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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.

We expand the theoretical background of the recently introduced outer measure spaces theory of Do and Thiele in harmonic and time-frequency analysis context. In the context of non-additive measures and integrals, we propose a certain framework for a natural extension of the basic ingredients of the theory (i.e., the concept of size, outer essential supremum and the corresponding super level measure) which, besides the covering the previously considered cases, permits us to introduce a further substantial extension of a class of non-additive integrals. All these notions are studied in detail and exemplified.

Let \(L_{0,1}(\mathbb {R}_+)\) be the Banach space endowed with the norm $$\displaystyle \|f\|{ }_{0,1}=\int _0^\infty f^*,\qquad f^*(x)=\underset {t\geq x}{\text{ess sup}}~|f(t)|, $$ while \(L_{0,\infty }(\mathbb {R}_+)\) with the norm \(\displaystyle \sup \limits _{x>0}\frac {1}{x}\int _0^x|f|\).

The doubly degenerate nutrient taxis model u t = ∇ ⋅ ( u v ∇ u ) − ∇ ⋅ ( u 2 v ∇ v ) + l u v , x ∈ Ω , t > 0 , v t = Δ v − u v , x ∈ Ω , t > 0 , is considered in smoothly bounded convex subdomains of the plane, with l ≥ 0 . It is shown that for any p > 2 and each fixed nonnegative u 0 ∈ W 1 , ∞ ( Ω ) , a smallness condition exclusively involving v 0 can be identified as sufficient to ensure that an associated no-flux type initial–boundary value problem with ( u , v ) | t = 0 = ( u 0 , v 0 ) admits a global weak solution satisfying ess sup t > 0 ‖ u ( ⋅ , t ) ‖ L p ( Ω ) ∞ . The proof relies on the use of an apparently novel class of functional inequalities which provide estimates from below for certain Dirichlet integrals involving possibly degenerate weight functions.

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.

The differential ${\mathcal{L}_{2,p}}$ gain of a linear, time-invariant, p-dominant system is shown to coincide with the ${\mathcal{H}_{\infty ,p}}$ norm of its transfer function G, defined as the essential supremum of the absolute value of G over a vertical strip in the complex plane such that p poles of G lie to right of the strip. The close analogy between the ${\mathcal{H}_{\infty ,p}}$ norm and the classical ${\mathcal{H}_\infty }$ norm suggests that robust H dominance of linear systems can be studied along the same lines as robust stability. This property can be exploited in the analysis and design of nonlinear uncertain systems that can be decomposed as the feedback interconnection of a linear, time-invariant system with bounded gain uncertainties or nonlinearities.

In many situations, such as the definition of the essential supremum, or definition of \(L_p\)-norms, or the weak type estimates of the Hardy-Littlewood operator, or the construction of certain nonadditive integrals, etc.

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 consider a supremal functional of the form F ( u ) = ess sup x ∈ Ω f ( x , D u ( x ) ) where Ω ⊆ R N is a regular bounded open set, u ∈ W 1 , ∞ ( Ω ) and f is a Borel function. Assuming that the intrinsic distances d F λ ( x , y ) ≔ sup { u ( x ) − u ( y ) : F ( u ) ≤ λ } are locally equivalent to the euclidean one for every λ > inf W 1 , ∞ ( Ω ) F , we give a description of the sublevel sets of the weak ∗ -lower semicontinuous envelope of F in terms of the sub-level sets of the difference quotient functionals R d F λ ( u ) ≔ sup x ≠ y u ( x ) − u ( y ) d F λ ( x , y ) . As a consequence we prove that the relaxed functional of positive 1-homogeneous supremal functionals coincides with R d F 1 . Moreover, for a more general supremal functional F (a priori non coercive), we prove that the sublevel sets of its relaxed functionals with respect to the weak ∗ topology, the weak ∗ convergence and the uniform convergence are convex. The proof of these results relies both on a deep analysis of the intrinsic distances associated to F and on a careful use of variational tools such as Γ -convergence.

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.

Let f be a continuous transformation on a compact, finite-dimensional manifold M, and ϕ a continuous function on M. This paper establishes the following formula: $$ess \sup \mathop {\lim \sup }\limits_{n \to \infty } \frac{1} {n}\phi _n (x) = \sup \left\{ {\left. {\int {\phi d\mu } } \right|\mu \in \mathcal{O}_f } \right\} \leqslant \mathop {\lim \sup }\limits_{n \to \infty } \frac{1} {n}ess \sup \phi _n (x),$$ where ess sup denotes the essential supremum taken against the Lebesgue measure, $$\phi _n (x) = \sum\limits_{i = 0}^{n - 1} {\phi (f^i x)}$$ and Of is the set of observable measures. Examples are provided to illustrate that the inequality could be an equality or strict. Moreover, if μ is the unique maximizing observable measure for ϕ, it is weakly statistical stable.

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.

We consider the last passage percolation on the complete graph. Let G n = ( [ n ] , E n ) be the complete graph on vertex set [ n ] = { 1 , 2 , … , n } , and i.i.d. sequence { X e : e ∈ E n } be the passage times of edges. Denote by W n the largest passage time among all self-avoiding paths from 1 to n . First, it is proved that W n ∕ n converges to constant μ , where μ is called the time constant and coincides with the essential supremum of X e . Second, when μ = ∞ , lower and upper bounds for W n ∕ n are given, in particular, for a large class of passage times with light tails, a weak law of large number for W n is obtained. Finally, when μ ∞ , it is proved that the deviation probability P ( W n ∕ n ≤ μ − x ) decays as fast as e − Θ ( n 2 ) .

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

Financial market models defined by a liquidation value process generalize the conic models of Schachermayer and Kabanov where the transaction costs are proportional to the exchanged volumes of traded assets. The solvency set of all portfolio positions that can be liquidated without any debt is not necessary convex, e.g. in presence of proportional transaction costs and fixed costs. Therefore, the classical duality principle based on the Hahn–Banach separation theorem is not appropriate to characterize the prices super hedging a contingent claim. Using an alternative method based on the concepts of essential supremum and maximum, we provide a characterization of European and American contingent claim prices under the absence of arbitrage opportunity of the second kind.

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.

Consider in a real Hilbert space H the differential equation (E) : p ( t ) u ′ ′ ( t ) + q ( t ) u ′ ( t ) ∈ A u ( t ) + f ( t ) , for a.a. t ∈ R + = [ 0 , ∞ ) , with the condition u ( 0 ) = x ∈ D ( A ) ¯ , where A : D ( A ) ⊂ H → H is a (possibly set-valued) maximal monotone operator, with [ 0 , 0 ] ∈ A (or, more generally, 0 ∈ R ( A ) ); p , q ∈ L ∞ ( R + ) , with ess inf p > 0 and either ess inf q > 0 or ess sup q 0 . Recall that equation (E) in the case p ≡ 1 , q ≡ 0 , f ≡ 0 , subject to u ( 0 ) = x and sup t ≥ 0 ‖ u ( t ) ‖ ∞ , was investigated in the early 1970s by V. Barbu, who derived in particular from his results a definition for the square root of the nonlinear operator A . Subsequently H. Brezis, N.H. Pavel, L. Veron and others have paid attention to equation (E) . In this paper we prove the existence and uniqueness of the solution to equation (E) subject to u ( 0 ) = x ∈ D ( A ) in the weighted space X = L b 2 ( R + ; H ) , where b ( t ) = a ( t ) / p ( t ) , a ( t ) = exp ( ∫ 0 t q ( s ) / p ( s ) d s ) , under our weak assumptions on p and q (see above) and f ∈ X . For x ∈ D ( A ) ¯ we prove the existence of a generalized solution. This is a classic solution if p ≡ 1 , q ≡ c ∈ R ∖ { 0 } . If p ≡ 1 , q ( t ) ≡ c ∈ R ∖ { 0 } , f ≡ 0 the solutions give rise to a nonlinear semigroup of contractions. If A is linear its infinitesimal generator G is given by G = − ( c / 2 ) I − ( c 2 / 4 ) I + A .

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.

We consider branching random walks in random environment (BRWRE) on ℕ with only one particle starting at the origin. Particles reproduce according to offspring distribution (which depends on its locations) and move one step to the right (with a probability in (0,1] which may depend on the location) or stay in the same site. We give an estimate to the minimal displacement of BRWRE at time n in the case where the essential supremum of mean number of offsprings which stay in the same place is equal to 1.

By catching the so-called strictly critical points, this paper presents an effective algorithm for computing the global infimum of a polynomial function. For a multivariate real polynomial f, the algorithm in this paper is able to decide whether or not the global infimum of f is finite. In the case of f having a finite infimum, the global infimum of f can be accurately coded in the Interval Representation. Another usage of our algorithm is to decide whether or not the infimum of f is attained when the global infimum of f is finite. In the design of our algorithm, the well-known Wu’s method plays an important role.

Given an independent and identically distributed sample of the distribution of an R d × R -valued random vector ( X , Y ) , the problem of estimation of the essential supremum of the corresponding regression function m ( x ) = E { Y | X = x } is considered. Estimates are constructed, which converge almost surely to this value whenever the dependent variable Y satisfies some weak integrability condition.

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.

For functions u, subharmonic in the plane, letand let N(r,u) be the integrated counting function. Suppose that $\mathcal{N}\colon[0,\infty)\rightarrow\mathbb{R}$ is a non-negative non-decreasing convex function of log r for which $\mathcal{N}(r)=0$ for all small r and $\limsup_{r\to\infty}\log\mathcal{N}(r)/\4\log r=\rho$, where 1 < ρ < 2, and defineA sharp upper bound is obtained for $\liminf_{r\to\infty}\mathcal{B}(r,\mathcal{N})/\mathcal{N}(r)$ and a sharp lower bound is obtained for $\limsup_{r\to\infty}\mathcal{A}(r,\mathcal{N})/\mathcal{N}(r)$.

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 $X\in L$.

We introduce a new random, non-gradient based, optimization scheme that is consistent for all measurable functions with essential supremum. Theorem 1 establishes the exponential rate of convergence for a large class of functions. Performance comparisons with Pure Random Search (PRS), three quasi-Newton-type optimization routines as well as numerous non-gradient based procedures are reported.

For a sequence of nondegenerate i.i.d. bounded random variables {Y n , n ≥ 1} defined on a probability space (Ω, ℱ, ℙ) and a constant b > 1, it is shown for that and that for almost every ω ∈ Ω, where S(F Y 1 ) and S(F V ) are the spectrums of the distribution functions of Y 1 and , respectively and l and L are the essential infimum of Y 1 and the essential supremum of Y 1, respectively. Examples are provided showing that, in general, the above two inclusions are proper.

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.

In this paper, the concept of supremum and infimum for sequence of sets are defined. By using this two new notion the concept of ideal supremum and ideal infimum for sequence of sets are introduced. Besides basic properties of these concepts, ideal convergence of sequence of sets is defined and the relations between them are investigated.

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.

The quantum effects for a physical system can be described by the set ɛ(ℌ) of positive operators on a complex Hilbert space ℌ that are bounded above by the identity operator I. The infimum problem of Hilbert space effects is to find under what condition the infimum A ∧ B exists for two quantum effects A and B e ɛ(ℌ). The problem has been studied in different contexts by R. Kadison, S. Gudder, M. Moreland, and T. Ando. In this note, using the method of the spectral theory of operators, we give a complete answer of the infimum problem. The characterizations of the existence of infimum A ∧ B for two effects A, B e ɛ(ℌ) are established.

An upper (lower) probability can be defined as a supremum (infimum) over a set of probability measures. The upper probability measure is also called a capacity. A capacity is usually studied under an assumption that it is 2-alternating, but many capacities are not 2-alternating. In this paper, we introduce a new definition of sub 2-alternating, defined for a capacity and its conjugate capacity. We study these inequalities for a certain natural capacity that is the supremum over a particular set of probability measures with a particular form of Radon Nikodym derivative with respect to Brownian motion. This capacity is not 2-alternating, but the pair of the capacity and its conjugate capacity is sub 2-alternating.

Let D be be the open unit disc, H∞0 the space of all bounded analytic functions in D and H∞k the set of all functions of the form f(z)/(z−z1)...(z−zk) where z1,...,zk ∈ D and f ∈ H∞0. Given {z1,...,zn},{w1,...,wn}, where zi ∈D,wi ∈ ** and zi≠zj if i≠j, we show for 0≤k≤n−1, under certain assumptions, how to construct the unique interpolating function Bk∈H∞k, Bk(zj)=wj, of minimal essential supremum norm on ∂D by solving an eigenvalue problem defined by the interpolation data. The function Bk is a scaled quotient of two finite Blaschke products.

In this paper we first give a simplified proof of the existence theorem of supremum and infimum in fuzzy number space E 1 established by Wu and Wu (J. Math. Anal. Appl. 210 (1997) 499) and improve the expressions of the supremum and infimum. As a straightforward corollary of this result, we obtain a necessary and sufficient condition under which, for a bounded sequence of fuzzy numbers { u n }, the pair of functions sup n u n − ( λ ) and sup n u n + ( λ ) can determine a fuzzy number. Secondly, we give a necessary and sufficient condition for a sequence of fuzzy numbers { u n } to be levelwise convergent in E 1 , and generalize some important theorems in real number spaces to fuzzy number spaces. Finally, we prove the existence of supremum and infimum for the level-continuous fuzzy-valued function on a closed interval and give a necessary and sufficient condition under which its supremum and infimum can be attained.