The Mathematical Theory Of Bridge Pdf

Mathematical Theory

The purpose of the mathematical theory of homogenization is to describe this limit process when the parameters which describe the fineness of the microscopic structure tend to zero.

From: Encyclopedia of Mathematical Physics , 2006

LIMITATIONS OF CLASSICAL LOGIC

AUGUST STERN , in Quantum Theoretic Machines, 2000

LOGIC INTERPOLATES, MIND EXTRAPOLATES

Mathematical and physical theories are commonly idealizations and approximations, which often disregard actual physical features. A ball is not a perfect sphere, there is no pointlike mass and no absolutely smooth surface without friction. In the realm of idealizations logic is no exception, a fact which has both its advantages and its drawbacks. But besides the flaws that are due to idealization, such as two-valuedness of classical logic, there are defects in logical theory that are fundamental in nature. This fact becomes clear when one attempts to employ classical logic to describe the intelligent processes in the brain. One soon discovers that while logic interpolates, mind extrapolates:

Our fascination with the elegance and effectiveness of logic in the realm of deduction is combined with our frustration at its impotence in the realm of induction.

What are logical interpolation and extrapolation? When, given two logical statements, we connect them by means of connectives, this in some abstract topological space amounts to interpolation between two logical points. Technically speaking we establish validity of an expression by computing some formula which must be well-formed. Logic doesn't extrapolate. Although the usage of the modal connectives possibly ⋄ and necessarily can be useful, they are not genuine computational connectives, and provide little support, if any, to the attempt to formulate and compute the unknown. In cognitive logic, circumstances are fundamentally different. A thought very often must run forward, into unknown and uncharted territory. The final destination point is not given but has to be found or created. The fundamental extrapolation mechanism enters the thought process, indicating that all attempts to solve the question of artificial intelligence in the framework of a logic which only interpolates are doomed to failure.

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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)

$\begin{array}{ll}{\Delta }_0\equiv 0,& \hfill {\Delta }_{\text{m}+1}\equiv \left({\Delta }_{\text{m}}+1\right),\hfill \end{array}$

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

$\begin{array}{l}\text{T}=\left\{\text{A}:\text{N}\models \text{A}\right\}\\ \cup \left\{{\Delta }_0\le \text{c},{\Delta }_1\le \text{c},{\Delta }_2\le \text{c},\dots \right\}.\end{array}$

Every finite subset S of T has a model, namely

${\text{N}}_{\text{s}}=(\mathbb{N},0,1,+,\cdot ,\text{m}),$

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

${\text{N}}_{\text{T}}=(\bar{\mathbb{N}},\bar{0},\bar{1},\overline{+},\bar{\cdot },\text{c}),$

and then $\bar{\text{N}}=(\bar{\mathbb{N}},\bar{0},\bar{1},\overline{+},\bar{\cdot })$ 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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Probability Theory

T. Rudas , in International Encyclopedia of Education (Third Edition), 2010

The Mathematical Theory of Probability: Axioms

The mathematical theory of probability is based on three fundamental assumptions or axioms. These are not derived or proved based on other considerations but are posited to capture the essence of probability. The axioms refer to the probabilities associated with events that may be observed. The observable events reflect the substantive problem at hand to which the probabilistic model is applied. For example, when 20 students take an exam, the relevant events may be associated with the number who pass it – that is, 0 or 1 or 2 or … or 20 if it is irrelevant, who passes and who does not. As the example illustrates – when the events of interest are selected – the irrelevant aspects of the problem at hand are disregarded. The fundamental requirement is that following the experiment, one has to be able to determine for every event whether or not that even occurred. If, for example, only Mary, John, and Jim passed the test, the event stating that three students passed has occurred and the others have not. Usually, the impossible event (that never occurs) and the sure event (that always occurs) are also included. There are certain operations that may be applied to events. The two most important operations are addition (union) and product (intersection). If A and B are two events, their sum

$A+B$

is also an event and it occurred if and only at least one of A and B occurred. The product

$AB$

is also an event and it occurred if and only if both of A and B occurred. With repeated application of these operations, one can generate events such as 'at least 5 passed'. When the intersection of two events is impossible,

$AB=Ø$

we say that the events exclude each other.

The above operations may be – and often have to be – applied infinitely many times, one after the other. For example, if one tosses a coin and the question of interest is how many times the coin needs to be tossed before the first tail occurs, this number does not have an upper bound in the sense that no larger value may be observed than the bound. Rudas (2004) gives an intuitive description of the precise meaning of applying the operations infinitely many times. One may think that such infinitely long experiments are not possible in practice, but many procedures and arguments in statistics are asymptotic. This implies that they give a good approximation for large samples but, strictly speaking, are only true for infinitely large samples.

The axioms relate these operations to probabilities.

First, the probability of any event is a number between 0 and 1

$0\le P\left(A\right)\le 1.$

Second, the probability of the impossible event is

$P\left(Ø\right)=0.$

Third, if two events exclude each other, then the probability of their sum is the sum of their probabilities

$P\left(A+B\right)=P\left(A\right)+P\left(B\right),\text{if}\text{<mi mathvariant="italic" is="true">AB</mi>}=Ø.$

This is, of course, the axiom of the greatest interest. It means that if, for example, no student can go to two classes during the same period, then the probability that a randomly selected (i.e., each student has the same chance of being selected) student in the school goes to the science class or the physical education (PE) class, is the sum of the probabilities that the student goes to the science class and that the student goes to the PE class. This is the additivity of probability, which is its fundamental property.

Note that all the assumed properties hold true for relative frequencies. This is because the axioms are selected to be the basis of a mathematical construction that is intuitively related to the relative frequencies. There are several properties that may be derived from the axioms. For example, that the probability of the certain event is 1 or that the property in the third axiom is also true for more than two events. For example, it is true for all subjects taught in school. In fact, usually one postulates a more general variant of the third axiom, namely that it is true for infinite sequences of events:

$P\left(A_1+A_2+\dots \right)=P\left(A_1\right)+P\left(A_2\right)+\dots ,\text{if}{A}_i{A}_j=Ø\text{for}\text{all}i\text{and}j$

where the meaning of the union of an infinite sequence of events is that at least one of them occurs and the meaning of the sum of an infinite sequence of probabilities is the limit of the finite partial sums (for details see Rudas, 2004).

A question of fundamental importance is whether the mathematical construction of probability that is based on the above axioms possesses the fundamental property which characterized the frequentist interpretation of probability. The question is of lesser importance in the case of subjective probability that is not necessarily supposed to follow any rules (Kopylov, 2008). It is possible to construct a model with an event that has a fixed probability and imagine an infinitely long sequence of independent observations of the experiment with the relative frequency of the event recalculated following each observation. One obtains a sequence of numbers indicating the relative frequency of the event during the first 1, 2, 3, 4, … repetitions of the experiment. Then, mathematical methods – based on the axioms – can be used to show that this sequence, indeed, converges (i.e., gets closer and closer as the number of repetitions increases) to the probability of the event. In other words, the basic conceptual assumption behind the frequentist view of probability is implied by the axioms.

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Discrete Boltzmann Equation Models for Mixtures

Victor Vedenyapin , ... Eugene Dulov , in Kinetic Boltzmann, Vlasov and Related Equations, 2011

12.9.4 Conclusions

1.

Mathematical theory is constructed for a movement of a big particle interacted physically or chemically with gas. Especially positive and negative photophoresia, electrophoresia, magnetophoresia, and thermophoresia got some explanation.

2.

Exact solutions are constructed for system of equations of rigid body motion. Are proposed ideal helix trajectories as asymptotes for any solution as time tends to infinity. This part of job was in principle carried out by Batisheva [20, 21, 197, 296, 299] [20] [21] [197] [296] [299] .

3.

Experiments of Felix Ehrenhaft were explained and spiral paths got their mathematical justification.

4.

Several new experiments are proposed. Especially to repeat Ehrenhaft ones and to compare diameters and steps of cones with exact formulae (12.9.6).

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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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Percolation Theory

V. Beffara , V. Sidoravicius , in Encyclopedia of Mathematical Physics, 2006

Introduction

Percolation as a mathematical theory was introduced by Broadbent and Hammersley (1957), as a stochastic way of modeling the flow of a fluid or gas through a porous medium of small channels which may or may not let gas or fluid pass. It is one of the simplest models exhibiting a phase transition, and the occurrence of a critical phenomenon is central to the appeal of percolation. Having truly applied origins, percolation has been used to model the fingering and spreading of oil in water, to estimate whether one can build nondefective integrated circuits, and to model the spread of infections and forest fires. From a mathematical point of view, percolation is attractive because it exhibits relations between probabilistic and algebraic/topological properties of graphs.

Figure 1. The behavior of θ(p) around the critical point (for bond percolation).

To make the mathematical construction of such a system of channels, take a graph $\mathcal{G}$ (which originally was taken as ${\mathbb{Z}}^d$ ), with vertex set $\mathcal{V}$ and edge set $\mathcal{E}$ , and make all the edges independently open (or passable) with probability p or closed (or blocked) with probability 1 − p. Write P _p for the corresponding probability measure on the set of configurations of open and closed edges – that model is called bond percolation. The collection of open edges thus forms a random subgraph of $\mathcal{G}$ , and the original question stated by Broadbent was whether the connected component of the origin in that subgraph is finite or infinite.

A path on $\mathcal{G}$ is a sequence v ₁, v ₂,… of vertices of $\mathcal{G}$ , such that for all i ≥ 1, v _i and v _i+1 are adjacent on $\mathcal{G}$ . A path is called open if all the edges {v _i , v _i+1} between successive vertices are open. The infiniteness of the cluster of the origin is equivalent to the existence of an unbounded open path starting from the origin.

There is an analogous model, called "site percolation," in which all edges are assumed to be passable, but the vertices are independently open or closed with probability p or 1 − p, respectively. An open path is then a path along which all vertices are open. Site percolation is more general than bond percolation in the sense that the existence of a path for bond percolation on a graph $\mathcal{G}$ is equivalent to the existence of a path for site percolation on the covering graph of $\mathcal{G}$ . However, site percolation on a given graph may not be equivalent to bond percolation on any other graph.

All graphs under consideration will be assumed to be connected, locally finite and quasitransitive. If $A,B\subset \mathcal{V}$ , then A ↔ B means that there exists an open path from some vertex of A to some vertex of B; by a slight abuse of notation, u ↔ v will stand for the existence of a path between sites u and v, that is, the event {u} ↔ {v}. The open cluster C{v} of the vertex v is the set of all open vertices which are connected to v by an open path:

$C(v)=\{u\epsilon \mathcal{V}:u\leftrightarrow v\}$

The central quantity of the percolation theory is the percolation probability:

$\theta (p):=P_p\{0\leftrightarrow \infty \}=P_p\left\{\right|C(0)|=\infty \}$

The most important property of the percolation model is that it exhibits a phase transition, that is, there exists a threshold value p _c ∈ [0, 1], such that the global behavior of the system is substantially different in the two regions p < p _c and p > p _c. To make this precise, observe that θ is a nondecreasing function. This can be seen using Hammersley's joint construction of percolation systems for all p ∈ [0, 1] on $\mathcal{G}$ : let $\left\{U\left(v\right),v\epsilon \mathcal{V}\right\}$ be independent random variables, uniform in [0,1]. Declare v to be p-open if U(v) ≤ p, otherwise it is declared p-closed. The configuration of p-open vertices has the distribution P _p for each p ∈ [0, 1]. The collection of p-open vertices is nondecreasing in p, and therefore θ(p) is nondecreasing as well. Clearly, θ(0) = 0 and θ(1) = 1 ( Figure 1 ).

The critical probability is defined as

$p_c:=p_c(\mathcal{G})=sup\{p:\theta (p)=0\}$

By definition, when p < p _c, the open cluster of the origin is P _p -a.s. finite; hence, all the clusters are also finite. On the other hand, for p > p _c there is a strictly positive P _p -probability that the cluster of the origin is infinite. Thus, from Kolmogorov's zero–one law it follows that

$P_p\left\{\right|C(v)|=\infty \text{for some}v\epsilon \mathcal{V}\}=1\text{for}p>p_c$

Therefore, if the intervals [0, p _c) and (p _c, 1] are both nonempty, there is a phase transition at p _c.

Using a so-called Peierls argument it is easy to see that $p_{\text{c}}\left(\mathcal{G}\right)>0$ for any graph $\mathcal{G}$ of bounded degree. On the other hand, Hammersley proved that $p_{\text{c}}\left({\mathbb{Z}}^d\right)<1$ for bond percolation as soon as d ≥ 2, and a similar argument works for site percolation and various periodic graphs as well. But for some graphs $\mathcal{G}$ , it is not so easy to show that $p_{\text{c}}\left(\mathcal{G}\right)<1$ . One says that the system is in the subcritical (resp. supercritical) phase if p < p _c (resp. p > p _c).

It was one of the most remarkable moments in the history of percolation when Kesten (1980) proved, based on results by Harris, Russo, Seymour and Welsh, that the critical parameter for bond percolation on ${\mathbb{Z}}^2$ is equal to 1/2. Nevertheless, the exact value of $p_{\text{c}}\left(\mathcal{G}\right)$ is known only for a handful of graphs, all of them periodic and two dimensional – see below.

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Introduction to the Mathematical Theory of Kinetic Equations

Victor Vedenyapin , ... Eugene Dulov , in Kinetic Boltzmann, Vlasov and Related Equations, 2011

Publisher Summary

This chapter discusses the mathematical theory of kinetic equations, the Boltzmann equation, the Vlasov–Poisson, and Vlasov–Maxwell systems, and describes their mathematical structure and introduces weak and classical solutions. To outline the importance of these classical results for modern applications, it discusses about semiconductor modeling and introduces a number of open relevant fundamental problems known for Vlasov–Poisson and Vlasov–Maxwell systems. The Vlasov equation describes the evolution of the system of particles in the force field F(t,x,p), which depends on time t, position x, and momentum p. For every particle with index j, a distinctive peculiarity of mathematical analysis of semiconductors is its connection with hierarchy of models of transport of charged particles in various mediums. It is explained by that transport processes have a practical interest for various scales of length and time generated by various physical effects. In particular, on the very small scale, using microelectronic technology, quantum, and/or kinetic effects are important.

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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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Preliminaries

N. Balakrishnan , ... Fotios S. Milienos , in Reliability Analysis and Plans for Successive Testing, 2021

2.2 Combinatorial analysis and generating functions

Combinatorial analysis, the mathematical theory of counting, plays a prominent role to the probability and statistical theory; many probability problems, especially those which related to experiments with finite sample spaces, can be readily solved only by counting the number of different ways that specific events can occur (e.g., Feller, 1968, Chapter 2; Charalambides, 2005, Chapter 1). This is the case in many of the problems studied in the current book, wherein the assessment of the condition of an equipment is usually described by a discrete random variable. For example, inspecting a specific system for n times, and assessing whether it works perfectly (success; 1) or not (failure; 0), the deduced sequence of outcomes belongs to the set

$\Omega =\{(x_1,\dots ,x_n):x_i\in \{0,1\},i=1,\dots ,n\},$

where $x_i$ denotes the outcome of the ith inspection, for $i=1,\dots ,n$ ; the number of different elements in Ω, i.e., the cardinality of Ω (symb. $|\Omega |$ ), is obviously equal to $|\Omega |=2^n$ .

Among the $2^n$ elements of Ω, there are $\left(\begin{array}{c}n\\ k\end{array}\right)$ (binomial coefficient) elements in which k inspections resulted in successes and the remaining $n-k$ , in failures, with $k⩽n$ . However, among the previous $\left(\begin{array}{c}n\\ k\end{array}\right)$ elements of Ω only $n-k+1$ of them have all the k successes next to each other, forming an uninterrupted sequence of successes of length k. For example, in the sequence 0101111000 the five ones form two uninterrupted sequences of successes, one of length four (starting from the fourth and ending at the seventh position), and one of length one (starting and ending at the second position). In probability and statistics, this kind of uninterrupted sequences of the same outcome is usually termed as run of specific length or size. Therefore, in the above sequence, apart from the two runs of successes, we also observe three runs of failures (zeros) of lengths one, one and three.

Given that the number of successes among the n inspections equals k, we know that only $n-k+1$ of the $2^n$ elements of Ω, provide us with a success run of length k; however, of special importance is sometimes the problem of counting the number of elements in Ω, wherein the maximum success run length is less than a specific value. This problem could be solved by the aid of probability generating functions. Let X be a nonnegative integer valued random variable; then the probability generating function of X (symb. $P(t)$ ), is defined by

$P(t)=E(t^X)=\sum _{j=0}^{\infty }P(X=j)t^j.$

One important property is that the probability generating function, of the sum of independent random variables $X_1,\dots ,X_n$ is given by the product of the probability generating functions of $X_i^{\prime }s$ . This means that

$P_S(t)=P_1(t)P_2(t)\cdots P_n(t),$

where $P_S(t)$ and $P_i(t),i=1,\dots ,n$ , are the probability generating functions of the sum $S=X_1+\dots +X_n$ , and $X_i,i=1,\dots ,n$ , respectively.

Suppose now that the number of successes among the n inspections equals k. The question is how to count the number of ways that these k successes can be distributed to the inspections such that no more than m being consecutive, with $m⩽k$ . To do so, it might be helpful to mention that the $n-k$ failures provide $n-k+1$ cells or urns, where the k successes should be distributed (see also Fig. 2.1). However, the capacity of each cell should be less than m, in order to have a maximum success run length less than m. Hence, suppose that each of the $n-k+1$ urns contains $m+1$ balls numbered $0,1,\dots ,m$ . A ball is drawn at random and independently from each urn and let $X_i$ denote the number on ball chosen from the i urn, with $i=1,\dots ,n-k+1$ . This means that $X_i\in \{0,1,\dots ,m\}$ for $i=1,\dots ,n-k+1$ and, the probability generating function of $X_i$ is given by

Figure 2.1. The n −k + 1 hypothesized cells or urns given that the number of successes among n inspections, equals k.

$P_i(t)=\sum _{j=0}^m\frac{1}{m+1}{t}^j=\frac{1}{m+1}\frac{1-t^{m+1}}{1-t},$

for $i=1,\dots ,n-k+1$ . So, in order to answer our question, we should count the number of different solutions of the equation

$X_1+\dots +X_{n-k+1}=k,$

which is given by the coefficient of $x^k$ in

${(1+x+\dots +x^m)}^{n-k+1}$

(see Charalambides, 2005). Thus we have

${(1+x+\dots +x^m)}^{n-k+1}={(1-x^{m+1})}^{n-k+1}{(1-x)}^{-(n-k+1)}$

and using the binomial theorem and its generalizations we get that the number of ways that these k successes can be distributed to the n inspections such that no more than m being consecutive (i.e., the coefficient of $x^k$ ) is given by

(2.2.1) $N(k,m,n)=\sum _{z=0}^{n-k+1}{(-1)}^z\left(\begin{array}{c}n-k+1\\ z\end{array}\right)\left(\begin{array}{c}n-z(m+1)\\ n-k\end{array}\right).$

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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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The Mathematical Theory Of Bridge Pdf

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