Blockchain Papers

Authors unavailable

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WILLIAM J. LEDGER, EDWIN P. PETERSON

No abstract is available for this record.

Martin C. McGuire

This paper addresses the question of whether, in the context of decentralization among levels of government, an output or performance subsidy is demonstrably superior to a cost subsidy as a tool for influencing the behavior of a local unit. Grants-in-aid for example are cost subsidies; are they therefore inherently inferior incentive devices? The answer suggested to this question is that a performance subsidy may be superior to, equal to, or inferior to a cost subsidy depending upon three factors: (1) the technology of local production, (2) the optimizing behavior of local officials, (3) the equity objectives of federal government.

Jesse Burkhead, Jerry Miner

Analysis of the pure theory of public expenditures reveals a pathological case of market failure. That is, goods with the “double polar” characteristics of joint supply and the impossibility of exclusion are such that their production will occur only under public organization (collective supply). Yet, in itself, the pure theory of public finance constitutes neither an economic rationale for the state nor an adequate economic theory of government expenditure. An economic theory of the state, consistent with the traditions of individualistic economics, may be developed around the assumption that the state aims to maximize societal economic welfare defined in terms of conformance with individual preferences. 1 If market-determined prices are presumed to be the primary way in which preferences are manifest, such a theory requires a thoroughgoing description of all sources of failure of decentralized markets coupled with an analysis of the potentialities of various government policies to deal with these failures. The analogous economic theory of public expenditure must elucidate the specific role of government budgetary outlays as one among various state actions intended to deal with market failure. Further, it must distinguish public expenditures whose purpose is to subsidize households and private market organizations from those government expenditures for goods and services which then are either provided through collective organization and supply or are sold by the state. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Bryan Ellickson

completely from the question of why households decide to live within a particular jurisdiction. Although we have no satisfactory thleory of urban local government, economists have not been reluctant to propose reforms in existing institutions. Advocates of metropolitan government suggest that decisions must be made at the metropolitan level if externalities are to be internalized and economies of scale realized. Tn contrast, the proponents of decentralization argue that further political fragmentation is required in order to provide greater variety in local public services. The only consensus, if any exists at all, is that present institutions of local government are inefficient. However, nowhere in the literature do we find an explanation of why, in view of this inefficiency, change is so rare. Annexations to the cen-tral city, relatively common at the turn of the century, ceased rather abruptly in most metropolitan areas after 1918. Subdivision of the larger political jurisdictions in our metropolitan areas does not appear at all likely. Thus, it seeins reasonab)le to ask of an adequate theory of metropolitan political economy an answer to the question: why are existing jurisdictional bourndaries so impervious to change? To answer this question we must investigate the impact of local governmental structure not only on allocative efficiency but on the extent of redistribution from rich to poor as well. In a recent study of suburbs in the Philadelphia metropolitan area, Williams, et al. [5] report that, when heavy expenditures were involved, wealthy communities were unwilling to enter into cooperative agreements with less wealthy communities. Only when their wealth was about the same would cities agree to engage in a jointly financed program. Across the United States, proposals for nietro-

Jaromír Kremen, Petr Demel

No abstract is available for this record.

Donald Ornstein, Louis Sucheston

A recent theorem of Orey [12] (see also [1], [6], [7], [13]) asserts that if $T$ is an $L_1$ operator induced on a discrete measure space by an irreducible recurrent aperiodic Markov matrix, then the condition (C) holds: $f \epsilon L_1, \int f = 0$ implies that $T^n f$ converges to zero in $L_1$. In an attempt to determine when (C) holds for more general operators, we at first prove the following (Theorem 1.1): Let $T$ be a positive linear contraction operator on $L_1$; if $T^nf$ and $T^{n+1}f$ intersect slightly, but uniformly in $f$ in the unit sphere of $L_1$, then $T^nf - T^{n+1}f$ converges to zero in norm. (C) follows if $T$ is conservative and ergodic (Corollary 1.3). In Section 2 we derive from this a simple proof of Orey's theorem. The main result of the paper is in Section 3 and could be called a "zero-two" theorem: Let $P(x, A)$ be a Markov kernel, and assume that there is a $\sigma$-finite measure $m$ such that for each $A, m(A) = 0$ implies $P(x, A) = 0$ a.e. and $m(A) > 0$ implies $\sum^\infty_{n=0} P^{(n)}(x, A) = \infty$ a.e. Then the total variation of the measure $P^{(n)}(x, \cdot) - P^{(n+1)}(x, \cdot)$ is either a.e. 2 for all $n$ or it converges a.e. to 0 as $n \rightarrow \infty$. In Section 4 it is shown that a version of the zero-two theorem essentially contains the Jamison-Orey generalization of Orey's theorem to Harris processes. Section 1 and Section 2 of this paper do not assume any knowledge of either operator ergodic theory or probability. Some known results in ergodic theory are applied in Section 3, but the proof of the main theorem does not depend on them.

William Michael Lloyd Holcombe

The theory of near-rings has arisen in a variety of ways. There is a natural desire to generalise the theory of rings and skew fields by relaxing some of their defining axioms. It has also been the hope of some mathematicians that certain problems in group theory, particularly
\ninvolving permutation groups and group representations, may perhaps be clarified by developing a coherent algebraic theory of near-rings. Moreover, there is an increasing recognition by mathematicians in many branches of the subject, both pure and applied, of the ubiquity of
\nnear-ring like objects.
\n
\nThe first steps in the subject were taken by Dickson and Zassenhans with their studies of 'near-fields', and by Wielandt with his classification of an important class of abstract near-rings. Papers by Frohlich, Blackett, Betsch and Laxton developed the theory considerably. Lately authors such as Beidleman, Ramakotaiah, Tharmanatram, Maxson, Malone and Clay have all added to our knowledge.
\n
\nThe history of the subject has been strongly influenced by our knowledge of ring theory, and although this has often been beneficial it must not be overlooked that a number of important problems in near-ring theory have no real parallel in the theory of rings. It is probably best to try to preserve a balance, and not to endeavour exclusively, either to generalise theorems from ring theory irrespective
\nof their usefulness, or to ignore the theory of rings and attempt to formulate a completely independent theory. In many cases our results are generalisations of theorems from ring-theory but at certain important junctures we will explicitly use the fact that we are dealing with a near-ring which is not a ring. This is a very interesting
\ndevelopment in the subject.
\n
\nWe proceed, in the first chapter, with a review of the terms and notation that will be used in this thesis.
\n
\nWhere definitions and concepts are of a specialized or technical nature and only used in one section, it seems more sensible to postpone introducing them until a more natural point in the proceedings.
\n
\nChapter 2 gives a summary of the results on the various radicals corresponding to the Jacobson radical for associative rings. Most of these results are well known and readily available in the literature. We also consider near-rings with one, or more, of these radicals zero.
\n
\nWe defined, in Chapter 1, three different types of primitive
\nnear-ring, which are all genuine generalisations of the ring theoretic concept. Of these three, the two most important are 2-primitive and 0-primitive near-rings. In Chapter 3, we examine 2-primitive near-rings with certain natural conditions imposed on them. A theorem is obtained
\nwhich could be considered to be the equivalent result for near-rings of the theorem classifying simple, artinian rings, due originally to Wedderburn and redeveloped by Jacobson.
\n
\nChapters 4 and 5 deal with 0-primitive near-rings satisfying
\ncertain conditions. Chapter 5 is a generalisation of Chapter 4, but we felt that the mathematical techniques involved would be clearer if the special case in Chapter 4 was expounded first. In these two chapters we classify a sizeable class of 0-primitive near-rings with identity.
\nand descending chain condition on right ideals.
\n
\nSeveral types of prime near-rings have been developed in the
\nliterature. In Chapter 6 we examine these and related concepts.
\n
\nIn the theory of rings, Goldies' classification of prime and
\nsemi-prime ring with ascending chain conditions, has been of immense importance. Whether such a result could be obtained in the theory of near-rings is a matter for conjecture, at the moment. We have made a start on the problem with the construction of a class of near-rings which
\nbehave in a very similar way to Prime rings with the Goldie chain conditions. This is the content of Chapter 7. The inspiration for its came mainly from the proof of Goldies' first theorem, due to C. Procesi, which is featured in Jacobson's book. (Jacobson [1]).
\n
\nChapter 8, is an attempt to initiate the development of a theory of vector groups and near-algebras which would play an important röle-in the future theory of near-rings, in a way, perhaps, similar to the Ale vector spaces and algebras play in ring theory. This may lead, in time, to results on 2-primitive near-rings with identity and a minimal right
\nideal, for example, or a Galois theory for certain 2-primitive nearrings. For the former problem, the experience of the semi-group theorists (Hoehake [1] etc. ) may prove useful.
\n
\nFinally a note on the numbering of results and definitions etc. If a reference is made, containing only two numbers, e. g. 1.12 then this means, "item 12 of section 1 of the present chapter". If a reference reads: 3.1.12, then this means "item 12 of section 1 of Chapter 3.

Anat Lerner

The formulation of the argument for distributional equality by William Breit and William Culbertson is an improvement on that of The Economics of Control and is more effective in class. Their generalization of the to the case of increasing marginal utility (of income) offset by a greater degree of diminishing marginal utility elsewhere, is also an improvement. Their point that Paul Samuelson did not escape the ''equal ignorance assumption is well taken. Ambiguity, being a case of lack of clarity is a charge that can never successfully be refuted. However, I would like to deny a switching of conclusions. Perhaps the ambiguity would have been avoided if I had added the following words in Roman type to the italicized sentence quoted: . . if it is desired to maximize the total satisfaction in a society, the rational procedure, in the absence of the knowledge that would enable us to equalize the marginal utilities, is to maximize the probable total satisfaction-i.e., to divide income on an equlitarian basis. The theorem on page 32 is not than the and mild one of page 29. It is the same proposition. The ingenous device of the 100 million coconut islands in one way does more than is claimed for it and in another way, does less. If it were possible to divide the total population into pairs which had the same utility functions, the equalization of income within each pair would never involve a wrong movement to be offset by a right oine. That is why there is certainty of improvement from equalization on every island. Furthermore, there would be an absolute maximization, with certainty, of the total satisfaction of the pair on each island from their joint income. On the other hand, the parable assumes that the combined incomes of the pairs have somehow already been equalized; that for every individual in the half of the total population with incomes less than the mean, his partner in the other half of the population (with an identical utility function) has an income greater than the mean by the exact amount that his is less than the mean. (This implies incidentally that no individual has an income as as twice the mean unless his partner has a zero income.) If this is not the case, some islands will be richer than others. We will then have to equalize the incomes of the islands before we could conduct Breit and Culbertson's experiment. The parable, therefore, while not necessary for the meek that income equalization maximizes the probable total satisfaction, is not sufficient for the bold proposition (to which I have never subscribed) that income equalization increases total satisfaction with absolute certainty. Breit and Culbertson's development of their parable reflects the same discomfort they have seen in others. The pair on the island are not satisfied with the proof that the equalization of the incomes has maximized their probable satisfaction. Sharing a widespread human craving for certainty, they want to be quite sure that they have at least increased their actual total satisfactions. This assurance is unfortunately not available as long as the utility functions are unknown. Breit and Culbertson also are seeking for a certainty of gain in a much bolder and more interesting regarding realized satisfactions instead of the maximization of a mere probability, and are accurately represented by the island pair they have invented. They have imagined a certainty of gain only by imagining the discovery of identicalutility twins. But the whole point of the * University of California, Berkeley.

Vasudha Chopra, Sukanya Das

Given the increasing demand pressure on water resources coupled with supply holdups and institutional failures, fresh-water resources are increasingly susceptible to depletion and could potentially add to water stress in India. A vast demand-supply gap necessitates water conservation, including recycling measures. India has a great potential in wastewater treatment, and one of the ways to address it is decentralisation of wastewater treatment given its environmental benefits. Based on the Contingent Valuation Method (CVM), this study assesses Delhi urban households’ willingness to pay for the Operation & Maintenance (O&M) costs of a local Wastewater Treatment Plants (WWTP) that supplies residential complexes treated water for toilet-flushing. The study found that if freshwater prices rise sufficiently for consumers, they may be willing to subsidise a decentralized WWTP to cover at least their non-potable water uses. In addition, the co-provision of such public goods can become an important supplement to urban municipal finance.

Jiarun Hu, Qian Luo, Jiaen Zhang

Security breaches of the cryptocurrency exchanges usually cause the price fluctuation in the market. Approximately one hundred cryptocurrency thefts, including hacks and scams, has occurred since 2012 to 2018, half of which are hacks of Bitcoins. Based on the thirty Bitcoin hacks, this study portrays the general price pattern during the hack. And it illustrates the link between the size of the hack and the subsequent price change of Bitcoin. The tests reveal that the larger the volume of the hack, the stronger the price drop. However, a similar obvious relationship does not exist for the recovery of the price. The study might be the first piece of research focus on the hacks and the price pattern in a short time period.

Vasilis Kostakis, Chris Giotitsas

The still raging financial crisis of 2007–2008 has enabled the emergence of several alternative practices concerning the production, circulation, and use of money. This essay explores the political economy of the Bitcoin ecosystem. Specifically, we examine the context in which this digital currency is emerging as well as its nature, dynamics, advantages, and disadvantages. We conclude that Bitcoin, a truly interesting experiment, exemplifies “distributed capitalism” and should be mostly seen as a technological innovation. Rather than providing pragmatic answers and solutions to the current views on the financial crisis, Bitcoin provides some useful and timely questions about the principles and bases of the dominant political economy. A ECONOMIA POLÍTICA DO BITCOINResumoO aquecimento da crise financeira de 2007-2008 permitiu o surgimento de várias práticas alternativas em matéria de produção, circulação e uso do dinheiro. Este ensaio explora a economia política do ecossistema Bitcoin. Especificamente, vamos examinar o contexto em que essa moeda digital está emergindo, bem como a sua natureza, dinâmica, vantagens e desvantagens. Concluímos que Bitcoin, uma experiência verdadeiramente interessante, exemplifica "capitalismo distribuído" e deve ser visto principalmente como uma inovação tecnológica. Em vez de fornecer respostas e soluções pragmáticas para os pontos de vista atuais sobre a crise financeira, Bitcoin fornece algumas perguntas úteis e oportunas sobre os princípios e as bases da economia política dominante.

James E. Bruno

Public school finance in the United States is highly decentralized and can be described as the sum of separately determined school finance programs established by each state. Since funds to support educational expenditures in a school district come from various sources, primarily state and local, it is necessary for states to derive schemes by which these resources can be combined to finance education at the local level. The administration and derivation of these resource allocation schemes is usually carried out at the state level.

Edmund S. Phelps, Karl Shell

No abstract is available for this record.

WILLIAM J. LEDGER

No abstract is available for this record.

Karl Borch

The behavioral assumptions which economists call “perfect competition,” imply that decentralized decision making under certain conditions leads to a social optimum. This is a central result of classical economic theory. The author discusses the result, and shows that it cannot be expected to hold when uncertainty is introduced. The point is illustrated by a simple example from business finance.

William C. Hsiao, Alan Maynard

On behalf of the American Association of Neurological Surgeons/Congress of Neurological Surgeons Joint Section on Disorders of the Spine and Peripheral Nerves, it is my great privilege to introduce these Guidelines for the Management of Acute Cervical Spine and Spinal Cord Injuries. These guidelines represent the initial installment of a more comprehensive guidelines initiative from the Joint Section on behalf of all practicing neurosurgeons and their patients. The Section is grateful to the small working group who devoted considerable time and effort to the generation of this outstanding document. We would like to formally recognize the Joint Section on Trauma for their important collaboration on this project. The Section would also like to acknowledge and thank the parent organizations, the American Association of Neurological Surgeons and the Congress of Neurological Surgeons, for their guidance of and support for this project, most notably through the efforts of the American Association of Neurological Surgeons/ Congress of Neurological Surgeons Guidelines Committee. The Section is also deeply indebted to Michael Apuzzo and the staff of Neurosurgery for their advice and editorial assistance in preparing this document for publication. The application of Neurosurgery’ s rigorous peer-reviewed editorial process has clearly enhanced the quality, balance, and stature of this document. Perhaps most importantly, Neurosurgery has provided an extraordinary vehicle for the widespread dissemination and ultimate incorporation of these guidelines to improve the care and enhance the outcomes of patients with traumatic cervical spine and spinal cord injuries. One of the truly important functions of organized neurosurgery is the generation of evidenced-based clinical practice guidelines. Properly developed, such guidelines can answer important questions, resolve uncertainty, identify areas of deficient knowledge and opportunities for future scientific investigation, standardize treatment, and improve the quality of care and the outcomes for patients. The now widely disseminated head trauma guidelines, for example, have clearly made a difference in the outcomes of patients with severe head injury. Guidelines development is a highly structured process with rigorous methodological criteria and exacting standards. It is a time-, labor-, and resource-intensive process that has served as a significant obstacle to more widespread guidelines development throughout neurosurgery. In the past, clinical practice guidelines have been developed by publicly supported epidemiologists and methodologists who understood study design, data analysis, and the guidelines process, but not the disease. This absence of context and clinical perspective significantly limited the value and relevance of their results. Alternatively, clinician-generated guidelines often took the form of methodologically flawed consensus panels and expert opinion, also of limited value. The Joint Spine Section recognized the importance of evidence-based clinical practice guidelines and the challenges of their development. The appropriate clinical expertise, strict adherence to established methodological standards for guidelines development, and considerable resource investment for the development, dissemination, and maintenance of the guidelines documents were deemed crucial to our guidelines initiative. Cervical spine and spinal cord injury was chosen as the initial guidelines topic because of the personal, social, and economic devastation of these injuries, their complex nature, and the high level of uncertainty, as reflected in wide practice variations, as to the value and indications for many of the aspects of evaluation and treatment. The clinical practice guidelines contained in this supplement to Neurosurgery represent a remarkable effort. They address the key issues related to the evaluation and management of these complex conditions that are relevant to the treating physician. In every chapter, the pertinent issues are succinctly stated, the published data are comprehensively presented in the evidentiary tables, and the evidence is thoroughly discussed and critically evaluated throughout the text. The linkage between the quality of the evidence and the strength of the recommendations was not a “black box” process but an open, deliberative exercise by skilled experts guided by a rigorous set of standards. Despite the strength and potential value of this document, it is important to acknowledge the inherent limitations of clinical practice guidelines. This, or any other, evidence-based clinical practice guidelines document does not represent the definitive source of knowledge on the stated topic. Rather, it represents recommendations of varying strength and certainty based on an analysis of the best available published data. These data, however, are often conflicting, flawed, or incomplete, and there are unavoidable elements of potential bias from subjectivity, perspective, and experience of the individuals and group involved in the analysis and interpretation of these data. In essence, proof is a relative term based on the interpretation of evidence. Furthermore, it is subject to different standards. A relevant example comes from the field of jurisprudence, where the standard of proof (i.e., guilt) for criminal trials is “beyond a reasonable doubt,” whereas the standard for civil courts must simply reflect “a preponderance of evidence” or “more likely than not.” These different standards evolved because of the perceived different consequences of a wrongful verdict. Moreover, as Stephen Haines likes to note, the verdict “not guilty” does not mean innocent; it merely says not proved. Such are the vagaries associated with the interpretation of even scientific evidence. Principled people can look at the same evidence and come to different conclusions subject to their own personal perspective, experience, and stake in the result. Nevertheless, the Joint Section and Guidelines Development Group went to great lengths to identify and avoid—or at least minimize—these potential problems. The working group adopted the most widely recognized and rigorous standards for guidelines development. A diverse panel of experts with expertise in spine, trauma, and epidemiology brought relevant clinical, scientific, and methodological competence to enhance both the analytical and the deliberative aspects of this process. Periodic outside reviews were routinely obtained for topics or areas of contention or uncertainty to add additional perspective and balance. Above all, the process was accountable and transparent at every stage. Ultimately, we offer these guidelines as a living document to those professionals who treat patients with traumatic spinal injury. We hope each practitioner will critically evaluate these guidelines and come to his or her own conclusion on how, whether, and when to implement its recommendations. It may be used either as a reference or as a basis for standardized protocols of evaluation and management of the patients with traumatic spinal injury. For clinical and basic science researchers, we hope that it will identify and catalyze scientific investigation in areas of deficient knowledge. As a Section, we stand firmly behind this important document and will continuously update the recommendations as new knowledge and understanding is developed. We sincerely believe that these guidelines can improve the care and enhance the outcomes of patients with traumatic injuries to the cervical spine and spinal cord.

Kurt Kreith

1. Introduction.The classical Sturmian theorem of ordinary differential equations deals with functions u(x) and v(x) which are, respectively, solutions of differential equations(1) Um-^^j+tumO,(2) Mv=-l{J^+yv = 0.Under the assumption that "F is larger than M" (in the sense that a(x) ä a(x) > 0 and c(x) £ y(x)) one can infer information about all solutions of (2) from knowledge about a particular nontrivial solution of (1)-i.e. if u(xx) = u(x2)=0 then every solution of (2) has a zero in [xx, x2].These ideas have been generalized to second order elliptic equations by several authors ([l]-[4]) considering elliptic operators and also by Protter [5] and Swanson [6] considering the nonselfadjoint case.Given a proper relation among the coefficients of F and M and that Lu=0 has a nontrivial solution with nodal domain Ü, then it can be shown that every solution of Mv = 0 has a zero in Í2.While all the above proofs of this fact make essential use of some sort of ordering among elliptic operators, the nature of this ordering is never defined in operator-theoretic terms.The results of §2 below suggest that it is an order relationship between certain resolvents of the differential operators F and M which underlies the separation properties characteristic of Sturmian theorems.It will be shown that quite general operator equations in a Banach space 3S satisfy a type of Sturmian theorem if the operators' resolvents satisfy prescribed positivity requirements with respect to a cone SP.In order to apply this theory to differential operators, one must first establish the corresponding positivity properties for their resolvents.This is done in §3 for sufficiently regular nonselfadjoint second order elliptic operators, and the general theory of §2 is then applied in the proof of two Sturmian theorems and the establishment of criteria for certain Green's functions to be positive.

Harry N. Scheiber

Historians have recently given much attention to the active, formative role of state governments in the American economy before the Civil War.1 The states exercised nearly exclusive control over many aspects of economic life, and in such areas as labor, banking, and corporation policy the federal government interfered relatively little. The consequence was considerable decentralization of power in policymaking, together with variations in policy from state to state.2 Perhaps in no policy area were variations so dramatic as in state legislation on banking. In some states banking was prohibited outright, while in others the state government itself established and operated banks, sometimes on a monopoly basis. Elsewhere, safety funds were established and banks required to meet minimum standards of specie reserve and the like; and in a few states, stringent regulatory policies were pursued, with public commissioners given considerable discretion in administering policy.3

Grace Wahba

Let $\{X(t), t = \cdots -1, 0, 1, \cdots\}$ be a $P$ dimensional zero mean stationary Gaussian time series, $X(t) = \begin{pmatrix}X_1(t)\\X_2(t)\\\vdots\\X_P(t)\end{pmatrix}$ we let $R(\tau) = EX(t)X' (t + \tau)$, where $R(\tau) = \{R_{ij}(\tau), i,j = 1, 2, \cdots P\}$, and $F(\omega) = (2\pi)^{-1} \sum^\infty_{\tau=-\infty}e^{-i\omega\tau}R(\tau)$. It is assumed that $\sum^P_{i,j=1} \sum^\infty_{\tau=-\infty} |\tau| |R_{ij}(\tau)| < \infty$, and hence $F(\omega)$ exists and the elements possess bounded derivatives. It is further assumed that $F(\omega)$ is strictly positive definite, all $\omega$. Knowledge of $F(\omega)$ serves to specify the process. $F(\omega)$, and $S$, the covariance matrix of $x = \begin{pmatrix}x_1 \\ x_2\ \\ vdots\\x_P\end{pmatrix}$, a Normal $(0, S)$ random vector are known to enjoy many analogous properties. (See [7].) To cite two examples, the hypothesis that $X_i(s)$ is independent of $X_j(t)$ for $i \neq j = 1, 2, \cdots P$, any $s, t$, is equivalent to the hypothesis that $F(\omega)$ is diagonal, all $\omega$, while the hypothesis that $x_i$ is independent of $x_j$, for $i \neq j = 1, 2, \cdots P$ is equivalent to the hypothesis that $S$ is diagonal. The conditional expectation of $x_1$, given $x_2, \cdots x_P$ is \begin{equation*}E(x_1\mid x_2, \cdots x_P) = S_{12}S^{-1}_{22}\begin{pmatrix}x_2 \\ \vdots \\ x_P\end{pmatrix}, S = \bigg(\begin{array}{c|c} S_{11} & S_{12} \\ \hline S_{21} & S_{22}\end{array} \bigg)\end{equation*}. The corresponding regression problem for stationary Gaussian time series goes as follows. If \begin{equation*}E\{X_1(t)\mid X_2(s), \cdots X_P(s), s = \cdots -1, 0, 1, \cdots\} = \sum^P_{j=2} \sum^\infty_{s=-\infty} b_j(t - s)X_j(s)\end{equation*} then $B(\omega)$, defined by $B(\omega) = (B_2(\omega), \cdots B_P(\omega)), B_j(\omega) = \sum^\infty_{s=-\infty} b_j(s)e^{i\omega s}$ satisfies \begin{equation*}B(\omega) = F_{12}(\omega)F_{22}^{-1}(\omega), \quad F(\omega) = \bigg(\begin{array}{c|c}f_{11}(\omega) & F_{12}(\omega) \\ \hline F_{21}(\omega) & F_{22}(\omega)\end{array} \bigg).\end{equation*} It is interesting to ask how well these and similar analogies carry over to sampling theory and hypothesis testing. Goodman [3] gave a heuristic argument to support the conclusion that $\hat{F}_X(\omega_k)$, a suitably formed estimate of the spectral density matrix $F(\omega_k)$ has the complex Wishart distribution. The question is met here by the following results. Firstly if $\hat{F}_X(\omega_l), l = 1, 2, \cdots M$ are estimates of the spectral density matrix, each consisting of averages of $(2n + 1)$ periodograms based on a record of length $T$, with the $\omega_l$ equally spaced and $(2n + 1)M \leqq \frac{1}{2} T$, then it is possible to construct, on the same sample space as $X(t), M$ independent complex Wishart matrices $\hat{F}{\bar{\bar{X}}}(\omega_l), l = 1, 2, \cdots M$ such that $\{\hat{F}_X(\omega_l), l = 1, 2, \cdots M\}$ converge simultaneously in mean square to $\{\hat{F}_{\bar{\bar{X}}}(\omega_l), l = 1, 2,\cdots M\}$, as $n, M$ get large. Secondly, it is legitimate to use the natural analogies from multivariate analysis to test hypotheses about time series. One example is presented, as follows. The likelihood ratio test statistic for testing $S$ diagonal is $|\hat{S}|/\mathbf{\prod}^P_{i=1} \hat{s}_{ii}$ where $\hat{S} = \{\hat{s}_{ij}$ is the sample covariance matrix. The analogous statistic $\psi$ for testing $X_i(s), X_j(t)$ independent, $i,j 1 = 2, \cdots P$ from a record of length $T$ is $\psi = \prod^M_{l=1} \lbrack|\hat{F}_X(\omega_l)|/\prod^P_{i=1} \hat{f}_{ii}(\omega_l)\rbrack$ where $\hat{F}_X(\omega_l) = \{\hat{f}_{ij}(\omega_l)\}$ are the sample spectral density matrices as above. Letting ${\bar{dbar{\psi}}} = \prod^M_{l=1} \lbrack|\hat{F}_{\bar{\bar{x}}}(\omega_l)|/\prod^P_{i=1} \hat{h}_{ii}(\omega_l)\rbrack$ where $\hat{F}_{\bar{\bar{x}}}(\omega_l) = \{\hat{h}_{ij}(\omega_l)\}$ are the independent complex Wishart matrices referred to above, we show $EC_{n,M} |\log \psi - \log {\bar{\bar{\psi}}} \rightarrow 0$ for large $n, M$, where $C_{n,M}$ are chosen to make the result non-trival. The method of proof applies to any statistic which is a product over $l$ of sufficiently smooth functions of the entries of $\hat{F}_X(\omega_l)$. Applications to estimation and testing in the regression problem will appear elsewhere [8]. The distribution theory of functions of complex Wishart matrices has been well investigated by a number of authors [3] [5] [6], and hence can be easily applied here to statistics like ${\bar{\bar{\psi}}}$. The results above are shown for $P = 2$, it is clear that the proofs extended to any (fixed) finite $P$. The proofs proceed as follows, via a theorem which has somewhat more general application. For each $T$, let $X$ be the $2 \times T$ random matrix $X = \binom{X_1}{X_2} = \begin{pmatrix}X_1(1), \cdots, X_1(T)\\X_2(1), \cdots, X_2(T)\end{pmatrix}$ and let the $2T \times 2T$ covariance matrix $\Sigma$ be given by $\Sigma = \begin{pmatrix}\sum_{11} \sum_{12} \\ \sum_{21} \sum_{22}\end{pmatrix}$ where $\Sigma_{ij} = EX_i'X_j. \{\hat{F}_X(\omega_l)\}$, the sample spectral density matrices described above based on a record of length $T$, are each of the form $\hat{F}_X(\omega_l) = T^{-1}XQX'$ where $Q$ is a $T \times T$ circulant matrix with largest eigenvalue $ = T(2n + 1)^{-1} \leqq \frac{1}{2}M < <T$. We define circulant matrices $\bar{\Sigma}_{ij}$ which approximate $\Sigma_{ij}$, and a random matrix $\bar{X}$ on the sample space of $X$, $\bar{X} = \binom{\bar{X}_1}{\bar{X}_2} = \begin{pmatrix}\bar{X}_1(1), \cdots, \bar{X}_1(T)\\\bar{X}_2(1), \cdots, \bar{X}_2(T)\end{pmatrix}$ with $E\bar{X}_i'\bar{X}_j = \bar\Sigma_{ij}$. The $2T$ eigenvalues of the block circulant matrix $\bar\Sigma = \begin{pmatrix}\bar\Sigma_{11} \bar\Sigma_{12} \\ \bar\Sigma_{21} \bar\Sigma_{22}\end{pmatrix}$ will be the $2T$ eigenvalues of the $T$ matrices $\{F(2\pi j/T),j = 1, 2, \cdots T\}$. The distribution of random matrices of the form $T^{-1}\bar{X}Q\bar{X}'$ where $Q$ is any circulant matrix are relatively simple to investigate due to the fact that all circulant matrices commute, and their eigenvalues may be exhibited as simple functions of the elements. Circulant quadratic forms in random vectors with circulant covariance matrices are well known in the literature, (See [1] and references cited there). Let $\hat{F}_{X,Q} = T^{-1}XQX'$ and $\hat{F}_{\bar{X},Q} = T^{-1}\bar{X}Q\bar{X}'$ where $Q$ is now any $T \times T$ (real or complex) quadratic form with largest absolute eigenvalue $\leqq q$. The main Theorem allows the replacement of $X$ by $\bar{X}$ in the analysis, and is, that under the assumptions on $F(\omega)$ and $R(\tau)$, for any $T$, \begin{equation*}\tag{1.1} E \operatorname{tr} (\hat{F}_{X,Q} - \hat{F}_{\bar{X},Q})(\hat{F}_{X,Q} - \hat{F}_{\bar{X}, Q})^{\ast'} \leqq cq^2/T^2\end{equation*} where $c$ is a constant depending only on $F(\omega)$ and $R(\tau)$. A lemma, essentially allowing the replacement of $F(\omega)$ by a suitably chosen step-function, together with the application of (1.1) gives the results concerning the $\{\hat{F}_X(\omega_l)\}$ an $\lambda$. Since $\hat{R}(\tau)$, the sample (circularized) autocorrelation function is also of the form $T^{-1}XQX'$ with $Q$ circulant we obtain an easy corollary on the distribution of $\{\hat{R}(\tau)\}$.

Alan S. Manne

For a self-financing business enterprise (or for an underdeveloped economy subject to constraints on the availability of foreign investment funds), three theorems are presented. Each result is based upon the assumption that the firm's investment opportunities follow constant returns-to-scale, and are of the “point input—stream output” type. Theorem 1 shows that if the enterprise is attempting to maximize a linear function of the cash dividends paid out, the optimization model cannot explain a readily observed phenomenon: both investment expenditures and also cash dividends at the same point in time. Theorems 2 and 3 explore the consequences of supposing that the maximand is a concave, nonlinear function of the cash dividends paid out, and that the optimal solution consists of positive investment expenditures over time. (The optimal policy may or may not call for positive dividends during each time period.) Then Theorem 2 shows that the optimal dual variable price ratios are determined uniquely by the set of investment opportunities available, and Theorem 3 shows that the optimal policy can be evaluated numerically through optimization of the original utility function subject to a specially constructed single linear equality constraint on the cash withdrawals. An economic decentralization interpretation is attached to this auxiliary maximization problem.

Benoît B. Mandelbrot, John W. Van Ness

Previous article Next article Fractional Brownian Motions, Fractional Noises and ApplicationsBenoit B. Mandelbrot and John W. Van NessBenoit B. Mandelbrot and John W. Van Nesshttps://doi.org/10.1137/1010093PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAbout[1] I. Adelman, Long cycles—fact or artifact?, Amer. Economic Rev., 60 (1965), 444–463 Google Scholar[2] William Feller, The asymptotic distribution of the range of sums of independent random variables, Ann. Math. Statistics, 22 (1951), 427–432 MR0042626 0043.34201 CrossrefISIGoogle Scholar[3] I. M. Gel'fand and , N. Ya. Vilenkin, Generalized functions. Vol. 4, Academic Press [Harcourt Brace Jovanovich Publishers], New York, 1964 [1977]xiv+384 MR0435834 Google Scholar[4] C. W. J. Granger, The typical spectral shape of an economic variable, Econometrica, 34 (1966), 150–161 CrossrefISIGoogle Scholar[5] G. A. Hunt, Random Fourier transforms, Trans. Amer. Math. 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Marshall W. Meyer

This study attempts to link the formal structure of bureaucratic organizations to decision-making processes, and in particular to centralization or decentralization of authority. Interview data were obtained from 254 city, county, and state departments of finance. These data show that, controlling for an organization's size, decision-making authority is more highly centralized as the number of subunits in an organization increases; but as the number of levels of supervision grows, there is greater decentralization and at the same time proliferation of rules that specify criteria to guide decisions. Marshall W. Meyer is lecturer on sociology in the department of social relations at Harvard University.

John A. Ross, Oliver D. Finnigan

The Korean family planning program from 1964 to 1968 is considered in its entirety. The program is financed primarily by Korean government appropriations (averaging about $2 million in past years from central and local budgets) but also is supported substantially by foreign agencies such as the Population Council SIDA and AID. The Planned Parenthood Federation of Korea working in cooperation with the government is financed by about $200000 yearly from Korean government and foreign sources. The program has offered the IUD condom and vasectomy to every couple without charge. Accomplishments include adoption of the loop or vasectomy by 1/3 of couples with wife below age 45 elicitation of the best response among illiterate and low-income families and in rural areas reduction of crude birth rate by roughly 10% as of 1968 by IUDs inserted through mid-1968 and launching a pill program for IUD drop-outs starting with an unknown method a population of 30 million was informed of it and their approval won. Approaches used in the program have included strong governmental policy; implementation through the existing health structure; decentralization of functions to local and provincial government levels; payment by government of physicians and fieldworkers involved as well as vasectomy acceptors for work lost; high fieldworker density; fixation of target quotas with positive and negative sanctions on workers; and full use of mass media. Problems to be dealt with involve funding doctorless and remote areas and termination rates. The problems have been obtaining the funds; areas that are doctorless and that are remote; the training of workers; funds for maintenance and supply systems; shortcomings of the present contraceptive methods; and the changing of field-worker duties.