arXiv research

Geometric framework for dissipative systems on Lie algebroids.

Develops theory of contact systems with nonholonomic constraints.

The paper integrates dissipative and curl forces using geometric methods.

Variational reduction simplifies Lagrangian systems with scaling symmetries.

Paper derives invariantised Euler-Lagrange equations for Herglotz problems.

The present paper extends the classical second-order variational problem of Herglotz type to the more general context of the Euclidean sphere S^n following variational and optimal control approaches. The relation between the Hamiltonian equations and the generalized Euler-Lagrange equations is established. This problem…

Injectivity of geodesic ray transform on specific Finsler manifolds proven.

We prove a trace formula for three-dimensional spherically symmetric Riemannian manifolds with boundary which satisfy the Herglotz condition: The wave trace is singular precisely at the length spectrum of periodic broken rays. In particular, the Neumann spectrum of the Laplace--Beltrami operator uniquely determines the…

Reduces equations for contact mechanical systems on Lie groups by exploiting symmetries.

Research decouples Lie algebroids using bicocycle double cross product theory.

This paper extends the evolution operator to contact mechanics, linking Lagrangian and Hamiltonian formulations.

Develop contact Tulczyjew formalism for dissipative dynamics on skew algebroids.

We study ray transforms on spherically symmetric manifolds with a piecewise $C^{1,1}$ metric. Assuming the Herglotz condition, the X-ray transform is injective on the space of $L^2$ functions on such manifolds. We also prove injectivity results for broken ray transforms (with and without periodicity) on such manifolds …

Suppose M_t is a smooth family of compact connected two dimensional submanifolds of Euclidean space E^3 without boundary varying isometrically in their induced Riemannian metrics. Then we show that the mean curvature integrals over M_t are constant. It is unknown whether there are nontrivial such bendings. The estimate…

Reduces symplectic Hamiltonian systems to contact systems, realizing Poincaré's dream.

In this paper we analyze the local and global boundary rigidity problem for general Riemannian manifolds with boundary $(M,g)$. We show that the boundary distance function, i.e., $d_g|_{\partial M\times\partial M}$, known near a point $p\in \partial M$ at which $\partial M$ is strictly convex, determines $g$ in a suita…

Research examines how Islamic banking principles spread among managers and scholars.

The paper establishes maximum principles and stochastic completeness for pseudo-Hermitian manifolds.

The h-principle helps solve complex geometric problems.

The study establishes uncertainty principles on harmonic manifolds of rank one.

Conditions for polynomial to be isometry of lattice, answering Hasse principles.

Shows flexible sheaves as fibrant objects for Gromov's h-principle.

A new method to break down insurance costs into risk and uncertainty.

Study on maximum principles for nonlinear equations on Riemannian manifolds.

Derives Fredholm criteria for isotypical components from a Simonenko principle.

Establishes a boundary maximum principle for varifolds with fixed contact angle.

Study proves Maximum Principles for unbounded Riemannian domains.

A pricing principle is introduced for non-attainable claims in incomplete markets.

Proves a principle for one-phase Bernoulli problem minimizers.

We show that a well known uncertainty principle for functions on the circle can be derived from an uncertainty principle for the Euclidean motion group.

New method proves $h$-principles for stable forms on manifolds.

We give a version of the comparison principle from pluripotential theory where the Monge-Ampère measure is replaced by the Bergman kernel and use it to derive a maximum principle

In this paper, we develop a new mathematical technique which allows us to express the joint distribution of a Markov process and its running maximum (or minimum) through the marginal distribution of the process itself. This technique is an extension of the classical reflection principle for Brownian motion, and it is o…

Note establishes a local maximum principle for Ricci flow under curvature conditions.

The Weyl principle holds in some Finsler settings despite general failure.

Along with fruitful applications of Deep Neural Networks (DNNs) to realistic problems, recently, some empirical studies of DNNs reported a universal phenomenon of Frequency Principle (F-Principle): a DNN tends to learn a target function from low to high frequencies during the training. The F-Principle has been very use…

Derives time-averaged active inference from control principles.

New principle for harmonic maps helps study higher-dimensional submanifolds.

In this paper we give three applications of a method to prove h-principles on closed manifolds. Under weaker conditions this method proves a homological h-principle, under stronger conditions it proves a homotopical one. The three applications are as follows: a homotopical version of Vassiliev's h-principle, the contra…

In this paper we prove two extensions of Hamilton's maximal principle for systems pf parabolic equations which sould be useful for the study of the Ricci flow and some other geometric evolution equations. One extension is a time-dependent maximum principle and the other is a time-dependent maximum principle subject to …

We present a new statistical learning paradigm for Boltzmann machines based on a new inference principle we have proposed: the latent maximum entropy principle (LME). LME is different both from Jaynes maximum entropy principle and from standard maximum likelihood estimation.We demonstrate the LME principle BY deriving …

New equivalences found linking parabolicity, comparison principle, and capacity on Riemannian manifolds.

3-manifolds study Hasse norm principle, akin to number fields.

Paper proves h-principles for symplectic structures and foliations.

Maps to manifolds transverse to certain distributions satisfy an $h$-principle.

Previous studies have shown that deep neural networks (DNNs) with common settings often capture target functions from low to high frequency, which is called Frequency Principle (F-Principle). It has also been shown that F-Principle can provide an understanding to the often observed good generalization ability of DNNs. …

Study shows strong min-max principle for phase transitions.

Why deep neural networks (DNNs) capable of overfitting often generalize well in practice is a mystery [#zhang2016understanding]. To find a potential mechanism, we focus on the study of implicit biases underlying the training process of DNNs. In this work, for both real and synthetic datasets, we empirically find that a…