336 research outputs found

    The Relativized Second Eigenvalue Conjecture of Alon

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    We prove a relativization of the Alon Second Eigenvalue Conjecture for all dd-regular base graphs, BB, with d≥3d\ge 3: for any ϵ>0\epsilon>0, we show that a random covering map of degree nn to BB has a new eigenvalue greater than 2d−1+ϵ2\sqrt{d-1}+\epsilon in absolute value with probability O(1/n)O(1/n). Furthermore, if BB is a Ramanujan graph, we show that this probability is proportional to n−η fund(B)n^{-{\eta_{\rm \,fund}}(B)}, where η fund(B){\eta_{\rm \,fund}}(B) is an integer depending on BB, which can be computed by a finite algorithm for any fixed BB. For any dd-regular graph, BB, η fund(B){\eta_{\rm \,fund}}(B) is greater than d−1\sqrt{d-1}. Our proof introduces a number of ideas that simplify and strengthen the methods of Friedman's proof of the original conjecture of Alon. The most significant new idea is that of a ``certified trace,'' which is not only greatly simplifies our trace methods, but is the reason we can obtain the n−η fund(B)n^{-{\eta_{\rm \,fund}}(B)} estimate above. This estimate represents an improvement over Friedman's results of the original Alon conjecture for random dd-regular graphs, for certain values of dd

    THE RHETORIC OF FRIEDMAN'S QUANTITY THEORY MANIFESTO

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    In 1956 when Friedman published his ""The Quantity Theory of Money: A Restatement"" he faced a daunting task because of the widespread hostility to the quantity theory. This paper looks at the rhetoric (in the non-pejorative sense of the term) that he used to overcome this obstacle, and at some of the characteristics of the essay that contribute to its persuasiveness.

    Epistemic systems and Flagg and Friedman's translation

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    In 1986, Flagg and Friedman \cite{ff} gave an elegant alternative proof of the faithfulness of G\"{o}del (or Rasiowa-Sikorski) translation (⋅)□(\cdot)^\Box of Heyting arithmetic HA\bf HA to Shapiro's epistemic arithmetic EA\bf EA. In \S 2, we shall prove the faithfulness of (⋅)□(\cdot)^\Box without using stability, by introducing another translation from an epistemic system to corresponding intuitionistic system which we shall call \it the modified Rasiowa-Sikorski translation\rm . That is, this introduction of the new translation simplifies the original Flagg and Friedman's proof. In \S 3, we shall give some applications of the modified one for the disjunction property (DP\mathsf{DP}) and the numerical existence property (NEP\mathsf{NEP}) of Heyting arithmetic. In \S 4, we shall show that epistemic Markov's rule EMR\mathsf{EMR} in EA\bf EA is proved via HA\bf HA. So EA\bf EA ⊢EMR\vdash \mathsf{EMR} and HA\bf HA ⊢MR\vdash \mathsf{MR} are equivalent. In \S 5, we shall give some relations among the translations treated in the previous sections. In \S 6, we shall give an alternative proof of Glivenko's theorem. In \S 7, we shall propose several(modal-)epistemic versions of Markov's rule for Horsten's modal-epistemic arithmetic MEA\bf MEA. And, as in \S 4, we shall study some meta-implications among those versions of Markov's rules in MEA\bf MEA and one in HA\bf HA. Friedman and Sheard gave a modal analogue FS\mathsf{FS} (i.e. Theorem in \cite{fs}) of Friedman's theorem F\mathsf{F} (i.e. Theorem 1 in \cite {friedman}): \it Any recursively enumerable extension of HA\bf HA which has DP\mathsf{DP} also has NPE\mathsf{NPE}\rm . In \S 8, we shall give a proof of our \it Fundamental Conjecture \rm FC\mathsf{FC} proposed in Inou\'{e} \cite{ino90a} as follows: FC:FS⟹F.\mathsf{FC}: \enspace \mathsf{FS} \enspace \Longrightarrow \enspace \mathsf{F}. This is a new type of proofs. In \S 9, I shall give discussions.Comment: 33 page

    THE INFLUENCE OF FRIEDMAN'S METHODOLOGICAL ESSAY

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    Many leading methodologists have described the central role that Milton Friedman''s 1953 essay (henceforth referred to as F53) has played in methodological discussions. (See for instance Daniel Hammond, 1998; Kevin Hoover; 2001; Roger Backhouse 2002.) However, it does not necessarily follow that it has had a great influence on the practice of economics, because practicing economists pay little attention to free-standing discussions of methodology; at best they learn their methodology by seeing it put to work on substantive problems.1 Arguably, Friedman and Schwartz''s (1963) A Monetary History of the United States, has had more influence on the methodology of practicing economist than did F53. The most pervasive methodological influence in macroeconomics in the last thirty years has been the insistence of new classical economists on reducing macroeconomics to microeconomics, and in this they paid no attention to philosophical debates about reductionism

    Why Friedman's Non-monotonic Reasoning Defies Hempel's Covering Law Model

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    In this paper we will show that Hempel's covering law model can't deal very well with explanations that are based on incomplete knowledge. In particular the symmetry thesis, which is an important aspect of the covering law model, turns out to be problematic for these explanations. We will discuss an example of an electric circuit, which clearly indicates that the symmetry of explanation and prediction does not always hold. It will be argued that an alternative logic for causal explanation is needed. And we will investigate to what extent non-monotonic epistemic logic can provide such an alternative logical framework. Finally we will show that our non-monotonic logical analysis of explanation is not only suitable for simple cases such as the electric circuit, but that it also sheds new light on more controversial causal explanations such as Milton Friedman's explanation of the business cycle

    Explanation, unification, and content

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    The following is an essay on the notion of scientific explanation as unification. In it a new notion of (logical) content is used to explicate Michael Friedman's notion of "k-atomicity," and to explicate the notion of the surplus content of hypothesis h relative to evidence e. From this basis an analysis of unification as theoretical reduction is advanced. A second notion of unification, unification as reconciling prima facie incompatible statements, is introduced again with the aid of this new notion of content. More generally, it is argued that rather than seek the essence of scientific explanation we should carefully catalog the various distinct explanatory virtues. Finally it is argue that in particular philosophical explanations put a high premium on the quality of unification through showing how to make compatible seemingly irreconcilable claims

    Investment, Consumption, and Hedging under Incomplete Markets

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    Entrepreneurs often face undiversifiable idiosyncratic risks from their business investments. We extend the standard real options approach to an incomplete markets environment and analyze their joint decisions of business investments, consumption/savings, and portfolio selection. For a lump-sum investment payoff and an agent with a su¡Àciently strong precautionary savings motive, an increase in volatility can accelerate investment, contrary to the standard real options analysis. When the agent can trade the market portfolio to partially hedge against investment risk, the systematic volatility is compensated via the standard CAPM argument, and the idiosyncratic volatility generates a private equity premium. Finally, when the investment payoff is a series of flows, the agent's idiosyncratic risk exposure alters both the implied option value and the implied project value, causing a reversal of the results in the lump-sum payoff case.real options, idiosyncratic risk, hedging, risk aversion, precautionary savings, incomplete markets

    Gap Embedding for Well-Quasi-Orderings1 1This research was supported in part by The Israel Science Foundation (grant no. 254/01).

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    AbstractGiven a quasi-ordering of labels, a labelled ordered tree s is embedded with gaps in another tree t if there is an injection from the nodes of s into those of t that maps each edge in s to a unique disjoint path in t with greater-or-equivalent labels, and which preserves the order of children. We show that finite trees are well-quasi-ordered with respect to gap embedding when labels are taken from an arbitrary well-quasi-ordering such that each tree path can be partitioned into a bounded number of subpaths of comparable nodes. This extends Kříž's result and is also optimal in the sense that unbounded incomparability yields a counterexample

    Curriculum Vitae (Feminae): Biography and Early American Women Lawyers

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    In this review, Carol Sanger examines the recent surge of interest in the lives of early women lawyers. Using Jane Friedman\u27s biography of Myra Bradwell, America\u27s First Woman Lawyer, as a starting point, Professor Sanger explores the complexities for the feminist biographer of reconciling for herself and for her subject conflicting professional, political, and personal sensibilities. Professor Sanger concludes that to advance the project of women\u27s history, feminist biographers ought not retreat to the comforts of commemorative Victorian biography, even for Victorian subjects, but should instead strive to present and accept early women subjects on their own complex terms

    Some Remarks on Monetary Policy in an Overlapping Generations Model

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    It is the purpose of this paper to show that certain results (derived from rational expectations monetary models where real balance services enter the utility function directly) —such as (1) an increase in the mean of the rate of growth of the money supply induces a welfare loss and (2) an increase in the variance of the rate of growth of the money supply may cause an increase in welfare—are not dependent upon the Friedman-Patinkin-Samuelson device of inserting real balances into the utility function
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