Related papers: Arnold-Thom gradient conjecture for the arrival ti…
We derive a second-order ordinary differential equation (ODE) which is the limit of Nesterov's accelerated gradient method. This ODE exhibits approximate equivalence to Nesterov's scheme and thus can serve as a tool for analysis. We show…
We prove the existence of classical solutions to the Dirichlet problem for a class of fully nonlinear elliptic equations of curvature type on Riemannian manifolds. We also derive new second derivative boundary estimates which allows us to…
Li-Vogelius and Li-Nirenberg gave a gradient estimate for solutions of strongly elliptic equations and systems of divergence forms with piecewise smooth coefficients, respectively. The discontinuities of the coefficients are assumed to be…
We prove the existence theorem for basic elements in the quasi-projective case, extending results of Eisenbud-Evans and Bruns from the affine case. We give several geometric applications. For example, we show that every local complete…
A new approach to the analytic theory of difference equations with rational and elliptic coefficients is proposed. It is based on the construction of canonical meromorphic solutions which are analytical along "thick paths". The concept of…
A surprising, gravity related Verlinde-conjecture, that generated immense interest, asserts that gravity is an emergent entropic force. We provided a classical proof of the assertion in [doi.org/j.physa.2018.03.019]. Here, we classically…
One of the most remarkable features of known nonstationary solutions to the incompressible Euler equations is the phenomenon known as the Taylor hypothesis, which predicts that coarse scale averages of the velocity carry the fine scale…
We introduce a new theory of generalised solutions which applies to fully nonlinear PDE systems of any order and allows for merely measurable maps as solutions. This approach bypasses the standard problems arising by the application of…
Starting from the geometric calculus based on Clifford algebra, the idea that physical quantities are Clifford aggregates ("polyvectors") is explored. A generalized point particle action ("polyvector action") is proposed. It is shown that…
We generalize and refine the hyperk\"ahler Arnold conjecture, which was originally established, in the non-degenerate case, for three-dimensional time by Hohloch, Noetzel and Salamon by means of hyperk\"ahler Floer theory. In particular, we…
Minimizing the sum of a convex function and a composite function appears in various fields. The generalized Levenberg--Marquardt (LM) method, also known as the prox-linear method, has been developed for such optimization problems. The…
In this paper, we propose a geometric framework to analyze the convergence properties of gradient descent trajectories in the context of linear neural networks. We translate a well-known empirical observation of linear neural nets into a…
It is well known that for analytic cost functions, gradient flow trajectories have finite length and converge to a single critical point. The gradient conjecture of R. Thom states that, again for analytic cost functions, whenever the…
Equations of motion for a general relativistic post-Newtonian Lagrangian approach mainly refer to acceleration equations, i.e. differential equations of velocities. They are directly from the Euler-Lagrangian equations, and usually have…
We study the critical points of the solution of second elliptic equations in divergence and diagonal form with a bounded and positive definite coefficient, under the assumption that the statement of the Hopf lemma holds (sign assumptions on…
Democritus and the early atomists held that "the material cause of all things that exist is the coming together of atoms and void. Atoms are eternal and have many different shapes, and they can cluster together to create things that are…
We consider linear, time-dependent and skew-adjoint perturbations of periodic transport equations on the one-dimensional torus. We describe the long-time behavior of solutions for all non-degenerate perturbations in resonant regime, proving…
We consider the asymptotic behavior of solutions to an obstacle problem for the mean curvature flow equation by using a game-theoretic approximation, to which we extend that of Kohn and Serfaty (2006). Kohn and Serfaty (2006) give a…
We give partial affirmative answers to Landis conjecture in all dimensions for two different types of linear, second order, elliptic operators in a domain $\Omega\subset \mathbb{R}^N$. In particular, we provide a sharp decay criterion that…
Recently V. Arnold introduced Strangeness and $J^{\pm}$ invariants of generic immersions of an oriented circle to $\R^2$. Here these invariants are generalized to the case of generic immersions of an oriented circle to an arbitrary surface…