arXiv subjects

Eugene Stepanov

Publications and source records attributed to Eugene Stepanov.

28 records · Page 2Linked to original sources

arXiv2019

The saga of a fish: from a survival guide to closing lemmas

In the paper by D.~Burago, S.~Ivanov and A.~Novikov, "A survival guide for feeble fish", it has been shown that a fish with limited velocity can reach any point in the (possibly unbounded) ocean provided that the fluid velocity field is incompressible, bounded and has vanishing mean drift. This result extends some known global controllability theorems though being substantially nonconstructive. We give a fish a different recipe of how to survive in a turbulent ocean, and show its relationship to structural stability of dynamical systems by providing a constructive way to change slightly the velocity field to produce conservative (in the sense of not having wandering sets of positive measure) dynamics. In particular, this leads to the extension of C.~Pugh's closing lemma to incompressible vector fields over unbounded domains. The results are based on an extension of the Poincaré recurrence theorem to some $σ$-finite measures and on specially constructed Newtonian potentials.

math.DS↗

arXiv2019

Structural transition states explored with minimalist coarse grained models: applications to Calmodulin

Transitions between different conformational states are ubiquitous in proteins, being involved in signaling, catalysis and other fundamental activities in cells. However, modeling those processes is extremely difficult, due to the need of efficiently exploring a vast conformational space in order to seek for the actual transition path for systems whose complexity is already high in the steady states. Here we report a strategy that simplifies this task attacking the complexity on several sides. We first apply a minimalist coarse-grained model to Calmodulin, based on an empirical force field with a partial structural bias, to explore the transition paths between the apo-closed state and the Ca-bound open state of the protein. We then select representative structures along the trajectory based on a structural clustering algorithm and build a cleaned-up trajectory with them. We finally compare this trajectory with that produced by the online tool MinActionPath, by minimizing the action integral using a harmonic network model, and with that obtained by the PROMPT morphing method, based on an optimal mass transportation-type approach including physical constraints. The comparison is performed both on the structural and energetic level, using the coarse-grained and the atomistic force fields upon reconstruction. Our analysis indicates that this method returns trajectories capable of exploring intermediate states with physical meaning, retaining a very low computational cost, which can allow systematic and extensive exploration of the multi-stable proteins transition pathways.

physics.chem-ph↗

arXiv2017

Self-contracted curves have finite length

A curve $θ$: $I\to E$ in a metric space $E$ equipped with the distance $d$, where $I\subset \R$ is a (possibly unbounded) interval, is called self-contracted, if for any triple of instances of time $\{t_i\}_{i=1}^3\subset I$ with $t_1\leq t_2\leq t_3$ one has $d(θ(t_3),θ(t_2))\leq d(θ(t_3),θ(t_1))$. We prove that if $E$ is a finite-dimensional normed space with an arbitrary norm, the trace of $θ$ is bounded, then $θ$ has finite length, i.e. is rectifiable, thus answering positively the question raised in~\cite{Lemenant16sc-rectif}.

math.MG↗

arXiv2016

A rough calculus approach to level sets in the Heisenberg group

We introduce novel equations, in the spirit of rough path theory, that parametrize level sets of intrinsically regular maps on the Heisenberg group with values in $\mathbb{R}^2$. These equations can be seen as a sub-Riemannian counterpart to classical ODEs arising from the implicit function theorem. We show that they enjoy all the natural well-posedness properties, thus allowing for a "good calculus" on nonsmooth level sets. We apply these results to prove an area formula for the intrinsic measure of level sets, along with the corresponding coarea formula.

math.DG↗

arXiv2016

Regularity for the optimal compliance problem with length penalization

We prove some regularity results for a connected set S in the planar domain O, which minimizes the compliance of its complement O§, plus its length. This problem, interpreted as to find the best location for attaching a membrane subject to a given external force f so as to minimize the compliance, can be seen as an elliptic PDE version of the average distance problem/irrigation problem (in a penalized version rather than a constrained one), which has been extensively studied in the literature. We prove that minimizers consist of a finite number of smooth curves meeting only by three at 120 degree angles, containing no loop, and possibly touching the boundary of the domain only tangentially. Several new technical tools together with the classical ones are developed for this purpose.

math.OC↗

arXiv2015

Three superposition principles: currents, continuity equations and curves of measures

We establish a general superposition principle for curves of measures solving a continuity equation on metric spaces without any smooth structure nor underlying measure, representing them as marginals of measures concentrated on the solutions of the associated ODE defined by some algebra of observables. We relate this result with decomposition of acyclic normal currents in metric spaces. As an application, a slightly extended version of a probabilistic representation for absolutely continuous curves in Kantorovich-Wasserstein spaces, originally due to S. Lisini, is provided in the metric framework. This gives a hierarchy of implications between superposition principles for curves of measures and for metric currents.

math.FA↗

arXiv2014

Structure of metric cycles and normal one-dimensional currents

We prove that every one-dimensional real Ambrosio-Kirchheim normal current in a Polish (i.e. complete separable metric) space can be naturally represented as an integral of simpler currents associated to Lipschitz curves. As a consequence a representation of every such current with zero boundary (i.e. a cycle) as an integral of so-called elementary solenoids (which are, very roughly speaking, more or less the same as asymptotic cycles introduced by S. Schwartzman)is obtained. The latter result on cycles is in fact a generalization of the analogous result proven by S. Smirnov for classical Whitney currents in a Euclidean space. The same results are true for every complete metric space under suitable set-theoretic assumptions.

math.DG↗

arXiv2014

Stationary configurations for the average distance functional and related problems

For a functional defined on the class of closed one-dimensional connected subsets of ${\mathbb R}^n$ we consider the corresponding minimization problem and we give suitable first order necessary conditions of optimality. The cases studied here are the average distance functional arising in the mass transportation theory, and the energy related to an elliptic PDE.

math.OC↗

arXiv2013

Asymptotic optimal location of facilities in a competition between population and industries

We consider the problem of optimally locating a given number $k$ of points in ${\mathbb R}^n$ for an integral cost function which takes into account two measures $φ^+$ and $φ^-$. The points represent for example new industrial facilities that have to be located, the measure $φ^+$ representing in this case already existing industries that want to be close to the new ones, and $φ^-$ representing private citizens who want to stay far away. The asymptotic analysis as $k\to\infty$ is performed, providing the asymptotic density of optimal locations.

math.OC↗

arXiv2013

Droplet condensation and isoperimetric towers

We consider a variational problem in a planar convex domain, motivated by statistical mechanics of crystal growth in a saturated solution. The minimizers are constructed explicitly and are completely characterized.

math.OC↗