arXiv research

New dynamics derived from Lie groups using 2nd order tangent groups.

This paper is devoted to study the Lie algebra of linear symmetries of a homogenous 2nd order ODE, by the method of Kushner, Lychagin and Robstov.

The artificial neural network is a popular framework in machine learning. To empower individual neurons, we recently suggested that the current type of neurons could be upgraded to 2nd order counterparts, in which the linear operation between inputs to a neuron and the associated weights is replaced with a nonlinear qu…

In 1896 Tresse gave a complete description of relative differential invariants for the pseudogroup action of point transformations on the 2nd order ODEs. The purpose of this paper is to review, in light of modern geometric approach to PDEs, this classification and also discuss the role of absolute invariants and the eq…

The study examines how market trade randomness influences price and return volatility.

We show that for two dimensional manifolds M with negative Euler characteristic there exists subsets of the space of smooth Riemannian metrics which are invariant and either parabolic or backwards-parabolic for the 2nd order RG flow. We also show that solutions exists globally on these sets. Finally, we establish the e…

The hybrid Monte Carlo algorithm (HMCA) is applied for Bayesian parameter estimation of the realized stochastic volatility (RSV) model. Using the 2nd order minimum norm integrator (2MNI) for the molecular dynamics (MD) simulation in the HMCA, we find that the 2MNI is more efficient than the conventional leapfrog integr…

Natural gradient descent is an optimization method traditionally motivated from the perspective of information geometry, and works well for many applications as an alternative to stochastic gradient descent. In this paper we critically analyze this method and its properties, and show how it can be viewed as a type of 2…

We solve the metrisability problem for the six Painlevé equations, and more generally for all 2nd order ODEs with Painlevé property, and determine for which of these equations their integral curves are geodesics of a (pseudo) Riemannian metric on a surface.

These are lecture notes of the Summer school on the geometry of differential equations held in Nordfjordeid, Norway in 1996. They cover geometric structures related to scalar second order ODEs, the construction of the associated Cartan connection, techniques for computing invariants of differential equations starting f…

Deep neural networks undergo hierarchical free-energy landscape transitions with increasing data size.

The equivalence problem for second order ODEs given modulo point transformations is solved in full analogy with the equivalence problem of nondegenerate 3-dimensional CR structures. This approach enables an analog of the Feffereman metrics to be defined. The conformal class of these (split signature) metrics is well de…

We prove new results on existence of solutions for the prescribed gaussian curvature problem on the euclidean sphere S^2. Those results are achieved by relating this problem with the holomorphic triples theory on Riemann surfaces. We think this approach might be applied to study some other semi-linear elliptic equation…

The n-dimensional Lorentzian manifolds with vanishing second covariant derivative of the Riemann tensor (2-symmetric spacetimes) are characterized and classified. The main result is that either they are locally symmetric or they have a covariantly constant null vector field, in this case defining a subfamily of Brinkma…

Recent advances in Bayesian learning with large-scale data have witnessed emergence of stochastic gradient MCMC algorithms (SG-MCMC), such as stochastic gradient Langevin dynamics (SGLD), stochastic gradient Hamiltonian MCMC (SGHMC), and the stochastic gradient thermostat. While finite-time convergence properties of th…

Paper formulates governing equations for membrane O surfaces.

Survey on advanced gauge theory concepts.

Simplified proof for Frank and Lieb's inequality on Heisenberg group.

In this paper, we consider nonlinear PDEs in a port-Hamiltonian setting based on an underlying jet-bundle structure. We restrict ourselves to systems with 1-dimensional spatial domain and 2nd-order Hamiltonian including certain dissipation models that can be incorporated in the port- Hamiltonian framework by means of a…

In this paper we consider the cohomology of a closed arithmetic hyperbolic 3-manifold with coefficients in the local system defined by the even symmetric powers of the standard representation of SL(2,C). The cohomology is defined over the integers and is a finite abelian group. We show that the order of the 2nd cohomol…

For each simple Lie algebra $\mathfrak{g}$ (excluding, for trivial reasons, type ${\sf C}$) we find the lowest possible degree of an invariant second-order PDE over the adjoint variety in $\mathbb{P}\mathfrak{g}$, a homogeneous contact manifold. Here a PDE $F(x^i,u,u_i,u_{ij})=0$ has degree $\le d$ if $F$ is a polynomi…

The experimental techniques have evolved to a stage where various examples of nanostructures with non-trivial shapes have been synthesized, turning the dynamics of a constrained particle and the link with geometry into a realistic and important topic of research. Some decades ago, a formalism to deduce a meaningful Ham…

We construct non-Kähler simply connected Calabi-Yau 3-folds with arbitrarily large 2nd Betti numbers by smoothing normal crossing varieties with trivial dualizing sheaves.

The paper studies conformal and projective structures using 2-frame bundles.

Higher topos theory applied to physics.

We prove some stability results for smooth H-minimal hypersurfaces immersed in a sub-Riemannian k-step Carnot group G. The main tools are the formulas for the 1st and 2nd variation of the H-perimeter measure.

We prove that for some knot-like objects one can easily recognize non-equivalence w.r.t. all Reidemeister moves by studying some equivalence classes modulo only 2nd Reidemeister moves. There are applications to virtual knots, graph-links and looped graphs.

In this work we show that the systems of balance equations (balance systems) of continuum thermodynamics occupy a natural place in the variational bicomplex formalism. We apply the vertical homotopy decomposition to get a local splitting (in a convenient domain) of a general balance system as the sum of a Lagrangian pa…

New stable HOIF estimators for statistical functionals.

We provide five examples of conformal geometries which are naturally associated with ordinary differential equations (ODEs). The first example describes a one-to-one correspondence between the Wuenschmann class of 3rd order ODEs considered modulo contact transformations of variables and (local) 3-dimensional conformal …

We establish new, optimal gradient continuity estimates for solutions to a class of 2nd order partial differential equations, $\mathscr{L}(X, \nabla u, D^2 u) = f$, whose diffusion properties (ellipticity) degenerate along the \textit{a priori} unknown singular set of an existing solution, $\mathscr{S}(u) := \{X : \nab…

A regression algorithm uses Green's function and covariance matrix for predictive distributions.

Quantum character varieties unify four construction methods.

I consider the geometry of the general class of scalar 2nd-order differential equations with parabolic symbol, including non-linear and non-evolutionary parabolic equations. After defining the appropriate $G$-structure to model parabolic equations, I apply Cartan techniques to determine local geometric invariants (quan…

We first discuss the problems in the theory of ordinary differential equations that gave rise to the concept of a flag system and illustrate these with the Cartan criterion for Monge equations (1st order) as well as the Cartan statement concerning the local equivalence of Monge-Ampère type equations (2nd order). Next, …

The paper bounds the handle number of sutured manifolds using Morse-Novikov numbers and tunnel numbers.

Study on SGD dynamics and scaling laws for training quadratic neural networks in high dimensions.

The Tick library simulates and learns Hawkes processes with latency effects.

Using Jeff Holman's comments in Quantitative Finance to illustrate 4 critical errors students should learn to avoid: 1) Mistaking tails (4th moment) for volatility (2nd moment), 2) Missing Jensen's Inequality, 3) Analyzing the hedging wihout the underlying, 4) The necessity of a numeraire in finance.

In stochastic gradient descent, especially for neural network training, there are currently dominating first order methods: not modeling local distance to minimum. This information required for optimal step size is provided by second order methods, however, they have many difficulties, starting with full Hessian having…

Formulae quantify gaps in geodesic quadrilaterals on manifolds.

This paper solves Yang-Baxter cohomology for cyclic biquandles.

Constructs non-Kähler Calabi-Yau manifolds with large Betti numbers.

We study biharmonic maps and f-biharmonic maps from a round sphere $(S^2, g_0)$, the latter maps are equivalent to biharmonic maps from Riemann spheres $(S^2, f^{-1}g_0)$. We proved that for rotationally symmetric maps between rotationally symmetric spaces, both biharmonicity and f-biharmonicity reduce to a 2nd order l…

After introducing the sub-Riemannian geometry of the Heisenberg group Hn, n \geq 1, we recall some basics about hypersurfaces endowed with the H-perimeter measure and horizontal Green's formulas. Then, we describe a class of compact closed hypersurfaces of constant horizontal mean curvature called "Isoperimetric Profil…

In this paper, we study conformal invariants that arise from nodal sets and negative eigenvalues of conformally covariant operators; more specifically, the GJMS operators, which include the Yamabe and Paneitz operators. We give several applications to curvature prescription problems. We establish a version in conformal…

Let (M,g) be a compact Riemannian manifold with boundary. This paper is concerned with the set of scalar-flat metrics which are in the conformal class of g and have the boundary as a constant mean curvature hypersurface. We prove that this set is compact for dimensions greater than or equal to 7 under the generic condi…

These lecture notes, which were designed for the Summer School "Heegaard-Floer Homology and Khovanov Homology" in Marseilles, 29th May - 2nd June, 2006, provide an elementary introduction to Khovanov homology. The intended audience is graduate students with some minimal background in low-dimensional and algebraic topol…