# Nonlinear Analysis and Optimization Seminar[ Edit ]

## Moderator: Simeon Reich

*Abstract:*

Abstract: Poincar\'{e}'s inequality, which is probably best known for its applications in PDEs and the calculus of variations, is one of the simplest examples of an inequality that lies at the crossroads of Analysis, Probability and Semigroup/Spectral theory. It can be understood as the functional inequality that arises from attempting to understand convergence of the so-called heat flow to its equilibrium state. This approach can be generalized to the setting of Markov semigroups, with a non-positive generator that posseses a spectral gap. A natural question that one can consider is: What happens if the generator does not have a spectral gap? Can we still deduce a rate of convergence from a functional setting? In this talk we will discuss a new approach to this question and see how an understanding of the way the spectrum of the generator behaves near the origin, in the form of a density of states estimate, can lead to weak Poincar\'{e} type inequalities, from which a quantitative estimation of convergence can be obtained. This talk is based on a joint work with Jonathan Ben-Artzi.

*Abstract:*

Abstract: Consider a polygon-shaped billiard table on which a ball can roll along straight lines and reflect off of edges infinitely. In work joint with Moon Duchin, Viveka Erlandsson and Chris Leininger, we have characterized the relationship between the shape of a polygonal billiard table and the set of possible infinite edge itineraries of balls travelling on it. In this talk, we will explore this relationship and the tools used in our characterization (notably a new rigidity result for flat cone metrics).

*Abstract:*

Let $(X,\|\cdot\|)$ be a uniformly convex Banach space and let $C$ be a bounded, closed and convex subset of $X$. Assume that $C$ has nonempty interior and is locally uniformly rotund. Let $T$ be a nonexpansive self-mapping of $C$. If $T$ has no fixed point in the interior of $C$, then there exists a unique point $\tilde{x}$ on the boundary of $C$ such that each sequence of iterates of $T$ converges in norm to $\tilde{x}$. We also establish an analogous result for nonexpansive semigroups. This is joint work with Aleksandra Grzesik, Wieslawa Kaczor and Tadeusz Kuczumow.

*Abstract:*

This talk will be devoted to probabilistic constructions appearing in statistics and geometry. I will introduce the classical notion of VC dimension and discuss how it arises naturally in several problems. One of the questions will be the so-called epsilon-approximation problem. That is, how well what you see in a small random sample approximates the real structure. In the last part of the talk, I will explain how a clever deterministic choice of points may improve standard guarantees provided by the random sampling.

*Abstract:*

This is the second of two talks concerning invariants of families of self-adjoint elliptic boundary value problems on a compact surface. With each such family, parameterized by points of a compact topological space $X$, one can associate an invariant reflecting the analytical and the spectral properties of the family. This invariant is called the analytical index. It takes values in the Abelian group $K^1(X)$, the definition of which I will also give in the talk. I will present the index theorem, which expresses the analytical index in terms of the topological data extracted from the family of boundary value problems.

*Abstract:*

In this talk we present a systematic study of regular quasi-nonexpansive operators in Hilbert space. We are interested, in particular, in weakly, boundedly and linearly regular operators. We show that the type of the regularity is preserved under relaxations, convex combinations and products of operators. Moreover, in this connection, we show that weak, bounded and linear regularity lead to weak, strong and linear convergence, respectively, of various iterative methods. This applies, in particular, to projection methods, which oftentimes are based on the above-mentioned algebraic operations applied to projections. This is joint work with Andrzej Cegielski and Simeon Reich.

*Abstract:*

With every family of self-adjoint Fredholm operators on a Hilbert space one can associate an invariant reflecting the analytical and the spectral properties of the family. In the case of a one-parameter family, the corresponding invariant is integer-valued and is called the spectral flow. It can be defined as the net number of eigenvalues of the operaor passing through zero with the change of parameter. In the general case, for a family parameterized by points of a compact space $X$, the corresponding invariant takes values in the Abelian group $K^1(X)$ and is called the family index. I intend to give two talks concerning the computation of these invariants for families of self-adjoint elliptic boundary value problems on a compact surface. In the first talk I will explain how to compute the spectral flow using the topological data extracted from a given one-parameter family of boundary value problems. The talk is based on the preprint arXiv:1703.06105 (math.AP). In the second talk I will show how this result can be generalized to an arbitrary base space $X$.

*Abstract:*

The problem of minimization of a separable convex objective function has various theoretical and real-world applications. One of the popular methods for solving this problem is the proximal gradient method (proximal forward-backward algorithm). A very common assumption in the use of this method is that the gradient of the smooth term in the objective function is globally Lipschitz continuous. However, this assumption is not always satisfied in practice, thus casting a limitation on the method. We discuss, in a wide class of finite and infinite-dimensional spaces, a new variant (BISTA) of the proximal gradient method which does not impose the above-mentioned global Lipschitz continuity assumption. A key contribution of the method is the dependence of the iterative steps on a certain decomposition of the objective set into subsets. Moreover, we use a Bregman divergence in the proximal forward-backward operation. Under certain practical conditions, a non-asymptotic rate of convergence (that is, in the function values) is established, as well as the weak convergence of the whole sequence to a minimizer. We also obtain a few auxiliary results of independent interest, among them a general and usefu lstability principle which, roughly speaking, says that given a uniformly continuous function on an arbitrary metric space, if we slightly change the objective set over which the optimal (extreme) values are computed, then these values vary slightly. This principle suggests a general scheme for tackling a wide class of non-convex and non-smooth optimization problems. This is a joint work with Alvaro De Pierro and Simeon Reich.

*Abstract:*

We develop new iterative methods for solving convex feasibility and common fixed point problems, based on the notion of coherence. We also present new concepts and results in Nonlinear Analysis related to the theory of coherence and Opial's demi-closedness principle. We investigate, in particular, the properties of relaxations, convex combinations and compositions of certain kinds of operators defined on a real Hilbert space, under static and dynamic controls, as well as other properties regarding the algorithmic structure of some operators. Our iterative techniques are applied, for example, to the study of various metric and subgradient projection methods. Furthermore, all the methods are presented in both weak and strong convergence versions.

*Abstract:*

Numerous optimization problems are solved using the tools of distributionally robust optimization. In this framework, the distribution of the problem's random parameter $z$ is assumed to be known only partially in the form of, for example, the values of its first moments. The aim is to minimize the expected value of a function of the decision variables $x$, assuming that Nature maximizes this expression using the worst-possible realization of the unknown probability measure of $z$. In the general moment problem approach, the worst-case distributions are atomic. We propose to model smooth uncertain density functions using sum-of-squares polynomials with known moments over a given domain. We show that in this setup, one can evaluate the worst-case expected values of the functions of the decision variables in a computationally tractable way. This is joint work with Etienne de Klerk (TU Delft) and Daniel Kuhn (EPFL Lausanne).

*Abstract:*

About 15 years ago, Bourgain, Brezis and Mironescu proposed a new characterization of BV and W^(1,q) spaces (for q > 1) using a certain double integral functional involving radial mollifiers. We study what happens when one changes the power of |x-y| in the denominator of the integrand from q to 1. It turns out that for q > 1 the corresponding functionals "see" only the jumps of the BV-function. We further identify the function space relevant to the study of these functionals as an appropriate Besov space. We also present applications to the study of singular perturbation problems of Aviles-Giga type.

*Abstract:*

A zone of width $\omega$ on the unit sphere is defined as the set of points within spherical distance $\omega/2$ of a given great circle. Zones can be thought of as the spherical analogue of planks. In this talk we show that the total width of any (finite) collection of zones covering the unit sphere is at least $\pi$, answering a question of Fejes T\'oth from 1973.This is a joint work with Alexandr Polyanskii.

*Abstract:*

In 1687 Sir Isaac Newton discovered that the area cut off from an oval in the plane by a straight line never depends algebraically on the line (the question was motivated by celestial mechanics). In 1987 V. I. Arnold proposed to generalize Newton's observation to higher dimensions and conjectured that all smooth bodies, with the exception of ellipsoids in odd-dimensional spaces, have an analogous property. The talk is devoted to the current status of this conjecture.

*Abstract:*

We focus on nonconvex and nonsmooth minimization problems with a composite objective, where the differentiable part of the objective is freed from the usual and restrictive global Lipschitz gradient continuity assumption. This long-standing smoothness restriction is pervasive in first order methods, and was recently circumvented for convex composite optimization by Bauschke, Bolte and Teboulle, through a simple and elegant framework which captures, all at once, the geometry of the function and of the feasible set. Building on this work, we tackle genuine nonconvex problems. We first complement and extend their approach to derive a full extended descent lemma by introducing the notion of smooth adaptable functions. We then consider a Bregman-based proximal gradient method for the nonconvex composite model with smooth adaptable functions, which is proven to globally converge to a critical point under natural assumptions on the problem's data, and, in particular, for semi-algebraic problems. To illustrate the power and potential of our general framework and results, we consider a broad class of quadratic inverse problems with sparsity constraints which arises in many fundamental applications, and we apply our approach to derive new globally convergent schemes for this class. The talk is based on joint work with Jerome Bolte (Toulouse), Marc Teboulle (TAU) and Yakov Vaisbroud (TAU).

*Abstract:*

A classical problem in geometry goes as follows. Suppose we are given two sets of $D$ dimensional data, that is, sets of points in $R^D$. The data sets are indexed by the same set, and we know that pairwise distances between corresponding points are equal in the two data sets. In other words, the sets are isometric. Can this correspondence be extended to an isometry of the ambient Euclidean space? In this form the question is not terribly interesting; the answer has long been known to be yes (see [Wells and Williams 1975], for example). But a related question is actually fundamental in data analysis: here the known points are samples from larger, unknown sets -- say, manifolds in $R^D$-- and we seek to know what can be said about the manifolds themselves. A typical example might be a face recognition problem, where all we have is multiple finite images of people's faces from various views. An added complication is that in general we are not given exact distances. We have noise and so we need to demand that instead of the pairwise distances being equal, they should be close in some reasonable metric. Some results on almost isometries in Euclidean spaces can be found in [John 1961; Alestalo et al. 2003]. This talk will consist of two parts. I will discuss various works in progress re this problem with Michael Werman (Hebrew U), Kai Diethelm (Braunschweig) and Charles Fefferman (Princeton). As it turns out the problem relates to the problem of Whitney extensions, interpolation in $R^D$ and bounds for Hilbert transforms. Moreover, for practical algorithms there is a natural deep learning framework as well for both labeled and unlabeled data.

*Abstract:*

The primitive equations are a fundamental model for many geophysical flows. They are derived from the Navier-Stokes equations by assuming a hydrostatic balance for the pressure term. These equations are known to be globally and strongly well-posed in the three-dimensional setting for arbitrarily large data belonging to $H^1$ by the seminal result of Cao and Titi. Here, I would like to consider the primitive equations in $L^p$-spaces using an evolution equation method. This yields several new results on global strong well-posedness for rough initial data as well as on the regularity of solutions. More precisely, one obtains well-posedness for anisotropic initial values in the scaling invariant space $L^{infty}(R^2;L^1(-h,0))$ and the analyticity of solutions in time and space.

*Abstract:*

Let $(M,d)$ be a metric space and let $Y$ be a Banach space. Suppose that for each point $x$ of $M$ we are given a compact convex subset $F(x)$ in $Y$ of dimension at most $m$. A ``Lipschitz selection'' for the family $\{F(x): x\in M\}$ is a Lipschitz map $f$ from $M$ into $Y$ such that $f(x)$ belongs to $F(x)$ for each $x\in M$. The talk explains how one can decide whether a Lipschitz selection exists. We discuss the following ``Finiteness Principle'' for the existence of a Lipschitz selection: Suppose that on every subset $M'$ of $M$ consisting of at most $2^{m+1}$ points, $F$ has a Lipschitz selection with Lipschitz constant at most $1$. Then $F$ has a Lipschitz selection on all of $M$. Furthermore, the Lipschitz constant of this selection is bounded by a certain constant depending only on $m$. The result is joint work with Charles Fefferman.

*Abstract:*

In this talk we present a certain extrapolation technique which we apply to some well-known projection, subgradient projection and other fixed point algorithms. All of them can be considered within the general string averaging framework. The analytical results show that under certain assumptions, the convergence can be linear, which is known to be the case for the extrapolated simultaneous projection method. This is joint work with Christian Bargetz, Victor I. Kolobov and Simeon Reich.

*Abstract:*

I intend to sketch well-known facts about ellipsoids, viewed as a particular case of symmetric convex sets, giving some background on the latter. The ambient spaces will be (finite or infinite dimensional) real linear spaces (some notions not depending on specifying a topology there).

*Abstract:*

Given two disjoint convex polyhedra, we look for a pair of points, one in each polyhedron, attaining the minimum distance between the sets. We propose a process based on projections onto the half-spaces defining the two polyhedra.

*Abstract:*

A well-known result says that the Euclidean unit ball is the unique fixed point of the polarity operator. This result implies that the only norm which can be defined on a finite-dimensional real vector space so that its induced unit ball be equal to the unit ball of the dual (polar) norm is the Euclidean norm. Motivated by these results and by relatively recent results in convex analysis and convex geometry regarding various properties of order reversing operators, we consider, in a real Hilbert space setting, a more general fixed point equation in which the polarity operator is composed with a continuous invertible linear operator. We show that if the linear operator is positive definite, then the considered equation is uniquely solvable by an ellipsoid. Otherwise, the equation can have several (possibly infinitely many) solutions or no solution at all. Our analysis yields a few by-products of possibly independent interest, among them results related to positive definite operators, to coercive bilinear forms and hence to partial differential equations, to infinite- dimensional convex geometry, and to a new class of linear operators (semi-skew operators) which is introduced here. This is joint work with Simeon Reich.

*Abstract:*

The validity, and invalidity, of the Entropy Method in Kac's many-particle model is a prominent problem in the field of Kinetic Theory. At its heart, it is an attempt to find a functional inequality, which is independent of the number of particles in the model, that will demonstrate an exponential rate of convergence to equilibrium. Surprisingly enough, a resolution of this method is still unavailable, and while the master equation for the process is simple, its reliance on the number of particles and the geometry of the appropriate sphere is remarkably strong. It seems that any significant advance in this problem always involves an interdisciplinary approach. In this talk I will present recent work with Eric Carlen and Maria Carvalho, where we have introduced new functional properties, and a notion of chaoticity, with which we have managed to considerably improve what is known about the entropy-entropy production ratio on Kac's sphere. Moreover, with that in hand, I will show how Kac's original hope to deduce a rate of decay for his model's limit equation from the many-particle model itself, is achieved.

*Abstract:*

There are subsets N of R^n for which one can find a real-valued Lipschitz function f defined on the whole of R^n but non-differentiable at every point of N. Of course, by the Rademacher theorem any such set N is Lebesgue null. However, due to a celebrated result of Preiss from 1990 not every Lebesgue null subset of R^n gives rise to such a Lipschitz function f.

In this talk I explain that a sufficient condition on a set N for such f to exist is being locally unrectifiable with respect to curves in a cone of directions. In particular, every purely unrectifiable set U possesses a Lipschitz function non-differentiable on U in the strongest possible sense. I also give an example of a universal differentiability set unrectifiable with respect to a fixed cone of directions, showing that one cannot relax the conditions.

This is joint work with David Preiss.

*Abstract:*

A recent result characterizes the fully order reversing operators acting on the class of lower semicontinuous proper convex functions in a real Banach space as certain linear deformations of the Legendre-Fenchel transform. Motivated by the Hilbert space version of this result and by the well-known result saying that this convex conjugate transform has a unique fixed point (namely, the normalized energy function), we investigate the fixed point equation in which the involved operator is fully order reversing and acts on the above-mentioned class of functions. It turns out that this nonlinear equation is very sensitive to the involved parameters and can have no solution, a unique solution, or infinitely many ones. Our analysis yields a few byproducts, such as results related to positive semi-definite operators and to functional equations and inclusions involving monotone operators. The talk is based on joint work with Alfredo N. Iusem (IMPA) and Simeon Reich (The Technion).

*Abstract:*

We describe the asymptotic behavior of critical points of $\int_\Omega [(1/2)|\nabla u|^2+W(u)/\varepsilon^2]$ when $\varepsilon \to 0$. Here $W$ is a Ginzburg-Landau type potential vanishing on a simple closed curve $\Gamma$. Unlike the case of the standard Ginzburg-Landau potential $W(u)=(1-|u|^2)^2/4$, studied by Bethuel, Brezis and H\'elein, we do not assume any symmetry of $W$ or $\Gamma$. This is a joint work with Petru Mironescu (Lyon I).

*Abstract:*

We establish metric convergence theorems for infinite products of possibly discontinuous operators defined on Hadamard spaces. This is joint work with Zuly Salinas.

*Abstract:*

In the first part of this talk we study sections of B = {x : |x_1| + ... + |x_n| < 1} with (n-1)-dimensional subspaces of R^n and present a new method of determining sections of maximal and minimal (n-1)-dimensional volume, using probabilistic methods. This part is based on joint work with A. Eskenazis and T. Tkocz. In the second part a similar problem for projections is studied using Fourier analytic methods on the discrete cube. This task boils down to the study of the optimal constant in the so-called Khinchine inequality. This part is based on articles of K. Ball and S. Szarek.

*Abstract:*

This talk is devoted to inequalities for best approximations and moduli of smoothness of functions and their derivatives in the spaces $L_p, p > 0.$ Namely, we consider the so-called direct inequalities (upper estimates of a best approximation (modulus of smoothness) of a function via the best approximation (modulus of smoothness) of the derivatives of the function) and the corresponding (weak) inverse inequalities. In the spaces $L_p, p \ge 1,$ both inequalities are well studied. In contrast, in the spaces $L_p, 0 < p < 1,$ there are only some partial positive results related to the inverse inequalities and some examples of functions for which the standard direct inequalities in $L_p, 0 < p < 1,$ are impossible. In my talk, first positive results related to the direct inequalities in the spaces $L_p, 0 < p < 1,$ will be presented. New (weak) inverse inequalities will also be discussed. These results are obtained for the approximation of functions by trigonometric polynomials, algebraic polynomials, and splines, as well as for periodic and non-periodic moduli of smoothness.

*Abstract:*

We propose a methodology for constructing decision rules for integer and continuous decision variables in multiperiod robust linear optimization problems. This type of problem finds application in, for example, inventory management, lot sizing, and manpower management. We show that by iteratively splitting the uncertainty set into subsets, one can differentiate the later-period decisions based on the revealed uncertain parameters. At the same time, the problem's computational complexity stays at the same level as for the static robust problem. This also holds in the nonfixed recourse situation. In the fixed recourse situation our approach can be combined with linear decision rules for the continuous decision variables. We provide theoretical results on how to split the uncertainty set. Based on this theory, we propose several heuristics. Joint work with Dick den Hertog (Tilburg University).

*Abstract:*

Please see event no. 428.

*Abstract:*

In this talk we find the optimal error bound (smallest possible estimate, independent of the starting point) for the linear convergence rate of the simultaneous projection method applied to closed linear subspaces in a real Hilbert space. We achieve this by computing the norm of an error operator which we also express in terms of the Friedrichs number. We compare our estimate with the optimal one provided for the alternating projection method by Kayalar and Weinert (1988). Moreover, we relate our result to the alternating projection formalization of Pierra (1984) in a product space. Finally, we adjust our results to closed affine subspaces and put them in context with recent dichotomy theorems. This is joint work with Simeon Reich.

*Abstract:*

The reality of the zeros of the product and cross-product of Bessel and modified Bessel functions of the first kind is studied. As a consequence, the reality of the zeros of two hypergeometric polynomials is obtained together with the number of the Fourier critical points of the normalized forms of the product and cross-product of Bessel functions. Moreover, the interlacing properties of the real zeros of these products of Bessel functions and their derivatives are also obtained. As an application some geometric properties of the normalized forms of the cross-product and product of Bessel and modified Bessel functions of the first kind are studied. For the cross-product and the product three different kinds of normalization are investigated and for each of the six functions the radii of starlikeness and convexity are precisely determined by using their Hadamard factorization. For these radii of starlikeness and convexity tight lower and upper bounds are given via Euler-Rayleigh inequalities. Necessary and sufficient conditions are also given for the parameters such that the six normalized functions are starlike and convex in the open unit disk. The properties and the characterization of real entire functions from the Laguerre-Polya class via hyperbolic polynomials play an important role. Some open problems are also stated, which may be of interest for further research.

*Abstract:*

Attached.

*Abstract:*

We study convex bi-level optimization problems for which the inner level consists of minimization of the sum of smooth and nonsmooth functions. The outer level aims at minimizing a smooth and strongly convex function over the optimal solution set of the inner problem. We analyze two first order methods and global sublinear rate of convergence of the methods is established in terms of the inner objective function values. The talk is based on two works: one with Amir Beck (Technion) and one with Shimrit Shtern (MIT).

*Abstract:*

Assuming that the absence of perturbations guarantees either weak or strong convergence to a common fixed point, we study the behavior of perturbed products of an infinite family of nonexpansive operators. Our main result indicates that the convergence rate of unperturbed products is essentially preserved in the presence of perturbations. This, in particular, applies to the linear convergence rate of dynamic string averaging projection methods, which we establish here as well. This is joint work with Christian Bargetz and Simeon Reich.

*Abstract:*

We study variational inequalities in a real Hilbert space, which are governed by a strongly monotone and Lipschitz continuous operator $F$ over a closed and convex set $C$. We assume that the set C can be outerly approximated by the fixed point sets of a sequence of certain quasi-nonexpansive operators called cutters. We propose an iterative method the main idea of which is to project at each step onto a particular super-half-space constructed by using the input data. Our approach is based on a method presented by Fukushima in 1986, which has recently been extended by several authors. We establish strong convergence in Hilbert space. To the best of our knowledge, Fukushima's method has so far been considered only in the Euclidean setting with different conditions on $F$. We also provide numerical illustrations of our theoretical results. This talk is based on joint work with Aviv Gibali and Rafal Zalas.

*Abstract:*

In this talk we study new algorithmic structures with Douglas--Rachford (DR) operators to solve convex feasibility problems. We propose to embed the basic two-set-DR algorithmic operator into the string-averaging projections and into the block-iterative projection algorithmic structures, thereby creating new DR algorithmic schemes that include the recently proposed cyclic DR algorithm and the averaged DR algorithm as special cases. We further propose and investigate a new multiple-set-DR algorithmic operator. Convergence of all these algorithmic schemes is studied by using properties of strongly quasi-nonexpansive operators and firmly nonexpansive operators. This is joint work with Yair Censor.

*Abstract:*

The interpretation of Einstein's equations as a geometric flow (the Einstein flow) allows to study the evolution of spacetimes from a dynamical point of view. Two types of initial data are mainly considered: Firstly, asymptotically flat data describing initial states of isolated self-gravitating systems and secondly, data on closed manifolds describing initial states for cosmological spacetimes. Studying the evolution of data under the flow we aim to understand its long-time behavior and the global geometry of its time-development. We are interested in the construction of static solutions (or static up to a time-rescaling) as potential attractors of the flow and their nonlinear stability, completeness and incompleteness properties of spacetimes and singularity formation. We present new methods to construct and study solutions by geometric and analytical tools as well as several results in the directions mentioned above. We consider in particular the case of matter models coupled to the Einstein equations, which turns out to provide several interesting phenomena and new classes of solutions.

*Abstract:*

In this talk we consider consistent convex feasibility problems in a real Hilbert space defined by a finite family of sets $C_i$. In particular, we are interested in the case where $C_i = Fix U_i = {z : p_i(z)=0}$, $U_i$ is a cutter and $p_i$ is a proximity function. Moreover, we make the following state-of-the-art assumption: the computation of $p_i$ is at most as difficult as the evaluation of $U_i$ and this is at most as difficult as projecting onto $C_i$. The considered double-layer fixed point algorithm, for every step $k$, applies two types of controls. The first one - the outer control - is assumed to be almost cyclic. The second one - the inner control - determines the most important sets from those offered by the first one. The selection is made in terms of proximity functions. The convergence results presented in this talk depend on the conditions which first, bind together sets, operators and proximity functions and second, connect inner and outer controls. We focus on weak, strong and linear convergence, and provide some useful estimates for designing stopping rules. The framework presented in this talk covers many known (subgradient) projection algorithms already existing in the literature; for example, those applied with (almost) cyclic, remotest set, most violated constraint, parallel and block iterative controls. This is a joint work with Victor Kolobov and Simeon Reich.

*Abstract:*

The organization and dynamics of the chromatin or DNA in the cell nucleus remains unclear. Two ensembles of data are accessible: many single particle trajectories of a DNA locus and the distribution of polymer loops across cell populations: What can be recovered about the geometrical organization of the DNA from these data? We will present our past efforts to study loop distributions by estimating the eigenvalues of Laplace's equation in high dimensions, when a tubular neighborhood of a sub-manifold is removed using the Chavel-Feldman asymptotic expansion. It is also possible to construct a polymer model with a prescribed anomalous exponent. These results are used to reconstruct the geometrical coarse-grained organization of the chromatin from a million-by-million matrix (Hi-C data) and to predict gene interactions.

*Abstract:*

Motivated by works of Karper, we propose a numerical scheme based on finite differences for the system of compressible Navier-Stokes equations and show its convergence to a weak solution of the problem. The proof follows the analytic proof of the existence of weak solutions to the CNS system developed by Lions and uses discrete results analogous to the more famous continuous ones. We present some difficulties occurring in the discrete case such as, for example, the importance of mixed derivatives which are not natural in the given setting.

*Abstract:*

The exact resolvent inclusion problem has various applications in nonlinear analysis and optimization, such as devising (proximal) algorithmic schemes aiming at minimizing convex functions and finding zeros of nonlinear operators. The inexact version of this problem allows error terms to appear and hence enables one to better deal with noise and computational errors, as well as superiorization. The question of existence and uniqueness of solutions to this problem has not yet been answered in a general setting. We show that if the space is a reflexive Banach space, the inducing function is fully Legendre, and the operator is maximally monotone with zeros, then the problem admits a unique and explicit solution. We use this result to significantly extend the scope of numerous known inexact algorithmic schemes (and corresponding convergence results). In the corresponding papers the question whether there exist sequences satisfying the schemes in the inexact case has been left open. As a byproduct we resolve, under certain assumptions, an open issue raised by Iusem, Pennanen and Svaiter (2003), and show, under simple conditions, the H\"{o}lder continuity of the protoresolvent.

This is joint work with Simeon Reich.

*Abstract:*

We study the structure of approximate optimal trajectories of linear control systems with periodic convex integrands and show that these systems possess a turnpike property. To have this property means, roughly speaking, that the approximate optimal trajectories are determined mainly by the integrand, and are essentially independent of the choice of the time interval and data, except in regions close to the endpoints of the time interval. We also show the stability of the turnpike phenomenon under small perturbations of the integrands and study the structure of approximate optimal trajectories in regions close to the endpoints of the time intervals.

*Abstract:*

A widely used method in the analysis of $C_0$-semigroups is to associate to a semigroup its so-called interpolation and extrapolation spaces. In the case of the shift semigroup acting on $L^{2}(\mathbb{R})$ the resulting chain of spaces consists of the classical Sobolev spaces. In 2013, Sven-Ake Wegner defined the universal interpolation space as the projective limit of the interpolation spaces and the universal extrapolation space as the completion of the inductive limit of the extrapolation spaces provided that this limit is Hausdorff. We use the notion of a dual space with respect to a pivot space to show that in the case of a $C_0$-semigroup on a reflexive Banach space $X$, where the generator $A$ satisfies $A^{-1} \in L(X)$, the universal extrapolation space always exists and the inductive limit of the extrapolation spaces itself is complete. This is joint work with Sven-Ake Wegner.

*Abstract:*

The Legendre transform (LET) is a product of a general duality principle: any smooth curve is, on the one hand, a locus of pairs which satisfy the given equation, and on the other, an envelope of a family of its tangent lines. An application of the LET to a strictly convex and smooth function leads to the Legendre identity (LEID). For strictly convex and three times differentiable functions, the LET leads to the Legendre invariant (LEINV). Although the LET has been known for more than 200 years, both the LEID and the LEINV are critical in modern optimization theory and methods. The purpose of this talk is to show the role the LEID and the LEINV play in both constrained and unconstrained optimization.

*Abstract:*

Let $D(A)$ be the domain of an $m$-accretive operator $A$ on a Banach space $E$. We provide sufficient conditions for the closure of $D(A)$ to be convex and for $D(A)$ to coincide with $E$ itself. Several related results and pertinent examples are also included. This is joint work with Jesus Garcia Falset and Omar Muniz Perez.

*Abstract:*

In this talk we intend to discuss two kinds of optimization problems with averaging of functions on their domains. Such problems are called averaged optimization problems. Necessary optimality conditions are derived and several important applications are considered. They include

1. Optimization of cyclic processes.

2. Generalization of the Pontryagin maximum principle for optimal control problems with terms of different types.

3. Completions of the Filippov problem of determining the sliding velocity for systems with discontinuous right-hand sides in the multidimensional case.

Please note the unusual time!

*Abstract:*

This talk is devoted to semigroups of composition operators and semigroups of holomorphic mappings on the right half-plane. We establish conditions under which these semigroups can be extended in their parameter to a sector given a priori. We show that the size of this sector can be controlled by the image properties of the infinitesimal generator, or, equivalently, by the geometry of the so-called associated planar domain. We also give a complete characterization of all composition operators acting on the Hardy space $H^p$ on the right half-plane. This is joint work with Fiana Jacobzon.

*Abstract:*

In my previous talk (which is not a prerequisite for this one) we have seen that the set of strict contractions on a bounded, closed and convex subset of a Banach space is a small subset of the space of all nonexpansive mappings. In this talk we show that the set of strict contractions on $\rho$-convex or more generally, $\rho$-star-shaped subsets of certain metric spaces $(X,\rho)$ is small in the sense that it is a $\sigma$-porous subset of the space of nonexpansive mappings. The class of metric spaces for which we can prove this result contains the class of hyperbolic spaces. This is joint work with Michael Dymond and Simeon Reich.

*Abstract:*

The notion of an orthogonal polynomial ensemble generalizes many important point processes arising in random matrix theory, probability and combinatorics. Perhaps the most famous example is that of the eigenvalue distributions of unitary invariant ensembles (such as GUE) of random matrix theory. Remarkably, the study of fluctuations of these point processes is intimately connected to the study of Jacobi matrices. This talk will review our recent joint work with Maurice Duits exploiting this connection to obtain central limit theorems for orthogonal polynomial ensembles.

*Abstract:*

In our previous work we used the notion of porosity to show that most of the nonexpansive self-mappings of bounded, closed and convex subsets of a Banach space are contractive and possess a unique fixed point, which is the uniform limit of all iterates. In this work we prove two variants of this result for nonexpansive self-mappings of closed and convex sets in a Banach space which are not necessarily bounded. As a matter of fact, it turns out that our results are true for all complete hyperbolic metric spaces. This is joint work with Alexander J. Zaslavski.

*Abstract:*

Abstract: Given a closed, convex and bounded subset $C$ of a Banach space $X$, we consider the space of all nonexpansive self-mappings of $C$ equipped with the metric of uniform convergence. We show that the subset of strict contractions is small in the sense that it is sigma-porous. In other words, a typical nonexpansive mapping has Lipschitz constant one. In the case where $X$ is a Hilbert space, this result was proved by F. S. De Blasi and J. Myjak in 1989. If time permits, I also plan to discuss possible generalizations of this result to more general metric spaces. This is joint work with Michael Dymond.

*Abstract:*

Given several subspaces $V_1,…,V_N$ in a Hilbert space and a point $x$, there are several algorithms for finding the orthogonal projection of $x$ on the intersection of $V_1,…,V_N$ given the orthogonal projections on each of the subspaces $V_1,…,V_N$.

I will only focus on one of these algorithms called the averaged projection method (which is a variant of the alternating projection method of von Neumann). It was shown that this algorithm either converges “very slowly” or “very quickly” and that one can assure quick convergence by bounding the (Friedrichs) angles between the subspaces (several results of this flavor were given by several authors).

In my talk, I will show how to generalize the notion of the Friedrichs angle for (linear) projections in Banach spaces in order to get a criterion for quick convergence of the averaged projection method in Banach spaces. This result does not require the projections to be of norm 1 (although the norm should be sufficiently close to 1) and therefore gives new results even in the Hilbert space setting for non-orthogonal (linear) projections.

*Abstract:*

We consider a broad class of regularized structured total least squares problems (RSTLS) encompassing many scenarios in image processing. This class of problems results in a nonconvex and often nonsmooth model in large dimension. To tackle this difficult class of problems we introduce a novel algorithm that blends proximal and alternating minimization methods by beneficially exploiting data information and structures inherently present in RSTLS. The proposed algorithm, which can also be applied to more general problems, is proven to globally converge to critical points, and is amenable to efficient and simple computational steps.

*Abstract:*

We use asymptotic centers and their variants to establish fixed point theorems for nonexpansive mappings in uniformly convex Banach spaces, and for holomorphic mappings in the Hilbert ball and its powers. This is joint work with Alexander J. Zaslavski.

*Abstract:*

We study the structure of approximate solutions of autonomous variational problems with a lower semicontinuous extended-valued integrand. In our recent research we showed that approximate solutions are determined mainly by the integrand, and are essentially independent of the choice of the time interval and the data, except in regions close to the endpoints of the time interval. In this talk we study the structure of approximate solutions in regions close to the endpoints of the time interval.

*Abstract:*

One of the true challenges in signal processing is to distinguish between different sources of variability. In this work we consider the case of multiple multimodal sensors measuring the same physical phenomenon, such that the properties of the physical phenomenon are manifested as a hidden common source of variability (which we would like to extract), while each sensor has its own sensor-specific effects. We will address the problem from a manifold learning standpoint and show a method based on alternating products of diffusion operators and local kernels, which extracts the common source of variability from multimodal recordings. The generality of the addressed problem sets the stage for the application of the developed method to many real signal processing problems, where different types of devices are typically used to measure the same activity In particular, we will show an application to sleep stage assessment. We demonstrate that through alternating-diffusion, the sleep information hidden inside multimodal respiratory signals can be better captured compared to single-modal methods.

This is joint work with Roy. R. Lederman.

*Abstract:*

The alternating direction method with multipliers (ADMM) is one of the most powerful and successful methods for solving various convex or nonconvex composite problems that arise in the fields of image and signal processing and machine learning. In the convex setting, numerous convergence results have been established for the ADMM as well as its varieties. However, there is much less study of the convergence properties of the ADMM in the nonconvex framework. In this talk we study the Bregman modification of ADMM (BADMM), which includes the conventional ADMM as a special case and can significantly improve the performance of the algorithm. Under some assumptions, we show that the iterative sequence generated by the BADMM converges to a stationary point of the associated augmented Lagrangian function. The obtained results underline the feasibility of the ADMM in applications in nonconvex settings. [This is a joint work with Fenghui Wang and Zongben Xu.]

Please note unusual day, time and place!

*Abstract:*

Given a finite number of closed and convex subsets of certain non-Hilbert spaces, the intersection of which is nonempty, we prove the convergence, either strong or weak, of methods for finding a point in that intersection. These methods involve infinite products of certain discontinuous operators as well as infinite products of their convex combinations. We study these infinite products on Banach spaces, the Hilbert ball and on CAT(0) spaces.

*Abstract:*

In a recent paper with L. Arosio (accepted in Trans. Amer. Math. Soc.) we introduced a universal way to construct a universal hyperbolic semi-model for every univalent self-map of the unit ball (that is, a holomorphic semiconjugation to a possibly lower dimensional ball which has the property that every other semi-conjugation to a hyperbolic space factorizes through this). This result has been very recently extended to not necessarily univalent mappings by L. Arosio. Let k be the dimension of the base space of the model for a holomorphic self-map f of the unit ball. Then (in a work in progress with L. Arosio) we proved that the map f has an f-absorbing open domain in the ball on which the rank of f is at least k. In case k=the dimension of the ball where f is defined, we proved that for each orbit the map f is eventually univalent on any given Kobayashi ball. These results can be seen as a generalization of some results of Pommerenke for the unit disc. Using such results, we can prove that every holomorphic map of the unit ball commuting with a hyperbolic self-map for which k is maximal, has to be either hyperbolic or has to fix at least a slice containing the Denjoy-Wolff point of f. When k< maximal dimension, there are examples of hyperbolic maps commuting with parabolic maps (in fact, semigroups). The aim of this talk is to explain these results.

Please note the unusual day, time and place!

*Abstract:*

I will define the category of partial differential equations. The objects of this category are partial differential equations (PDE) or systems of such equations, and the morphisms are some special surjective maps from the space of independent and dependent variables of the source equation to the space of independent and dependent variables of the target equation. The definition of the morphisms is dictated by the desire to ensure that the pullback by a morphism of any solution of the target equation is a solution of the source equation. I will illustrate the general definition by some simple examples, namely, the morphisms from first order PDE (in particular, PDE describing holomorphic submanifolds of a complex manifold) and the morphisms from nonlinear heat equations.

*Abstract:*

In this talk we describe the limiting behavior of the Kobayashi distance and use our results to construct families of domains which are used in a fixed point theorem for holomorphic self-mappings of Cartesian products of bounded and convex domains in a complex Banach space. We also obtain a Wolff-Denjoy type theorem. This is joint work with Monika Budzy\'nska and Simeon Reich. Please note unusual day and place!

*Abstract:*

Convex functions which are given as a sum of marginal convex functions have been studied in analysis and optimization for a long time. In my talk I will focus on discrete and continuous gradient flows of such functions and present some of the new results in this area.

*Abstract:*

We consider degenerate second order partial differential equations satisfying Hormander's hypoellipticity condition. This talk will focus on the geometric properties of this family of degenerate PDEs, and on the geometric aspects of the strong maximum principle and Harnack inequalities. Applications of the Harnack inequalities to the study of the asymptotic behavior of the solutions will be discussed.

*Abstract:*

The convex feasibility problem (CFP) is defined as the problem of finding a point lying in the intersection of given closed and convex sets C_{i}, i=1,2,...,m, which are called constraints. A typical example of such a problem is solving a system of linear inequalities. In this talk I will discuss an algorithmic approach to solving the CFP reformulated as the common fixed point problem (CFPP) for quasi-nonexpansive mappings U_{i}, i=1,...,m, satisfying the relation Fix(U_{i})=C_{i}. The solution strategy, roughly speaking, exploits the underlying assumption that the computation of each U_{i} is simple. The presented method, called Modular String Averaging, is based on three simple concepts: relaxation, convex combination and composition of the input operators U_{i}. The algorithm, despite its simple description, covers a wide class of methods commonly used for CFPs and CFPPs. It will be shown that under certain additional assumptions, a similar algorithm can be adapted to variational inequalities. This talk is partially based on my PhD thesis and on some recent results obtained jointly with Simeon Reich.

*Abstract:*

The most famous example in game theory shows that individual rationality can be incompatible with collective welfare. I.e, a Nash Equilibrium can be inefficient. In this talk I will discuss the issue of genericity of inefficiency of Nash equilibria. In the case of two players the result I present is particularly neat: a fully mixed strong equilibrium can exist only if the game is strictly competitive.

*Abstract:*

Many problems in science and engineering involve, as part of their solution process, the consideration of a minimization of a composite function F=f+g where f is smooth, g possibly not, and both are convex.The talk will discuss, in generalized settings, two proximal forward-backward algorithms aiming at solving this task. The first is FISTA, a popular accelerated method suggested by Beck-Teboulle. We consider it in Hilbert space and allow error terms which satisfy a certain decay rate. The notion of inexactness we discuss seems to be simpler than the ones discussed in related works but, interestingly, very similar decay rates of the error terms yield very similar non-asymptotic convergence rates (in the function values). Our derivation also sheds some light on the somewhat mysterious origin of some relevant parameters. In the the second method, which is non-accelerated, the setting is closed and convex subsets of reflexive Banach spaces where the proximal operation is based on a (strongly convex) Bregman divergence. Now, in contrast to previous works, the gradient of f may not be globally Lipschitz continuous. Under certain assumptions a non-asymptotic rate of convergence is established, as well as weak convergence of the whole sequence.This is a joint work with Alvaro De Pierro.

*Abstract:*

The numerical range of holomorphic mappings arises in many aspects of nonlinear analysis, finite and infinite dimensional holomorphy, and complex dynamical systems. In particular, this notion plays a crucial role in establishing exponential and product formulas for semigroups of holomorphic mappings, the study of flow invariance and range conditions, geometric function theory in finite and infinite dimensional Banach spaces, and in the study of complete and semi-complete vector fields and their applications to starlike and spirallike mappings, and to Bloch (univalence) radii for locally biholomorphic mappings.In this talk, which is based on joint work with Filippo Bracci, Marina Levenshtein and Simeon Reich, we establish lower and upper bounds for the numerical range of holomorphic mappings in Banach spaces. In addition, we study and discuss some geometric and quantitative analytic aspects of fixed point theory, nonlinear resolvents of holomorphic mappings, Bloch radii, as well as radii of starlikeness and spirallikeness

*Abstract:*

We design a bounded feedback control steering a controllable linear system to equilibrium in finite time. The procedure amounts to solving a few linear-algebraic problems, including linear matrix inequalities. We solve these problems in an efficient way. The resulting steering time to equilibrium is of the same order of magnitude as the minimal one.

*Abstract:*

If one seeks an associative algebra which corresponds canonically to a Euclidean space E (or to any vector space with a quadratic form Q) - canonically means that we refrain from "choosing", say a basis/coordinate system - an option is the Clifford algebra of the space, defined as the associative algebra generated by E, with the relations that the square in the algebra of each vector v in E equals Q(v)1. This algebra contains a plethora of interesting members and structures. Focusing mainly on the Euclidean 4-space, we shall describe its basics and try to stroll in its garden and pick some flowers. In particular, we shall encounter objects mentioned in Maria Elena Luna-Elizarraras' lecture: the 2-dim extension of the reals by a k satisfying k^{2}=1, and bi-quaternions

*Abstract:*

If one seeks an associative algebra which corresponds canonically to a Euclidean space E (or to any vector space with a quadratic form Q) - canonically means that we refrain from "choosing", say a basis/coordinate system - an option is the Clifford algebra of the space, defined as the associative algebra generated by E, with the relations that the square in the algebra of each vector v in E equals Q(v)1. This algebra contains a plethora of interesting members and structures. Focusing mainly on the Euclidean 4-space, we shall describe its basics and try to stroll in its garden and pick some flowers. In particular, we shall encounter objects mentioned in Maria Elena Luna-Elizarraras' lecture: the 2-dim extension of the reals by a k satisfying k^{2}=1, and bi-quaternions.

*Abstract:*

We consider the projected gradient (PG) method for nonnegative least squares (NNLS). The NNLS problem plays a key role in statistical learning theory in general and in Support Vector Machines (SVM) in particular. In contrast to active set and interior point methods, which for a long time were the main tools for solving NNLS, the PG does not require solving at each step a linear system of equations. It rather requires matrix by vector multiplication as the main operation per step. Therefore the critical issue is the convergence rate of the PG methods. We established convergence rates and complexity bounds for PG methods under various assumptions on the input data.

*Abstract:*

We focus on convergence of numerical schemes for non-smooth non-convex optimization in finite dimension. In this realm, most results are given for "prox-friendly" data: the non-smooth part is simple enough to be handled through efficiently computable operators. This excludes many applications for which dedicated methods have been proposed. We focus on Sequential Quadratic Programming ideas (SQP) for general Non-Linear Programs (NLP). Despite their widespread usage, these methods lack satisfactory convergence analysis. This work constitutes a step toward obtaining such theoretical guarantees.The core algorithmic feature is the sequential minimization of tractable "tangent" upper-approximation models. This is typical of the SQP framework. Under semialgebraicity of the data, we propose an original convergence analysis for such processes. This is applied to analyze asymptotic properties of the Moving Ball Method (Auslender-Shefi-Teboulle 2010) and the Extended Sequential Quadratic Method (Auslender 2013). This work was conducted in collaboration with Jerome Bolte (Toulouse 1, France). The presentation is based on the preprint http://arxiv.org/pdf/1409.8147.pdf.

*Abstract:*

Numerical optimization algorithms are increasingly an amalgam of iterative algorithms within iterative algorithms. The theory for convergence and complexity of these schemes is often presented in the ideal world of exact arithmetic and exact evaluation of functions. When experimental results from implementations are presented two obvious issues are often swept under the rug: how do you decide to stop the inner iteration, and what is the effect of inexact evaluation of the operators on overall convergence? Put more bluntly, what relation, if any, does an actual numerical implementation have to the theory? We attempt to tackle these questions through the development and application of the theory of regularity, not only of the underlying functions and operators, but of the set of solutions themselves. Convexity is but one type of regularity which the theory encompasses, but in nonconvex settings the theory is local. We have focused in particular on local linear convergence as a way to provide meaningful stopping with reasonable error bounds. We survey progress over the last several years on sufficient conditions for local linear convergence and discuss challenges and prospects for further progress.

*Abstract:*

In large-scale streaming models, sketching is a function that reduces data size but preserves a certain property to enable queries. In this talk, which is based on a talk by Edo Liberty at Yahoo!, we present the frequent directions algorithm for sketching a large data matrix A while preserving A^{T}A

*Abstract:*

Riesz products have been popular among old-school harmonic analysts. (For good reasons!) I will discuss their constructions and features, mainly in the dyadic group setting. I will then construct amalgams of Riesz products, and use these to derive an upgrade of Littlewood's classical 4/3-inequality.

*Abstract:*

The global structure of solutions to the Einstein equations can be analyzed by studying the evolution of geometric quantities under the Einstein flow. An approach to the general problem of understanding the global geometry of space time is the non-linear stability problem, which considers initial data close to explicitly known solutions and analyzes their future development. The stability question is essential in Mathematical Relativity and is only understood for a few explicit solutions. In this talk we present results on the stability of solutions to the Einstein equations in 2+1 dimensions in the presence of Vlasov matter without symmetry assumptions. We discuss a technique of constructing geometric energies for distribution functions of this type of matter and how these are used to provestability. In addition, we construct future complete and stable solutions to the Einstein flow on the 2-sphere - a result which is shown to be an exclusive feature of Vlasov matter in 2+1 dimensions.

*Abstract:*

We study existence and turnpike properties of approximate solutions for a class of dynamic discrete-time two-player zero-sum games without using convexity-concavity assumptions. We describe the structure of approximate solutions which is independent of the length of the interval for all sufficiently large intervals and show that approximate solutions are determined mainly by the objective function, and are essentially independent of the choice of interval and endpoint conditions.

*Abstract:*

We show that solutions of certain second order elliptic differential equations cannot vanish on the boundary of arbitrarily small domains. Our first example is the Schrodinger equation -Delta u = V(x) u on a bounded domain D of R^{n}. We show that the measure of D is bounded above by a constant that depends on the L^{p} norm of V. If D is a slab of R^{n}, we consider the equation -Delta u = (V(x) + s) u, with V bounded and compactly supported in D. We show that ||V|| is bounded above by a constant that depends on the support of V and the distance of s and the integers. Most of our estimates are optimal in some sense. This is a joint work with J. Edward, S. Hudson, M. Leckband and X. Li from the Florida International University.

*Abstract:*

In this talk we discuss certain basic problems concerning the family of univalent harmonic mappings in the plane. In particular, our emphasis will be on old and new methods for constructing univalent harmonic mappings with tools from the theory of conformal mappings. At the end, a few open problems will be discussed.

*Abstract:*

Classical primal-dual based methods have recently attracted a revived interest for solving structured convex minimization problems arising in signal/image processing and machine learning. In this talk, we focus on the interplay between optimization problems, their saddle point representation, and global rate of convergence analysis of these methods. In particular, we introduce a new algorithm for a class of saddle point problems which allows to efficiently address an important class of convex models. We prove its global rate of convergence. Numerical examples illustrate its relevance and performance.This is joint work with Yoel Drori and Marc Teboulle.

*Abstract:*

It follows from Banach's fixed point theorem that every nonexpansive self-mapping of a bounded, closed and convex set in a Banach space has approximate fixed points. This is no longer true, in general, if the set is unbounded. Nevertheless, we show that there exists an open and everywhere dense set in the space of all nonexpansive self-mappings of any closed and convex (not necessarily bounded) set in a Banach space (endowed with the natural metric of uniform convergence on bounded subsets) such that all its elements have approximate fixed points. A corresponding result for nonexpansive set-valued mappings will also be presented.This is joint work with Alexander J. Zaslavski.

*Abstract:*

The nonlocal isoperimetric problem arises as a limit of the Ohta-Kawasaki model for diblock copolymers. As a variational problem, it takes the form of a competition between a local term favoring low surface area and a nonlocal term favoring high oscillation. In this talk I will survey some of the previous activity on this problem (of which there is a lot) and then focus on a recent result by Massimiliano Morini and me in the setting of thin domains, where we can identify the global minimizer for all values of the coefficient of nonlocality

*Abstract:*

A design centering problem consists in maximizing some measure function of a parametrized body while it is inscribed in a container set.An example is the lapidary cutting problem which deals with the maximization of the volume of a faceted colored gemstone which is cut from an irregularly shaped rough stone.The Fraunhofer ITWM developed a mathematical and a software tool for solving the problem with high-precision cuttings and maximal volume yield.This tool is used in practice in automated gemstone production. In this talk I will explain the mathematical formulation and techniques for solving this problem.

*Abstract:*

We establish three theorems which show that most of the bounded holomorphic self-mappings of a star-shaped domain in a complex Banach space map it strictly inside itself. According to the Earle-Hamilton fixed point theorem, each such mapping has a unique fixed point. This is joint work with Alexander J. Zaslavski.