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Dynamic optimization is widely used in many fields of economics, finance, and business management. Typically one searches for the optimal time path of one or several variables that maximizes the value of a specific objective function given certain constraints. While there exist some analytical solutions to deterministic dynamic optimization problems, things become much more complicated as soon as the environment in which we are searching for optimal decisions becomes uncertain. In such cases researchers typically rely on the technique of dynamic programming. This chapter introduces the principles of dynamic programming and provides a couple of solution algorithms that differ in accuracy, speed, and applicability. Chapters 8 to 11 show how to apply these dynamic programming techniques to various problems in macroeconomics and finance. To get things started we want to lay out the basic idea of dynamic programming and introduce the language that is typically used to describe it. The easiest way to do this is with a very simple example that we can solve both ‘by hand’ and with the dynamic programming technique. Let’s assume an agent owns a certain resource (say a cake or a mine) which has the size a0. In every period t = 0, 1, 2, . . . ,∞ the agent can decide how much to extract from this resource and consume, i.e. how much of the cake to eat or how many resources to extract from the mine.We denote his consumption in period t as ct. At each point in time the agent derives some utility from consumption which we express by the so-called instantaneous utility function u(ct). We furthermore assume that the agent’s utility is additively separable over time and that the agent is impatient, meaning that he derives more utility from consuming in period t than in any later period.We describe the extent of his impatience with the time discount factor 0 < β < 1.

This chapter studies second order and Gaussian fields on the background spaces which are the continuum limits of the finite graphs. For this random processes in Hilbert spaces are examined. Orthogonal expansions such as Karhunen–Loeve are examined, with spectral representations of the processes established. Gaussian processes induced by differential operators representing physical processes in the world are studied.

By now you are likely aware that unconditional education (UE) is a practice of optimization. That is, the aim is to provide just the right amount of intervention to get the job done, but never unnecessary excess. Chapter 1 introduced the key principles that drive UE: efficiency, intentional relationship building, cross-sector responsibility, and local decision-making. Much of the rest of this book has addressed what happens in schools when these principles are absent. However, in reviewing early UE implementation pitfalls, most, if not all, missteps can be traced back to an overzealous application of these principles without adequate consideration for a just-right approach. This chapter will explore these common missteps and trace the surprising ways in which an over-application of the principles of UE can unintentionally replicate the very practices of exclusion it was designed to address. The previous chapters have proposed that healthy and trusting relationships play a central role when it comes to both personal and organizational learning. While the cultivation of relationships takes time, once established, the presence of relational trust can accelerate efforts. In schools highly impacted by trauma, an initial investment in relationship building is in fact a prerequisite for any successful transformation to take hold. The work of creating trauma-informed schools necessitates that we acknowledge these experiences and create plans to address the vicarious trauma often felt by school staff themselves. In some cases, even this is not enough. Organizational trauma—in which interactions within the entire building or district itself evidence the weight of working in resource-strapped environments—is common in public schools. It is often the case that years of unhealthy competition, inadequate funding, and failed initiatives and promises have overwhelmed an organization’s protective structures and rendered it less resilient for the hard work required to bring about the exact change the organization needs in order to heal and thrive (Vickers & Kouzmin, 2001). Not all public schools operate as traumatized systems, yet the conditions within many schools, particularly those serving a high percentage of students who belong to systematically oppressed groups, are most vulnerable.

The reverse search starts from a set of desired properties and asks for substances that possess them. Theoretical knowledge and past experience should be relied upon to suggest where to look, since it is the fastest and least expensive approach. When theoretical knowledge and past experience have been exhausted, then random searches may be the only way to make progress, if the problem is sufficiently important and there is enough budget and patience. Table 7.1 compares some of the requirements and the pros and cons of the guided search and the random search The best strategy on how to spend resources of time and money most efficiently can be considered a problem in operations research, under the topic of “optimal resource allocation.” The best way to use the limited resources of money and time effectively may be a mixed strategy, with some guided and some random searches. Even a random search has to start somewhere. At the beginning, there should be a plan on what territories to cover and how to cover them. The plan can be deterministic, which is completely planned out in advance and executed accordingly. The plan can also be adaptive: after the arrival of each batch of results and preliminary evaluations, the plan would evolve to take advantage of the new information and understanding gained. Even a random search must begin at a starting point and stake out the most promising directions for initial explorations. In most cases, there is a lead compound that has some of the desired properties, but which is deficient in others, and serves as the starting point of the random search to find better compounds in this neighborhood. The historic cases in section 1.2 involve the modification of an existing product, such as vulcanizing raw rubber and adding an acetyl group to salicylic acid. One explores around the lead compound by using small amounts of additives, blending with other material, changing processing conditions and temperature, and changing structure by chemical reactions.

Programming is a process of optimization; taking a specification, which tells us what we want, and transforming it into an implementation, a program, which causes the target system to do exactly what we want. Conventionally, this optimization is achieved through manual design. However, manual design can be slow and error-prone, and recently there has been increasing interest in automatic programming; using computers to semiautomate the process of refining a specification into an implementation. Genetic programming is a developing approach to automatic programming, which, rather than treating programming as a design process, treats it as a search process. However, the space of possible programs is infinite, and finding the right program requires a powerful search process. Fortunately for us, we are surrounded by a monotonous search process capable of producing viable systems of great complexity: evolution. Evolution is the inspiration behind genetic programming. Genetic programming copies the process and genetic operators of biological evolution but does not take any inspiration from the biological representations to which they are applied. It can be argued that the program representation that genetic programming does use is not well suited to evolution. Biological representations, by comparison, are a product of evolution and, a fact to which this book is testament, describe computational structures. This chapter is about enzyme genetic programming, a form of genetic programming that mimics biological representations in an attempt to improve the evolvability of programs. Although it would be an advantage to have a familiarity with both genetic programming and biological representations, concise introductions to both these subjects are provided. According to modern biological understanding, evolution is solely responsible for the complexity we see in the structure and behavior of biological organisms. Nevertheless, evolution itself is a simple process that can occur in any population of imperfectly replicating entities where the right to replicate is determined by a process of selection. Consequently, given an appropriate model of such an environment, evolution can also occur within computers.

By the end of this chapter you should be able to:… ● Understand the responsibilities and accountability of the student and trained nurse with regards to medicines management ● Understand the reasons for policy to support medicines management ● Interpret the role of the nurse in relation to policies and standards for medicines management ● Understand the role of the nurse in relation to key standards and drivers for the safer administration and management of medicines…. The aims of this chapter are to support you to interpret the responsibility that you already carry as a student and will carry as a registrant when giving medicines to patients and to help you to understand what is meant by accountability and how this relates to your role now and in the future in the management of medicines. Medicines management occurs wherever there is a patient and is carried out in a variety of settings which include:… ● acute hospitals ● community hospitals ● care homes, both residential care homes and nursing homes ● the patient’s own home ● schools ● community clinics…. The National Patient Safety Agency (NPSA, 2004 ) has produced guidance for organizations on supporting patient safety. They suggested the implementation of seven steps as follows:… 1 Build a safety culture. 2 Lead and support your staff. 3 Integrate your risk management activity. 4 Promote reporting. 5 Involve and communicate with patients and the public. 6 Learn and share safety lessons. 7 Implement solutions to prevent harm…. When interpreted in relation to medicines management and nursing care this means that the employing organization has a duty of care to its employees and patients to ensure that medicines are dispensed, supplied, and administered safely and that procedures are in place to support this. Managers need to be made aware of anything that might prevent this, and must ensure that checks are in place to prevent harm from occurring. The clear and prompt reporting of concerns, risk, and errors to management is pivotal to patient safety and medicines management in nursing and, from an organizational point of view, patient consultation and involvement is vital. Lessons must be shared in a ‘low blame culture’ and changes made to support the reduction of risk and potential harm. For more on communication and on risk reduction please see Chapters 1 and 10.

In this chapter we develop simple methods for solving numerical problems. We start with linear equation systems, continue with nonlinear equations and finally talk about optimization, interpolation, and integration methods. Each section starts with a motivating example from economics before we discuss some of the theory and intuition behind the numerical solution method. Finally, we present some Fortran code that applies the solution technique to the economic problem. This section mainly addresses the issue of solving linear equation systems. As a linear equation system is usually defined by a matrix equation, we first have to talk about how to work with matrices and vectors in Fortran. After that, we will present some linear equation system solving techniques.

The discussion in the Chapters 3 and 4 centred around static optimization problems.The static general equilibrium model of Chapter 3 features an exogenous capital stock and Chapter 4 discusses investment decisions with risky assets, but in a static context. In this chapter we take a first step towards the analysis of dynamic problems. We introduce the life-cycle model and analyse the intertemporal choice of consumption and individual savings. We start with discussing the most basic version of this model and then introduce labour-income uncertainty to explain different motives for saving. In later sections, we extended the model by considering alternative savings vehicles and explain portfolio choice and annuity demand. Throughout this chapter we follow a partial equilibrium approach, so that factor prices for capital and labour are specified exogenously and not determined endogenously as in Chapter 3. This section assumes that households can only save in one asset. Since we abstract from bequest motives in this chapter, households do save because they need resources to consume in old age or because they want to provide a buffer stock in case of uncertain future outcomes.The first motive is the so-called old-age savings motive while the second is the precautionary savings motive. In order to derive savings decisions it is assumed in the following that a household lives for three periods. In the first two periods the agent works and receives labour income w while in the last period the agent lives from his accumulated previous savings. In order to derive the optimal asset structure a2 and a3 (i.e. the optimal savings), the agent maximizes the utility function . . . U(c1, c2, c3) = u(c1) + βu(c2) + β2u(c3) . . . where β denotes a time discount factor and u(c) = c1−1/γ /1−1/γ describes the preference function with γ ≥ 0 measuring the intertemporal elasticity of substitution.

Rényi divergence is a natural way to measure the rate of information flow in contexts like Bayesian updating. This chapter shows how Monte Carlo integration can be used to measure Rényi divergence when (as is often the case) only kernels of the relevant probability densities are available. The chapter further demonstrates that Rényi divergence is central to the convergence and efficiency of Monte Carlo integration procedures in which information flow is controlled. It uses this perspective to develop more flexible approaches to the controlled introduction of information; in the limited set of examples considered here, these alternatives enhance efficiency.

The core of data science is our fundamental understanding about data intelligence processes for transforming data to decisions. One aspect of this understanding is how to analyze the cost-benefit of data intelligence workflows. This work is built on the information-theoretic metric proposed by Chen and Golan for this purpose and several recent studies and applications of the metric. We present a set of extended interpretations of the metric by relating the metric to encryption, compression, model development, perception, cognition, languages, and media.

Mathematical tools necessary to the argument are presented and discussed. The focus is on concepts borrowed from the convex analysis and variational analysis literatures. The chapter starts by introducing the notions of a correspondence, upper hemi-continuity, and lower hemi-continuity. Superdifferential and subdifferential correspondences for real-valued functions are then introduced, and their essential properties and their role in characterizing global optima are surveyed. Convex sets are introduced and related to functional concavity (convexity). The relationship between functional concavity (convexity), superdifferentiability (subdifferentiability), and the existence of (one-sided) directional derivatives is examined. The theory of convex conjugates and essential conjugate duality results are discussed. Topics treated include Berge's Maximum Theorem, cyclical monotonicity of superdifferential (subdifferential) correspondences, concave (convex) conjugates and biconjugates, Fenchel's Inequality, the Fenchel-Rockafellar Conjugate Duality Theorem, support functions, superlinear functions, sublinear functions, the theory of infimal convolutions and supremal convolutions, and Fenchel's Duality Theorem.

Three generic economic optimization problems (expenditure (cost) minimization, revenue maximization, and profit maximization) are studied using the mathematical tools developed in Chapters 2 and 3. Conjugate duality results are developed for each. The resulting dual representations (E(q;y), R(p,x), and π‎(p,q)) are shown to characterize all of the economically relevant information in, respectively, V(y), Y(x), and Gr(≽(y)). The implications of different restrictions on ≽(y) for the dual representations are examined.

A computational theory of meaning tries to understand the phenomenon of meaning in terms of computation. Here we give an analysis in the context of Kolmogorov complexity. This theory measures the complexity of a data set in terms of the length of the smallest program that generates the data set on a universal computer. As a natural extension, the set of all programs that produce a data set on a computer can be interpreted as the set of meanings of the data set. We give an analysis of the Kolmogorov structure function and some other attempts to formulate a mathematical theory of meaning in terms of two-part optimal model selection. We show that such theories will always be context dependent: the invariance conditions that make Kolmogorov complexity a valid theory of measurement fail for this more general notion of meaning. One cause is the notion of polysemy: one data set (i.e., a string of symbols) can have different programs with no mutual information that compresses it. Another cause is the existence of recursive bijections between ℕ and ℕ2 for which the two-part code is always more efficient. This generates vacuous optimal two-part codes. We introduce a formal framework to study such contexts in the form of a theory that generalizes the concept of Turing machines to learning agents that have a memory and have access to each other’s functions in terms of a possible world semantics. In such a framework, the notions of randomness and informativeness become agent dependent. We show that such a rich framework explains many of the anomalies of the correct theory of algorithmic complexity. It also provides perspectives for, among other things, the study of cognitive and social processes. Finally, we sketch some application paradigms of the theory.

This chapter offers an introduction to the methods and main models used in dynamic macroeconomics. After reviewing key concepts such as lifetime utility maximization and the period-by-period and intertemporal budget constraints, first-order conditions for intertemporal optimization (the Euler equation and the labour-leisure choice) are developed. These methods are applied in developing the workhorse Ramsey model, with discussion of related concepts such as dynamic efficiency and market equilibrium versus the command optimum. An extension of the Ramsey model incorporates adjustment costs in investment and develops the user cost of capital. Furthermore, the Sidrauski model, with its implications for monetary economies, is reviewed. Finally, the discussion turns to another workhorse dynamic model, the overlapping-generations model and its implications. As an application of this model, the properties of various methods of funding social insurance are discussed.