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The quest for optimal/stable paths in graphs concerns a few practical or theoretical areas. Taking part in the quest, this paper adopts an abstract, general, equilibrium-oriented approach: it uses (quasi-arbitrary) arc-labelled digraphs, and assumes little about the structure of the sought paths and the definition of equilibrium, i.e. optimality/stability. The paper gives both a sufficient condition and a necessary condition for equilibrium existence for every ˵graph″, pinpoints the difference between these conditions, and shows coincidence when optimality relates to a total order. These results are applied to network routing. [ABSTRACT FROM AUTHOR]

All things began in order, so shall they end, and so shall they begin again; according to the ordainer of order and mystical mathematics of the city of heaven. Until now, our discussion has concerned isolated atoms and clusters of very few atoms. Considering that a single cubic centimetre of a typical piece of condensed matter, a block of stainless steel for example, contains approximately 1023 atoms, we see that there is still a long way to go in the description of actual substances. We must now inquire how large numbers of atoms are arranged with respect to one another, so as to produce the great variety of known materials. It will transpire that the observed arrangements are dictated both by the forces between atoms and the prevailing physical conditions. Before going into such details, however, it will be useful to examine the simple geometrical aspects of the question. It is frequently adequate to regard atoms as being spherical. The two-dimensional equivalent of a sphere is a circle, so we can begin by considering the arrangement of circles on a piece of paper. The circles could be drawn at random positions, well spread out and with close encounters occurring only by occasional chance. This is a reasonable representation of the instantaneous situation in a simple monatomic gas. A variant of this drawing would replace the single circles by small groups of circles, and we would then have a molecular gas. [ABSTRACT FROM AUTHOR]

Introduction During one of our very last discussions Wassily Leontief asked me: “What are you doing these days?” I replied that I reconcile input-output analysis and neoclassical economics. He leant back, thought, looked me straight in the eyes, and said: “Should be easy.” Yet input-output analysis and neoclassical economics seem hard to mix. The resentment between the two schools of economics is a two-way affair. Neoclassical economists consider input-output analysis a futile exercise in central planning. The relationship between the delivery of a bill of final goods and its requirements in terms of gross output and factor inputs is considered mechanical, with little or no attention paid to the role of the price mechanism in the choice of techniques (Leontief, 1941). True, input-output analysis is used to relate prices to factor costs, but here too the analysis is considered mechanical as input-output coefficients are presumed to be fixed. To make things worse, the quantity and value analyses are perceived to be disjunct, with no interaction between supply and demand. Conversely, input-output economists consider neoclassical economics a futile exercise in marginal analysis that fails to grasp the underlying structure of the economy. Firms supply up to the point that marginal revenue equals marginal cost and set the price accordingly. But does not marginal cost depend on all prices in the system, including the one of the product under consideration? And if the answer is yes, should not we take into account the interindustry relations, i.e. apply input-output analysis? [ABSTRACT FROM AUTHOR]

The group of writers labelled MMM consists of Metzler (1960 [1973: Chapter 8]), Meade (1951a and 1951b), and Mundell (1968). The first two wrote about 1950. Mundell was later but he winds up this era with the beginnings of what is known as the Mundell-Fleming model. Mundell was concerned primarily with policy recommendations, and the Mundell-Fleming literature, which I do not propose to survey extensively, is generally interpreted as being based on a text-book type of IS-LM analysis. Meade's important work has for some reason dropped out of the limelight; it is, like most of his work, too detailed and “complete” to serve as a useful hand-book. Presumably, also, the fact that it puts “wages and incomes” policy on the same footing as “financial” (fiscal plus monetary) policy made it somewhat alien to the spirit of western, particularly anglophone, economics in the post-World War II period. Metzler's work is on an entirely different analytical footing from the others. It is called “Keynesian” but is much more Wicksellian. Unlike the others, it is not at all policy-oriented. The role of the central bank is simply to buy and sell foreign exchange in order to peg the exchange rate; it is therefore applicable to any regime where exchange rates are fixed and the money supply varies with the balance of payments. I shall deal with it first. Metzler Metzler's contribution in this area is the paper, “The Process of International Adjustment under Conditions of Full Employment: A Keynesian View.” [ABSTRACT FROM AUTHOR]

Keynes himself, as noted previously, nowhere presented a formal, precise statement of the application of the General theory model to an open economy. The same point is made, inter alia, by Nurkse (1947a [1947:264]). (And when Moggridge (1986: 57) defines the topics Keynes dealt with over time, he adopts a list drawn up by Williamson in the policy-oriented context of monetary reform.) When the applications did come, they came in a great rush. I shall not cover all of them, but have chosen from among them on the basis of seminal or path-breaking quality in the sense of stimulating other research and applications, frequent citations in the literature, or inherent interest. We find the following types of works: Surveys. These include those of Metzler, in his survey article (1948 [1973: Chapter 1]); Iversen (1935), assessing where the theory of international economics is, or should be, going after Keynes; and Haberler (1937 [1941]), in the international portions of Prosperity and depression. Nurkse (1944) includes a brief and cogent summary statement. More narrow direct applications of the keynesian theory to the theory of international adjustment. The two striking examples here are Machlup (1943) and the two Metzler papers (1942a and 1942b [1973: Chapters 10 and 2]), which I have labeled “Early Metzler”. Both are presented as explicitly analytical works, neither econometric nor prescriptive. Both deal with the dynamics of the adjustment; Metzler, in particular, places great emphasis on the discussion of stability conditions. […] [ABSTRACT FROM AUTHOR]

A science experiment related to equilibrium is presented to demonstrate that a state of balance between opposing forces in nature is called equilibrium. A list of materials is provided, along with preparation and safety considerations, specific procedures, and discussion topics. It is concluded that the expanding air drives some of the air out of the test tube, when the water in the test tube is heated.

In this paper, we establish sufficient conditions for simplifying the checking that an isolated equilibrium point of autonomous neural network is exponentially stable. Meanwhile, these conditions are generalized to convergent properties of all equilibrium points of arbitrary order neural network in an open region of the state space. The explicit lower bound on the exponential convergence rate to an equilibrium point is also estimated. [ABSTRACT FROM AUTHOR]

Chapter 5 of the book "Equations of State and PVT Analysis," by Tarek Ahmed is presented. It discusses the concepts of equilibrium ratios and flash calculations in determining the state of equilibrium in the hydrocarbon liquid phase and gas phase that occur in petroleum production and notes that the chief variables are system temperature, system pressure and composition. It presents methods like the correlations of Wilson's and Standing's for predicting the equilibrium ratios of hydrocarbon mixtures.

Life and writings Alfred Marshall (1842–1924) was not among the protagonists of the 1871–4 ‘marginalist revolution’: his first major writings belong to the end of the 1870s, and his main contribution, his Principles of economics, appeared in 1890, nearly two decades after the works of Jevons, Menger and Walras. Marshall himself was averse to considering the new road taken by economic analysis as a ‘revolution’ or a clear-cut break with the past: in his opinion, this was instead a step forward, although certainly an important one, relative to the classical economists’ (in particular Ricardo's and the Ricardians’) approach. Indeed, his personal contribution, in his own opinion, consisted in the synthesis between the great tradition inherited from the past and the new yeast of the subjective approach. Yet, we must recognise that Marshall contributed more than anyone else, possibly at least in part against his own intentions, to ‘shunt the car of economic science’ in the direction of that approach (which Hicks, Stigler and Samuelson preferred to call ‘neoclassical’, rather than ‘marginalist’ or ‘subjectivist’, in order to stress that the turn-around implied an important element of continuity with the past) which still today dominates the teaching and thinking of economists all over the world. Marshall was born in London, on 26 July 1842, to a modest bourgeois family. His father, authoritarian if not tyrannical within the family, was a modest clerk of the Bank of England. [ABSTRACT FROM AUTHOR]

The invisible hand of the market Among contemporary economists the idea is widespread that general economic equilibrium theory is to be identified with theory tout court, and is to be taken as a yardstick by which any other theory can be considered as a particular case. To anyone sharing this viewpoint, the history of economic thought appears as the path of progressive development and consolidation of this theory. Along this route, in interpreting classical economists the economic issue they dealt with is identified in the functioning of the ‘invisible hand of the market’. The latter would ensure not simply a sufficiently regular working of the economy but, more than this, a systematic tendency towards an equilibrium with perfect equality between supply and demand for each commodity (market clearing), even in the presence of many commodities and many economic agents. As a matter of fact, such an extreme idea cannot be attributed to the economists of the classical period; it was originally developed by only one of the ‘schools’ which concurred in the so-called marginalist revolution, the ‘Lausanne school’, founded by Léon Walras. In order to clarify this point, let us first consider which elements enter the view of the economic system underlying the general economic equilibrium approach; we shall then see whether these elements were present among classical economists or in the other marginalist ‘schools’. [ABSTRACT FROM AUTHOR]

First writings: money and banking Piero Sraffa (1898–1983) is one of the leading intellectuals of the twentieth century: not only for his strictly economic contributions, but also for his influence on others, from Antonio Gramsci to Ludwig Wittgenstein. In the field of economic sciences, Sraffa's cultural project is an extremely ambitious one: ‘to shunt the car of economic science’ in a direction opposite to that indicated by Jevons, one of the protagonists of the ‘marginalist revolution’. With his writings, in fact, Sraffa aims to expose the weak points of the marginalist approach as developed by, for instance, Jevons, Menger, Walras, Marshall, Böhm-Bawerk, Hayek and Pigou and at the same time to repropose the classical approach of Adam Smith, David Ricardo and, in certain respects, Karl Marx. In order to better understand its nature and impact, it may be useful to follow the gradual development of this cultural project, from the first writings on money and banking to the edition of Ricardo's works and the small but dense volume on Production of commodities by means of commodities (1960). Piero Sraffa was born in Turin on 5 August 1898. His father, Angelo Sraffa (1865–1937), was a well-known professor of commercial law and subsequently (from 1917 to 1926) Dean of the Bocconi University in Milan. Following his father as he moved from one university seat to another, the young Sraffa studied in Parma, Milan and Turin. [ABSTRACT FROM AUTHOR]

Life and writings John Maynard Keynes was born in Cambridge, England, on 5 June 1883, the first son of John Neville Keynes (1852–1949) and Florence Brown (1861–1958). His father, a pupil of Marshall, was a scholar of logic and economics, author of The scope and method of political economy (1891), but had preferred an administrative career to prospects of a professorship, reaching the top of the Cambridge University administration; his mother was one of the first female graduates of that university, and the first woman to be elected mayor of Cambridge. Maynard's curriculum was in keeping with the highest standards of the bourgeoisie: secondary school at Eton, university at King's College, Cambridge. Here he studied mathematics and classical humanities; he was also elected into the elitist secret society of the Apostles, devoted to ‘the pursuit of truth’. In a generation shortly preceding that of Keynes, another Apostle, the philosopher George Edward Moore (1873–1958), had rejected the utilitarian identification between ‘to be good’ and ‘to do good’, proposing an ethics of inner self-searching for truth and personal coherence. In the climate of cultural renewal characterising the Edwardian period, to Keynes and his friends this meant a radical reappraisal of Victorian culture and ethics, manifested also in their personal conduct (marked by extreme intellectualism and the pursuit of aesthetic pleasures), while, departing from Moore, they rejected the idea of general rules of conduct, substituted by confidence in the ability of the ‘elect’ to evaluate case by case what the right behaviour would be. [ABSTRACT FROM AUTHOR]

The ‘marginalist revolution’: an overview The term ‘marginalist revolution’ is commonly utilised to indicate a sudden change of direction in economic science, with the abandonment of the classical – and, more precisely, Ricardian – approach, and the shift to a new approach based on a subjective theory of value and the analytical notion of marginal utility. The outbreak of the ‘revolution’ is commonly located in the years between 1871 and 1874, when the main writings were published of the leaders of the Austrian marginalist school, Carl Menger (1840–1921), of the British school, William Stanley Jevons (1835–82), and of the French (Lausanne) school, Léon Walras (1834–1910). In fact, 1871 saw the appearance of both the Principles of pure economics by Menger and The theory of political economy by Jevons, while Walras brought out his Elements of pure economics in 1874. Let us, however, once again reiterate that the ‘marginalist revolution’ had had important precursors, as we will see again below. Moreover, the differences between the Austrian imputation approach, the French general economic equilibrium and Marshallian partial equilibriums were quite important, as far as both method and the basic view of the functioning of the economy were concerned. [ABSTRACT FROM AUTHOR]

Carl menger The founder of the ‘Austrian school’ was born in Poland, then part of the Austro-Hungarian empire, in 1840. He attended university in Vienna and Prague, and went on to take his doctorate in Cracow. His first job was as a journalist, and by 1871, when he published the book he owes his fame to, the Principles of political economy, he had become a civil servant. Thanks to this book he had a rapid academic career, obtaining the Habilitation (the qualification to teach) and a teaching appointment (Privatdozent), and by 1873 he was already professor. In 1876–8 he was made tutor to Crown Prince Rudolf of Austria, and from 1878 to 1903 he held the chair of political economy at the University of Vienna. Until his death, in 1921, he worked on a second edition of his Principles, which was published posthumously by his son Karl in 1923. The Principles of political economy are hardly what a modern reader would expect of a key text for the marginalist approach, and the differences to Jevons and Walras are significant. Menger had studied law, which, in the continental European tradition, implied an approach with a strong emphasis on history and great pains taken over the definition/illustration of concepts, often proving pedantic and prolix. Menger thus appeared quite distant from the project – shared by Jevons and Walras – to construct economic theory as a quantitative science, to be developed in mathematical terms. [ABSTRACT FROM AUTHOR]

The debates of the time In the century stretching between William Petty's writings and Adam Smith's, economic thinking proceeded in many directions. It is impossible here to consider them all with the attention they deserve: some authors and research currents will simply be ignored, others will receive only brief mention, while only a few will be treated in more detail. It is important to stress just how rich the debate on economic phenomena was during this period, moving forward on various planes, linking up with ethical or philosophical aspects in general or more immediate issues of political choices, and constituting the background from which certain personalities emerged to prominence from the point of view of our account. The contributions of the most important authors would be difficult if not impossible to understand if wholly isolated from the cultural context in which they took shape, and which they helped to enrich. This holds true for the period here considered in a measure that may be difficult to appreciate for those accustomed to the extreme specialisation in research characterising our times. Actually, in the seventeenth and eighteenth centuries the figure of the economist was still far from clearly defined: reflections on economic phenomena were part of general reflections on society and man, and the same authors would in the course of time range over a vast field of issues. [ABSTRACT FROM AUTHOR]

Various multi-agent models of wealth distributions defined by microscopic laws regulating the trades, with or without a saving criterion, are reviewed. We discuss and clarify the equilibrium properties of the model with constant global saving propensity, resulting in Gamma distributions, and their equivalence to the Maxwell-Boltzmann kinetic energy distribution for a system of molecules in an effective number of dimensions Dλ, related to the saving propensity λ [M. Patriarca, A. Chakraborti, and K. Kaski, Phys. Rev. E 70 (2004) 016104]. We use these results to analyze the model in which the individual saving propensities of the agents are quenched random variables, and the tail of the equilibrium wealth distribution exhibits a Pareto law f(x) ∝ x−α−1 with an exponent α = 1 [A. Chatterjee, B. K. Chakrabarti, and S. S. Manna, Physica Scripta T106 (2003) 367]. Here, we show that the observed Pareto power law can be explained as arising from the overlap of the Maxwell-Boltzmann distributions associated to the various agents, which reach an equilibrium state characterized by their individual Gamma distributions. We also consider the influence of different types of saving propensity distributions on the equilibrium state. [ABSTRACT FROM AUTHOR]

I believe that I have succeeded in discovering the force (Kraft), also the law of the effect of this force, that makes possible the coexistence of the human race and that governs inexorably the progress of mankind. And just as the discoveries of Copernicus have made it possible to determine the paths of the planets for any future time, I believe that my discoveries enable me to point out to any man with unfailing certainty the path he must follow in order to accomplish the purpose of his life. The truth is, most persons, not excepting professional economists, are satisfied with very hazy notions. How few scholars of the literary and historical type retain from their study of mechanics an adequate notion of force!” Most people with some academic training in economics are aware that the rise of the theory that commands the greatest allegiance in the United States (which we shall call neoclassical economics) was located in the 1870s in several European countries. A little more familiarity with the conventional histories of economic thought will foster the impression that neoclassical theory was “simultaneously discovered” by an Englishman (William Stanley Jevons), a Frenchman (Leon Walras), and an Austrian (Carl Menger), and that after its improvement by a host of others, it eventually displaced all other competing schools of thought in those countries. [ABSTRACT FROM AUTHOR]

What became known as the keynesian theory of international payments was not, to my knowledge, ever spelled out by Keynes. The marginal propensity to import, and then the multiplier, began to appear in the late 1930s and early 1940s, associated with the names of Harrod (1933), Machlup (1939, 1943), Metzler (1942a, 1942b), Neisser and Modigliani (1953), Paish (1936), and others. A detailed account of these is presented in Chapters 14 and 15 on “the keynesians.” The full-blown integration of keynesian macroeconomic theory with endogenous capital mobility, monetary changes, and some notion of balance of payments equilibrium (even of a short- or medium-run nature) did not appear until the 1950s, with Meade, Mundell, Fleming, and, with great qualification, Metzler, whom I discuss in Chapter 16. Keynes himself, after the Treatise and the Macmillan Committee testimony, which were roughly coeval, wrote nothing of importance on international economics except for the proposals and correspondence regarding the postwar monetary plans. These, however, constitute a major body of work expressing his views on the international monetary system and the problems he thought were likely to arise in the postwar period. While they are less formally expressed, more diffuse, and more “politically” oriented than the earlier writings I have discussed, they do reveal an analytical stance which is highly recognizable and worth exploring. In the letters and drafts (and redrafts) of plans for the postwar monetary system (Keynes 1940–4 [1980]) several themes are prevalent throughout. [ABSTRACT FROM AUTHOR]

A preface for the book "Drug-Acceptor Interactions: Modeling Theoretical Tools to Test and Evaluate Experimental Equilibrium Effects" is presented.