Search

The surface area of a set which is only observed as a binary pixel image is often estimated by a weighted sum of pixel configurations counts. In this paper we examine these estimators in a design based setting—we assume that the observed set is shifted uniformly randomly. Bounds for the difference between the essential supremum and the essential infimum of such an estimator are derived, which imply that the variance is in $$O(t^2)$$ O ( t 2 ) as the lattice distance t tends to zero. In particular, it is asymptotically neglectable compared to the bias. A simulation study shows that the theoretically derived convergence order is optimal in general, but further improvements are possible in special cases.

Let Lip([0,1]) be the Banach space of all Lipschitz complex-valued functions f on [0,1], equipped with one of the norms: fσ=|f(0)|+f′L∞ or fm=max|f(0)|,f′L∞, where ·L∞ denotes the essential supremum norm. It is known that the surjective linear isometries of such spaces are integral operators, rather than the more familiar weighted composition operators. In this paper, we describe the topological reflexive closure of the isometry group of Lip([0,1]). Namely, we prove that every approximate local isometry of Lip([0,1]) can be represented as a sum of an elementary weighted composition operator and an integral operator. This description allows us to establish the algebraic reflexivity of the sets of surjective linear isometries, isometric reflections, and generalized bi-circular projections of Lip([0,1]). Additionally, some complete characterizations of such reflections and projections are stated.

Inspired by biological phenomena with effects of switching off (maybe just for a while), we investigate non-autonomous reaction-diffusion inclusions whose multi-valued reaction term may depend on the essential supremum over a time interval in the recent past (but) pointwise in space. The focus is on sufficient conditions for the existence of pullback attractors. If the multi-valued reaction term satisfies a form of inclusion principle standard tools for non-autonomous dynamical systems in metric spaces can be applied and provide new results (even) for infinite time intervals of delay. More challenging is the case without assuming such a monotonicity assumption. Then we consider the parabolic differential inclusion with the time interval of delay depending on space and extend the approaches of norm-to-weak semigroups to a purely metric setting. This provides completely new tools for proving pullback attractors of non-autonomous dynamical systems in metric spaces.

Let $\gamma \in (0,2)$, let $h$ be the planar Gaussian free field, and consider the $\gamma$-Liouville quantum gravity (LQG) metric associated with $h$. We show that the essential supremum of the Hausdorff dimension of the boundary of a $\gamma$-LQG metric ball with respect to the Euclidean (resp. $\gamma$-LQG) metric is $2 - \frac{\gamma}{d_\gamma}\left(\frac{2}{\gamma} + \frac{\gamma}{2} \right) + \frac{\gamma^2}{2d_\gamma^2}$ (resp. $d_\gamma-1$), where $d_\gamma$ is the Hausdorff dimension of the whole plane with respect to the $\gamma$-LQG metric. For $\gamma = \sqrt{8/3}$, in which case $d_{\sqrt{8/3}}=4$, we get that the essential supremum of Euclidean (resp. $\sqrt{8/3}$-LQG) dimension of a $\sqrt{8/3}$-LQG ball boundary is $5/4$ (resp. $3$). We also compute the essential suprema of the Euclidean and $\gamma$-LQG Hausdorff dimensions of the intersection of a $\gamma$-LQG ball boundary with the set of metric $\alpha$-thick points of the field $h$ for each $\alpha\in \mathbb R$. Our results show that the set of $\gamma/d_\gamma$-thick points on the ball boundary has full Euclidean dimension and the set of $\gamma$-thick points on the ball boundary has full $\gamma$-LQG dimension., Comment: 63 pages, 5 figures; final version, to appear in CMP

This paper first studies the measure theoretic pressure of measures that are not necessarily ergodic. We define the measure theoretic pressure of an invariant measure (not necessarily ergodic) via the Carath\'{e}odory-Pesin structure described in \cite{Pes97}, and show that this quantity is equal to the essential supremum of the free energy of the measures in an ergodic decomposition. To the best of our knowledge, this formula is new even for entropy. Meanwhile, we define the measure theoretic pressure in another way by using separated sets, it is showed that this quantity is exactly the free energy if the measure is ergodic. Particularly, if the dynamical system satisfies the uniform separation condition and the ergodic measures are entropy dense, this quantity is still equal to the the free energy even if the measure is non-ergodic. As an application of the main result, we find that the Hausdorff dimension of an invariant measure supported on an average conformal repeller is given by the zero of the measure theoretic pressure of this measure. Furthermore, if a hyperbolic diffeomorphism is average conformal and volume-preserving, the Hausdorff dimension of any invariant measure on the hyperbolic set is equal to the sum of the zeros of measure theoretic pressure restricted to stable and unstable directions., Comment: 23 pages

We give a simple necessary and sufficient condition for the existence of star supremum for two arbitrary operators on a Hilbert space. It is shown that the results of Hartwig (1979) [15] , and Janowitz (1983) [18] , when applied to the ring of bounded operators on a Hilbert space, require much simpler conditions. We also give a complete answer to a question stated by Hartwig and Drazin (1982) in [16] , regarding the extremal values of range and null-space of the star infimum.

In this work we prove continuity of solutions with respect to initial conditions and couple parameters and we prove joint upper semicontinuity of a family of global attractors for the problem \begin{eqnarray} &\frac{\partial u_{s}}{\partial t}(t)-\textrm{div}(D_s|\nabla u_{s}|^{p_s(x)-2}\nabla u_{s})+|u_s|^{p_s(x)-2}u_s=B(u_{s}(t)),\;\; t>0,\\ &u_{s}(0)=u_{0s}, \end{eqnarray} under homogeneous Neumann boundary conditions, $u_{0s}\in H:=L^2(\Omega),$ $\Omega\subset\mathbb{R}^n$ ($n\geq 1$) is a smooth bounded domain, $B:H\rightarrow H$ is a globally Lipschitz map with Lipschitz constant $L\geq 0$, $D_s\in[1,\infty)$, $p_s(\cdot)\in C(\bar{\Omega})$, $p_s^-:=\textrm{ess inf}\;p_s\geq p,$ $p_s^+:=\textrm{ess sup}\;p_s\leq a,$ for all $s\in \mathbb{N},$ when $p_s(\cdot)\rightarrow p$ in $L^\infty(\Omega)$ and $D_s\rightarrow\infty$ as $s\rightarrow\infty,$ with $a,p>2$ positive constants.

This paper deals with the following variational problem with variable exponents { − div ( ( 1 + | ∇ u | p ( x ) 1 + | ∇ u | 2 p ( x ) ) | ∇ u | p ( x ) − 2 ∇ u ) = f ( x , u ) , in Ω , u = 0 , on ∂ Ω , where Ω ⊆ R N ( N ≥ 2 ) be a bounded domain with the smooth boundary ∂Ω, p : Ω ‾ → R is a Lipschitz continuous function with 1 p − : = ess inf x ∈ Ω ⁡ p ( x ) ≤ p ( x ) ≤ p + : = ess sup x ∈ Ω ⁡ p ( x ) N and f ∈ C ( Ω × R , R ) is superlinear but does not satisfy the usual Ambrosetti-Rabinowitz type condition. Under three different superlinear conditions on f at infinity, we prove that the above equation has at least a nontrivial solution. Moreover, the existence of infinite many solutions is proved for odd nonlinearity. Especially, some new tricks are introduced to show the boundedness of Cerami sequences.

For several decades, the no-arbitrage (NA) condition and the martingale measures have played a major role in the financial asset's pricing theory. We propose a new approach for estimating the super-replication cost based on convex duality instead of martingale measures duality: Our prices will be expressed using Fenchel conjugate and bi-conjugate. The super-hedging problem leads endogenously to a weak condition of NA called Absence of Immediate Profit (AIP). We propose several characterizations of AIP and study the relation with the classical notions of no-arbitrage. We also give some promising numerical illustrations., 33 pages 6 figures

In this paper we show that if the supremal functional $$F(V,B)= \mathop{\rm ess sup} \limits_{x \in B} f(x,DV (x))$$ is sequentially weak* lower semicontinuous on $${W^{1,\infty}(B, \mathbb{R}^d)}$$ for every open set $${B \subseteq \Omega}$$ (where $${\Omega}$$ is a fixed open set of $${\mathbb{R}^N}$$ ), then $${f(x,\cdot)}$$ is rank-one level convex for a.e $${x \in \Omega}$$ . Next, we provide an example of a weak Morrey quasiconvex function which is not strong Morrey quasiconvex. Finally we discuss the L p -approximation of a supremal functional F via $${\Gamma}$$ -convergence when f is a non-negative and coercive Caratheodory function.

We provide a representation for strong-weak continuous dynamic risk measures from Lp into Lpt spaces where these spaces are equipped respectively with strong and weak topologies and p is a finite number strictly larger than one. Conversely, we show that any such representation that admits a compact (with respect to the product of weak topologies) sub-differential generates a dynamic risk measure that is strong--weak continuous. Furthermore, we investigate sufficient conditions on the sub-differential for which the essential supremum of the representation is attained. Finally, the main purpose is to show that any convex dynamic risk measure that is strong-weak continuous can be approximated by a sequence of convex dynamic risk measures which are strong--weak continuous and admit compact sub-differentials with respect to the product of weak topologies. Throughout the arguments, no conditional translation invariance or monotonicity assumptions are applied.

Let $\mathcal{M}(n,D)$ be the space of closed $n$-dimensional Riemannian manifolds $(M,g)$ with $diam(M) \leq D$ and $| \sec^M | \leq 1$. In this paper we consider sequences $(M_i,g_i)$ in $\mathcal{M}(n,D)$ converging in the Gromov-Hausdorff topology to a compact metric space $Y$. We show on the one hand that the limit space of this sequence has at most codimension $1$ if there is a positive number $r$ such that the quotient $\frac{vol(B^{M_i}_r(x))}{inj^{M_i}(x)}$ can be uniformly bounded from below by a positive constant $C(n,r,Y)$ for all points $x \in M_i$. On the other hand, we show that if the limit space has at most codimension $1$ then for all positive $r$ there is a positive constant $C(n,r,Y)$ bounding the quotient $\frac{vol(B^{M_i}_r(x))}{inj^{M_i}(x)}$ uniformly from below for all $x \in M_i$. The proof uses results about the structure of collapse in $\mathcal{M}(n,D)$ by Cheeger, Fukaya and Gromov. In addition, we derive, for a submersion $M \rightarrow Y$ with uniformly bounded fundamental tensors, an upper bound on the injectivity radius of the fiber $F_p$, with $p \in Y$, which is proportional to the injectivity radius of $M$ at some $x \in F_p$, if the injectivity at $x$ is sufficiently small relative to the injectivity radius of $Y$. As a conclusion, we derive a uniform lower bound on the volume and a bound on the essential supremum of the sectional curvature for the closure of the space consisting of all manifolds in $\mathcal{M}(n,D)$ with $C \leq \min_{x \in M}\frac{vol(B^{M}_r(x))}{inj^{M}(x)}$ for fixed positive numbers $r$ and $C$., Comments are welcome

In this note, we study the sharp weighted estimate involving one supremum. In particular, we give a positive answer to an open question raised by Lerner and Moen \cite{LM}. We also extend the result to rough homogeneous singular integral operators., Comment: 5 pages, typos corrected. Accepted for publication in C. R. Acad. Sci. Paris, Ser. I

We prove that if f : ℝN → ℝ̄ is quasiconvex and U ⊂ ℝN is open in the density topology, then supU ƒ = ess supU f ; while infU ƒ = ess supU ƒ if and only if the equality holds when U = RN: The first (second) property is typical of lsc (usc) functions, and, even when U is an ordinary open subset, there seems to be no record that they both hold for all quasiconvex functions.This property ensures that the pointwise extrema of f on any nonempty density open subset can be arbitrarily closely approximated by values of ƒ achieved on “large” subsets, which may be of relevance in a variety of situations. To support this claim, we use it to characterize the common points of continuity, or approximate continuity, of two quasiconvex functions that coincide away from a set of measure zero.

In the first part of the paper, we study concepts of supremum and maximum as subsets of a topological space X endowed by preference relations. Several rather general existence theorems are obtained for the case where the preferences are defined by countable semicontinuous multi-utility representations. In the second part of the paper, we consider partial orders and preference relations “lifted” from a metric separable space X endowed by a random preference relation to the space L0(X) of X-valued random variables. We provide an example of application of the notion of essential maximum to the problem of the minimal portfolio super-replicating an American-type contingent claim under transaction costs.

Inspired by the theory of financial markets with transaction costs, we study a concept of essential supremum in the framework where a random partial order in R d is lifted to the space L 0 ( R d ) of d -dimensional random variables. In contrast to the classical definition, we define the essential supremum as a subset of random variables satisfying some natural properties. Applications of the introduced notion to a hedging problem under transaction costs and set-valued dynamic risk measures are given.

The paper is devoted to the subdifferential study and applications of the supremum of uniformly Lipschitzian functions over arbitrary index sets with no topology. Based on advanced techniques of variational analysis, we evaluate major subdifferentials of the supremum functions in the general framework of Asplund (in particular, reflexive) spaces with no convexity or relaxation assumptions. The results obtained are applied to deriving new necessary optimality conditions for nonsmooth and nonconvex problems of semi-infinite and infinite programming.

In this paper, we present the full characterization of the iterativity condition for the mean-value principle under the cumulative prospect theory. It turns out that the premium principle is iterative for exactly six pairs of probability distortion functions. Some of the corresponding premium principles are the classical mean-value principle, essential infimum or essential supremum of the random loss. Moreover, from the proof of the main theorem of this paper, it follows that the iterativity of the mean-value principle is equivalent to the iterativity of the generalized Choquet integral.

We study a class of branching processes in which a population consists of immortal individuals equipped with a fitness value. Individuals produce offspring with a rate given by their fitness, and offspring may either belong to the same family, sharing the fitness of their parent, or be founders of new families, with a fitness sampled from a fitness distribution. Examples that can be embedded in this class are stochastic house-of-cards models, urn models with reinforcement, and the preferential attachment tree of Bianconi and Barabasi. Our focus is on the case when the fitness distribution has bounded support and regularly varying tail at the essential supremum. In this case there exists a condensation phase, in which asymptotically a proportion of mass in the empirical fitness distribution of the overall population condenses in the maximal fitness value. Our main results describe the asymptotic behaviour of the size and fitness of the largest family at a given time. In particular, we show that as time goes to infinity the size of the largest family is always negligible compared to the overall population size. This implies that condensation, when it arises, is non-extensive and emerges as a collective effort of several families none of which can create a condensate on its own. Our result disproves claims made in the physics literature in the context of preferential attachment trees., Comment: 1 figure

We review subbadditivity properties of Shannon entropy, in particular, from the Shearer's inequality we derive the "infimum rule" for actions of amenable groups. We briefly discuss applicability of the "infimum formula" to actions of other groups. Then we pass to topological entropy of a cover. We prove Shearer's inequality for disjoint covers and give counterexamples otherwise. We also prove that, for actions of amenable groups, the supremum over all open covers of the "infimum fomula" gives correct value of topological entropy.

The purpose of this paper is to present two algorithms for global minimization of multivariate polynomials. For a multivariate real polynomial f, we provide an effective algorithm for deciding whether or not the infimum of f is finite. In the case of f having a finite infimum, the infimum of f can be accurately coded as (h; a, b), where h is a real polynomial in one variable, a and b is two rational numbers with a < b, and (h, a, b) stands for the only real root of h in the open interval ]a, b[. Moreover, another algorithm is provided to decide whether or not the infimum of f is attained when the infimum of f is finite. Our methods are called “nonstandard”, because an infinitesimal element is introduced in our arguments.

Let $L$ be a linear space of real bounded random variables on the probability space $(\Omega,\mathcal{A},P_0)$. There is a finitely additive probability $P$ on $\mathcal{A}$, such that $P\sim P_0$ and $E_P(X)=0$ for all $X\in L$, if and only if $c\,E_Q(X)\leq\text{ess sup}(-X)$, $X\in L$, for some constant $c>0$ and (countably additive) probability $Q$ on $\mathcal{A}$ such that $Q\sim P_0$. A necessary condition for such a $P$ to exist is $\bar{L-L_\infty^+}\,\cap L_\infty^+=\{0\}$, where the closure is in the norm-topology. If $P_0$ is atomic, the condition is sufficient as well. In addition, there is a finitely additive probability $P$ on $\mathcal{A}$, such that $P\ll P_0$ and $E_P(X)=0$ for all $X\in L$, if and only if $\text{ess sup}(X)\geq 0$ for all $X\in L$.

The complexity of infinite words is considered from the point of view of a transformation with a Mealy machine that is the simplest model of a finite automaton transducer. We are mostly interested in algebraic properties of the underlying partially ordered set. Results considered with the existence of supremum, infimum, antichains, chains and density aspects are investigated.

If X is a spectrally positive stable process of index α ∈ (1, 2) whose Levy measure has density on (0,∞), and it is known that as x → ∞. It is also known that S 1 has a continuous density, s say. The point of this note is to show that as x → ∞.

The present article is devoted to the study of a constrained weighted total variation minimization problem, which may be viewed as a relaxation of a generalized Cheeger problem and is motivated by landslide modeling. Using the fact that the set of minimizers is invariant by a wide class of monotone transformations, we prove that level sets of minimizers are generalized Cheeger sets and obtain qualitative properties of the minimizers: they are all bounded and all achieve their essential supremum on a set of positive measure.

For conformal hyperbolic flows, we establish explicit formulas for the Hausdorff dimension and for the pointwise dimension of an arbitrary invariant measure. We emphasize that these measures are not necessarily ergodic. The formula for the pointwise dimension is expressed in terms of the local entropy and of the Lyapunov exponents. We note that this formula was obtained before only in the special case of (ergodic) equilibrium measures, and these always possess a local product structure (which is not the case for arbitrary invariant measures). The formula for the pointwise dimension allows us to show that the Hausdorff dimension of a (nonergodic) invariant measure is equal to the essential supremum of the Hausdorff dimension of the measures in an ergodic decomposition.

Let $W$ denote $d$-dimensional Brownian motion. We find an explicit formula for the essential supremum of Hausdorff dimension of $W(E)\cap F$, where $E\subset(0,\infty)$ and $F\subset \mathbf {R}^d$ are arbitrary nonrandom compact sets. Our formula is related intimately to the thermal capacity of Watson [Proc. Lond. Math. Soc. (3) 37 (1978) 342-362]. We prove also that when $d\ge2$, our formula can be described in terms of the Hausdorff dimension of $E\times F$, where $E\times F$ is viewed as a subspace of space time., Published in at http://dx.doi.org/10.1214/14-AOP910 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)

Given a Carnot-Caratheodory metric space (R", d X ) generated by vector fields {X i } m i=1 satisfying Hormander's condition, we prove in Theorem A that any absolute minimizer u ∈ W 1,∞ X (Ω) to F(v, Q) = ess sup x∈Ω f(x, Xv(x)) is a viscosity solution to the Aronsson equation - Σ X i (f(x, X u (x)))f pi (x, X u (x)) = 0, in Ω, under suitable conditions on f. In particular, any AMLE is a viscosity solution to the subelliptic oo-Laplacian equation Δ (X) ∞ u:= - Σ X i uX j uX i X j u = 0, in Ω. If the Carnot-Caratheodory space is a Carnot group G and f is independent of the x-variable, we establish in Theorem C the uniqueness of viscosity solutions to the Aronsson equation A(Xu,(D 2 u)*):= - Σ f pi (Xu)f pj (Xu)X i X j u = 0, in Ω, u = Φ, on ∂Ω, under suitable conditions on f. As a consequence, the uniqueness of both AMLE and viscosity solutions to the subelliptic oo-Laplacian equation is established on any Carnot group G.

In this paper it is shown that supremum and infimum of fuzzy numbers can be characterized by membership function method via endograph metric more directly than by using the levelwise metric, it is proved that the endograph metric is approximative with respect to orders on fuzzy number spaces, also, the endograph metric is computable. The result in this paper shows that the fuzzy-number-valued integrals of the kind based on concept like Riemann sum in calculus are computable.

In this paper we study the lower semicontinuity problem for a supremal functional of the form F(u,Ω) = ess sup f(x, u(x),Du(x)) with respect to the strong convergence in L ∞ (Ω), furnishing a comparison with the analogoustheory developed by Serrin for integrals. A sort of Mazur's lemma for gradients of uniformly converging sequences is proved.

We examine the dimensions of the intersection of a subset $E$ of an $m$-ary Cantor space $\mathcal{C}^m$ with the image of a subset $F$ under a random isometry with respect to a natural metric. We obtain almost sure upper bounds for the Hausdorff and upper box-counting dimensions of the intersection, and a lower bound for the essential supremum of the Hausdorff dimension. The dimensions of the intersections are typically $\max\{\dim E +\dim F -\dim \mathcal{C}^m, 0\}$, akin to other codimension theorems. The upper estimates come from the expected sizes of coverings, whilst the lower estimate is more intricate, using martingales to define a random measure on the intersection to facilitate a potential theoretic argument., Comment: Accepted version, Proc. Amer. Math. Soc

We characterize the convergence of fuzzy sets in the supremum metric given by the supremum of the Hausdorff distances of the alpha-cuts of the fuzzy sets. We do it by dividing this metric into its lower and upper quasi-pseudometric parts. This characterization is given in the more general context with no assumption on the fuzzy sets. Furthermore, motivated from the theory of Convex Analysis, we also provide some results about the behavior of the convergence in the supremum metric with respect to maximizers. (C) 2014 Elsevier B.V. All rights reserved., The second and third authors thank the support of the Ministry of Economy and Competitiveness of Spain under grant MTM2012-37894-C02-01.

We study the lower semicontinuity properties and existence of a minimizer of the functional F( u)= ess sup x∈� f (x,u(x),Du(x)) on W 1,∞ (� ; R m ). We introduce the notions of Morrey quasiconvexity, polyquasiconvexity, and rank-one quasiconvexity, all stemming from the notion of quasiconvexity (= convex level sets) of f in the last variable. We also formally derive the Aronsson-Euler equation for such problems.  2001 Editions scientifiques et medicales Elsevier SAS

We consider a problem of minimization of convex function f ( x ) over the convex region R where the objective function and the feasible region have a common direction of recession. In cases when one of these directions is not in the constancy space of the objective function, then the minimal solution is not achieved even if the function f ( x ) is bounded below over the region R . Many algorithms, if applied to this class of programmes, do not guarantee convergence to the global infimum. Our approach to this problem leads to derivation of the equation of the feasible parametrized curve C ( t ), such that the infimum of the logarithmic penalty function along this curve is equal to the global infimum of the objective function over the region R . We show that if all functions defining the program are analytic, then C ( t ) is also an analytic function. The equation of the curve can be successfully used to determine the global infimum (in particular, unboundedness) of the convex constrained programmes in cases when the application of classical methods, such as the steepest descent method, fails to converge to the global infimum.

We give some sufficient conditions of deterioration of smoothness under the operation of convolution. We show that the convolution of two probability densities which are restrictions to R of entire functions can possess infinite essential supremum on each interval. © 1998 Academic Press.

In this paper, we prove that the bounded set of fuzzy numbers must exist supremum and infimum and give the concrete representation of supremum and infimum. As an application, we obtain that the continuous fuzzy-valued function on a closed interval exists supremum and infimum and give the precise representation. We also show that the bounded fuzzy-valued function on a closed interval can define the lower and upper sums and the lower and upper integrals of Riemann and Riemann–Stieltjes by the usual way.

An absolutely convergent double series representation for the density of the supremum of $\alpha$-stable Levy process is given in [3, Theorem 2] for almost all irrational $\alpha$. This result cannot be made stronger in the following sense: the series does not converge absolutely when $\alpha$ belongs to a certain subset of irrational numbers of Lebesgue measure zero (see [6, Theorem 2]). Our main result in this note shows that for every irrational $\alpha$ there is a way to rearrange the terms of the double series, so that it converges to the density of the supremum. We show how one can establish this stronger result by introducing a simple yet non-trivial modification in the original proof of [3,Theorem 2]., Comment: 6 pages

Possibility measures have been conceived by Zadeh [1], and developed by, e. g., Dubois, Prade and others, as very simple, if compared with, e.g., probability theory, but still nontrivial and reasonable uncertainty measures. The relatively poor descriptional and operational abilities of possibility measures seem to be closely related to the standard linear ordering relation and the corresponding supremum and infimum operations defined over the unit interval of real numbers. Having discussed, very briefly, the possibilities how to overcome these limitations, we propose and investigate possibility measures taking their values in the well-known Cantor subset of the unit interval but defined with respect to a nonstandard operation of supremum resulting from a nonstandard partial ordering of real numbers from the Cantor set. Such a nonstandard possibility measure is proved to be a sufficient tool to define an infinite sequence of probability measures defined over countable sets, consequently, the distribution of each continuous real-valued random variable defined over a general abstract probability space can be defined by an appropriate nonstandard possibility measure.

International audience; The aim of the paper is to provide a linearization approach to the $\mathbb{L}^{\infty}$-control problems. We begin by proving a semigroup-type behaviour of the set of constraints appearing in the linearized formulation of (standard) control problems. As a byproduct we obtain a linear formulation of the dynamic programming principle. Then, we use the $\mathbb{L}^{p}$ approach and the associated linear formulations. This seems to be the most appropriate tool for treating $\mathbb{L}^{\infty}$ problems in continuous and lower semicontinuous setting.

Let \(\Phi \) be a continuous \(n\times n\) matrix-valued function on the unit circle \(\mathbb T \) such that the \((k-1)\)st singular value of the Hankel operator with symbol \(\Phi \) is greater than the \(k\)th singular value. In this case, it is well-known that \(\Phi \) has a unique superoptimal meromorphic approximant \(Q\) in \(H^{\infty }_{(k)}\); that is, \(Q\) has at most \(k\) poles in the unit disc \(\mathbb D \) (in the sense that the McMillan degree of \(Q\) in \(\mathbb D \) is at most \(k\)) and \(Q\) minimizes the essential suprema of singular values \(s_{j}\left((\Phi -Q)(\zeta )\right)\!, j\ge 0\), with respect to the lexicographic ordering. For each \(j\ge 0\), the essential supremum of \(s_{j}\left((\Phi -Q)(\zeta )\right)\) is called the \(j\)th superoptimal singular value of degree \(k\) of \(\Phi \). We prove that if \(\Phi \) has \(n\) non-zero superoptimal singular values of degree \(k\), then the Toeplitz operator \(T_{\Phi -Q}\) with symbol \(\Phi -Q\) is Fredholm and has index $$ \mathrm{ind}T_{\Phi -Q}=\dim \ker T_{\Phi -Q}=2k+\dim \mathcal E , $$ where \(\mathcal E =\{ \xi \in \ker H_{Q}: \Vert H_{\Phi }\xi \Vert _{2}=\Vert (\Phi -Q)\xi \Vert _{2}\}\) and \(H_{\Phi }\) denotes the Hankel operator with symbol \(\Phi \). This result can in fact be extended from continuous matrix-valued functions to the wider class of \(k\)-admissible matrix-valued functions, i.e. essentially bounded \(n\times n\) matrix-valued functions \(\Phi \) on \(\mathbb T \) for which the essential norm of the Hankel operator \(H_{\Phi }\) is strictly less than the smallest non-zero superoptimal singular value of degree \(k\) of \(\Phi \).

We discuss existence and multiplicity of solutions of the periodic problem for the curvature-like equation\begin{equation*}-\Big( u'/{ \sqrt{a^2+{u'}^2}}\Big)' = f(t,u) \end{equation*} by means of variational techniques in the space of bounded variation functions. As $a= 0$ is allowed,both the prescribed curvature equation and the $1$-Laplace equation are considered. We are concerned with the case where the right-hand side $f$ of the equation interacts with the beginning of the spectrum of the $1$-Laplace operator with periodic boundary conditions on $[0,T]$, being mainly interested in the situation where $\supess{[0,T]\times\RR}{f(t,s)} $ may differ from $-\infess{[0,T]\times \RR}{f(t,s)}$.

Let X i i =1,2,3,... be a sequence of independent and identically distributed random variable which belong to the domain of attraction of a stable law of index α. Wrie \(S_0=0,\quad S_n\sum\nolimits^{n}_{i}=1 \,{\mathbf{X}}_i, n\geqq 1,\) and \(M_n={\rm max}_{0\leqq k \leqq n} S_k.\) In the case where the X i are such that \(\sum\nolimits^{\infty}_{1}\, n^{-1}\mathbf{P}{\rm r}\left(S_n > 0\right) 0\right)=\infty\) (the case of oscillation of the random walk generated bu the S n) and the only result available with case α=2 (Erdos and Kac [5]) and the case where the X i themselves have a symmetric stable distribution (Darling [4]). In this paper we obtain a general limit therome for M n in the case of oscillation. Specifically, if {B n,n=1,2,3,...} is a monotone sequence of constants such that B -1 n S n converges in distribution to the stable with charactistic function $$\begin{array}{ccc} {\rm exp}\left\{-\lambda \mid t\mid^{\infty}\left(1+\,i\beta \,{\rm sgn}\, t\, {\rm tan}\frac{\pi \alpha}{2}\right)\right\},\\ \lambda > 0,0 < \alpha \leqq 2, \beta = 0\, {\rm if} \,\alpha =1, \mid \beta \mid < 1\,{\rm if}\, \alpha < 1 \,{\rm if}\, 1 < \alpha \leqq 2,{\rm we\, shall\, find}\\ H\left(x\right)={\rm \lim \limits_{n\rightarrow \infty}}\, \mathbf{P}{\rm r}\left(\mathbf{B}^{-1}_{n} M_n\leqq x\right)\end{array}.$$ (1) In connection with the parameter restrictions, we note that the stable law with characteristic function (1) is one-sided if \(\alpha < 1, | \beta | = 1\) (e.g., Lukacs [12], page 106) so that the random walk generated by the S n does not oscillate ([9], Lemma). The case \(\alpha = 1, \beta \neq 0\) introduces a normalization complication and is not amenable to tratment by the methods of this paper.

The quantum effects for a physical system can be described by the set E (H) of positive operators on a complex Hilbert space H that are bounded above by the identity operator. While a general effect may be unsharp, the collection of sharp effects is described by the set of orthogonal projections P (H) ⊆E (H). Under the natural order, E (H) becomes a partially ordered set that is not a lattice if dim H≥2. A physically significant and useful characterization of the pairs A,B∈E (H) such that the infimum A∧B exists is called the infimum problem. We show that A∧P exists for all A∈E (H), P∈P (H) and give an explicit expression for A∧P. We also give a characterization of when A∧ (I-A) exists in terms of the location of the spectrum of A. We present a counterexample which shows that a recent conjecture concerning the infimum problem is false. Finally, we compare our results with the work of Ando on the infimum problem. © 2005 American Institute of Physics.

In the beginning of the 90’s Kovacik and Rakosnik [17] introduced variable exponent Lebesgue and Sobolev spaces. In fact, generalized Lebesgue and Sobolev spaces are special cases of so-called Orlicz-Musielak spaces, and in this form their investigation goes back a bit further, to Orlicz [20], Hudzik [15], and Musielak [18], see also Sharapudinov [23]. During the last couple of years Lebesgue and Sobolev spaces with variable exponent have been studied at an increasing pace by Diening [4, 5], Edmunds and Rakosnik [6, 7], Fan, Shen and Zhao [9, 10], Cruz-Uribe, Fiorenze and Neugebauer [3], Kokilasvili and Samko [16], and Nekvinda [19], among others. One area where these spaces have found applications is the study of electrorheological fluids, as described in the book of Růžicka [22]. A mathematical application is the study of variational integrals with non-standard growth, see the papers by Acerbi and Mingione [1, 2]. Sobolev capacity for fixed exponent spaces has found a great number of uses (e.g. the monographs by Evans and Gariepy [8] and Heinonen, Kilpelainen, and Martio [14]). It was introduced into the study of variable exponent spaces in [12] and has been applied to the investigation of zero boundary values of Sobolev functions in [13]. In [12] we required the assumption 1 < ess inf p ≤ ess sup p < ∞ of the variable exponent p to guarantee that our set-function is indeed a Choquet capacity. This is unsatisfactory, since there is no reason to expect this condition to be of relevance in this context. In this paper we show that the lower inequality needs to hold only locally. In particular we show in Corollary 4.2 that if the exponent p is continuous, then zero capacity sets enjoy the usual subadditivity property.

We study quasilinear elliptic equations of Leray–Lions type in W 1, p ( Ω ), maximum principles, nonexistence and existence of solutions, the control of lower (upper) bound for essential supremum (essential infimum) of solutions, sign-changing solutions, local and global oscillation of solutions, geometry of domain, generating singularities of solutions, and lower bounds on constants appearing in Schauder, Agmon, Douglis, and Nirenberg estimates.

We introduce supsets and infsets for subsets of a partially ordered topological linear space. These notions generalize the usual notions of supremum and infimum in Riesz spaces. We shall investigate properties of supsets and infsets in this paper.

We consider the parabolic Anderson problem $\partial_t u=\kappa\Delta u+\xi u$ on $(0,\infty)\times \Z^d$ with random i.i.d. potential $\xi=(\xi(z))_{z\in\Z^d}$ and the initial condition $u(0,\cdot)\equiv1$. Our main assumption is that $\esssup\xi(0)=0$. Depending on the thickness of the distribution $\prob(\xi(0)\in\cdot)$ close to its essential supremum, we identify both the asymptotics of the moments of $u(t,0)$ and the almost-sure asymptotics of $u(t,0)$ as $t\to\infty$ in terms of variational problems. As a by-product, we establish Lifshitz tails for the random Schr\"odinger operator $-\kappa\Delta-\xi$ at the bottom of its spectrum. In our class of $\xi$ distributions, the Lifshitz exponent ranges from $d/2$ to $\infty$; the power law is typically accompanied by lower-order corrections., Comment: 40 pages, LaTeX 2e+times, version published in Ann. Probab

Let T 1,T 2,.. .,T d be commuting automorphisms of the probability space (X,μ). We call the Z d action (X, T, μ) a dynamical system and write \({T^n}x = T_1^{{n_1}}T_2^{{n_2}} \ldots T_2^{{n_d}}(x)\) for n ∊ Z d . Denote by χ the crossed product of the von Neumann algebra A = L ∞(X,M(N,C)) with the dynamical system (X,T,μ). The group Z d acts on A by automorphisms f↦ f(T n ) where f(T n )(x) = f(T n x) and the algebra χ is obtained by completing the algebra of all polynomials in the variables τ1,..., τ d with coefficients in A $$K = {\sum\limits_{n \in F \subset {{\Bbb Z}^d}} {{K_n}\tau } ^n},{(KL)_n} = \sum\limits_{l + m = n} {{K_l}{L_m}} ({T^l}){\tau ^n}$$ with respect to the norm |‖K|‖ =| ‖K(x)|| |∞ where K(x) is the bounded linear operator on l 2(Z d ) defined by (K(x)u) n = ∑ m K m (x)u n+m and where || · || is the operator norm on B(l 2(Z d )) and | . |∞ the essential supremum norm. With the involution on χ defined by $$ \sum\limits_{n} {{{K}_{n}}{{\tau }^{n}}} )* = \sum\limits_{n} {K_{n}^{*}} ({{T}^{{ - n}}}){{\tau }^{{ - n}}} $$ χ becomes a von Neumann algebra. It has the trace $$tr(K) = {\smallint _X}Tr({L_0}(x)d\mu (x),$$ where Tr denotes the usual trace on the matrix algebra M(N, C). We call the elements in χ discrete random operators.

In this paper we solve a general extremal problem for a nonnega- tive operator in Hilbert space. We show that it contains the classical infimum problems of Szego and Kolmogorov for bounded weight functions on the circle and also prove some new prediction theorems.