A Perfect Zero-Knowledge interactive proof system convinces a verifier that a string is in a language without revealing any additional knowledge in an information-theoretic sense. We show that for any language that has a perfect zero-knowledge proof system, its complement has a short interactive protocol. This result implies that there are not any perfect zero-knowledge protocols for NP-complete languages unless the polynomial time hierarchy collapses. This paper demonstrates that knowledge complexity can be used to show that a language is easy to prove.
A perfect zero-knowledge interactive proof is a protocol by which Alice can convince Bob of the truth of some theorem in a way that yields no information as to how the proof might proceed (in the sense of Shannon's information theory). We give a general technique for achieving this goal for any problem in NP (and beyond). The fact that our protocol is perfect zero-knowledge does not depend on unproved cryptographic assumptions. Furthermore, our protocol is powerful enough to allow Alice to convince Bob of theorems for which she does not even have a proof. Whenever Alice can convince herself probabilistically of a theorem, perhaps thanks to her knowledge of some trap-door information, she can convince Bob as well without compromising the trap-door in any way. This results in a non-transitive transfer of confidence from Alice to Bob, because Bob will not be able to subsequently convince someone else that the theorem is true. Our protocol is dual to those of [GMW1, BC].
Abstract. Usually, a proof of a theorem contains more knowledge than the mere fact that the theorem is true. For instance, to prove that a graph is Hamiltonian it suffices to exhibit a Hamiltonian tour in it; however, this seems to contain more knowledge than the single bit Hamiltonian/non-Hamiltonian. In this paper a computational complexity theory of the "knowledge " contained in a proof is developed. Zero-knowledge proofs are defined as those proofs that convey no additional knowledge other than the correctness of the proposition in question. Examples of zero-knowledge proof systems are given for the languages of quadratic residuosity and quadratic nonresiduosity. These are the first examples of zeroknowledge proofs for languages not known to be efficiently recognizable. Key words, cryptography, zero knowledge, interactive proofs, quadratic residues AMS(MOS) subject classifications. 68Q15, 94A60 1. Introduction. It is often regarded that saying a language L is in NP (that is, acceptable in nondeterministic polynomial time) is equivalent to saying that there is a polynomial time "proof system " for L. The proof system we have in mind is one where on input x, a "prover " creates a string a, and the "verifier " then computes on x and a in time polynomial in the length of the binary representation of x to check that
We prove that the scale map of the zero-crossings of almost all signals filtered by a gaussian of variable size determines the signal uniquely, up to a constant scaling. Exceptions are signals that are antisymmetric about all their zeros (for instance infinitely periodic gratings). Our proof provides a method for reconstructing almost all signals from knowledge of how the zero-crossing contours of the signal, filtered by a gaussian filter, change with the size of the filter. The proof assumes that the filtered signal can be represented as a polynomial of finite, albeit possibly very high, order. The result applies to zero- and level-crossings of signals filtered by gaussian filters. The theorem is extended to two dimensions, that is to images. These results imply that extrema (for instance of derivatives) at different scales are a complete representation of a signal.
Stewart C. Myers President of American Finance Association 1983 This paper's title is intended to remind you of Fischer Black's well-known note on âThe Dividend Puzzle,â which he closed by saying, âWhat should the corporation do about dividend policy? We don't know.â 6 I will start by asking, âHow do firms choose their capital structures?â Again, the answer is, âWe don't know.â The capital structure puzzle is tougher than the dividend one. We know quite a bit about dividend policy. John Lintner's model of how firms set dividends 20 dates back to 1956, and it still seems to work. We know stock prices respond to unanticipated dividend changes, so it is clear that dividends have information contentâthis observation dates back at least to Miller and Modigliani (MM) in 1961 28. We do not know whether high dividend yield increases the expected rate of return demanded by investors, as adding taxes to the MM proof of dividend irrelevance suggests, but financial economists are at least hammering away at this issue. By contrast, we know very little about capital structure. We do not know how firms choose the debt, equity or hybrid securities they issue. We have only recently discovered that capital structure changes convey information to investors. There has been little if any research testing whether the relationship between financial leverage and investors' required return is as the pure MM theory predicts. In general, we have inadequate understanding of corporate financing behavior, and of how that behavior affects security returns. I do not want to sound too pessimistic or discouraged. We have accumulated many helpful insights into capital structure choice, starting with the most important one, MM's No Magic in Leverage Theorem (Proposition I) 31. We have thought long and hard about what these insights imply for optimal capital structure. Many of us have translated these theories, or stories, of optimal capital structure into more or less definite advice to managers. But our theories don't seem to explain actual financing behavior, and it seems presumptuous to advise firms on optimal capital structure when we are so far from explaining actual decisions. I have done more than my share of writing on optimal capital structure, so I take this opportunity to make amends, and to try to push research in some new directions. A static tradeoff framework, in which the firm is viewed as setting a target debt-to-value ratio and gradually moving towards it, in much the same way that a firm adjusts dividends to move towards a target payout ratio. An old-fashioned pecking order framework, in which the firm prefers internal to external financing, and debt to equity if it issues securities. In the pure pecking order theory, the firm has no well-defined target debt-to-value ratio. Recent theoretical work has breathed new life into the pecking order framework. I will argue that this theory performs at least as well as the static tradeoff theory in explaining what we know about actual financing choices and their average impacts on stock prices. I have arbitrarily, and probably unfairly, excluded âmanagerialâ theories which might explain firms' capital structure choices.1 I have chosen not to consider models which cut the umbilical cord that ties managers' acts to stockholders' interests. I am also sidestepping Miller's idea of âneutral mutation.â2 He suggests that firms fall into some financing patterns or habits which have no material effect on firm value. The habits may make managers feel better, and since they do no harm, no one cares to stop or change them. Thus someone who identifies these habits and uses them to predict financing behavior would not be explaining anything important. The neutral mutations idea is important as a warning. Given time and imagination, economists can usually invent some model that assigns apparent economic rationality to any random event. But taking neutral mutation as a strict null hypothesis makes the game of research too tough to play. If an economist identifies costs of various financing strategies, obtains independent evidence that the costs are really there, and then builds a model based on these costs which explains firms' financing behavior, then some progress has been made, even if it proves difficult to demonstrate that, say, a type A financing strategy gives higher firm value than a type B. (In fact, we would never see type B if all firms follow value-maximizing strategies.) There is another reason for not immediately embracing neutral mutations: we know investors are interested in the firm's financing choices, because stock prices change when the choices are announced. The change might be explained as an âinformation effectâ having nothing to do with financing per seâbut again, it is a bit too easy to wait until the results of an event study are in, and then to think of an information story to explain them. On the other hand, if one starts by assuming that managers have special information, builds a model of how that information changes financing choices, and predicts which choices will be interpreted by investors as good or bad news, then some progress has been made. So this paper is designed as a one-on-one competition of the static tradeoff and pecking-order stories. If neither story explains actual behavior, the neutral mutations story will be there faithfully waiting. A firm's optimal debt ratio is usually viewed as determined by a tradeoff of the costs and benefits of borrowing, holding the firm's assets and investment plans constant. The firm is portrayed as balancing the value of interest tax shields against various costs of bankruptcy or financial embarassment. Of course, there is controversy about how valuable the tax shields are, and which, if any, of the costs of financial embarassment are material, but these disagreements give only variations on a theme. The firm is supposed to substitute debt for equity, or equity for debt, until the value of the firm is maximized. Thus the debt-equity tradeoff is as illustrated in Fig. 1. Costs of adjustment. If there were no costs of adjustment, and the static tradeoff theory is correct, then each firm's observed debt-to-value ratio should be its optimal ratio. However, there must be costs, and therefore lags, in adjusting to the optimum. Firms can not immediately offset the random events that bump them away from the optimum, so there should be some cross-sectional dispersion of actual debt ratios across a sample of firms having the same target ratio. The static-tradeoff theory of capital structure. Large adjustment costs could possibly explain the observed wide variation in actual debt ratios, since firms would be forced into long excursions away from their optimal ratios. But there is nothing in the usual static tradeoff stories suggesting that adjustment costs are a first-order concernâin fact, they are rarely mentioned. Invoking them without modelling them is a cop-out. Any cross-sectional test of financing behavior should specify whether firms' debt ratios differ because they have different optimal ratios or because their actual ratios diverge from optimal ones. It is easy to get the two cases mixed up. For example, think of the early cross-sectional studies which attempted to test MM's Proposition I. These studies tried to find out whether differences in leverage affected the market value of the firm (or the market capitalization rate for its operating income). With hindsight, we can quickly see the problem: if adjustment costs are small, and each firm in the sample is at, or close to its optimum, then the in-sample dispersion of debt ratios must reflect differences in risk or in other variables affecting optimal capital structure. But then MM's Proposition I cannot be tested unless the effects of risk and other variables on firm value can be adjusted for. By now we have learned from experience how hard it is to hold âother things constantâ in cross-sectional regressions. Of course, one way to make sense of these tests is to assume that adjustment costs are small, but managers don't know, or don't care, what the optimal debt ratio is, and thus do not stay close to it. The researcher then assumes some (usually unspecified) âmanagerialâ theory of capital structure choice. This may be a convenient assumption for a cross-sectional test of MM's Proposition I, but not very helpful if the object is to understand financing behavior.3 But suppose we don't take this âmanagerialâ fork. Then if adjustment costs are small, and firms stay near their target debt ratios, I find it hard to understand the observed diversity of capital structures across firms that seem similar in a static tradeoff framework. If adjustment costs are large, so that some firms take extended excursions away from their targets, then we ought to give less attention to refining our static tradeoff stories and relatively more to understanding what the adjustment costs are, why they are so important, and how rational managers would respond to them. But I am getting ahead of my story. On to debt and taxes. Debt and taxes. Miller's famous âDebt and Taxesâ paper 27 cut us loose from the extreme implications of the original MM theory, which made interest tax shields so valuable that we could not explain why all firms were not awash in debt. Miller described an equilibrium of aggregate supply and demand for corporate debt, in which personal income taxes paid by the marginal investor in corporate debt just offset the corporate tax saving. However, since the equilibrium only determines aggregates, debt policy should not matter for any single taxpaying firm. Thus Miller's model allows us to explain the dispersion of actual debt policies without having to introduce non-value-maximizing managers.4 Trouble is, this explanation works only if we assume that all firms face approximately the same marginal tax rate, and that is an assumption we can immediately reject. The extensive trading of depreciation tax shields and investment tax credits, through financial leases and other devices, proves that plenty of firms face low marginal rates.5 Given significant differences in effective marginal tax rates, and given that the static tradeoff theory works, we would expect to find a strong tax effect in any cross-sectional test, regardless of whose theory of debt and taxes you believe. Figure 2 plots the net tax gain from corporate borrowing against the expected realizable tax shield from a future deduction of one dollar of interest paid. For some firms this number is 46 cents, or close to it. At the other extreme, there are firms with large unused loss carryforwards which pay no immediate taxes. An extra dollar of interest paid by these firms would create only a potential future deduction, usable when and if the firm earns enough to work off prior carryforwards. The expected realizable tax shield is positive but small. Also, there are firms paying taxes today which cannot be sure they will do so in the future. Such a firm values expected future interest tax shields at somewhere between zero and the full statutory rate. In the âcorrectedâ MM theory 28 any tax-paying corporation gains by borrowing; the greater the marginal tax rate, the greater the gain. This gives the top line in the figure. In Miller's theory, the personal income taxes on interest payments would exactly offset the corporate interest tax shield, provided that the firm pays the full statutory tax rate. However, any firm paying a lower rate would see a net loss to corporate borrowing and a net gain to lending. This gives the bottom line. There are also compromise theories, advanced by D'Angelo and Masulis 12, Modigliani 30 and others, indicated by the middle dashed line in the figure. The compromise theories are appealing because they seem less extreme than either the MM or Miller theories. But regardless of which theory holds, the slope of the line is always positive. The difference between (1) the tax advantage of borrowing to firms facing the full statutory rate, and (2) the tax advantage of lending (or at least not borrowing) to firms with large tax loss carryforwards, is exactly the same as in the âextremeâ theories. Thus, although the theories tell different stories about aggregate supply and demand of corporate debt, they make essentially the same predictions about which firms borrow more or less than average. The net tax gain to corporate borrowing. So the tax side of the static tradeoff theory predicts that IBM should borrow more than Bethlehem Steel, other things equal, and that General Motors' debt-to-value ratio should be more than Chrysler's. Costs of financial distress. Costs of financial distress include the legal and administrative costs of bankruptcy, as well as the subtler agency, moral hazard, monitoring and contracting costs which can erode firm value even if formal default is avoided. We know these costs exist, although we may debate their magnitude. For example, there is no satisfactory explanation of debt covenants unless agency costs and moral hazard problems are recognized. The literature on costs of financial distress supports two qualitative statements about financing behavior.6 Risky firms ought to borrow less, other things equal. Here âriskâ would be defined as the variance rate of the market value of the firm's assets. The higher the variance rate, the greater the probability of default on any given package of debt claims. Since costs of financial distress are caused by threatened or actual default, safe firms ought to be able to borrow more before expected costs of financial distress offset the tax advantages of borrowing. Firms holding tangible assets-in-place having active second-hand markets will borrow less than firms holding specialized, intangible assets or valuable growth opportunities. The expected cost of financial distress depends not just on the probability of trouble, but the value lost if trouble comes. Specialized, intangible assets or growth opportunities are more likely to lose value in financial distress. Firms prefer internal finance. They adapt their target dividend payout ratios to their investment opportunities, although dividends are sticky and target payout ratios are only gradually adjusted to shifts in the extent of valuable investment opportunities. Sticky dividend policies, plus unpredictable fluctuations in profitability and investment opportunities, mean that internally-generated cash flow may be more or less than investment outlays. If it is less, the firm first draws down its cash balance or marketable securities portfolio.7 If external finance is required, firms issue the safest security first. That is, they start with debt, then possibly hybrid securities such as convertible bonds, then perhaps equity as a last resort. In this story, there is no well-defined target debt-equity mix, because there are two kinds of equity, internal and external, one at the top of the pecking order and one at the bottom. Each firm's observed debt ratio reflects its cumulative requirements for external finance. The pecking order literature. The pecking order hypothesis is hardly new.8 For example, it comes through loud and clear in Donaldson's 1961 study of the financing practices of a sample of large corporations. He observed 13 that âManagement strongly favored internal generation as a source of new funds even to the exclusion of external funds except for occasional unavoidable âbulgesâ in the need for funds.â These bulges were not generally met by cutting dividends: Reducing the âcustomary cash dividend payment⊠was unthinkable to most managements except as a defensive measure in a period of extreme financial distressâ (p. 70). Given that external finance was needed, managers rarely thought of issuing stock: Though few companies would go so far as to rule out a sale of common under any circumstances, the large majority had not had such a sale in the past 20 years and did not anticipate one in the foreseeable future. This was particularly remarkable in view of the very high Price-Earnings ratios of recent years. Several financial officers showed that they were well aware that this had been a good time to sell common, but the reluctance still persisted. (pp. 57â58). Of course, the pecking order hypothesis can be quickly rejected if we require it to explain everything. There are plenty of examples of firms issuing stock when they could issue investment-grade debt. But when one looks at aggregates, the heavy reliance on internal finance and debt is clear. For all non-financial corporations over the decade 1973â1982, internally generated cash covered, on average, 62 percent of capital expenditures, including investment in inventory and other current assets. The bulk of required external financing came from borrowing. Net new stock issues were never more than 6 percent of external financing.9 Anyone innocent of modern finance who looked at these statistics would find the pecking order idea entirely plausible, at least as a description of typical behavior. Writers on âmanagerial capitalismâ have interpreted firms' reliance on internal finance as a byproduct of the separation of ownership and control: professional managers avoid relying on external finance because it would subject them to the discipline of the capital market.10 Donaldson's 1969 book was not primarily about managerial capitalism, but he nevertheless observed that the financing decisions of the firms he studied were not directed towards and that to explain decisions would have to start by the âmanagerial of corporate finance. This is given the of finance theory in the it is not so that financing by a pecking order against interests. financing with I to the pecking order story because I could think of no theoretical for it that would in with the theory of modern finance. An could be made for internal financing to avoid issue costs, and if external finance is needed, for debt to avoid the still higher costs of But issue costs in do not seem large enough to the costs and benefits of leverage in the static tradeoff story. However, recent work based on information gives predictions in line with the pecking order The is based on a paper by and although I will down that paper's to the firm has to in order to some valuable investment be this net value and be what the firm will be if the opportunity is The firm's what and are, but investors in capital markets do they see only a of values The information is as from the information capital markets are and MM's Proposition I in the sense that the stock of debt to assets is if information to investors is constant. The to by a security issue is the of the firm's investment There is also a the firm may have to sell the securities for less than they are really the firm issues stock with an aggregate market when of will consider debt issues in a However, the the are really That is, is what the new will be other things equal, when investors the special and I managers might in this The one we think makes the most sense is the or value of the firm's That is, the about the value of the in the firm. investors know the will do In the investors who any stock issue will assume that the is not on their and will the they are to If the information is is and the firm will always even if the only good for the funds is to them in the If the information is the firm may a investment opportunity than issue Thus, given and and given that stock is the greater the per the less value is given to new and the less The cost of relying on external We usually think of the cost of external finance as administrative and costs, and in some cases of the new securities. information the of a different of the that the firm will choose not to and will therefore a This cost is if the firm can enough internally-generated cash to its opportunities. The advantages of debt over equity If the firm external it is off issuing debt than equity securities. The rule is, safe securities before This is explaining that the firm issues and if the of its investment is greater than or to the by which the new are if or if For example, suppose the investment but in order to that the firm must issue that are really It will go ahead only if is at least If it is only the firm to the for the value of the firm is by but the are The could have this by the firm's cash that is The only he can do now is to the security issue to For example, if could be cut to the investment could be without the value of The way to is to issue the safest securities whose future value changes least when the information is to the Of course, is so it is loose to of the it. However, there are cases in which the value of is always less for debt than for For example, if the firm can issue debt, is and the firm never a valuable investment Thus, the to issue debt is as good as cash in the if default risk is the value of will be less for debt than for equity if we make the of Thus, if the has information it is to issue debt than This assumes that new or debt would be if the managers' information is so that any security issue would be In this the firm want to make as large as to take advantage of new If stock would seem than debt The rule seems to debt when investors the and equity, or some other when they The trouble with this strategy is you in investors' If you know the firm will issue equity only when it is and debt you will to equity unless the firm has its is, unless the firm has so much debt that it would face costs in issuing Thus investors would the firm to follow a pecking this is too The model just would need of out before it could actual behavior. I have it just to how models based on information can predict the two of the pecking order the for internal the for debt over equity if external financing is I will now what we know about financing behavior and try to make sense of this in of the two I with about financing behavior, and then a few from evidence or personal Of even based on good statistics have been to away under so with external investment are by debt issues and internally-generated stock issues a relatively as has this is what many managers they are to This is what the pecking order hypothesis in the first However, it might also be explained in a static tradeoff theory by adding significant costs of equity issues and the tax of capital gains to This would make external equity relatively It would explain why companies target dividend low enough to avoid having to make stock It would also explain why a firm whose debt ratio target not immediately issue back debt, and a more debt-to-value ratio. Thus firms might take extended excursions their debt that the static tradeoff hypothesis as usually rarely this of adjustment But the costs of seems small. It is thus hard to explain extended excursions a firm's debt target by an static tradeoff firm could quickly issue debt and back if personal income taxes are important in explaining firms' apparent for internal equity, then difficult to explain why external equity is not strongly is, why most firms gradually to lower target payout ratios and the cash to of security Firms try to stock issues when security prices are Given that they external they are more likely to issue stock than stock prices have than they have For example, past stock were one of the variables in study of firms' choices between new debt and new equity and have similar behavior in the This is to static tradeoff If firm value the debt-to-value ratio and firms ought to issue debt, not equity, to their capital The is to the pecking order There is no reason to that the information is more when stock prices are if there were such a investors would have learned it by and would the firm's issue There is no way firms can take advantage of of new equity in a rational against and growth opportunities. Firms holding valuable intangible assets or growth opportunities to borrow less than firms holding tangible assets. For example, and a significant relationship between of investment in and research and and the of borrowing. They also a significant positive relationship between the rate of capital and and the of borrowing. the same by a different for a firm's and growth opportunities was the difference between the market value of its debt and equity securities and the cost of its tangible assets. The higher this he the less the firm's debt-to-value ratio. There is plenty of evidence that the of borrowing is determined not just by the value and risk of the firm's but also by the type of assets it For example, without this the static tradeoff theory would specify all target debt ratios in of not book Since many firms have market values far in of book values if book values are in current we ought to see at least a few such firms operating at very high book debt of we do This to make as as we that book values reflect assets-in-place assets and values reflect and growth opportunities as well as Thus, firms do not set target book debt ratios because the values are for the values of assets in Masulis has that stock prices on average, when a firm
Isolation of the real roots of a polynomial equation is the process of finding real, disjoint intervals such that each contains exactly one real root and every real root is contained in some interval. This process is quite important because, as J. B. J. Fourier pointed out, it constitutes the first step toward the solution of general equations of degree greater than four, the second step being the approximation of roots to any desired degree of accuracy. In the beginning of the 19th century F. D. Budan and J. B. J. Fourier presented two different (but equivalent) theorems which enable us to determine the maximum possible number of real roots that an equation has within a given interval. Budan's theorem appeared in 1807 in the memoir Nouvelle methode pour la resolution des equations numeriques [10, p. 219], whereas Fourier's theorem was first published in 1820 in Le Bulletin des sciences par la Societe Philomatique de Paris, pp. 156, 181 [10, p. 223]. Due to the importance of these two theorems, there was a great controversy regarding priority rights. In his book (1859) Biographies of distinguished scientific men, p. 383, F. Arago informs us that Fourier deemed it necessary to have recourse to the certificates of early students of the Polytechnic School or Professors of the University in order to prove that he had taught his theorem in 1796, 1797 and 1803 [10]. Based on Fourier's proposition, C. Sturm presented in 1829 an improved theorem whose application yields the exact number of real roots which a polynomial equation without multiple zeros has within a real interval; thus he solved the real root isolation problem. Since 1830 Sturm's method has been the only one widely known and used, and consequently Budan's theorem was pushed into oblivion. To our knowledge, Budan's theorem can be found only in [16] and [61 whereas Fourier's proposition appears in almost all texts on the theory of equations. We feel that Budan's theorem merits special attention because it constitutes the basis of Vincent's forgotten theorem of 1836 which, in turn, is the foundation of our method for the isolation of the real roots of an equation [1], a method which far surpasses Sturm's in efficiency [2], [3]. In the discussion which follows we present separately, and without proofs, the classical theorems by Fourier and Budan and we indicate how they lead to the corresponding real root isolation methods. Some empirical results are also presented for comparison.
This thesis is an attempt to generalise to the odd orthogonal group Î_K, over an Infinite field K not of characteristic two, the work of Schur [S], and more recently Green [G], on the general linear group G_K using the approach of Weyl [W] in characteristic zero. The special feature here is that we treat Î_K as merely a group of matrices defined by the vanishing of polynomials in its coefficients (the classical view) rather than a group generated by elements derived from an associated Lie algebra, the approach used initially by Chevalley and adopted by most authors in recent times. \n \nAfter generalising Green's [G] Schur algebra for G_K to Î_K in §0 we prove in §1 Chevalley's famous theorem on the 'Big Cell' in G_K and then, by an easy extension, prove it for the Big Cell in Î_K. Chevalley's original proof uses representations of Lie algebras, ours requires nothing but a little knowledge of the coordinate ring K_+[G] of all 'polynomial' functions on G_K . We define K[Î], the coordinate ring of Î_K, to be the space of all polynomial functions on G_K restricted to Î_K and in §2 give a generating set of the kernel of the restriction map Ï_K:K_+ [G]âK[Î]. This generalises Weyl's result in characteristic zero. In §3 we use this result to show that the family, or 'scheme', of rings K[Î] (K varying over all infinite fields not of characteristic two) is 'defined over Z' ; in fact K[Î] is naturally isomorphic to K Ξ Z[Î_Q], where Z[Î_Q] is the subring of Q[Î] spanned by 'monomial' functions. This enables us to formulate a 'modular' representation theory for Î which connects polynomial representations of Î_Q with those of Î_K. \n \nIn §4 we investigate the Schur algebras of Î_Q following Weyl [W] and in §5 find a complete set of irreducibles for each of them, once again following the lead of Weyl. In §6 we attempt to 'reduce' these modules modulo p to obtain 'Weyl' modules for Î_K, a task only partially completed.
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Advanced Algebra and Geometry
Finite Group Theory Research
Advanced Differential Equations and Dynamical Systems
THE MAJOR INTEREST in the recent literature on economies has lain in the results about but finite economies that have been derived from the results proved for economies. It is thus important to find simple yet general proofs for economies. In this article we wish to provide a simple yet general proof of the existence of a competitive equilibrium in an infinite, nonstandard economy with production. The simplicity of our proof comes from the fact that nonstandard analysis can deal with large and small quantities very much as ordinary analysis deals with finite quantities.2 As a result, it is possible to follow very closely the proof of existence for an economy with a finite number of traders, such as that in G. Debreu's classic Theory of Value [6]. In fact, if one is willing to believe that nonstandard analysis permits us to manipulate quantities as claimed above, then no further knowledge of nonstandard analysis is required in order to follow the proof. As examples of the simplicity of nonstandard analysis, it may be pointed out that no analogue of the Fatou-Schmeidler lemma [7, p. 69], a fairly difficult mathematical theorem, is required; nor is it necessary to prove separately that preserves upper-semicontinuity, a proof that Aumann [1] has recently simplified, because integration in the nonstandard model consists of an infinite summation, hence an appeal to 1.9.4 of Debreu [6] suffices to establish this point. As our main objective is to obtain results about but finite economies, it is a welcome bonus to find out that no further effort is needed to obtain these desired theorems. This arises because of the following property of nonstandard analysis. Consider a sequence of real numbers {an} which tends to zero. If we could extend this sequence to the integers, it would surely be a necessary property of the values of {an} at the integers that they are all infinitely close to zero. What makes nonstandard analysis powerful is that the above line of reasoning can be reversed, so to speak. Suppose we have a sequence which
We study the following two related questions:\n- What are the minimal computational resources required for general secure multiparty computation in the presence of an honest majority?\n- What are the minimal resources required for two-party primitives such as zero-knowledge proofs and general secure two-party computation?\nWe obtain a nearly tight answer to the first question by presenting a perfectly secure protocol which allows n players to evaluate an arithmetic circuit of size s by performing a total of O(s log s log2 n) arithmetic operations, plus an additive term which depends (polynomially) on n and the circuit depth, but only logarithmically on s. Thus, for typical large-scale computations whose circuit width is much bigger than their depth and the number of players, the amortized overhead is just polylogarithmic in n and s. The protocol provides perfect security with guaranteed output delivery in the presence of an active, adaptive adversary corrupting a (1/3 Âż ") fraction of the players, for an arbitrary constant Âż > 0 and succiently large n. The best previous protocols in this setting could only offer computational security with a computational overhead of poly(k; log n; log s), where k is a computational security parameter, or perfect security with a computational overhead of O(n log n). We then apply the above result towards making progress on the second question. Concretely, under standard cryptographic assumptions, we obtain zero-knowledge proofs for circuit satisfiability with 2 -k soundness error in which the amortized computational overhead per gate is only polylogarithmic in k, improving over the Âż(k) overhead of the best previous protocols. Under stronger cryptographic assumptions, we obtain similar results for general secure two-party computation.
Zero-field and high-field optical detection of magnetic resonance (ODMR), electron paramagnetic resonance (EPR), and optical spectroscopy experiments have been performed on several systems in order to further basic knowledge of the structure, reactions, and response to radiation of atoms, molecules, and ions in their ground and/or excited electronic states. Particularly noteworthy results for the present contract year include the determination of the complete magnetic and optical properties of the lowest triplet states of 1-chloro, 1-bromo, and 1-iodonaphthalene, the development of a microscopic model for the intramolecular heavy-atom effect in the /sup 3/(..pi..,..pi..*) states of aromatic molecules, a detailed analysis of the angular dependence of the hyperfine and quadrupole structure in triplet 1-bromonaphthalene, observation of proton hyperfine structure in the hf ODMR spectra of short-lived triplet states, a definitive paper on the relative importance of spin delocalization and second-order spin-orbit coupling effects in /sup 3/(n,..pi..*) benzophenone (a phototype photochemical system), a detailed analysis of the level-anticrossing spectra of several triplet state benzophenones which exhibit hyperfine structure in the cross-relaxation region (thus permitting the determination of key magnetic parameters in the complete absence of perturbing microwave or radiofrequency fields), optical detection of ground-state NQR transitions in host crystal molecules, the observation of strong radiofrequency transitions near avoided crossing points in Zeeman energy level diagrams of photoexcited triplet states, the construction of zero-field ODMR, ODENDOR, and hf ODENDOR spectrometers, measurements of the activation parameters for ring interconversions of several free radicals containing five- and six-membered rings, and experimental proof that the triplet state of trimethylenemethane (a key reactive intermediate in organic chemistry) is the ground state.
Abstract We consider a container of fluid with a rod inserted in the centre. As the rod rotates the surface of the fluid forms a curved surface whose shape correctly balances the forces of gravity, internal stress, atmospheric pressure, and surface tension. The surface of the fluid is depressed in the neighbourhood of the rod in the case of a Newtonian fluid, but the fluid may climb along the rod in the case of non-Newtonian fluids. In recent work by D. D. Joseph and R. S. Fosdick this free surface problem has been treated quantitatively by virtue of a formal perturbation series in which the solution is developed in powers of the angular velocity of the rod. The purpose of the present paper is to give a rigorous proof of convergence in the case of a Newtonian fluid. The method here may possibly be applied to other free surface problems - for example, surface waves of a viscous fluid. The convergence proof provides, to our knowledge, the first rigorous existence theorem for a free surface problem in the theory of viscous fluids. The proof also raises a novel problem in the theory of elliptic systems of partial differential equations. By means of the implicit function theorem, the question of convergence is reduced to that of obtaining a priori estimates for an elliptic boundary value problem. That problem is formulated on a domain with a ridge where the fluid surface meets the rod. In addition, the type of boundary conditions prescribed differ on the free surface and on the rod. In general, the solution to such a mixed boundary value problem is not smooth at the ridge, even if the boundary data is smooth. The solution will be smooth, however, if the boundary data satisfy certain consistency conditions which make it compatible with the given set of partial differential equations. The consistency conditions in question here are a set of linear relations between various derivatives of the boundary and inhomogeneous data. It is shown that if one makes the assumption that the wetting angle of the surface is zero, the consistency conditions are invariant under the full nonlinear equations of the free surface problem. This makes it possible to consider the original problem on a smaller function space - namely the subclass of functions satisfying the appropriate consistency conditions - and in this subclass one can apply the implicit function theorem and obtain the required a priori estimates. The solution thus obtained is regular up to the ridge.
Navier-Stokes equation solutions
Elasticity and Material Modeling
Advanced Numerical Methods in Computational Mathematics
A review of optimal control theory for linear systems with quadratic cost functions is presented. Some of the theoretical and practical limitations are discussed with special reference to distributed parameter systems. First a procedure is described for finding the optimal control by constructing a sequence of controllers that converges to the optimal; this method is valid for systems of infinite dimension provided that the operators in the state differential equation satisfy certain conditions. The proof is carried out both for the finite and infinite time interval and the connection is shown with the Riccati equation. The main problem in implementation is that one needs complete knowledge of the state at all times in order to build the optimal controller, this is almost certainly impossible for distributed parameter systems. When the state cannot be measured completely it is proved that an optimal control is realisable for time invariant finite dimensional systems. \n \nThe problems of finding this control are then investigated and computational methods discussed. If the optimal control with complete knowledge of the state cannot be implemented, a method is presented whereby one can find bounds on the possible increase in the value of the cost function arising from the use of some sub-optimal control; several examples are considered. The constrained optimal control depends on the initial state and new optimisation criteria must be put forward to deal with the case in which the initial state is unknown; the most common consist of minimising the cost that can result from the worst initial state. It is then shown how the controllers designed according to these criteria may be improved by using one's limited observation at time zero to place some constraints on the initial state. The Liapunov matrix equation plays an important part in calculating the cost of any control so reducing the computational effort in its solution is useful. It is shown how this can be done and it is of special relevance for distributed parameter systems with their states expressed as an infinite series of eigenfunctions; the results are applied to a diffusion equation example. \n \nFinally, it is shown how optimal control theory may be applied to the design of proportional-integral-derivative controllers. This is done from two standpoints and the resulting controllers are shown to be identical, though the second method of proof is valid for infinite dimensional systems. The results are then applied to a simple example and to a distributed population dynamics system. The practicality of the methods of the thesis are applied to a system with realistic parameters; recommendations are made as to the best approaches. \n
B. Balamurugan, T. Poongodi, M. R. Manu, S. Karthikeyan · 5 authors
The moving image archive of the US Agency for International Development (USAID) includes a copy of the film The Double Day (1975), cataloged in the series Moving Images Relating to International Development Programs and Activities, 1979â1991, a collection of more than eight hundred titles âcreated to provide information on assistance programs supported by the Agency for International Development (AID).â1 Yet, The Double Day does not, in fact, directly depict or engage with any specific development or aid initiative. Instead, the filmâdirected by US-based Brazilian filmmaker Helena Solberg as part of the International Womenâs Film Project collective and described as âthe first Latin American feminist documentaryââexamines the gendered dynamics of paid and unpaid labor through the testimonies of women from Argentina, Bolivia, Venezuela, and Mexico.2 Its presence in the USAID archive is likely a consequence of its funding history, having received support from the Inter-American Foundation, a USAID-affiliated entity; the development agencies of Denmark, Norway, and Sweden; the United Nations Development Program; and US philanthropist Calvin Cafritz.3These transnational funding structures not only enabled the filmâs production but also determined its archival destination, which renders legible its place within the history of international development.4 The Double Dayâs institutional trajectory reflects the shifting configurations of aid, gender, and media during a historic moment when women were being repositioned at the center of what Arturo Escobar has described as developmentâs âregimes of visuality.â5 Especially relevant to The Double Dayâs production and exhibition was the international institutional framework of Women in Development (WID). Emerging in the early 1970s and culminating in United Nationsâ proclamation of 1975 as International Womenâs Year, WID emphasized womenâs participation in the global economy as both an index and mechanism of development. Indeed, The Double Day premiered at the World Conference of the International Womenâs Year, held in Mexico City.6 Within this context, the film forms part of a broader trajectory of media use by international organizations that intensified during the 1970sâas best exemplified by Media Habitat, a collection of 236 documentary films commissioned by the United Nations to represent urban and rural development initiatives for the 1976 Habitat Conference on Human Settlements in Vancouver. As a policy-shaping initiative, Media Habitat primarily featured films from the Global South intended not only to illustrate but also to help codify standardized audiovisual markers of âunderdevelopmentâ that determined access to the emerging global economic order and to international aid.7Framed in relation to these international institutions and their operations, The Double Day could similarly be considered âdevelopment mediaââexemplifying the type of nonfiction media produced and distributed outside of the commercial film circuits whose aspects and subcategories have been variously described in scholarship as nontheatrical, useful, sponsored, institutional, industrial, educational, or nonprofessional/amateur.8 Scholars working in this area have emphasized the institutional contexts of such mediaâs production and exhibition infrastructures as shaping its instrumentalized effects. From a feminist perspective, such an approach is crucial to grappling with the broader question of how âgender impacts [these worksâ] shape, content, and trajectories.â9 Yet we also argue that, taken in isolation, the institutional and infrastructural contexts are insufficient to account for the complex relationship between media and development, potentially not only limiting our understanding of the reach and impact of development but also distorting our interpretive conclusions. For example, to categorize The Double Day as âdevelopment mediaâ is to overlook the filmâs place within Solbergâs directorial oeuvre, as well as within the histories of both transnational womenâs filmmaking and radical Latin American documentary cinema to which it simultaneously belongs. Such exclusive framing is especially limiting given that women globally were disproportionately engaged in nonfiction production throughout the twentieth centuryâsometimes by political choice but more often due to structural exclusions from fiction filmmaking. Even in nonfiction historiography, however, institutional media has remained particularly marginal, reinforcing hierarchies that separate such works from the aesthetic and authorial frameworks through which film history has been constructed.10 This marginalization not only tends to erase womenâs contributions but also presumes a âweakâ or derivative authorship, rendering these films unworthy of the interpretive attention needed to apprehend their aesthetic and political complexity.11 The same dynamic is likely to structure assumptions about âdevelopment mediaâ as well.Categorizing The Double Day exclusively within this category would further prompt us to assume top-down institutional analyses that have been characteristic of both institutional media methodologies and the scholarship on development at large. This, in turn, would risk obscuring this filmâs radical Marxist approach to womenâs labor as well as its concrete contribution to activism and its attendant grassroot structures. In Mexico City, The Double Day became a catalyst for feminist solidarity in practice when one of the filmâs protagonists, Bolivian activist and trade unionist Domitila Barrios de Chungara, was invited to participate in the Tribune of Non-Governmental Organizations held alongside the official UN conference.12 There, Barrios de Chungara challenged Western feminist priorities by reframing the debate around labor, class, and imperialism, helping to articulate a shared Third World feminist agenda that significantly departed from the developmentalist vision of the United Nations and USAID.13 Seen through the lens of activist media, The Double Day helped forge transnational solidarity networks by enabling information exchange across the diverse voices that shaped its makingâfrom the women featured in the film to the activists who circulated itârevealing a considerably more dynamic interplay between institutional and grassroots or contingent media practices.Moreover, the filmâs Latin American contextâreflected in Solbergâs formation in Brazil as the only woman in Cinema Novo, its focus on women from across the region, and its premiere and key reception in Mexico Cityârequires grappling with the regional specificities of the very notion of development in its multiple iterations.14 Far from being an epistemological and political framework imposed solely by the Global North, both the practices of development and the theoretical foundations of developmentalism (understood as a broad and polysemic set of discourses) were shaped through the active participation of Latin American economists.15 Within this iteration, underdevelopment, as a constitutive notion of developmentalism, became central to a distinctly critical strand, which by the late 1960s became known as the dependency theory.16 This same approach is reflected in some of the best-known Latin American radical film manifestos of the time, arising precisely from the same milieus to which Solberg belonged.17These various considerations of the filmâs history illustrate the methodological challenges confronting feminist scholars seeking to assess the impact of development on media projects, theories, and practices. To disregard the developmentalist context of such works by emphasizing their political aesthetics and affects risks reproducing a romanticized narrative of heroic resistance (albeit from a feminist perspective). Yet to engage exclusively with their institutional and material infrastructures risks naturalizing developmentalismâs political and epistemological foundations at the expense of the goals and beliefs of the many women who participated in these projects. The contradictions and ambivalences that animate such histories call for feminist frameworks capable of holding both institutional complicity and radical possibility in view.This challenge resonates with ongoing debates about the politics of the archive and what Allyson Field has termed âthe practice of informed speculation.â18 As she reminds us, feminist, queer, and decolonial methodologies have long taught us to âpress at the limitsâ of the archive to âinoculate our scholarship against our evidenceâs afflictions.â19 The concern that the evidence we draw on in our analysis reproduces the very structures and blind spots of the dominant ideology and therefore shapes and delimits our interpretation becomes particularly urgent when engaging the developmentalist media corpus. Informed speculation offers an alternative by inviting the experimental, creative, and speculative rewriting of history, mobilizing the archive âin a project that runs counter to the original purpose, or the imperative to preserve, or the conditions that led to erasure.â20 Yet, as Field cautions, such speculative gestures must remain grounded in a deep and âintimate familiarity with the archiveâ that we are working with and against. Building on this imperative, we suggest that the developmentalist archive, in particular, demands expansion and critical reconsideration in ways that unsettle the very disciplinary frameworks through which it has been studied as well as the larger institutional contexts for such knowledge production.Our focus on The Double Day in the opening of this introduction thus foregrounds the entanglements of institutional and grassroots forces, local and international contexts, structural and interpersonal relations, and creative and economic factors that have shaped not only this film but the broader ecosystem of development media projectsâand their preservationâover time. Addressing such a constellation involves transgressing methodologically entrenched divisions between political economy and aesthetics, between material infrastructures and affective regimes; reckoning with divergent periodizations across film history and world economics; and situating these within the local specificities of womenâs movements and international institutional programs. It also demands attentiveness to the coexistence of multiple, and sometimes competing, understandings of developmentâeach historically, geographically, and ideologically situated.We imagine this special issue as an opening toward a critical dialogue, not only about how such an approach might be enacted in practice but also about the far-reaching ways development paradigms have shaped both our objects of study and the contours of the field itself. The decision to center institutionally sponsored films across all the essays in this issue is deliberate and enables us to highlight institutional critique as a vital methodological imperative within our analytical framework. Created within the frameworks of international organizations, state agencies, or NGOs, these filmsâ histories make legible the institutional logics that underwrite their production. Yet our critique does not stop at these specific entities. Rather, we argue for a broader interrogation of the political conditions and institutional infrastructures that shape media and knowledge production more generally. This includes contemporary corporations embedded in the digital platform economy, from streaming services to the rapid expansion of AI. Equally critical is a reflexive examination of academia itself, where departments of economics, political science, and centers for development have played a formative role in producing and legitimizing developmentalist theories and policy frameworks. While the humanities and arts have at times offered critical alternatives, they have also frequently mirrored and reinforced many of the same developmentalist assumptions. A feminist analysis of the nexus between development and media must therefore unsettle not only dominant archives but also the institutional and disciplinary foundations of our own scholarly practices.As scholars, we share the complex position of navigating the same tensions between institutional complicity and emancipatory aspiration as many of the media-makers whose work we study. Mirroring our subjects is also the transnational, collective mode of this special issueâs own production as it emerges from an ongoing informal working group we have sustained over several years. While relying on institutional and disciplinary affordancesâsuch as university funding for conferences or access to academic publishing platformsâwe have been working toward creating a community that exceeds, and often resists, the prevailing logics of our academic institutions. Our aim has been to create a space for shared inquiry and mutual support that pushes back against disciplinary siloing and technocratic neoliberal assessment modes of both labor and knowledge productionâand this certainly extends to our experience collaborating with the journal editors throughout the publication process. Our goal has been to examine both the persistence and variability of developmentalism, understood as what Gustavo Esteva calls a âpowerful but fragile semantic constellation,â as a conceptual formation that has historically inspired, legitimized, and mobilized media projects across Asia, Africa, and Latin America.21 And gendered biopolitics, from population control to gender mainstreaming, have remained integral to development policies and media practices, recurring across formats from institutional newsreels to film festivals.We share the conviction that, far beyond the history of nonfiction institutional media, development (as both a broad ideological project and a network of material and institutional practices) and developmentalism (as a set of discourses and theoretical models associated with development) have exerted a far-reaching influence on film and media cultures at large. As such, they must be treated as a major force in shaping global film and media systems and also the many ongoing assumptions behind their critical discourses. The discipline of communication studies was founded on modernization theory governed by Cold War goals of dissemination of Western liberal democracy around the world, while âan area studies framework allowed compartmentalizing Western and non-Western outcomes of technologies that were always claimed to be universal.â22 Despite critique from postcolonial and critical race studies, many of these frameworks have remained foundational for media theory.23 In historical scholarship, as we increasingly move beyond âmodernityâ as a dominant conceptual anchor, engaging with practices and discourses of development opens more precise analytical pathways. These film and mediaâs entanglements with the logics of and economic and the associated with as well as their conceptual underdevelopment, and impact in as diverse as and of media aesthetic and paradigms in film and film and funding are developmentalist assumptions to the they to underwrite the narrative and logics of and global media from the structure of the film to the cultures of argue that a examination of the historical entanglements between film and developmentalist aesthetics, modes of as well as infrastructures and critical the and that our contemporary media is a of the historical contours of development as a field of inquiry embedded within a of some of the methodologies by the featured in this special a of this complex history, to the larger at in media, development, and gender all its development a it as an of be with practices as divergent as and in In as we work on this the of the of the USAID the ideological that have the history of international the and of this on the global its a from development as a global practice and as an institutional and which over the has as a with its own media Yet or does not erase the historical impact of these the contradictions they have reflected and over the as the of development several crucial to the economic theories of development initiatives to and through the of the development an international policy framework at from the Global to the of the Global the by the political role of and the of the International the on was challenged by dependency theory and at development as global structural the neoliberal of the the Programs by the and World imposed and as conditions for and forms of assistance became known as the major markers for international development development has moving beyond economic to such as the Human Development and the Development and from international organizations such as the United Nations to a broader network of These frameworks increasingly of and liberal of and with what in many ways to various while the of the to developmentâs epistemological more scholarship has toward a of its and material the field has to a of and critical by and scholars, which development as a dynamic of conceptual and political to these the studies in this special issue call for sustained analysis in place of In we draw from a of and that in of of gender and to this special issue engage that and from Mexico to contemporary specific local historical of global distinctly feminist methodologies as well as attention to studies that this issue a in the global history of development on and documentary work in Mexico how early women infrastructures as of while US films an early of state and transnational on a outside the Global as the of developmentalist both the and markers of development media, its to the and Latin central grounded in archival also womenâs in transnational and the methodological of archives on study of the series to a a digital media not by but by women not as but as subjects and within circuits of neoliberal and the aesthetics and of how developmentâs has models of to contemporary neoliberal paradigms of and analysis how digital and gender and practices, discourses on and their of these essays the of some of the historical of developmentalist as the question of the role of the state within such on a notion of economic as a and the dominant of development that the state as the for and through infrastructural projects. This understanding of development was shared across both of the as well as in the Third The United and the increasingly mobilized international in the of the modes of international at the postcolonial of policy and ideological These programs were at the expansion of their and of while with the and decolonial of The conceptual between modernization and development that the Cold in of and hierarchies of that long and As an of postcolonial their role as of international aid concern about the of Western development projects. In alternative across the Global South that to structural between and through policies of and In many they were further the global structures. In Latin for example, this approach was through organizations such as the United Nations for Latin and the both developmentalist projects and their have taken many forms historically, and they have and they have been to a of projects, and postcolonial and development has been and in ways and with frequently political While many of these alternative frameworks the embedded in dominant Western they often technocratic and assumptions. gender and gender as for the of to be shaped by This becomes particularly in the media that of development where women frequently a crucial associated with and women were as both the subjects and of their participation in the and labor often as the index of In this study of from the 1970s how women were as of this as these about womenâs urban and recurring and of a within the own of and critique that official examination of archives and enables a feminist interrogation of their mobilizing a of the in the of contribution to this issue similarly the of a dynamic within as well as across ideological the of womenâs and the media of the and as active in labor and as integral to the technocratic women became of as their labor remained a of and analysis and archival with attention to the specific of and It also transnational that as a and affective a nexus of and practices through which gender, media, and developmentalism one film not as a but as part of the of alongside and work a broader concern of this special media as a of developmentalist projects. The debates the World and which international such as the United Nations and a critical historical for such primarily by postcolonial and of the for the of media and communication infrastructures in as to counter Western The a between and economic how systems of media and reinforced global hierarchies of and in an agenda that supported and alternative media infrastructures across postcolonial contexts, the framework gender as an analytical This âgender in the 1970s by of the Women in Development in relation to The Double its political critique of with its broader to center women in development policy through and This was also in of media, which was shaped by its international reach and audiovisual production and and and Yet, as WID institutional it also to the and of the womenâs development programs often by in These were by a of studies, and on and the to of womenâs presence and through the of their economic and In this on the filmmaker this history directly by the and of for women by the WID framework during the with and agencies the of and against Women in at the of the neoliberal turn, when institutional media often the only to access and support to projects. The an by this dynamic as of the from the Global South are to engage with the developmentalist assumptions funding models that their similarly the as a in developmentalist media history by the of and centers on the Film a between of and the attention toward the and labor by and work was foundational the of and the of audiovisual aid, as both a of development projects and a critical for their the in this issue argue for a with media, gender, and developmentalism as an field of that is and shaped by feminist and methodologies and that challenge the entanglements between academic knowledge production and and technocratic development And while methodological these dynamics at the structural the of feminist in media and cinema by this the very a
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.
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.
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.