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The existence of ideal objects, such as maximal ideals in nonzero rings, plays a crucial role in commutative algebra. These are typically justified using Zorn's lemma, and thus pose a challenge from a computational point of view. Giving a constructive meaning to ideal objects is a problem which dates back to Hilbert's program, and today is still a central theme in the area of dynamical algebra, which focuses on the elimination of ideal objects via syntactic methods. In this paper, we take an alternative approach based on Kreisel's no counterexample interpretation and sequential algorithms. We first give a computational interpretation to an abstract maximality principle in the countable setting via an intuitive, state based algorithm. We then carry out a concrete case study, in which we give an algorithmic account of the result that in any commutative ring, the intersection of all prime ideals is contained in its nilradical.

For many investment decisions, the decision-maker wishes to pursue several targets, rather than a single target as the earlier chapters have assumed. Such a decision-making problem is typical in strategic investment decision-making as, for example, when installing a new plant in a new location, using new technology and/or manufacturing a new product. Non-monetary measures have to be considered and a number of complex alternatives need to be analysed. This chapter describes a number of multi-criteria methods, such as: the utility value analysis, the analytic hierarchy process, the multi-attribute utility theory and PROMETHEE.

Uncertainty plays a significant role in all investment decision-making. In the previous chapter the analysis of single investment projects under conditions of uncertainty was discussed. This chapter now expands to methods and models for analysing investment programmes in such environments. When investment programmes are planned, often many (or even an infinite number of) investment alternatives exist, considerably complicating analytical models and/or evaluations that attempt to take account of uncertainty. Limitations on the ranges of uncertain conditions or investment alternatives then become necessary. The chapter includes a description of portfolio selection and of flexible planning.

The discounted cash flow methods described in this chapter are classified as ‘dynamic’ investment appraisal methods, which, unlike the static methods described in Chap. 2, explicitly consider more than one time period and acknowledge the time value of money. Investment projects can be described as streams of (expected) cash inflows and outflows over the whole course of their economic life, i.e. over different time periods. Methods described and discussed in this chapter are the net present value method, the annuity method, the internal rate of return method and the dynamic payback period method. All of them are subject to a set of assumptions that are also discussed in this part.

Investment decision-making under uncertainty requires a lot of additional considerations. This chapter picks this up and describes decision theory, the risk-adjusted analysis, the sensitivity analysis, the decision-tree method and options pricing models. All these cover a wider range within uncertain environment situations for single investment projects.

Decisions about investment programmes often involve simultaneous choices about types and numbers of investment projects. Additionally, models used for simultaneous decision-making might need to accommodate choices within a range of company areas such as financing, production, sales, human resources and tax policy. In this chapter, the finance and production areas—because of their relevance and close connections with investment decisions—are selected to illustrate ways of supporting investment decision-making in a broader sense than has been discussed previously. In the sections of this chapter some models are presented in detail, their practical relevance is discussed, and problems with their practical application are analysed. A static and two multi-tier models of simultaneous investment and financing/production decisions are thoroughly described in this chapter, illustrated by examples.

Investment decisions are of vital importance to all companies, since they determine both their potential to succeed and their ultimate cost structure. Investments usually entail high initial cash outflows and thus tie up substantial funds. Sound investment decisions are important, therefore. Yet, due to a highly complex and rapidly changing business environment they remain a challenging management task. Effective appraisal methods are valuable tools to support investment decisions. The following chapter explains characteristics of investment projects and discusses investment planning, provides an overview of concepts and methods and the didactical approach of this book. The decision models are then used throughout the book.

This chapter describes methods that do not assume a perfect capital market—i.e. the following methods use differing debt and credit interest rates instead of a uniform discount rate: the compound value method, the critical debt interest rate method and the visualisation of financial implication (VoFI) method. As in the previous chapters the methods are described including a discussion of their assumptions and their applicability.

This chapter examines some further applications of the investment appraisal methods already discussed: the inclusion of taxes in investment appraisals; the assessment of foreign investments; and the use of selected investment appraisal methods for determining optimum economic lives, replacement times and investment timing (under the assumption of certainty). All these applications require specialised procedures described in the chapter. Optimum economic life decisions are important and different target measures, points in time and the number and type of subsequent projects have to be considered. The use of different investment appraisal methods for this decision problem is shown in this chapter illustrated by a number of examples

In Chap. 14 it was shown how a matrix A and a vector r can be multiplied to give a new vector r' (provided the dimensions of the vector and the matrix fit): r' = Ar.

Consider a steady flow of water through a pipe. The water is assumed to be incompressible, i.e. it has a uniform density (for which we will use the symbol ρ), the velocity of each particle having a constant value v = ds/dt.

In the preceding chapter ″Fourier series″ we showed that an arbitrary periodic function with period T can be described as the sum of trigonometric functions with multiples of the period T.

Theoretical biology is incomplete without a comprehensive theory of evolution, since evolution is at the core of biological thought. Evolution is visualized as a migration process in genotype or sequence space that is either an adaptive walk driven by some fitness gradient or a random walk in the absence of (sufficiently large) fitness differences. The Darwinian concept of natural selection consisting in the interplay of variation and selection is based on a dichotomy: All variations occur on genotypes whereas selection operates on phenotypes, and relations between genotypes and phenotypes, as encapsulated in a mapping from genotype space into phenotype space, are central to an understanding of evolution. Fitness is conceived as a function of the phenotype, represented by a second mapping from phenotype space into nonnegative real numbers. In the biology of organisms, genotype–phenotype maps are enormously complex and relevant information on them is exceedingly scarce. The situation is better in the case of viruses but so far only one example of a genotype–phenotype map, the mapping of RNA sequences into RNA secondary structures, has been investigated in sufficient detail. It provides direct information on RNA selection in vitro and test-tube evolution, and it is a basis for testing in silico evolution on a realistic fitness landscape. Most of the modeling efforts in theoretical and mathematical biology today are done by means of differential equations but stochastic effects are of undeniably great importance for evolution. Population sizes are much smaller than the numbers of genotypes constituting sequence space. Every mutant, after all, has to begin with a single copy. Evolution can be modeled by a chemical master equation, which (in principle) can be approximated by a stochastic differential equation. In addition, simulation tools are available that compute trajectories for master equations. The accessible population sizes in the range of \(10^7\le N\le 10^8\) molecules are commonly too small for problems in chemistry but sufficient for biology.

The purpose of this chapter is to consider some of the important applications of integration as applied to problems in physics and engineering. Its objective is twofold. Firstly, it demonstrates the practical use of the integral calculus to readers who are particularly interested in applications. Secondly, other readers may use this chapter as a reference when practical problems are encountered.