Convexity properties of the condition number II

In our previous paper [SIMAX 31 n.3 1491-1506(2010)], we studied the condition metric in the space of maximal rank matrices. Here, we show that this condition metric induces a Lipschitz-Riemann structure on that space. After investigating geodesics in such a nonsmooth structure, we show that the inverse of the smallest singular value of a matrix is a log-convex function along geodesics (Theorem 1). We also show that a similar result holds for the solution variety of linear systems (Theorem 31). Some of our intermediate results, such as Theorem 12, on the second covariant derivative or Hessian of a function with symmetries on a manifold, and Theorem 29 on piecewise self-convex functions, are of independent interest. Those results were motivated by our investigations on the com- plexity of path-following algorithms for solving polynomial systems.Comment: Revised versio

Temporal parameters of the public space: Chelyabinsk «Kirovka»

Статья посвящена рассмотрению проблемы городского центра как особой социокультурной среды, специфического урбанистического хронотопа, в контексте которого выстраивается пространственно-временная коммуникация прошлого и настоящего, провинциального и столичного, делового и досугового. В качестве эмпирической базы анализа выступила конкретная рекреационная зона – пешеходная улица г. Челябинска – «Кировка». В рамках статьи она рассмотрена сквозь призму исторического ландшафта, коммеморативных и магических практик и художественной компоненты.Article is devoted to consideration of a problem of the city center as special sociocultural environment, specific urbanistic chronotope in the context of which existential communication past and present, provincial and capital, business and leisure is built. The concrete recreational zone – a pedestrian street of Chelyabinsk – Kirovka has acted as empirical base of the analysis. Within article she is considered through a prism of a historical landscape, the kommemorativnykh and magic the practician and art components

Convex Dynamics and Applications

This paper proves a theorem about bounding orbits of a time dependent dynamical system. The maps that are involved are examples in convex dynamics, by which we mean the dynamics of piecewise isometries where the pieces are convex. The theorem came to the attention of the authors in connection with the problem of digital halftoning. \textit{Digital halftoning} is a family of printing technologies for getting full color images from only a few different colors deposited at dots all of the same size. The simplest version consist in obtaining grey scale images from only black and white dots. A corollary of the theorem is that for \textit{error diffusion}, one of the methods of digital halftoning, averages of colors of the printed dots converge to averages of the colors taken from the same dots of the actual images. Digital printing is a special case of a much wider class of scheduling problems to which the theorem applies. Convex dynamics has roots in classical areas of mathematics such as symbolic dynamics, Diophantine approximation, and the theory of uniform distributions.Comment: LaTex with 9 PostScript figure

Polynomial-Time Amoeba Neighborhood Membership and Faster Localized Solving

We derive efficient algorithms for coarse approximation of algebraic hypersurfaces, useful for estimating the distance between an input polynomial zero set and a given query point. Our methods work best on sparse polynomials of high degree (in any number of variables) but are nevertheless completely general. The underlying ideas, which we take the time to describe in an elementary way, come from tropical geometry. We thus reduce a hard algebraic problem to high-precision linear optimization, proving new upper and lower complexity estimates along the way.Comment: 15 pages, 9 figures. Submitted to a conference proceeding

Shadowing by non uniformly hyperbolic periodic points and uniform hyperbolicity

We prove that, under a mild condition on the hyperbolicity of its periodic points, a map $g$ which is topologically conjugated to a hyperbolic map (respectively, an expanding map) is also a hyperbolic map (respectively, an expanding map). In particular, this result gives a partial positive answer for a question done by A. Katok, in a related context

The multipliers of periodic points in one-dimensional dynamics

It will be shown that the smooth conjugacy class of an $S-$unimodal map which does not have a periodic attractor neither a Cantor attractor is determined by the multipliers of the periodic orbits. This generalizes a result by M.Shub and D.Sullivan for smooth expanding maps of the circle

The past phenomenon in system of emotivny perception

The relation at the right time including to the past, is an important component of a picture of the world in any historical type of society. Studying of perception of the past promotes deeper understanding of actual sociocultural space. It also causes extremely high research interest in this perspective. Owing to the complexity and ambiguity studying of a phenomenon of the past is carried out in different contexts — subjective and objective, collective and individual, rational and emotional. This article is also devoted to the last aspect.Отношение ко времени, в том числе и прошлому, является важной составляющей картины мира в любом историческом типе общества. Изучение восприятия прошлого способствует более глубокому пониманию актуального социокультурного пространства. Это и обусловливает чрезвычайно высокий исследовательский интерес к данной проблематике. В силу своей сложности и неоднозначности изучение феномена прошлого осуществляется в разных контекстах — субъективном и объективном, коллективном и индивидуальном, рациональном и эмоциональном. Именно последнему аспекту и посвящена данная статья

Stochastic stability at the boundary of expanding maps

We consider endomorphisms of a compact manifold which are expanding except for a finite number of points and prove the existence and uniqueness of a physical measure and its stochastical stability. We also characterize the zero-noise limit measures for a model of the intermittent map and obtain stochastic stability for some values of the parameter. The physical measures are obtained as zero-noise limits which are shown to satisfy Pesin?s Entropy Formula

Fast-slow partially hyperbolic systems versus Freidlin-Wentzell random systems

We consider a simple class of fast-slow partially hyperbolic dynamical systems and show that the (properly rescaled) behaviour of the slow variable is very close to a Friedlin--Wentzell type random system for times that are rather long, but much shorter than the metastability scale. Also, we show the possibility of a "sink" with all the Lyapunov exponents positive, a phenomenon that turns out to be related to the lack of absolutely continuity of the central foliation.Comment: To appear in Journal of Statistical Physic

Simultaneous Continuation of Infinitely Many Sinks Near a Quadratic Homoclinic Tangency

We prove that the $C^3$ diffeomorphisms on surfaces, exhibiting infinitely many sinksnear the generic unfolding of a quadratic homoclinic tangency of a dissipative saddle, can be perturbed along an infinite dimensional manifold of $C^3$ diffeomorphisms such that infinitely many sinks persist simultaneously. On the other hand, if they are perturbed along one-parameter families that unfold generically the quadratic tangencies, then at most a finite number of those sinks have continuation