bol loops having an at most $9$-dimensional semi-simple Lie group as the group topologically generated by their left translations. We show that all these Bol loops are is…

A skew loop is a closed curve without parallel tangent lines. We prove: The only complete surfaces in euclidean 3-space with a point of positive curvature and no skew loops are the quadrics. In particular, ellipsoids are the only closed surfaces without skew loops. We also prove results about skew loops on cylinders an…

Introduces string structures linking to loop spaces.

Study shows looping a 6-manifold over a 4-manifold results in a product of loops on spheres.

Develops a new method for minimal Lagrangian surfaces in complex quadrics.

The 1-d Schrodinger flow on 2-sphere, the Gauss-Codazzi equation for flat Lagrangian submanifolds in C^n, and the space-time monopole equation are all examples of geometric soliton equations. The linear systems with a spectral parameter (Lax pair) associated to these equations satisfy the reality condition associated t…

New theory defines invariants for framed 3-manifolds using trivalent graphs with self-loops.

Hydrological storm events are a primary driver for transporting water quality constituents such as turbidity, suspended sediments and nutrients. Analyzing the concentration (C) of these water quality constituents in response to increased streamflow discharge (Q), particularly when monitored at high temporal resolution …

Geodesic loops on tetrahedra are studied in spherical and hyperbolic spaces.

Algorithm creates polynomials for knotted surfaces, with bounds on degree.

New definition of twisted 1-loop invariant using Ptolemy coordinates.

Study path spaces and their homology, extending loop products and coproducts.

We study symmetry properties of quaternionic Kähler manifolds obtained by the HK/QK correspondence. To any Lie algebra $\mathfrak{g}$ of infinitesimal automorphisms of the initial hyper-Kähler data we associate a central extension of $\mathfrak{g}$, acting by infinitesimal automorphisms of the resulting quaternionic Kä…