Preface xi 1 Simple Applications 1 1.1 Introduction 1 1.2 Compartment systems 1 1.3 Springs and masses 5 1.4 Electric circuits 9 1.5 Notes 13 1.6 Exercises 13 2 Properties of Linear Systems 19 2.1 Introduction 19 2.2 Basic linear algebra 20 2.2.1 Vector spaces 20 2.2.2 Matrices 22 2.2.3 Vector spaces of functions 24 2.3 First-order systems 25 2.3.1 Introduction 25 2.3.2 First-order homogeneous systems 26 2.3.3 The Wronskian 30 2.3.4 First-order nonhomogeneous systems 32 2.4 Higher-order equations 34 2.4.1 Linear equations of order n 34 2.4.2 Nonhomogeneous linear equations of order n. . 37 2.5 Notes 40 2.6 Exercises 40 3 Constant Coefficients 49 3.1 Introduction 49 3.2 Properties of the exponential of a matrix 54 3.3 Nonhomogeneous systems 57 3.4 Structure of the solution space 57 3.4.1 A special case 57 3.4.2 The general case 61 vn
With increasing pressure for shorter delivery schedules, space is a critical resource at construction sites. Current industry practice lacks a formalized approach or a tool to help project managers analyze spatial conflicts between activities prior to construction. Consequently, time-space conflicts occur frequently and significantly impact construction processes. Time-space conflicts have three characteristics which impede the detection and analysis of time-space conflicts prior to construction: (1) They have a temporal aspect, (2) They have different forms creating different problems, (3) Multiple types of spatial conflicts can exist between a pair of conflicting activities. This research formalizes time-space conflict analysis as a classification task and addresses these challenges by automatically (1) detecting conflicts in four dimensions, (2) categorizing the conflicts according to a taxonomy of time-space conflicts developed, and (3) prioritizing the multiple types of conflicts between the same pair of conflicting activities. This research extends previous research on construction space management by developing a taxonomy of time-space conflicts and by defining an approach for the analysis of time-space conflicts prior to construction.
Abstract. The differential operators iD and −D 2 − p are constructed on certain finite directed weighted graphs. Two types of inverse spectral problems are considered. First, information about the graph weights and boundary conditions is extracted from the spectrum of −D 2 . Second, the compactness of isospectral sets for −D 2 − p is established by computation of the residues of the zeta function.
This paper presents a direct analog of the Borg-Levinson theorem on the recovery of a potential from the sequence of eigenvalues and norming constants for differential equations of the formon the unit interval subject to various boundary conditions. This result is used to show that even zonal Schrödinger operators and Laplace operators on spheres are uniquely determined by a subsequence of their eigenvalues.
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