Mathematical Theory And Modeling Pdf

COMMUNITY DIVERSITY AND STABILITY: CHANGING PERSPECTIVES AND CHANGING DEFINITIONS

Anthony R. Ives , in Ecological Paradigms Lost, 2005

8.1 INTRODUCTION

Mathematical theory has been applied to numerous topics in community ecology. These include questions such as how many species can coexist within communities ( MacArthur 1972), are there rules dictating the structure of food webs (Pimm 1982), and what explains the relative abundances of species (Preston 1962). Here I focus on the relationship between community diversity and community stability. Because perspectives on this topic have changed dramatically over the last 50 years, this provides a good topic to investigate if paradigm shifts have occurred in theoretical community ecology. In his companion chapter, Kevin McCann gives a summary of the history of this question and his view of current breakthroughs and future challenges to theory.* My goal is largely orthogonal to McCann's. Even though it appears that successive hypotheses about the relationship between diversity and stability have replaced each other, I want to show that it is not so much that the failure of one hypothesis led to its replacement by another. Instead, I think the dominant definition of stability has changed over the last 50 years, and these changes in definition have understandably led to changing conclusions about the relationship between diversity and stability. Although these changes have occurred within the theoretical literature, investigations of the relationship between diversity and stability have proceeded somewhat independently within the empirical literature. The changes within theoretical ecology and the contrast between theory and empiricism make it difficult to draw any conclusion about the general relationship between diversity and stability, if indeed there is one.

The chapter is organized into four parts. The first is a brief history of perspectives on the relationship between diversity and stability. This partly repeats the history in McCann's chapter, although my account of the more recent history expands beyond strict theory into experimental perspectives on diversity and stability. I then present a simple model that illustrates several different relationships between diversity and stability, or more properly, relationships between diversity and different definitions of stability. The goal here is to show that different perspectives on the relationship between diversity and stability are complementary rather than at odds. My intent is not to try to resolve conflict and end with a chorus of Kumbaya, but is instead to shift the debate towards selecting a suitable definition, or definitions, of stability for particular situations and needs of the researchers. Then, I describe the history of the sometimes rocky relationship between theory and empiricism stemming from different definitions of stability. Finally, I give some examples of specific ideas that I think can be tested, or at least explored, to derive insights into the relationship between diversity and stability, regardless of the preferred definition of stability.

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Mathematical Logic

Yiannis N. Moschovakis , in Encyclopedia of Physical Science and Technology (Third Edition), 2003

II.F Model Theory

The mathematical theory of structures starts with the following basic result:

Compactness and Skolem-Löwenheim Theorem.

If every finite subset of a set of sentences T has a model, then T has a countable model.

For an impressive application, let (in the vocabulary of arithmetic)

$\begin{array}{l} Δ_{0} \equiv 0, & Δ_{m + 1} \equiv (Δ_{m} + 1), \end{array}$

so that the numeral Δ_m is about the simplest term which denotes the number m, add a constant c to the language, and let

$\begin{array}{l} T = \{A : N ⊨ A\} \\ \cup \{Δ_{0} \leq c, Δ_{1} \leq c, Δ_{2} \leq c, \dots\} . \end{array}$

Every finite subset S of T has a model, namely

$N_{s} = (N, 0, 1, +, \cdot, m),$

where the object m which interprets c is some number bigger than all the numerals which occur in formulas of S. So T has a countable model

$N_{T} = (\overset{―}{N}, \overset{―}{0}, \overset{―}{1}, \overset{―}{+}, \overset{―}{\cdot}, c),$

and then

\overset{―}{N} = (\overset{―}{N}, \overset{―}{0}, \overset{―}{1}, \overset{―}{+}, \overset{―}{\cdot})

is a structure for the vocabulary of arithmetic which satisfies all the first-order sentences true in the "standard" structure N but is not isomorphic with N—because it has in it some object c which is "larger" than all the interpretations of the numerals Δ₀.Δ₁, …. It follows that, with all its expressiveness, First-Order Logic does not capture the isomorphism type of complex structures such as N.

These nonstandard models of arithmetic were constructed by Skolem in the 1930s. Later, in the 1950s, Abraham, Robinson constructed by the same methods nonstandard models of analysis, and provided firm foundations for the classical Calculus of Leibnitz with its infinitesimals and "infinitely large" real numbers.

Model Theory has advanced immensely since the early work of Tarski, Abraham Robinson and Malcev. Especially with the contributions of Shelah in the 1970s and, more recently, Hrushovsky, it has become one of the most mathematically sophisticated branches of logic, with substantial applications to algebra and number theory.

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Electromagnetism

Daniel R. Stump , in Encyclopedia of Energy, 2004

6 The Maxwell Equations of the Electromagnetic Field

The mathematical theory of electromagnetism was developed and published in 1864 by James Clerk Maxwell. He described the known electric and magnetic effects in terms of four equations relating the electric and magnetic fields and their sources—charged particles and electric currents. The development of this theory was a supreme achievement in the history of science. Maxwell's theory is still used today by physicists and electrical engineers. The theory was further developed in the 20th century to account for the quantum theory of light. But even in quantum electrodynamics, Maxwell's equations remain valid although their interpretation is somewhat different from the classical theory. In any case, Maxwell's theory continues today to be an essential part of theoretical physics.

A knowledge of calculus and vectors is necessary for a full understanding of the Maxwell equations. However, the essential structure of the theory can be understood without going into the mathematical details. Each equation is expressed most powerfully as a partial differential equation relating variations of the electric and magnetic fields, with respect to variations of position or time, and the charge and current densities in space.

Gauss's Law. Gauss's law, written as a field equation, is ∇·E=ρ/ε ₀. (The symbol ∇· denotes the divergence operator.) Here ρ(x, t) is the charge per unit volume in the neighborhood of x at time t; ε ₀ is a constant of nature equal to 8.85×10⁻¹² C²/Nm². Gauss's law relates the electric field E(x, t) and the charge density. The solution for a charged particle q at rest is E(x)=kq e/r ², where r is the distance from the charge to x, e is the direction vector, and k=1/(4πε ₀); this is the familiar inverse square law of electrostatics. Electric field lines diverge at a point charge.

Gauss's Law for Magnetism. The analogous equation for the magnetic field is ∇·B=0. There are no magnetic monopoles, particles that act as a point source of B(x, t). Unlike the electric field lines, which may terminate on charges, the magnetic field lines always form closed curves because magnetic charges do not exist. There is no divergence of magnetic field lines.

Faraday's Law. The field equation that describes Faraday's law of electromagnetic induction is ∇×E=−∂ B/∂t. The quantity ∇×E, called the curl of E(x, t), determines the way that the vector field E curls around each direction in space. Also, ∂ B/∂t is the rate of change of the magnetic field. This field equation expresses the fact that a magnetic field that varies in time implies an electric field that curls around the change of the magnetic field. It is equivalent to Faraday's statement that the rate of change of magnetic field flux through a surface S is equal to an electromotive force (EMF) around the boundary curve of S.

The Ampère-Maxwell Law. In a system of steady electric currents, the magnetic field is constant in time and curls around the current in directions defined by the right-hand rule. The field equation that expresses this field B (Ampère's law) is ∇×B=μ ₀ J, where J(x) is the current per unit area at x and μ ₀ is a constant equal to 4π×10⁻⁷ Tm/A. (The units are T=tesla for magnetic field and A=ampere for electric current.) But Ampère's law is incomplete, because it does not apply to systems in which the currents and fields vary in time. Maxwell deduced from mathematical considerations a generalization of Ampère's law,

$\nabla \times B = μ_{0} J + μ_{0} ϵ_{0} \frac{\partial E}{\partial t},$

in which the second term on the right side is called the displacement current. The displacement current is a necessary term in order for the system of four partial differential equations to be self-consistent. The Ampère-Maxwell law implies that B curls around either electric current (J) or changing electric field (∂ E/∂t). The latter case is analogous to electromagnetic induction but with E and B reversed; a rate of change in one field induces circulation in the other field.

Maxwell's introduction of the displacement current was a daring theoretical prediction. At that time there was no experimental evidence for the existence of displacement current. Laboratory effects predicted by the displacement current are very small, and their observation was not possible with the apparatus available at that time. However, the Maxwell equations, including the displacement current, make a striking prediction—that light consists of electromagnetic waves. The fact that Maxwell's theory explains the properties of light, and other forms of electromagnetic radiation, provides the evidence for the existence of the displacement current.

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Emergence of Polygonal Shapes in Oil Droplets and Living Cells: The Potential Role of Tensegrity in the Origin of Life

Richard Gordon , ... Stoyan K. Smoukov , in Habitability of the Universe Before Earth, 2018

Appendix Overview of Tensegrity Structures

Tensegrity structures were originally conceived as manmade systems of isolated components of two kinds: stiff parts (bars) that could bear compression with little distortion, and taut or prestressed parts (cables) attached to them that either hold the stiff elements from moving or additionally compress them to some extent (usually not to the point of buckling). Such simplest tensegrity structure is shown in Fig. 17. Tensegrity structures may be compounded into and actually originated in fine art (Snelson & Heartney, 2013) (Fig. 18). We can also find them in nature (Fig. 19).

In the mathematical theory of tensegrity, a third kind of element (or "member") is sometimes allowed that acts in the opposite manner to the cable. This element, called a strut, is straight when compressed and floppy when stretched. In simple mathematical terms, then, the three kinds of elements have the following behaviors:

Bar: $t = k (l - l_{0})$

Cable: $t = k (l - l_{0})$ for $l \geq l_{0}$ , $t = 0$ otherwise

Strut: $t = - k (l - l_{0})$ for $l \leq l_{0}$ , $t = 0$ otherwise

where t is the internal energy of an element and l its length. The "initial" length of an element is l ₀. Note that when $l = l_{0}$ , $t = 0$ , meaning that the element then has zero prestress. The spring constant k and the zero prestress lengths l ₀ could be different for each kind of element, or even for each individual element. In one mathematical idealization, k approaches infinity (Fig. 20). Of course, these simple Hookean springs could be generalized to more complicated constitutive relationships (Rimoli, 2016).

Snapping between configurations is possible (Gordon, 1999), as between the boat and chair shapes of cyclohexane (Wikipedia, 2016a) (Fig. 21). Sometimes small deviations in the length of elements can result in a large change in the shape of the whole configuration (Connelly & Gortler, 2015) (Fig. 22).

Tensegrity systems have been classified as follows:

A tensegrity [system] that has no contacts between its rigid bodies [bars] is a class 1 tensegrity system, and a tensegrity system with as many as k rigid bodies in contact is a class k tensegrity system
(Skelton & de Oliveira, 2009).

In our illustrations, the following are Class 1 tensegrity systems: Figs. 1, 18B and C, 19, 23. Class 2: Fig. 18A. Class 5: Fig. 21. Class 30: Fig. 22. Some systems with k > 1 are called "mechanisms" (Connelly & Gortler, 2015).

When we deal with cells or organisms, we are closer to continuum mechanics than these tensegrity models suggest. Contrary to the open spaces in between human and spider-made tensegrity structure elements, in cells we are confronted with what has been called "the crowded cytoplasm" (Gnutt & Ebbinghaus, 2016). Similarly, the nucleus is crowded (Nakano et al., 2014), with 2 m of double-stranded DNA (Greulich, 2005) compacted into a human cell nucleus of 8 μm diameter (Greeley et al., 1978). Crowding itself has effects on the mechanical properties of cytoskeletal tensegrity structures (Zhou et al., 2009).

We can look at the Wurfel, for instance, a toddler's toy that we have used as a toy model for changes in gene expression in a cell nucleus (Gordon & Gordon, 2016a; Gordon, 1999), as a space filling tensegrity structure. A Wurfel consists of a set of wooden cubes connected via an elastic band through them that forms a closed loop. The elastic band enters and exits each cube at right angles, pulling them together face to face (Fig. 24). Each pair of connected blocks can be regarded as a strut. There is a discrete set of equivalent energy ground states of a Wurfel (Tromp & Gordon, 2006). However, if we replace the cubes by spheres, we still have a structure of stiff elements held together under tension ("if one imagines hard spherical billiard balls, the centers of any two touching balls form a natural strut" (Connelly & Guest, 2015)), but with a continuum set of ground states (Fig. 25). Cube and sphere-based Wurfels with many ground states have analogies in folded proteins. While most proteins have single, nondegenerate ground states (Khatib et al., 2011), some have many or even a continuum of ground states. The latter are called disordered proteins (Uversky, 2013).

If we allow a set of hard spheres to have attractive forces (nonzero prestress) between all near neighbors, we effectively have a hard sphere model for condensed matter (Camp, 2003) or dispersions when the spheres do not always touch (Gonzalez et al., 2014). This too could be considered a tensegrity structure, albeit a changing one as it flows or its atoms or molecules undergo Brownian motion relative to one another. Thus, any drop of liquid, with its molecules regarded as the stiff elements, is a tensegrity structure. Indeed, tensegrity models for the rigidity of packings of balls have been studied (Connelly, 2008; Connelly et al., 2014), which represent a step towards molecular tensegrity modeling of liquids and solids.

In most liquids, we have to deal only with interactions between near neighbors, to get an accurate picture of the statistics of their structure, such as the radial distribution function (Cockayne, 2008; Gotoh, 2012). Nevertheless, there can be a long-range order imposed by the network of elements under tension, as in packings and crystallization. Alternatively, the structure of long elements, i.e., elements that are far from spherical, can also lead to long-range effects and long-range order. A remarkable example is the case of microtubules, which are so long and thin that one would expect them to buckle like wet spaghetti (Gordon & Gordon, 2016a). However, when supported along their length by attached intermediate filaments, it takes the order of 10⁴ times more compressive force to buckle them (Brodland & Gordon, 1990).

While this calculation has been used to justify microtubules as the stiff elements in a tensegrity model for cytoplasm (Ingber et al., 1994), there is some circular reasoning in doing so, since what anchors those particular intermediate filaments at their other ends (cf. pinning in Fig. 18) has not been worked out. Also, the multiple attachments (nodes) along the microtubule make for a structure that differs from standard tensegrity modeling, in which the elements are allowed to rotate freely to any angle about the "joints" or nodes. This is because each long, polymeric structure in the cytoplasm has a stiffness, characterized by a persistence length (Fig. 5.30 in (Gordon & Gordon, 2016a)), so that the amount of bending at each node would be constrained, and any bending would add to the energy (prestress) of the whole structure.

A further step in generalizing tensegrity structures was taken with the invention of tensegrity robots, in which element lengths are manipulated to make a tensegrity structure change shape (Piazza, 2015) and, for example, move over a rough planetary landscape by shifting the center of gravity of the robot or pushing against terrain (Bruce et al., 2014; SunSpiral, 2015) (Fig. 23). If one combines such force generation by the elements with the change of neighbors in dispersions, we approach a tensegrity model for cytoskeleton dynamically changing via motor molecules, polymerization, and depolymerization, and changing connections (nodes) via bifunctional attachment proteins (Perera et al., 2016). Getting beyond the stick and string tensegrity model for cytoskeleton has just begun (Ingber et al., 2014), for example, by looking at the bistable configurations of the cell state splitter (Gordon & Gordon, 2016a).

However, a tensegrity structure that can make and break connections and grow and dissolve elements is prone to instability and collapse. Indeed, such collapses may be important steps in cell differentiation (Gordon & Gordon, 2016a). A computer simulation framework for investigating such cytoskeletal instability phenomena is under construction based on PushMePullMe (Senatore, 2017).

In zero gravity, the taut strings of a simple tensegrity structure (Fig. 17), for example, would hold the structure together, but need not be under any prestress. If prestressed, the configuration would look much the same, except that the stiff rods would be slightly compressed and the strings slightly stretched or slightly buckled. In the biological literature, the prestress is assumed to be nonzero and essential to the maintenance of the structure (Ingber et al., 2014; Shen & Wolynes, 2005). In the mathematical literature on tensegrity structures, the concept of prestress includes allowing its value to be zero (Connelly & Guest, 2015). Thus, there is a conceptual contradiction here. Of course, mathematically, any small deviation from the equilibrium structure may generate a small prestress, usually driving the structure back towards its equilibrium shape, unless that equilibrium state is metastable, degenerate (Fig. 25), or sensitive to small perturbations (Fig. 22). Nevertheless, the presumption that nonzero prestress is essential to structure maintenance in biology is mathematically incorrect. While nonzero prestress may be present in most biological tensegrity structures, that does not imply the structure would collapse at zero prestress. Thus, the assumption "that the forces required for such a strained assembly in the cell are generated by nonequilibrium polymerizations and movements of motor proteins…" may be wrong, when cytoskeletal structure does not require such forces for its stability. For example, while microtubules may undergo frequent elongation and shortening, a process called dynamic instability (Gordon & Gordon, 2016a), they can also be stabilized against such behavior (van der Vaart et al., 2009) and thus act more like simple tensegrity bars. Prestress may be important in building cytoskeletal structures, but sometimes it may not be necessary for maintenance of those structures. These distinctions are important here, because in modeling a tensegrity origin of life, we cannot assume that continuous nonequilibrium, energy requiring processes of nascent cytoskeletal molecules, generating and maintaining prestress, played any role in the abiotic precursors to life.

Most tensegrity modeling ignores the buckling of elements under stress. Buckling can be quite important in cytoskeleton, varying from smooth Eulerian buckling (Brodland & Gordon, 1990) to kinking. Kinking in effect splits an element into two elements with a new node at the kink. However, kinks come in two kinds: stationary and propagating. If a cytoskeletal microtubule or microfilament is bent into a ring (Gordon & Brodland, 1987), so long as the radius of curvature is comparable to its persistence length, we can anticipate that the ring will be circular. Epithelia commonly have microfilament rings, but the cells are generally close-packed in a plane and polygonal in shape. Whether or not individual microfilaments in the bundle forming the polygonal ring end at the corners or are kinked there has apparently not yet been investigated. As the cell state splitter also has an intermediate filament ring (Gordon & Gordon, 2016a; Martin & Gordon, 1997), the same question arises for its components. An epithelial cell is an example of confinement of a cytoskeletal structure (Gürsoy et al., 2014; Koudehi et al., 2016; Pinot et al., 2009; Soares e Silva et al., 2011; Vetter et al., 2014). Bundles of microfilaments confined to liposomes exhibit kinks and polygonal shapes (Tsai & Koenderink, 2015). Computer simulations have not yet revealed polygonal shapes, perhaps because spherical boundary conditions were imposed (Koudehi et al., 2016).

Details have been worked out for kinking of carbon nanotubes (Iijima et al., 1996; Wang et al., 2016a; Zeng et al., 2004). Analogies have been made between kinking of nanotubes and cytoskeleton (Cohen & Mahadevan, 2003).

Propagating kinks in cytoskeletal rings were discovered by Robert Jarosch (1956, 1957) in cytoplasm squeezed from Chara foetida and Kiyoko Kuroda in cytoplasm dripped out of cut Nitella cells (Kuroda, 1964) (Fig. 16). Kuroda observed:

…triangles, quadrangles, pentagons, hexagons and other polygons…. [Each] consists of a pair of straight lines running in parallel close to each other, their both ends being joined together by tiny circular arcs with a radius of approximately 1 μ. Of various polygons observed, pentagons and hexagons are found most frequently. The distribution of angles, measured in about 300 specimens, shows the sharp peak between 110° and 120°…. Corners of the polygon propagate as waves along the fibril all in the same direction with the same speed. Since the angle of each corner is also kept constant, the polygon maintains its definite shape while corners propagate successively in one direction. On pulling with two microneedles, the polygon is split into finer fibrils
(Kuroda, 1968).

These dynamic rings were later shown to consist of microfilaments (Higashi-Fujime, 1980), which are presumably the "finer fibrils." Whether the kink propagates with sliding or kinking of the individual microfilaments has not been investigated. Kink bends can propagate along microtubules (Tuszynski et al., 2005, 2009). It is worth noting that for shaped droplets: "A very large majority of the interior angles of the polygons are seen in experiments to have measures close to 60° or 120°…" (Haas et al., 2016). Electron microscopy of triangular Archaea shows unexpected 90° corners, made up by a slight rounding of the edges (Nishiyama et al., 1992, 1995; Takao, 2006) (Fig. 1D). All of these cases suggest specific molecular configurations at corners that warrant investigation and modeling.

This does not exhaust the phenomena we should anticipate in the tensegrity behavior of cytoskeleton and its precursors in protocells. A long molecule such as DNA, when supercoiled, exhibits nonlinear phenomena similar to that of a twisted rubber band (Marko & Neukirch, 2012). Supercoiling of microtubules, which are chiral, may alter the binding of motor molecules such as dynein (Gordon & Gordon, 2016a; Gordon, 1999) and perhaps attached bifunctional molecules.

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Time and Methods in Environmental Interfaces Modelling

D.T. Mihailović , ... D. Kapor , in Developments in Environmental Modelling, 2017

2.3 Basics of Mathematical Theory of General Systems

Following Mesarovic's Mathematical Theory of General Systems ( Mesarovic and Takahara, 1972), if we observe interactions of agents with their surrounding environment, such a system can be defined as a set of interacting objects $S \subseteq O_{1} \times O_{2} \times O_{3} \times \dots \times O_{n}$ . If we denote the population of agents under consideration as $p = {p_{1}, p_{2}, p_{3}, \dots, p_{n}}$ and a set of external influences as $E = {e_{1}, e_{2}, e_{3}, \dots e_{n}}$ (these influences can be either other agents or extra-systemic influences), then the state of such formed systems at any particular moment in time can be defined as the Cartesian product $s \subseteq P \times E$ . Because our system is a dynamical network of interactions where at each moment the hierarchical status of network elements can vary significantly, we have to define state of the population P as a mapping $ω : e \to p,$ $e \in E, p \in P$ . Both e and p are defined as temporal sequences of events such that $E = {e : T \to I}$ and $P = {e : T \to R}$ , where T is a set of time points t, I is a set of external stimuli on a particular agent such that at each time system receives stimulus i(t) and R is a set of responses, r(t). Furthermore, both P and E are formal systems. Therefore, the occurrence of p and the occurrence of e at some particular time point t are governed not only by mapping ω but also by the internal rules of these systems, which are partially independent. Thus, it is obvious that changes in an environment induce appropriate responses in agents through the model of coupled input/output pairs. In real systems, the reverse situation is also possible such that some external changes can be influenced by the activity of organisms. It is clear that a critical factor in building an evolvable model as described above is choosing the appropriate structure for the mapping $I \to R$ . When dealing with models usually developed as prediction tools, it is sufficient to assume the attitude of analyzing a "black box." Therefore, we can propose a function that should summarize all available experimental data and obtain a set of more or less accurate predictions for various initial conditions. However, in such a case we will neglect the real meaning of the nature of mappings within E and P. Taking a slightly closer look at these relations, we can see that a somewhat hidden problem is that of how I is generated from the wholeness of external changes and what is the connection between generating I with a constitution of the corresponding R. Although this connection can be efficiently represented using the FCA (Ganter and Wille, 1997), its evolvability demands a more advanced formal treatment to be fully comprehended.

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Measurement Theory and Engineering

Patrick Suppes , in Philosophy of Technology and Engineering Sciences, 2009

Publisher Summary

The formal or mathematical theory of representation has as its primary goal such an enrichment of the understanding, although there are other goals of representation of nearly as great importance—for instance, the use of numerical representations of measurement procedures to make computations more efficient. A conceptual analysis of measurement can properly begin by formulating the two fundamental problems of any measurement procedure. The first problem is that of representation, justifying the assignment of numbers to objects or phenomena. What one must show is that the structure of a set of phenomena under certain empirical operations and relations is the same as the structure of some set of numbers under corresponding arithmetical operations and relations. Solution of the representation problem for a theory of measurement does not completely lay bare the structure of the theory, for there is often a formal difference between the kind of assignment of numbers arising from different procedures of measurement. This is the second fundamental problem, determining the scale type of a given procedure. The scale type is based on the proof of an invariance theorem for the representation. This is another way of stating the second fundamental problem.

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Simple Models and Complex Interactions

Greg Dwyer , in Population Dynamics, 1995

I. Introduction

The application of formal mathematical theory to interspecific interactions has a well-known history, dating from the work of Lotka and Volterra in the 1920s ( Kingsland, 1985). The usefulness of such classical mathematical theory for understanding the population dynamics of herbivorous insects, however, has at times been questioned (Onstad, 1991; Strong, 1986). This chapter is intended to demonstrate that theory can be useful both qualitatively and quantitatively. I consider first a variety of mathematical models that have been used to reach qualitative conclusions, and then I describe my own work using a mathematical model to make quantitative predictions.

Theoreticians using differential equation models usually focus on either stability or what is known in mathematics as complex dynamics (Drazin, 1992), which means oscillatory behavior that includes limit cycles and chaos. The question of biological interest is, typically, under what conditions will populations of the species in a model be stable, or alternatively show complex dynamics? Whether mathematical stability properties are ecologically meaningful has been controversial (Murdoch et al., 1985); in fact, stability (Connell and Sousa, 1983), limit cycles (Gilbert, 1984), and chaos (Hassell et al., 1976; Berryman and Millstein, 1989) have all been questioned for their relevance to real ecological systems. Work with long time series of a wide variety of different animals, however, has suggested that many animal populations, at least, experience complex dynamics (Turchin and Taylor, 1992), in turn reemphasizing a need for simple mechanistic models.

One of the root causes of criticism of mathematical models by field ecologists is model simplicity (Onstad, 1991). That is, the models often consider only a small fraction of the biological detail that field ecologists believe is important, and this is sometimes used as an argument in favor of complex simulation models (Logan, 1994). Simple models, however, have the advantage that they allow the relationship between biological mechanism and population dynamics to be much more easily understood. Moreover, the idea of simplifying a situation is a standard research strategy, and is just as often used when designing models as when designing experiments. In Section II, an attempt is made to make clear the value of even the simplest models by briefly reviewing nonintuitive qualitative results from a variety of models of interspecific interactions among insects. In Section III, the aim is to show that simplicity is not necessarily a barrier to quantitative accuracy.

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Viscous and/or Heat Conducting Compressible Fluids

Eduard Feireisl , in Handbook of Mathematical Fluid Dynamics, 2002

1.5 Bibliographical comments

An elementary introduction to the mathematical theory of fluid mechanics can be found in the book by Chorin and Marsden [ 12]. More extensive material is available in the monographs by Batchelor [7], Meyer [77], Serrin [93], or Shapiro [94]. A more recent treatment including the so-called alternative models is presented by Truesdell and Rajagopal [105].

A rigorous mathematical justification of various models of viscous heat conducting fluids is given by Šilhavý [96]. Mainly mathematical aspects of the problem are discussed by Antontsev et al. [4], Málek et al. [68], and more recently by Lions [61,62].

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Seismic hazard assessment from the perspective of disaster prevention

Jens-Uwe Klugel , ... Jens-Uwe Klügel , in Earthquakes and Sustainable Infrastructure, 2022

2.2 Derivation of safety factor

The purpose of a safety factor in seismic hazard analysis for a disaster-resilient design of critical infrastructures is that the infrastructure will not be damaged, or its critical functions will not fail due to a surprisingly strong earthquake during its lifetime. Such surprising, extreme events exceeding the imagination of people are called "Black Swan" events in analogy to the popular book by Taleb (2007). In common sense, the term "Black Swan" event is applied to an "event that comes as a surprise, has a major impact, and is often inappropriately rationalized after the fact with the benefit of hindsight" (Wikipedia, last visited September 27th, 2015). The theory of "Black Swans" was originally developed for financial risk assessment by (Taleb, 2007) to explain the disproportionate role of large impact, hard to predict, and rare events that are beyond the realm of normal expectations in science, finance, and technology. In Klügel (2015), the following definition for a "Black Swan" event with respect to natural hazards is given: A "Black Swan" event is a large impact hazard, which is caused by a rare event or a rare combination of events and circumstances, which has a magnitude beyond usual scientific expectations.

Despite the problems for accurately predicting rare extreme events, a general assessment of their magnitude and of the likelihood of their occurrence during the limited lifetime of a critical infrastructure is possible. First, one must note that a surprising event is an event that exceeds all previous historical observations in magnitude. That means that in a mathematical sense a "Black Swan" event represents an all-time record event. To be very surprising, the magnitude of the event must be significantly higher than was observed for the last registered record. The derivation of the safety factor is based on the approximate evaluation of an upper bound for the next record earthquake. By replacing the site intensity defined as the result of the evaluation of the three paths in Fig. 2.2 by the site intensity resulting from the prediction of an upper bound of the next record earthquake, we arrive finally at the design site intensity with incorporation of the safety factor. For an approximate estimate of the next record earthquake event, we make use of some important insights from the mathematical theory of records, see Nevzorov (2001), and of asymptotic properties of heavy-tail distributions, see Cook and Nieboor (2011).

For the applicability of these mathematical theories to seismic hazard analysis, we make the following assumptions:

(1): The seismicity in the region considered follows a common, but unknown mechanism (iid—assumption).
(2): The occurrence of extreme seismic events follows a heavy-tail distribution. This is equivalent to the usual power-law assumption.

These assumptions are not unusual in seismic hazard analysis and for our purpose it is sufficient to require an approximate compliance with these assumptions. We are interested in some general and some asymptotic properties of the theory of records and heavy-tail distributions, which we use for an approximate prediction of the upper bound of the next record earthquake. The properties used are the following:

(1): The expected number of records (denoted as $N_{n}$ ) in a sequence of n observations is given by

(2.1) $E (N_{n}) = \sum_{j = 1}^{n} \frac{1}{j}$

(2): The ratio between the largest and the second largest record in a record counting process for a superheavy-tailed distribution converges asymptotically to a factor of 2. The probability that the next record value will exceed the second largest (the previous) record value by a factor of 2 or more is equal to:

(2.2) $\frac{2}{n}$

From these general properties, we can derive a set of interesting conclusions. The longer the historical period of observations, the more reliable we can predict an upper bound of the next record earthquake and the more robust the selected design site intensity will be. For example, if we have a seismic catalog including observations of 100 registered and ranked sequentially by magnitude record earthquakes in the region of interest, the probability that the largest recorded earthquake will be exceeded by a factor of 2 or more is just 2% according to Eq. 2.2. The occurrence of a new record earthquake is a rare event as shown by Eq. 2.1, if we have a reasonably complete catalog of strong earthquakes (high value of n).

By using the properties described, we can predict approximately the magnitude and the possible impact of a future extreme record earthquake exceeding historical experience and the seismological knowledge base on seismogenic nodes or mapped faults. Based on the property (2), we assume that the previously registered record earthquake will be exceeded by the new record approximately by a factor of 2 in the hazard characteristic defining its magnitude. The question remains, what relevant hazard characteristic must be considered for applying this factor of 2 to. For achieving a disaster-resilient design, the answer is straightforward. We want to avoid damage. Damaging is a nonlinear process for which it is necessary to perform external work. To perform work, you need energy. So, the amount of seismic energy, as well as the form how it is transmitted to a structure (response spectrum), represent the most appropriate parameters to apply our insights. Seismic energy can be characterized very well by the Arias intensity (Arias, 1970):

(2.3) $I_{a} = \frac{π}{2 g} \int_{0}^{\infty} a {(t)}^{2} d t$

So, the unexpected "Black Swan" event would be an event that doubles the Arias Intensity at the site in comparison to the controlling earthquakes expected by scientists and researchers after the evaluation of paths 1 to 3 in Fig. 2.2. In compliance with our terminology, we call the evaluated next "record event" a "Black Swan" event. By evaluating the maximum site intensity from this event, we obtain the design site intensity including a technically meaningful safety factor. For the calculation of Arias intensity, we may use an empirical prediction equation as for example (Travasarou et al., 2003) or we may use simulated time-histories directly.

It is important to note that the current DSHA practice of adding half a magnitude unit as a safety margin to the MCE for design purposes or of one intensity unit to the site intensity leads to similar results for the safety factor. Therefore, the current practice in DSHA is well founded in the mathematical theory of records and of heavy-tail distributions.

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Fundamentals of Discrete Element Methods for Rock Engineering

Lanru Jing , Ove Stephansson , in Developments in Geotechnical Engineering, 2007

Publisher Summary

This chapter presents the fundamental aspects of mathematical theories relating to the mechanical behavior of rock fractures and rock masses, including the most commonly encountered criteria for shear strength and constitutive models for rock fractures, and constitutive models for rock masses based on the theory of elasticity, elastoplasticity, and the crack tensor concept. Although tremendous efforts have been made to develop constitutive models for rock fractures, the presently available models still have significant limitations in predicting fracture behavior with an adequate level of confidence. The major difficulty is the lack of unique and quantitative representation of fracture surface roughness, confident prediction of surface damage evolution during a general deformation process, and its impact on the thermo-hydro-mechanical coupling processes and properties of rock fractures. Other difficulties include the models for large-scale features, such as faults or fracture zones with large widths, time-scale dependence and hydro-mechanical coupling effects.

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https://www.sciencedirect.com/science/article/pii/S0165125007850036