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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 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 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

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.

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 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.

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.

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 $\limsup_{r\to\infty}\mathcal{A}(r,\mathcal{N})/\mathcal{N}(r)$.

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 → ∞.

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.

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.

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.

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.

We consider a convex function f(x) with unbounded level sets. Many algorithms, if applied to this class of functions, do not guarantee convergence to the global infimum. Our approach to this problem leads to a derivation of the equation of a parametrized curve x(t), such that an infimum of f(x) along this curve is equal to the global infimum of the function on /Bbb Rn. We also investigate properties of the vectors of recession, showing in particular how to determine a cone of recession of the convex function. This allows us to determine a vector of recession required to construct the minimizing trajectory.

This chapter provides the foundations for the general theory of stochastic processes and optimal stopping problems. In Section 5.1, we elaborate on the concepts of s-algebras and information, probability spaces, uniform integrability, conditional expectations and essential supremum or infimum at an advanced level of probability theory. Then stochastic processes and the associated filtrations are introduced in Section 5.2, which intuitively explains the meaning of information flow. Section 5.3 deals with the concept of stopping times, with focus on the s-algebras at stopping times. Section 5.4 provides a concise introduction to the concept and fundamental results of martingales. The emphasis is focused on Doob’s stopping theorem and the convergence theorems of martingales, as well as their applications in studying the path properties of martingales. The materials in this chapter are essential in the context of stochastic controls, especially for derivation of dynamic policies.

In earlier paper [4], we introduced generalized supremum and infimam for a subset A of a partially ordered linear space E generalizing the notion of supremum and infimum in Riesz space and considered properties of generalized supremum and infimum. In this paper we shall introduce optimal set for a subset A of E in order to make new optimization theory in the direction of order relations.

We consider a class of quasilinear elliptic equations of Leray-Lions type that allow us to control the ess inf and ess sup of solutions. Precisely, for arbitrary given four constants m 0 m 1 ≤ m 1 m 0 , we find some sufficient conditions such that for each solution u e W 1, p (Ω) satisfying m 0 ≤ u ( x ) ≤ M 0 on ∂Ω, we have m 0 ≤ ess inf u m 1 and m 1 u ≤ M 0 . The main consequence of this result is the lower bound of oscillation of solutions. It enables us to generate singular solutions and to obtain a lower bound on the constants in Schauder and Agmon, Douglis, Nirenberg estimates.

The problem of approximation of a fuzzy subset of a normed space is considered. We study the error of approximation, which in this case is characterized by an L -fuzzy number. In order to do this we define the supremum of an L -fuzzy set of real numbers as well as the supremum and the infimum of a crisp set of L -fuzzy numbers. The introduced concepts allow us to investigate the best approximation and the optimal linear approximation. In particular, we consider approximation of a fuzzy subset in the space L p m of differentiable functions in the L q -metric. We prove the fuzzy counterparts of duality theorems, which in crisp case allows effectively to solve extremal problems of the classical approximation theory.

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.

In this paper, we give a necessary and sufficient condition under which a pair of usual functions of λ on [0, 1] (supn(un)λ−, supn(un)λ+) can be determined a fuzzy number, and give a condition under which the supremum and infimum for a sequence of fuzzy numbers preserve the approximation properties for a metric D. In addition, we obtain the criterion that the continuous fuzzy-valued function on a closed interval can attain its supremum and infimum.

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.

It is well known that any distributive poset (short for partially ordered set) has an isomorphic representation as a poset (Q, ⊆) such that the supremum and the infimum of any finite setF ofP correspond, respectively to the union and intersection of the images of the elements ofF. Here necessary and sufficient conditions are given for similar isomophic representation of a poset where however the supremum and infimum of also infinite subsetsI correspond to the union and intersection of images of elements ofI.

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

Mathematical morphology is a nonlinear image processing methodology based on the computation of supremum (dilation operator) and infimum (erosion operator) in local neighborhoods called structuring elements. This chapter deals with definition of supremum and infimum operators for positive definite symmetric (PDS) matrices, which are the basic ingredients for the extension mathematical morphology to PDS matrices-valued images. The problem is tackled under three different paradigms. Firstly, total orderings using lexicographic cascades of eigenvalues as well as kernelized distances to matrix references are studied. Secondly, by decoupling the shape and the orientation of the ellipsoid associated to each PDS matrix, the supremum and infimum can be obtained by using a marginal supremum/infimum for the eigenvalues and a geometric matrix mean for the orthogonal basis. Thirdly, an estimate of the supremum and infimum associated to the Löwner ellipsoids are computed as the asymptotic cases of nonlinear averaging using the original notion of counter-harmonic mean for PDS matrices. Properties of the three introduced approaches are explored in detail, including also some numerical examples. Chapter contents:- Total Orderings for Sup-Inf Input-Preserving Sets of PDS Matrices- Partial Spectral Ordering for PDS Matrices and Inverse Eigenvalue Problem- Asymptotic Nonlinear Averaging Using Counter-Harmonic- Mean for PDS Matrices- Application to Nonlinear Filtering of Matrix-Valued Images

In this chapter we prove Theorem 4.2, an estimate about the tail distribution of the supremum of an appropriate class of Gaussian random variables with the help of a method, called the chaining argument. We also investigate the proof of Theorem 4.1 which can be considered as a version of Theorem 4.2 about the supremum of partial sums of independent and identically distributed random variables.

This chapter contains an estimate about the supremum of a nice class of normalized sums of independent and identically distributed random variables together with an analogous result about the supremum of an appropriate class of onefold random integrals with respect to a normalized empirical distribution. We also compare these results with their natural Gaussian counterpart.

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.

Let D ≌ RN , 0 0}, where ess sup ƒ denotes the essential supremum of ƒ. These properties can be used for computing ess sup ƒ. As example, two algorithms are stated. If the function ƒ is dense, or lower semicontinuous, or if −ƒ is robust, then sup ƒ = ess sup ƒ. In this case, the algorithms mentioned can be applied for determining the supremum of ƒ, i.e., also the global maximum of ƒ if it exists.

An elementary algorithm for computing the infimum of two projections in a Hilbert space is examined constructively. It is shown that in order to obtain a constructive convergence proof for the algorithm, one must add some hypotheses such as Markov's principle or the locatedness of a certain range; and that in the finite-dimensional case, the existence of both the infimum and the supremum of the two projections suffices for the convergence of the algorithm.

We consider the vector optimization problem $$\text{ minimize }f : X \rightarrow \overline{Y }\text{ with respect to } \leq \text{ over }S \subset X.$$ (VOP) The extended partially ordered vector space \({\bigl (\overline{Y },\leq \bigr )}\) is regarded to be a subset of the complete lattice \((\mathcal{I},\preccurlyeq )\) which is defined as follows: The convex and pointed ordering cone C that induces the partial ordering ≤ on Y is assumed to satisfy ∅≠int C≠Y. The infimal set of a set A ⊂ Y is defined by $$\mathrm{Inf\,}A := \mathrm{wMin\,}\mathrm{cl\,}(A + C),$$ where some additional considerations with respect to the elements ± ∞ will be added later. A set A satisfying A = Inf A is called self-infimal.

For a subset

It is well known [1] that any distributive poset (short for partially ordered set) has an isomorphic representation as a poset (Q, (−) such that the supremum and the infimum of any finite setF ofp correspond, respectively, to the union and intersection of the images of the elements ofF. Here necessary and sufficient conditions are given for similar isomorphic representation of a poset where however the supremum and infimum of also infinite subsetsI correspond to the union and intersection of images of elements ofI.

This paper shows that the Choquet integral can represent some useful quantities such as supremum, infimum, essential supremum, essential infimum, mean, median, α-quantile (100α-percentile), and L-estimators including α-trimmed mean

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.

In this paper we discuss the standard H2 control problem for linear, finite-dimensional, time-invariant systems without any assumptions on the system parameters. We give an explicit formula for the infimum over all internally stabilizing strictly proper compensators and give a characterization when the infimum is attained.

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 C.-X. Wu and C. Wu (1997) the existences of supremum and infimum of a set of fuzzy numbers were proved by using levelwise method. In T.-H. Fan and G.-J. Wang (2004), we showed that the supremum and infimum can be finitely approximated via endograph metric. In this paper, we give a method to calculate supremum and infimum of fuzzy sets in a way that resembles the Riemann sum in calculus.

In this paper we study the quantity \(\mathbb{E} \sup_{{t \in T}}X_{t},\) where X t is some random process. In the case of the Gaussian process, there is a natural sub-metric d defined on T. We find an upper bound in terms of labelled-covering trees of (T,d) and a lower bound in terms of packing trees (this uses the knowledge of packing numbers of subsets of T). The two quantities are proved to be equivalent via a general result concerning packing trees and labelled-covering trees of a metric space. Instead of using the majorizing measure theory, all the results involve the language of entropy numbers. Part of the results can be extended to some more general processes which satisfy some concentration inequality.

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 consider the maximization of a ratio f of an arbitrary quadratic function and an affine function over a closed and unbounded set. The behavior of unbounded feasible sequences is studied in order to derive a) conditions under which f attains a finite supremum and b) conditions which guarantee that its supremum is finite. We first consider a function f where the quadratic form is semidefinite. We then obtain results for the case where the quadratic function is the product of two affine functions.

We consider a constrained maximization problem with a linear fractional function f over any closed and unbounded set X. Starting the analysis from the behavior of unbounded sequences in X, we find necessary and sufficient conditions for the supremum to be finite. We point out that when f does not have maximum value, the supremum is not necessarily attained along a recession direction. It is known that this fundamental property holds in the classical fractional problem with a polyhedral feasible region and we give a new proof of this result by using our approach and the Representation Theorem.

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.

It is known that for (bounded) self-adjoint operators A,B on a Hilbert space H the infimum A ⋀ B, with respect to the order induced by the cone of positive (semi-definite) operators, exists only when A and B are comparable, that is, A ≥ B or A ≥ B. In this paper we present a necessary and sufficient condition for that, given A,B ≥ 0, the iniimum considered in the positive cone exists.