Formal Identities Research Articles

Variations on a theme by Mark Kac

Mark Kac's theorem on the mean recurrence time in a stationary stochastic process in discrete time with discrete states is taken as the starting point for a series of variations, most of which are formulated in terms of 0–1 processes. Whereas the original theorem deals with the mean recurrence time of a given state under the condition that the state is realized at time 0, this condition is dropped in part of the variations; two others refer to the variance of the recurrence time and two to the Poincare cycle of a dynamical system. Most variations consist in inequalities and formal identities for the mean first-arrival time and subsequent recurrence times for the given state.

First layer formulas for characters of 𝑆𝐿(𝑛,𝐶)

Some problems concerning the decomposition of certain characters of S L ( n , C ) SL(n,\,{\mathbf {C}}) are studied from a combinatorial point of view. The specific characters considered include those of the exterior and symmetric algebras of the adjoint representation and the Euler characteristic of Hanlon’s so-called “Macdonald complex.” A general recursion is given for computing the irreducible decomposition of these characters. The recursion is explicitly solved for the first layer representations, which are the irreducible representations corresponding to partitions of n n . In the case of the exterior algebra, this settles a conjecture of Gupta and Hanlon. A further application of the recursion is used to give a family of formal Laurent series identities that generalize the (equal parameter) q q -Dyson Theorem.

Can systems theory generate testable hypotheses?: From talcott parsons to living systems theory

AbstractIt is useful for systems theories to distinguish between abstracted and concrete systems. Parsons and his followers have emphasized the first sort; Miller and others have emphasized the second. Scientific work can be conducted in terms of either sort of system, but concrete systems theory has elicited more quantitative research. In addition it brings social science closer to the natural sciences. Abstracted and concrete theories also differ in their categorization of subsystem processes. Recognition of the similarity of subsystems at various biological and social levels of systems facilitates the development of cross‐level hypotheses and quantitative research to test the existence of formal identities across levels.Few cross‐level formal identities have yet been investigated empirically, but such interdisciplinary fundamental research can confirm or disconfirm very powerful integrative systems principles. In recent years interest in such cross‐level research has been rising. Ten cross‐level studies are described.

The Incidence of a Tax on Pure Rent: The Old Reason for the Old Answer

The purpose of this note is to clarify the formal identities linking the problems analyzed by Barro (1974), Feldstein (1977), and Calvo, Kotlikoff, and Rodriguez (1979) (henceforth CKR) and to reaffirm the traditional proposition that in both the short run and the long run a tax on pure rent is never shifted, provided one adopts the traditional procedure in tax theory of analyzing the incidence of a compensated tax. Any uncompensated tax is equivalent to a compensated tax together with a lump-sum redistribution of income, and there is really nothing surprising about the fact that lump-sum redistributions of income may alter relative prices, as the literature on the transfer problem in international trade theory demonstrates. Actually, situations can be found (e.g., Gorman 1953; Barro 1974; Calvo et al. 1979, p. 874, n. 7) in which lump-sum redistributions of income do not affect relative prices. Even when they do, the incidence effect is often small relative to the size of the redistribution. The analyses of CKR and Feldstein give the impression that the classical proposition on the incidence of a tax on pure rent does not apply in a dynamic economy: As Feldstein points out, the classical proposition that a tax on pure rent is unshifted requires that the supplies of nonland factors of production are unaltered by the introduction of the tax on land rents. This static assumption of fixed factor supplies is inappropriate to the analysis of tax incidence in a dynamic economy in which the supply of capital reflects the economic choice between consumption and saving (Calvo et al. 1979, p. 869).

Adding and using structure in general arrays

Examples of pre-APL general arrays suggest uses for APL II. Two structure-creating functions are proposed. Using numerical or logical control arguments, they partition a level of an array regularly or irregularly, with or without reuse of items. The resulting arrays reduce string search to item search in two distinct ways. They can be used on arrays of any rank to simplify search for abstractly defined patterns of relationships between nearby elements or to construct a tessellation by repetition of a given pattern. The relationships between these techniques and programming techniques using operators is noted. Formal identities relate the two proposed functions to APL I functions and to each other.

Gauge dependence in the Yang-MillsSmatrix

There are formal proofs that $S$-matrix elements are gauge invariant in gauge field theories. These proofs break down unless infrared singularities of the $S$ matrix are handled with care. In a similar fashion, formal on-shell Ward identities can break down. Dimensional regularization of infrared singularities seems to preserve all formal relations.

The nature of living systems

The nature of living systems

Binomial posets, Möbius inversion, and permutation enumeration

A unified method is presented for enumerating permutations of sets and multisets with various conditions on their descents, inversions, etc. We first prove several formal identities involving Möbius functions associated with binomial posets. We then show that for certain binomial posets these Möbius functions are related to problems in permutation enumeration. Thus, for instance, we can explain “why” the exponential generating function for alternating permutations has the simple form ( 1 + sin x) ( cos x) . We can also clarify the reason for the ubiquitous appearance of e x in connection with permutations of sets, and of ξ( s) in connection with permutations of multisets.

The nature of living systems

The nature of living systems

Living systems. II. The organism.

General systems behavior theory is concerned with seven levels of living systems-cell, organ, organism, group, organization, society, and supranational system. The following article is an exposition of the basic concepts in this integrative theoretical approach. It is a condensation of a more detailed statement (Miller, 1965a; 1965b; 1965c). The second article is an analysis in terms of this conceptual system of present knowledge concerning one level of living system-the organism. In order to emphasize the cross-level formal identities among levels of living systems, a major consideration of general behavior systems theory, this article follows exactly the same outline, with identical subheadings and section numbers, as other articles written by the author. These deal with the lower levels of living systems-cell and organ (Miller, 1971a)-as well as higher levels-group (Miller, 1971b), organization (Miller, 1972), society, and supranational system. (All the articles will be published together as chapters of the author's forthcoming book, Living Systems.) The primary intent of this article, in addition to its analysis of the content it covers, is to show that the structures and processes of organisms and of living systems at all the levels are directly comparable. Since anatomists and physiologists are usually laymen in organization theory or international relations, psychologists are commonly laymen in economics, and social scientists are ordinarily laymen in cellular biology, all parts of the book, including the two following articles, are necessarily written for intelligent laymen rather than experts, even though the articles deal with many technical topics. Some statements in them will seem to experts to be too elementary to be worth repeating. If a fact is fundamental and may not be known to specialists in other fields it is stated here, even if it is elementary to the experts. The complex division of labor of modern science, often characterized by pluralistic insularity, requires this. The multitude of detailed and specialized experiments and studies that have been carried out provide the substance of the scientific investigation of organisms. Their findings constitute the trees. But an overview of these results and of the relationships among them-a view of the forest-is also essential. Such a telescopic rather than a microscopic view may suggest the proper balance for research on various aspects of organisms and clarify the priorities for future efforts. The Following articles view organisms broadly. They deal with all living systems at this level-protistans, fungi, and plants, as well as animals. Why such a range? To make manifest the continuity or organismic life in structure, process, and history (including evolution). To demonstrate how structure and process go together, so that anatomists, physiologists, and psychologists undestand the common task they share and how each field can enrich the others. Most of the ideas presented in these articles are not new even though the conceptual integration is original with the author. The concepts are derived from many aspects of science and have been developed by many workers, including various former associates of the author at the Universities of Chicago and Michigan, and systems scientists in many other places. The articles necessarily select for discussion only a few researches out of the vast published repertoire, and this selection has necessarily been arbitrary. Experts in each special field might agree on other studies as more important. Some of the author's statements may be wrong and his analysis ill advised. It so he would appreciate corrections-it is hard to cover such a wide range and still make no errors. The constant reference in the second article to cross-level hypotheses stated in the first article has the purpose of showing that propositions possibly valid at other levels may also apply to organism. At each level there are scientists who apply system theory in their investigations. They are systems theorists but not necessarily general systems theorists. They are general systems theorists only if they accept the more daring and controversial position that-though every living system and every level is obviously unique-there are important formal identities of large generality across levels. Such cross-level similarities can potentially be evaluated quantitatively, applying the same model to data collected at two or more levels. This possibility is the chief reason why the author has used the same outline with identically numbered sections to analyze the present knowledge about each of the seven levels of living systems. The following survey of what is known about organisms as systems, therefore, is to be read as a single segment of a larger, integrated whole.

Living systems. I. The nature of living systems.

General systems behavior theory is concerned with seven levels of living systems-cell, organ, organism, group, organization, society, and supranational system. The following article is an exposition of the basic concepts in this integrative theoretical approach. It is a condensation of a more detailed statement (Miller, 1965a; 1965b; 1965c). The second article is an analysis in terms of this conceptual system of present knowledge concerning one level of living system-the organism. In order to emphasize the cross-level formal identities among levels of living systems, a major consideration of general behavior systems theory, this article follows exactly the same outline, with identical subheadings and section numbers, as other articles written by the author. These deal with the lower levels of living systems-cell and organ (Miller, 1971a)-as well as higher levels-group (Miller, 1971b), organization (Miller, 1972), society, and supranational system. (All the articles will be published together as chapters of the author's forthcoming book, Living Systems.) The primary intent of this article, in addition to its analysis of the content it covers, is to show that the structures and processes of organisms and of living systems at all the levels are directly comparable. Since anatomists and physiologists are usually laymen in organization theory or international relations, psychologists are commonly laymen in economics, and social scientists are ordinarily laymen in cellular biology, all parts of the book, including the two following articles, are necessarily written for intelligent laymen rather than experts, even though the articles deal with many technical topics. Some statements in them will seem to experts to be too elementary to be worth repeating. If a fact is fundamental and may not be known to specialists in other fields it is stated here, even if it is elementary to the experts. The complex division of labor of modern science, often characterized by pluralistic insularity, requires this. The multitude of detailed and specialized experiments and studies that have been carried out provide the substance of the scientific investigation of organisms. Their findings constitute the trees. But an overview of these results and of the relationships among them-a view of the forest-is also essential. Such a telescopic rather than a microscopic view may suggest the proper balance for research on various aspects of organisms and clarify the priorities for future efforts. The Following articles view organisms broadly. They deal with all living systems at this level-protistans, fungi, and plants, as well as animals. Why such a range? To make manifest the continuity or organismic life in structure, process, and history (including evolution). To demonstrate how structure and process go together, so that anatomists, physiologists, and psychologists undestand the common task they share and how each field can enrich the others. Most of the ideas presented in these articles are not new even though the conceptual integration is original with the author. The concepts are derived from many aspects of science and have been developed by many workers, including various former associates of the author at the Universities of Chicago and Michigan, and systems scientists in many other places. The articles necessarily select for discussion only a few researches out of the vast published repertoire, and this selection has necessarily been arbitrary. Experts in each special field might agree on other studies as more important. Some of the author's statements may be wrong and his analysis ill advised. It so he would appreciate corrections-it is hard to cover such a wide range and still make no errors. The constant reference in the second article to cross-level hypotheses stated in the first article has the purpose of showing that propositions possibly valid at other levels may also apply to organism. At each level there are scientists who apply system theory in their investigations. They are systems theorists but not necessarily general systems theorists. They are general systems theorists only if they accept the more daring and controversial position that-though every living system and every level is obviously unique-there are important formal identities of large generality across levels. Such cross-level similarities can potentially be evaluated quantitatively, applying the same model to data collected at two or more levels. This possibility is the chief reason why the author has used the same outline with identically numbered sections to analyze the present knowledge about each of the seven levels of living systems. The following survey of what is known about organisms as systems, therefore, is to be read as a single segment of a larger, integrated whole.

Living systems: the organization.

General systems behavior theory is concerned with seven levels of living systems—cell, organ, organism, group, organization, society, and supranational system. An exposition of the basic concepts in this theory appeared in “Living Systems: Basic Concepts,” Behavioral Science, 1965, 10, 193–237. (See also “Living Systems: Structure and Process,” and “Living Systems: Cross-Level Hypotheses,” Behavorial Science, 1965, 10, 337–411.) A condensation of the basic concepts also appeared in “The Nature of Living Systems,” Behavorial Science, 1971, 16, 277–301. The following is an analysis in terms of this conceptual system of present knowledge concerning one level of living system—the organization. In order to emphasize the cross-level formal identities among levels of living systems, this article follows exactly the same outline as other articles written by the author on the cell (“Living Systems. II. The Cell,” Currents in Modern Biology, 1971, 4, 78–206), the organ (“Living Systems. III. The Organ,” Currents in Modern Biology, 1971, 4, 207–254), the organism, and the group (“Living Systems: The Group,” Behavorial Science, 1971, 16, 302–398). Their subheadings and section numbers are identical. All these articles will also be published as chapters of the author's forthcoming book Living Systems (New York: John Wiley & Sons, 1972). Other chapters in that book, also following the same outline, analyze current knowledge about higher levels of living systems—society and supranational system. Since anatomists and physiologists are usually laymen in organization theory or international relations, psychologists are commonly laymen in economics, and social scientists are ordinarily laymen in cellular biology, all parts of the book, including the article published here, are necessarily written for intelligent laymen rather than experts, even though the articles deal with many technical topics. Some statements in them will seem to experts to be too elementary to be worth repeating. If a fact is fundamental and may not be known to specialists in other fields, it is stated here, even if it is elementary to the experts. The complex division of labor of modern science, often characterized by pluralistic insularity, requires this. The multitude of detailed and specialized experiments and studies that have been carried out provide the substance of the scientific investigation of organizations. Their findings constitute the trees. But an overview of these results and of the relationships among them—a view of the forest—is also essential. Such a telescopic rather than a microscopic view may suggest the proper balance for research on various aspects of organizations and clarify the priorities for future efforts. Many ideas presented here are not original with the author, though the arrangement is. Unless several persons have wrestled with an idea it is not often fundamental. The author has necessarily selected only a few researches to discuss out of the vast published repertoire, and so his selection has necessarily been arbitrary. Experts in each special field might agree on other studies as more important. Some of the author's statements may be wrong and his analysis ill advised. If so he would appreciate corrections—it is hard to cover such a wide range and still make no errors. The discussion of the processes of each subsystem ends with a number of examples of the variables of that subsystem which can be observed and measured. These variables make concrete the content of science at this level. They appear also to be common for a particular subsystem at multiple levels of living systems. Measurement of these variables, therefore, can be one way to determine whether cross-level formal identities exist. Throughout the text there are numerous references to cross-level hypotheses. These are mentioned for a similar purpose—to show that propositions possibly valid at other levels may also apply to organizations. These hypotheses, numbered to indicate the section of the article to which they apply, appear on pages 2–5. Many of them have been shown in other articles to be relevant to other levels of living systems as well. At each level there are scientists who apply systems theory in their investigations. They are system theorists but not necessarily general systems theorists. They are general systems theorists only if they accept the more daring and controversial position that though every living system and every level is obviously unique-there are important formal identities of large generality across levels. These can potentially be evaluated quantitatively, applying the same model to data collected at two or more levels. This possibility is the chief reason why the author has used the same outline with identically numbered sections to analyze the present knowledge about each of the seven levels of living systems. The following survey of what is known about organizations as systems, therefore, is to be read as a single segment of a larger, integrated whole.

Anomalies in Ward Identities for Three-Point Functions

We study the Ward identities for three-point functions of vector, axial-vector, scalar, and pseudoscalar densities constructed in a free-quark model. Divergences in the integral representations for the two- and three-point functions have the effect that not all of the formal Ward identities are satisfied in the model. After making full use of the ambiguities inherent in the definition of linearly divergent objects, as well as adding known polynomials to certain of the three-point functions, we find that only the Ward identities for three axial-vector densities and for one axial-vector and two vector densities remain unsatisfied. Arguments are presented that for $n$-point functions with $n>3$, the Ward identities are satisfied.

Sector Income Accounting and Analysis for Latin American and Caribbean Economies--More Appropriate Equations

AR E not Latin American and Caribbean economies, as a broad group, sufficiently different in their structure and dynamics to justify, or, in fact, logically require recognition of this fact by national income statisticians and analysts? And if the way in which these economies work is, indeed, sufficiently distinctive, what are the most important, relevant characteristics that should influence national income research, statistics and analysis? What are the principal innovations that are suggested by the peculiarities? This paper offers answers to these questions. I shall point out briefly the main features of the structure of these economies, indicate the principal autonomous stimuli experienced by them, suggest a simple system of sector accounts that would (i) bring together meaningful information on the most relevant magnitudes and (2) reveal relative changes in these from one period to another, and finally, discuss how the information might be used, suggesting also some of the equations that might be derived to guide the formulation of policy as well as for theoretical study. It is not my argument, however, that the aggregative measures which have become familiar are without use for the study of Latin American economies. For certain purposes, summary national account systems of the form recommended by the United Nations Statistical Office are undoubtedly needed. But, as will be explained, the structure of each of the Latin American economies is such that important aggregate-relating parameters which are fairly stable in advanced countries, or subject to change within relatively narrow limits, change greatly in the short run, even in the economies we discuss. The general argument here, in other words, is that much of the information that is needed for shortand medium-term forward looking analysis in Latin American economies is lost in the conventional national accounts. I assume that it is not necessary for me to demonstrate or prove in these pages the now familiar propositions that the equations of an accounting system are simply the formal economic identities written out in a particular way, and that the same estimates of the values of transactions in relation to one another would be less intelligible if they were not presented in an accounting form.