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In this paper, we are concerned with the existence of the maximum and minimum iterative solutions for a tempered fractional turbulent flow model in a porous medium with nonlocal boundary conditions. By introducing a new growth condition and developing an iterative technique, we establish new results on the existence of the maximum and minimum solutions for the considered equation; at the same time, the iterative sequences for approximating the extremal solutions are performed, and the asymptotic estimates of solutions are also derived. [ABSTRACT FROM AUTHOR]

By means of the image space analysis, a necessary and sufficient condition is established for the existence of vector extrema. In the scalar case, the sufficient part is shown to shrink to a known one. [ABSTRACT FROM AUTHOR]

Based on recent results on image space analysis, the paper aims at describing a fixed point approach to vector optimization problems. Possible extensions to the bi-level vector optimization are discussed. [ABSTRACT FROM AUTHOR]

We propose a new notion called “extremal depth” (ED) for functional data, discuss its properties, and compare its performance with existing concepts. The proposed notion is based on a measure of extreme “outlyingness.” ED has several desirable properties that are not shared by other notions and is especially well suited for obtaining central regions of functional data and function spaces. In particular: (a) the central region achieves the nominal (desired) simultaneous coverage probability; (b) there is a correspondence between ED-based (simultaneous) central regions and appropriate pointwise central regions; and (c) the method is resistant to certain classes of functional outliers. The article examines the performance of ED and compares it with other depth notions. Its usefulness is demonstrated through applications to constructing central regions, functional boxplots, outlier detection, and simultaneous confidence bands in regression problems. Supplementary materials for this article are available online. [ABSTRACT FROM AUTHOR]

In this paper, we prove that every biderivation of certain triangular rings is the sum of an extremal biderivation and an inner biderivation, using the notion of maximal left ring of quotients. As a consequence, we show that every biderivation of the ring of allupper triangular matrices over a unital ring () is the sum of an extremal biderivation and an inner biderivation, which extends two results of Benkovič and Ghosseiri, respectively. [ABSTRACT FROM PUBLISHER]

Considering extreme quantiles is a popular way to understand the tail of a distribution. While they have been extensively studied for univariate distributions, much less has been done for multivariate ones, primarily because there is no universally accepted definition of what a multivariate quantile or a multivariate distribution tail should be. In this paper, we focus on extreme geometric quantiles. In Girard and Stupfler () Intriguing properties of extreme geometric quantiles, their asymptotics are established, both in direction and magnitude, under suitable integrability conditions, when the norm of the associated index vector tends to one. In this paper, we study extreme geometric quantiles when the integrability conditions are not fulfilled, in a framework of regular variation. [ABSTRACT FROM AUTHOR]

The notion of a phantom distribution function (phdf) was introduced by O'Brien (Ann. Probab. 15, 281-292 ()). We show that the existence of a phdf is a quite common phenomenon for stationary weakly dependent sequences. It is proved that any α-mixing stationary sequence with continuous marginals admits a continuous phdf. Sufficient conditions are given for stationary sequences exhibiting weak dependence, what allows the use of attractive models beyond mixing. The case of discontinuous marginals is also discussed for α-mixing. Special attention is paid to examples of processes which admit a continuous phantom distribution function while their extremal index is zero. We show that Asmussen (Ann. Appl. Probab. 8, 354-374 ) and Roberts et al. (Extremes. 9, 213-229 ) provide natural examples of such processes. We also construct a non-ergodic stationary process of this type. [ABSTRACT FROM AUTHOR]

The extremal index θ, a measure of the degree of local dependence in the extremes of a stationary process, plays an important role in extreme value analyses. We estimate θ semiparametrically, using the relationship between the distribution of block maxima and the marginal distribution of a process to define a semiparametric model. We show that these semiparametric estimators are simpler and substantially more efficient than their parametric counterparts. We seek to improve efficiency further using maxima over sliding blocks. A simulation study shows that the semiparametric estimators are competitive with the leading estimators. An application to sea-surge heights combines inferences about θ with a standard extreme value analysis of block maxima to estimate marginal quantiles. [ABSTRACT FROM AUTHOR]

The asymptotic behavior of stochastic gradient algorithms is studied. Relying on results from differential geometry (the Lojasiewicz gradient inequality), the single limit-point convergence of the algorithm iterates is demonstrated and relatively tight bounds on the convergence rate are derived. In sharp contrast to the existing asymptotic results, the new results presented here allow the objective function to have multiple and non-isolated minima. The new results also offer new insights into the asymptotic properties of several classes of recursive algorithms which are routinely used in engineering, statistics, machine learning and operations research. [ABSTRACT FROM AUTHOR]

We found that minimum ℓ2-norm and smoothness-constrained continuous solutions of the linear inverse problem of potential field data are harmonic and biharmonic, respectively. In the case of a discrete distribution, the minimum ℓ2-norm and smoothness-constrained solutions become biased toward being harmonic or biharmonic, respectively. As a result, the estimated discrete distribution of density or magnetization contrast tends to be smooth and to satisfy the maximum principle, which forces the solution maxima and minima to lie on any boundary of the discretized region. The above findings were illustrated with 2D numerical examples. The harmonic or biharmonic bias is brought forth when the strengths of the minimum ℓ2-norm or the smoothness constraint are enhanced (relative to all other constraints) by approximating the continuous case (a large number of discretizing cells relative to the number of independent observations) and/or by using a regularizing parameter. We discovered that, by inspecting the rearranged normal equations, it is possible to qualify three different possibilities of designing nonharmonic estimators. Then we found that all three possibilities have in fact already been implemented in the literature, reinforcing, in this way, the validity of the theoretical demonstration. [ABSTRACT FROM AUTHOR]

The notions of upper and lower exhausters represent generalizations of the notions of exhaustive families of upper convex and lower concave approximations (u.c.a., l.c.a.). The notions of u.c.a.'s and l.c.a.'s were introduced by Pshenichnyi (Convex Analysis and Extremal Problems, Series in Nonlinear Analysis and its Applications, ), while the notions of exhaustive families of u.c.a.'s and l.c.a.'s were described by Demyanov and Rubinov in Nonsmooth Problems of Optimization Theory and Control, Leningrad University Press, Leningrad, . These notions allow one to solve the problem of optimization of an arbitrary function by means of Convex Analysis thus essentially extending the area of application of Convex Analysis. In terms of exhausters it is possible to describe extremality conditions, and it turns out that conditions for a minimum are expressed via an upper exhauster while conditions for a maximum are formulated in terms of a lower exhauster (Abbasov and Demyanov (), Demyanov and Roshchina (Appl Comput Math 4(2): 114-124, ), Demyanov and Roshchina (), Demyanov and Roshchina (Optimization 55(5-6): 525-540, )). This is why an upper exhauster is called a proper exhauster for minimization problems while a lower exhauster is called a proper one for maximization problems. The results obtained provide a simple geometric interpretation and allow one to construct steepest descent and ascent directions. Until recently, the problem of expressing extremality conditions in terms of adjoint exhausters remained open. Demyanov and Roshchina (Appl Comput Math 4(2): 114-124, ), Demyanov and Roshchina (Optimization 55(5-6): 525-540, ) was the first to derive such conditions. However, using the conditions obtained (unlike the conditions expressed in terms of proper exhausters) it was not possible to find directions of descent and ascent. In Abbasov () new extremality conditions in terms of adjoint exhausters were discovered. In the present paper, a different proof of these conditions is given and it is shown how to find steepest descent and ascent conditions in terms of adjoint exhausters. The results obtained open the way to constructing numerical methods based on the usage of adjoint exhausters thus avoiding the necessity of converting the adjoint exhauster into a proper one. [ABSTRACT FROM AUTHOR]

We say that a graph G is maximal Kp-free if G does not contain Kp but if we add any new edge e ∈ E(Ḡ) to G, then the graph G + e contains Kp. We study the minimum and maximum size of non-(p - 1)-partite maximal Kp-free graphs with n vertices. We also answer the interpolation question: for which values of n and m are there any n-vertex maximal Kp-free graphs of size m? [ABSTRACT FROM AUTHOR]

The Grötzsch and Teichmüller well-known extremal problems for the moduli of plane doubly connected domains are extended to the case of domains on Riemann surfaces. [ABSTRACT FROM AUTHOR]

We propose a new approach to the solution of certain extremal problems in the geometric theory of functions of a complex variable. [ABSTRACT FROM AUTHOR]

Abstract: The problem of finding the maximal hyperplane section of , where , has been open for a long time. It is known that the answer depends on both p and n. In this paper, using the well-known equivalence between hyperplane sections and the isotropic constant of a body, we give an upper bound estimate for the volume of hyperplane sections of normalized -balls that does not depend on n and p. In addition, on the basis of results of Meyer, Pajor and Schmuckenschläger, we show further the corresponding extremal body and hyperplane section when this volume attains its minimum. [ABSTRACT FROM AUTHOR]

The Hosoya index and the Merrifield-Simmons index are typical examples of graph invariants used in mathematical chemistry for quantifying relevant details of molecular structure. In recent years, quite a lot of work has been done on the extremal problem for these two indices, i.e., the problem of determining the graphs within certain prescribed classes that maximize or minimize the index value. This survey collects and classifies these results, and also provides some useful auxiliary results, tools and techniques that are frequently used in the study of this type of problem. [ABSTRACT FROM AUTHOR]

Necessary conditions of extremum (from the times of Fermat and Lagrange till our times) for extremal problems where smoothness is interlaced with convexity, and some type of regularity takes place, correspond to a unique general principle, which is due to Lagrange. This report is devoted to the Lagrange principle in the theory of optimization. [ABSTRACT FROM AUTHOR]

We consider nonpredictable stopping times 0 = inf{t ≦ 1 Bt = max0≦s≦1 Bs}, g = sup{t ≦ 1 Bt = O} for the Brownian motion B = (Bt)0≦t≦1. The main results of the paper concern solving the following conditional-extremal problems: in classes of Markov times MαB(θ) = {τ ≦ 1 P {r < θ} ≦ α}, MαB(g) = { σ ≦ 1: P { σ < g} ≦ α}, where 0 < α < 1, to describe a structure of optimal stopping times τα*(θ) and σα(g), for which E [τα*(0) - 0]+ = infτεMαB(θ) E (τ - θ)+, E [σα*(g) -g)+ = infτεMαB(θ) E (τ - θ)+, E. We also consider the problems of the structure of sonic special stopping times in the classes MαB(θμ) and MαB(gμ) for the case of Brownian motion with drift Bμ = (Btμ)0≦t≦1, where Btμ = μt + Bt. [ABSTRACT FROM AUTHOR]

We consider auctions in which the winning bid is the smallest bid that is unique. Only the upper-price limit is given. Neither the number of participants nor the distribution of the offers are known, so that the problem of placing a bid to win with maximum probability looks, a priori, ill-posed. Indeed, the essence of the problem is to inject a (final) minimum into a random subset (of unique offers) of a larger random set. We will see, however, that here no more than two external (and almost compelling) arguments make the problem meaningful. By appropriately modeling the relationship between the number of participants and the distribution of the bids, we can then maximize our chances of winning the auction and propose a computable algorithm for placing our bid. [ABSTRACT FROM AUTHOR]

This paper presents a canonical duality theory for solving a general nonconvex quadratic minimization problem with nonconvex constraints. By using the canonical dual transformation developed by the first author, the nonconvex primal problem can be converted into a canonical dual problem with zero duality gap. A general analytical solution form is obtained. Both global and local extrema of the nonconvex problem can be identified by the triality theory associated with the canonical duality theory. Illustrative applications to quadratic minimization with multiple quadratic constraints, box/integer constraints, and general nonconvex polynomial constraints are discussed, along with insightful connections to classical Lagrangian duality. Criteria for the existence and uniqueness of optimal solutions are presented. Several numerical examples are provided. [ABSTRACT FROM AUTHOR]

For an arbitrary fixed segment [α, β] ⊂ R and given r ∈ N, A r, A0, and p > 0, we solve the extremal problem on the set of all functions x∈ L r ∞ such that ∥ x( r)∥∞ ≤ A r and L( x) p ≤ A0, where In the case where p = ∞ and k ≥ 1, this problem was solved earlier by Bojanov and Naidenov. [ABSTRACT FROM AUTHOR]

Abstract: By the extremal number we denote the maximum number of edges in a graph of order and girth at least . The set of such graphs is denoted by . In 1975, Erdős mentioned the problem of determining extremal numbers in a graph of order and girth at least five. In this paper, we consider a generalized version of the problem for any value of girth by using the hybrid simulated annealing and genetic algorithm (HSAGA). Using this algorithm, some new results for have been obtained. In particular, we generate some graphs of girth and 8 which in some cases have more edges than corresponding cages. Furthermore, future work will be described regarding the investigation of structural properties of such extremal graphs and the implementation of HSAGA using parallel computing. [Copyright &y& Elsevier]

Abstract.  We propose a unifying framework for studying extremal problems related to graph minors. This framework relates the existence of a large minor in a given graph to its expansion properties. We then apply the developed framework to several extremal problems and prove in particular that: (a) Every $$K_{s,s^\prime}$$-free graph G with average degree r ($$2 \leq s \leq s^\prime$$ are constants) contains a minor with average degree $$cr^{1+ {\frac{1}{2(s-1)}}}$$, for some constant $$c = c(s, s^\prime) > 0$$; (b) Every C2k-free graph G with average degree r (k ≥ 2 is a constant) contains a minor with average degree $$cr^{\frac{k+1}{2}}$$, for some constant c = c(k) > 0. We also derive explicit lower bounds on the minor density in random, pseudo-random and expanding graphs. [ABSTRACT FROM AUTHOR]

Abstract: In this note by saying that a 0–1 matrix avoids a pattern given as a 0–1 matrix we mean that no submatrix of either equals or can be transformed into by replacing some 1 entries with 0 entries. We present a new method for estimating the maximal number of the 1 entries in a matrix that avoids a certain pattern. Applying this method we give a linear bound on the maximal number of the 1 entries in an by matrix avoiding pattern and thereby we answer the question that was asked by Gábor Tardos. Furthermore, we use our approach on patterns related to . [Copyright &y& Elsevier]

Abstract: Given , we obtain the global estimate for the weak solution of a boundary-value problem for an elliptic equation with BMO nonlinearity in a Reifenberg domain, assuming that the nonlinearity has sufficiently small BMO seminorm and that the boundary of the domain is sufficiently flat. [Copyright &y& Elsevier]

Abstract: We derive a novel method to determine the parameters for regularized super-resolution problems, addressing both the traditional regularized super-resolution problem with single- and multiple-parameters and the simultaneous super-resolution problem with two parameters. The proposal relies on the joint maximum a posteriori (JMAP) estimation technique. The classical JMAP technique provides solutions at low computational cost, but it may be unstable and presents multiple local minima. We propose to stabilize the JMAP estimation, while achieving a cost function with a unique global solution, by assuming a gamma prior distribution for the hyperparameters. The resulting fidelity is similar to the quality provided by classical methods such as GCV, L-curve and Evidence, which are computationally expensive. Experimental results illustrate the low complexity and stability of the proposed method. [Copyright &y& Elsevier]

Natural examples of emergent behaviour, in groups due to interactions among the group's individuals, are numerous. Our aim, in this paper, is to use complex emergent behaviour among agents that interact via pair-wise attractive and repulsive potentials, to solve the local minima problem in the artificial potential based navigation method. We present a modified potential field based path planning algorithm, which uses agent internal states and swarm emergent behaviour to enhance group performance. The algorithm is used successfully to solve a reactive path planning problem that cannot be solved using conventional static potential fields due to local minima formation. Simulation results demonstrate the ability of a swarm of agents to perform problem solving using the dynamic internal states of the agents along with emergent behaviour of the entire group. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to develop a fairly large number of sets of global parametric sufficient optimality conditions under various generalized (η, ρ)-invexity assumptions for a discrete minmax fractional programming problem involving arbitrary norms. [ABSTRACT FROM AUTHOR]

Abstract: In modern numerical simulation of prospecting and exploiting oil–gas resources and environmental science, it is important to consider a numerical method for nonlinear convection-dominated diffusion problems. Based on actual conditions, such as the three-dimensional characteristics of large-scale science-engineering computation, we present a kind of characteristic finite volume element method. Some techniques, such as calculus of variations, commutating operators, the theory of prior estimates and techniques, are adopted. Suboptimal order error estimate in norm and optimal order error estimate in norm are derived to determine the errors for the approximate solution. Numerical results are presented to verify the performance of the scheme. [Copyright &y& Elsevier]

Let X be a connected separable linear order, a connected separable metric space, or a connected, locally connected complete metric space. We show that every continuous function f : X → ℝ with the property that every x E X is a local maximum or minimum of f is in fact constant. We provide an example of a compact connected linear order X and a continuous function f : X → ℝ that is not constant and yet every point of X is a local minimum or maximum of f. [ABSTRACT FROM AUTHOR]

The article offers suggestions on how the change point methods may be applied to detect changes in annual maximal and minimal temperatures. The data sets that have been examined were obtained from a CD-ROM that was a part of the book edited by Camuffo and Jones. Several illustrations are presented to show the periods of measurement together for the annual minima and maxima and to show the behavior of the series under study. It also presents the results for temperature series under study. It also shows that the approximate critical values obtained by the permutation principle vary only little.

This article aims to test and compare the performances of several estimators of the extremal coefficient using simulations of various spatial model. Section 2 of the article summarizes the multivariate extreme values definitions and some properties of the coefficient. Then Section 3 introduces the four estimators. They are compared in Section 4 through a large simulation study. Two applications with real data in the climate field are investigated in Section 5. The first one is related to maxima temperatures and the second one is related to maxima precipitations. Finally, Section 6 provides a conclusion and some perspectives.

The article presents research on the application of Gauss-Lobato formulae and the external problems with polynomials. Some general concepts on quadrature formulae were recalled in the investigation of Gauss-Lobatto type's quadrature furmulae. It considers the formulae of numerical integration which are called the Gauss-Jacobi-Lobatto quadrature formulae. The zeroes of the orthogonal polynomial on an interval which are real, distinct and located in the interval were also recalled in the investigation.

Abstract: In this paper, we establish the boundedness of maximal function on Morrey spaces related to the Monge–Ampère equation. [Copyright &y& Elsevier]

Abstract: In this paper we study the centered and non-centered Hardy–Littlewood maximal functions, M and , for a wide class of non-doubling measure μ defined on cusped manifolds. In particular, we give a class of manifolds with exponential volume growth on which M is of weak type and is bounded on for all . [Copyright &y& Elsevier]

Abstract: For a family of weight functions invariant under a finite reflection group, the boundedness of a maximal function on the unit sphere is established and used to prove a multiplier theorem for the orthogonal expansions with respect to the weight function on the unit sphere. Similar results are also established for the weighted space on the unit ball and on the standard simplex. [Copyright &y& Elsevier]

Abstract: Let be the minimum of the product under the conditions that and . In our previous paper [J. Kaneko, On an extremal problem of Selberg, J. Approx. Theory 142 (2006) 129–137], we showed that the following estimates hold. provided . In this note, we prove that the limit of as exists and is expressed by the (unique) solution of some simultaneous transcendental equations. By using this expression we obtain numerically. [Copyright &y& Elsevier]

In this paper, we present a useful bicriteria model for analyzing the time–cost tradeoffs that can be achieved when undertaking a project consisting of a single task. Some related mathematical results are also presented. The model assumes that several different processes may be used to undertake the task, defining different operating modes, and also assuming that managers may change the process in use after the task is started, depending on the way the task is developing. The model is easy to use, and it can be applied to a large number of real-life situations when the objectives are to minimize cost and time. We present some ideas for an algorithm aimed at identifying sets of relevant strategies from which the decision-maker may choose a suitable strategy. We also present several mathematical properties that may be useful for such an algorithm. [ABSTRACT FROM AUTHOR]

In this note, we derive first and second-order necessary (resp. sufficient) conditions for local minimum (resp. strict local minimum) points of optimization problems governed by set-valued maps. This allows us to present and to solve a generalized Lagrange claim in a multivalued setting. [ABSTRACT FROM AUTHOR]

This note presents a brief and partial review of the work of Broom, Cannings and Vickers 1. It also presents some simple examples of an extension of the their formalism to non-symmetric matrices. [ABSTRACT FROM AUTHOR]

Abstract: We prove smoothing estimates for Schrödinger equations with , real and bounded from below. We then bootstrap these estimates to obtain optimal Strichartz and maximal function estimates, all of which turn out to be identical to the constant coefficient case. We also provide counterexamples showing to be in a sense a minimal requirement. Finally, we provide an application to sharp well-posedness for a generalized Benjamin–Ono equation. [Copyright &y& Elsevier]

Some extremal problems between the generalized Hua domain of the first type and the Unit ball are studied. The extremal mapping and extremal value in explicit formulas are also obtained. [ABSTRACT FROM AUTHOR]

Let (X(s,t), s = (s1,s2) ϵ R2, t ϵ R) be a stationary random field defined over a discrete lattice. In this paper, we consider a set of domain of attraction criteria giving the notion of extremal index for random fields. Together with the extremal-types theorem given by Leadbetter and Rootzen (1997), this will give a characterization of the limiting distribution of the maximum of such random fields. [ABSTRACT FROM AUTHOR]

We use the concept of generalized inverses to show that the extrema of the function G(x) = ∥Ax - b∥2 are minima, A being a rectangular matrix not necessarily of full rank. [ABSTRACT FROM AUTHOR]

Abstract: We obtain a comparison of the level sets for two maximal functions on a space of homogeneous type: the Hardy–Littlewood maximal function of mean values over balls and the dyadic maximal function of mean values over the dyadic sets introduced by M. Christ in [M. Christ, A theorem with remarks on analytic capacity and the Cauchy integral, Colloq. Math. 60/61 (1990) 601–628]. As applications to the theory of weights on this setting, we compare the standard and the dyadic Muckenhoupt classes and we give an alternative proof of reverse Hölder type inequalities. [Copyright &y& Elsevier]

Abstract: We give a complete solution to the extremal problem posed by N.G. Tchebotaröv in the mid 1920s, and we establish explicit parametric formulae for the extremals. To cite this article: P. Tamrazov, C. R. Acad. Sci. Paris, Ser. I 341 (2005). [Copyright &y& Elsevier]

In 1958, the author proved the maximum principle. Pshenichni had written that the proof was sensational, using topology to obtain a result of variational calculus. Later the author worked out the tent method [Boltyanski, V., 1975, Uspehi Mathematika Nauk , 30 , 3-65.] as a general way to solve extremal problems. In fact, main ideas of the method were contained in [Boltyanski, V.G., 1958, Doklady Akademii Nauk SSSR , 119 (6), 1070-1073.]. We give here a short survey of the tent method and the idea of the proof of the maximum principle. [ABSTRACT FROM AUTHOR]

First-order and second-order necessary conditions of optimality for an impulsive control problem that remain informative for abnormal control processes are presented and derived. One of the main features of these conditions is that no a priori normality assumptions are required. This feature follows from the fact that these conditions rely on an extremal principle which is proved for an abstract minimization problem with equality constraints, inequality constraints, and constraints given by an inclusion in a convex cone. Two simple examples illustrate the power of the main result. [ABSTRACT FROM AUTHOR]

Examines the quadratic extremality conditions for broken extremals in the general problem of the calculus of variations. Abstract scheme for obtaining higher-order conditions in smooth extremal problems with constraints; Creation of the quadratic theory of conditions for Pontryagin minimum in the general problem of the calculus of variations that does not contain the local relation g(x,u,t) = 0; Decoding of the basic constant for the set of so-called "Pontryagin sequences".