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The maximal covering location problem has been shown to be a useful tool in siting emergency services. In this paper we expand the model along two dimensions - workload capacities on facilities and the allocation or multiple levels of backup or prioritized service for all demand points. In emergency service facility location decisions such as ambulance sitting, when all of a facility's resources are needed to meet each call for service and the demand cannot be queued, the need for a backup unit may be required. This need is especially significant in areas of high demand. These areas also will often result in excessive workload for some facilities. Effective siting decisions, therefore, must address both the need for a backup response facility for each demand point and a reasonable limit on each facility's workload. In this paper, we develop a model which captures these concerns as well as present an efficient solution procedure using Lagrangian relaxation. Results of extensive computational experiments are presented to demonstrate the viability of the approach. [ABSTRACT FROM AUTHOR]

Let $G$ be a group, let $A$ be an Abelian group, and let $n$ be an integer such that $n\backslash ge\; -1$. In the paper, the sets $\backslash Phi\_n(G,A)$ of functions from $G$ into $A$ of degree not greater than $n$ are studied. In essence, these sets were introduced by Logachev, Sal'nikov, and Yashchenko. We describe all cases in which any function from $G$ into $A$ is of bounded (or not necessarily bounded) finite degree. Moreover, it is shown that if $G$ is finite, then the study of the set $\backslash Phi\_n(G,A)$ is reduced to that of the set $\backslash Phi\_n(G/O^p(G),A\_p)$ for primes $p$ dividing $|G/G\text{'}|$. Here $O^p(G)$ stands for the $p$-coradical of the group $G$, $A\_p$ for the $p$-component of $A$, and $G\text{'}$ for the commutator subgroup of $G$. [ABSTRACT FROM AUTHOR]

The concept of a strict extended partial order (SEPO) has turned out to be very useful in explaining (resp. rationalizing) non-binary choice functions. The present paper provides a general account of the concept of extended binary relations, i.e. relations between subsets and elements of a given universal set of alternatives. In particular, we define the concept of a weak extended partial order (WEPO) and show how it can be used in order to represent rankings of Opportunity sets that display a ‘preference for opportunities’. We also clarify the relationship between SEPOs and WEPOs, which involves a non-trivial condition, called ‘strict properness’. Several characterizations of strict (and weak) properness are provided, based on which we argue for properness as an appropriate condition demarcating ‘choice based’ preference. [ABSTRACT FROM AUTHOR]

The following problem is considered: obtain the asymptotic properties of a function $u$ from the asymptotic properties of the integral $\backslash int\_0^r\{u(t)\}\backslash ,dt$. As is well known, this can be done under additional constraints on the function $u(t)$. In this paper, we obtain a theorem in which these constraints are weaker than in other well-known versions of such theorems. [ABSTRACT FROM AUTHOR]

In this paper, we study the spaces $B\_\{pq\}^s(G)$ and $L\_\{pq\}^s(G)$ of functions with positive exponent of smoothness $s>\; \backslash nomathbreak\; 0$, defined on a domain $G\backslash subset\; \backslash nomathbreak\; \backslash Bbb\; R^n$. For a domain $G$ with specific geometric properties, we establish the embedding $B\_\{pp\}^s(G)=L\_\{pp\}^s(G)\backslash subset\; L\_q(G)$, 1

The paper developed the method of multivalent vector guiding functions (MVGF). The notions of full collection of the strict MVGF's, full and sharp collection of the generalized MVGF's, and the regular MVGF were introduced and considered. These methods were extended to the problem of periodic trajectories of the controlled system obeying a differential inclusion. [ABSTRACT FROM AUTHOR]

An extremely simple proof of the K-K-M-S Theorem is given involving only Brouwer's fixed point theorem and some elementary calculus. A function is explicitly given such that a fixed point of it yields an intersection point of a balanced collection of sets together with balancing weights. Moreover, any intersection point of a balanced collection of sets together with balancing weights corresponds to a fixed point of the function. Furthermore, the proof can be used to show π-balanced versions of the K-K-M-S Theorem, with π-balancedness as introduced in Billera (1970). The proof makes clear that the conditions made with respect to π by Billera can be even weakened. [ABSTRACT FROM AUTHOR]

Numerical representations of choice functions allow the expression of a problem of choice as a problem of finding maxima of real-valued functions, which requires that less information be defined and which is easier to work with. In this paper, the existence of numerical representations of choice functions by imposing assumptions directly on the choice function is obtained. In particular, it is proved that (IIA) is a necessary and sufficient condition for a choice function to be representable, if the set of alternatives is countable, while the conjunction of (IIA) and a ‘continuity condition’ is sufficient to ensure it in separable topological spaces. [ABSTRACT FROM AUTHOR]

The article examines the coefficients of the expansion of smooth functions of several variables in the bases of the spaces Lp([0,1]d). It discusses the normalized basis in Lp([0,1]d) and the series of Fourier coefficients in the trigonometric system. It also discusses the Faber-Schauder system, Haar system, and Rademacher function.

Research on student learning in higher education suggests that threshold concepts within various disciplines have the capacity to transform students’ understanding. The present study explores students’ understanding in relation to particular threshold concepts in mathematics—integral and limit—and tries to clarify in what sense developing an understanding of those threshold concepts involves a transformation of understanding in relation to ways of thinking in mathematics. Drawing on data collected in interviews with students taking a basic course on calculus the analysis offers an initial characterisation of students’ understandings as algorithmic. It then proceeds to construct a more fine-grained theoretical account for how these understandings develop in the course of the interview, suggesting that the transformative aspects of threshold concepts may be conceptualised in terms of shifts in students’ contextualisations allowing the development of conceptions at different levels of abstraction simultaneously interacting to shape students’ awareness of the ways of thinking and practising in the subject. [ABSTRACT FROM AUTHOR]

This study is part of a project concerned with the analysis of how students work with two-variable functions. This is of fundamental importance given the role of multivariable functions in mathematics and its applications. The portion of the project we report here concentrates on investigating the relationship between students’ notion of subsets of Cartesian three-dimensional space and the understanding of graphs of two-variable functions. APOS theory and Duval’s theory of semiotic representations are used as theoretical framework. Nine students, who had taken a multivariable calculus course, were interviewed. Results show that students’ understanding can be related to the structure of their schema for R3 and to their flexibility in the use of different representations. [ABSTRACT FROM AUTHOR]

The article examines the existence of solutions for weighted p(t)-Laplacian system multipoint boundary value problems. The existence of solutions is given via Leray-Schauder degree when the nonlinearity term f (t,.,.) satisfies general growth condition. The study of differential equations and variational problems with variable exponent growth conditions is an interesting topic as many results have been obtained on these problems. Banach spaces are used in this study to denote positive constants to satisfy the growth condition of the nonlinearity term f.

The article analyzes the boundary asymptotic behavior of solutions for weighted p(x)-Laplacian equations that take infinite value on a bounded domain. It was found that the boundary asymptotic of solutions of the Laplacian function is continuous and positive. The p(x)-Laplacian possesses more complicated inhomogeneous nonlinearities, thus, some special techniques are needed. Its main difficulty arises from the lack of compactness. The uniqueness and asymptotic behavior of solutions for problem is a nonnegative function which is allowed to vanish on the boundary.

The article provides an overview of the study which investigates the Meromorphic Harmonic Functions with Respect to k-Symmetric Points. It has introduced a class of meromorphic harmonic function with respect to k-symmetric points by D[j][m]. It has obtained the coefficient bounds, distortion theorems, extreme points, convolution conditions, and convex combinations for the functions belonging to this class. The results of the calculations of various theorems are also provided.

The article presents a research regarding the association between Jordan-von Neumann-type additive functional equations and functional inequalities. It proves the generalized Hyers-Ulam stability of the following functional inequalities: ρf(x)+f(y)+f(z)ρ ≤ ρ 2f((x+y+z)/2)ρ, ρf(x)+f(y)+f(z) ρ ≤ ρf(x+y+z)ρ, ρ f(x)+f(y)+2f(z)ρ ρ2f((x+y)/2+z)ρ in the spirit of the Rassias stability method for approximately homomorphisms. According to the report, the inequality that was brought in for the first time by Rassias provided a lot of influence in the development of a generalization relating to the Hyers-Ulam stability concept.

A general concept of two-scale convergence is introduced and two-scale compactness theorems are stated and proved for some classes of sequences of bounded functions in L2(Ω) involving no periodicity assumptions. Further, the relation to the classical notion of compensated compactness and the recent concepts of two-scale compensated compactness and unfolding is discussed and a defect measure for two-scale convergence is introduced. [ABSTRACT FROM AUTHOR]

New dimension functions for topological spaces are introduced in the spirit of Nagata’s approach. Expressions for the new functions in terms of covering dimension include the Bruijning—Nagata and Hashimoto—Hattori formulas. [ABSTRACT FROM AUTHOR]

For dynamic systems described by differential equations with smooth functions in the state space, the size of the parameter space is minimized such that the input-output image of the system is preserved by finding the complete set of algebraic invariants for a fixed structure of the equations of the system. A relationship between the constructs of the algebra of functions used in the problem and the mathematical constructs used in the differential geometric approach is established. A method for finding the algebraic invariants—the analogs of Markov parameters of linear systems—is described. [ABSTRACT FROM AUTHOR]

Suppose that Δ is the set of functions x: I → ℝ on a finite interval I such that the divided differences [ x; t0,..., t s] of order s ∈ ℕ of these functions are nonnegative for all collections from ( s +1) different points t0,..., t s ∈ I. For all s ∈ ℕ and 1 ≤ p ≤ ∞, we establish exact orders of best approximations by splines with free nodes and rational functions in the metrics of L p for classes $$\Delta _ + ^s B_p : = \Delta _ + ^s \cap B_p$$ , where B p is the unit ball in L p. We also establish the asymptotics of pseudodimensional widths in L p of these classes of functions. [ABSTRACT FROM AUTHOR]

Abstract Given a function on Rn with many multiple local minima we approximate it from below, via concave minimization, with a piecewise-linear convex function by using sample points from the given function. The piecewise-linear function is then minimized using a single linear program to obtain an approximation to the global minimum of the original function. Successive shrinking of the original search region to which this procedure is applied leads to fairly accurate estimates, within 0.57%, of the global minima of synthetic nonconvex piecewise-quadratic functions for which the global minima are known exactly. [ABSTRACT FROM AUTHOR]

For z ∈ ∂Bn, the boundary of the unit ball in \Bbb {C}^n \ {\rm let} \ \Lambda(2)=\{ \lambda : \vert \lambda \vert \leqslan/math>. If f \in \Bbb{O}(B^n) then we call E(f)=\{ z \in \partial B^n: \int_{\Lambda(z)} \vert f(z) \vert ^2 d\Lambda(z)=\infty \} the exceptional set for f. In this note we give a tool for describing such sets. Moreover we prove that if E is a Gδ and Fσ subset of the projective (n − 1)-dimensional space \Bbb{P}^{n-1}=\Bbb{P}(\Bbb{C}^n) then there exists a holomorphic function f in the unit ball Bn so that E(f) = E. [ABSTRACT FROM AUTHOR]

Estimation of the frequency and spatial dependent boundary traction vector from measured vibration responses in a vibrating structure is addressed. This problem, also referred to as the inverse problem, may in some circumstances be ill-conditioned. Here a technique to overcome the ill-conditioning is proposed. A subset of a set of available eigenmodes is chosen such that the problem becomes well-conditioned enough. It is shown that the ill-conditioning originates from the fact that not all eigenmodes are orthogonal over the surface where the traction vector is sought. Consequently, by choosing a set of eigenmodes orthogonal over the surface of interest, the problem becomes well-conditioned. The calculated traction vector is shown to converge to the true one in the sense of a L2-norm on the boundary of the body. The proposed technique is verified, using numerical simulation of measured responses, with good agreement. [ABSTRACT FROM AUTHOR]