DOCSLIB.ORG

Mathematical Logic and Its Applications 2020 • Vassily Lyubetsky and Vladimir Kanovei Mathematical Logic and Its Applications 2020

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Mathematical Logic and Its Applications 2020 • Vassily Lyubetsky and Vladimir Kanovei Mathematical Logic and Its Applications 2020 Mathematical Logic and Its and Mathematical Logic Applications 2020 • Vassily Lyubetsky and Vladimir Kanovei Mathematical Logic and Its Applications 2020 Edited by Vassily Lyubetsky and Vladimir Kanovei Printed Edition of the Special Issue Published in Mathematics www.mdpi.com/journal/mathematics Mathematical Logic and Its Applications 2020 Mathematical Logic and Its Applications 2020 Editors Vassily Lyubetsky Vladimir Kanovei MDPI • Basel • Beijing • Wuhan • Barcelona • Belgrade • Manchester • Tokyo • Cluj • Tianjin Editors Vassily Lyubetsky Vladimir Kanovei Department of Mechanics and Institute for Information Mathematics of Moscow Transmission Problems of the Lomonosov State University Russian Academy of Sciences Russia Russia Editorial Office MDPI St. Alban-Anlage 66 4052 Basel, Switzerland This is a reprint of articles from the Special Issue published online in the open access journal Mathematics (ISSN 2227-7390) (available at: https://www.mdpi.com/journal/mathematics/special issues/math-logic-2020). For citation purposes, cite each article independently as indicated on the article page online and as indicated below: LastName, A.A.; LastName, B.B.; LastName, C.C. Article Title. Journal Name Year, Volume Number, Page Range. ISBN 978-3-0365-0778-1 (Hbk) ISBN 978-3-0365-0779-8 (PDF) © 2021 by the authors. Articles in this book are Open Access and distributed under the Creative Commons Attribution (CC BY) license, which allows users to download, copy and build upon published articles, as long as the author and publisher are properly credited, which ensures maximum dissemination and a wider impact of our publications. The book as a whole is distributed by MDPI under the terms and conditions of the Creative Commons license CC BY-NC-ND. Contents About the Editors .............................................. vii Preface to ”Mathematical Logic and Its Applications 2020” ..................... ix Vladimir Kanovei and Vassily Lyubetsky Models of Set Theory in which NonconstructibleReals First Appear at a Given Projective Level Reprinted from: Mathematics 2020, 8, 910, doi:10.3390/math8060910 ................. 1 Vladimir Kanovei and Vassily Lyubetsky 1 On the Δn Problem of Harvey Friedman Reprinted from: Mathematics 2020, 8, 1477, doi:10.3390/math8091477 ................ 47 Vladimir Kanovei and Vassily Lyubetsky On the ‘Definability of Definable’Problem of Alfred Tarski Reprinted from: Mathematics 2020, 8, 2214, doi:10.3390/math8122214 ................ 77 Konstantin Gorbunov and Vassily Lyubetsky Linear Time Additively Exact Algorithm for Transformation of Chain-Cycle Graphs for Arbitrary Costs of Deletions and Insertions Reprinted from: Mathematics 2020, 8, 2001, doi:10.3390/math8112001 ................113 Alexei Kanel-Belov, Alexei Chilikov, Ilya Ivanov-Pogodaev, Sergey Malev, Eugeny Plotkin, Jie-Tai Yu and Wenchao Zhang Nonstandard Analysis, Deformation Quantization and Some Logical Aspects of (Non)Commutative Algebraic Geometry Reprinted from: Mathematics 2020, 8, 1694, doi:10.3390/math8101694 ................143 Irina Alchinova and Mikhail Karganov Physiological Balance of the Body: Theory, Algorithms, and Results Reprinted from: Mathematics 2021, 9, 209, doi:10.3390/math9030209 .................177 v About the Editors Vassily Lyubetsky is a Doctor of physical and mathematical sciences, professor, and a principal researcher. He is also head of the Laboratory of Mathematical Methods and Models in Bioinformatics at the Institute for Information Transmission Problems of the Russian Academy of Sciences (IITP, Moscow), and Professor of the Faculty of Mechanics and Mathematics of the Moscow State University of the Department of Mathematical Logic and Theory of Algorithms. He has published more than 200 scientific papers and 9 books, and the Guest Invited Editor for the Regular Special Issue ”Molecular Phylogenomics” of the ”Biomed Research International journal (Molecular Biology)”. Vladimir Kanovei is a principal researcher at the Institute for Information Transmission Problems of the Russian Academy of Sciences (IITP, Moscow). He was awarded his Ph.D. by the Moscow State University, and his D.Sc. by the Steklov Mathematical Institute of the Russian Academy of Sciences. His research interests in mathematics include mathematical logic, descriptive set theory, forcing, and nonstandard analysis. He is an author of about 300 scientific publications, including 7 monographs. vii Preface to ”Mathematical Logic and Its Applications 2020” This Special Issue contains articles representing three directions: Descriptive set theory (DTM), exact polynomial complexity algorithms (EPA), and applications of mathematical logic and algorithm theory (Appl). We will say a few words about each of the directions. In accordance with the classical description of Nicolas Luzin, DTM considers simple properties of simple sets of real numbers R. “Simple” sets are Borel sets (the smallest family containing n open and closed sets in R and closed with respect to the operations of countable union and countable intersection) and projective sets (the smallest family containing Borel sets and closed n m with respect to the operations of projecting from R to R , m<n, and the complement to the whole space). The question of what is a “simple” property is more complicated, but it is not important, since in fact we study a small list of individual properties, including the 1 Lebesgue measurability, Baire property , and the individual definability of a set, function, or real. The latest means that there is a formula that holds for a given real number and for no others. This depends on the class of formulas allowed. Such a natural class consists of formulas of the form ∀x1 ∃y1 ∀x2 ∃y2 ... ∀xn ∃yn ψ(x1,y1,...,xn,yn,x), where the variables x1,y1,...,xn,yn,x run through the whole R, and the elementary part ψ(x1,y1,...,xn,yn,x) is any arithmetic formula (which contains any quantifiers over the natural numbers, as well as equalities and inequalities that connect the superpositions of operations from the semiring of natural numbers). To date, the development of DTM leads to a non-trivial general cultural conclusion: every real number is definable (using countable 2 ordinals )orrandom; in the latter case it does not possess any non-trivial properties. This implies that 3 there are absolutely undecidable statements ; as well as surprising connections between seemingly very different absolutely undecidable ones. For example, the measurability implies the Baire property for a wide class of sets. The first three articles belong to this direction. In particular, they solve the well-known problem (1948) of A. Tarski on the definability of the notion of definability itself, and prove the statement (1975) of H. Friedman. The EPA section contains an article contributing a solution for the meaningful combinatorial and, at first glance, complicated algorithmic problem of optimization of the functional given on paths of passing from one graph to another. It is solved by an algorithm of linear complexity, being at the same time exact. The latter means that for any input data, that is for any ordered pair of graphs A and B, accompanied by costs of elementary graph transformations, the algorithm produces exactly the minimal value of the above functional (i.e., the minimum distance between A and B and the minimum path itself from A to B). Here the complexity of the problem turned into the logical complexity of this, albeit linear, algorithm. Our goal was to draw attention to the search for, and possible discussion of, algorithmic problems that seem to require exhaustive search but are actually solved by exact algorithms of low polynomial complexity. This ensures their practical significance when working with large data (terabyte and larger sizes). The Appl section contains two articles. First of them is devoted to the application of non-standard analysis (and other logical methods) to the problems of isomorphism in algebra and mathematical physics (the Jacobian and M. Kontsevich’s conjectures, and algorithmic undecidability). The second is devoted to the application of logical and algorithmic approaches to the problem of theoretical medicine — a quantitative description of the balance and the adaptive resource of a human ix that determines his resistance to external influences. Applied problems in which logic and theory of algorithms have shown their usefulness could be of interest. The Editorial Board of Mathematics (WoS: Q1) has announced the preparation of the issue “Mathematical Logic and Its Applications 2021”; contributions in these directions and especially in other ones of this huge mathematical area, including various applications, are invited. 1) The Baire property of a set X says that there is an open set U such that the symmetric difference X Δ U is a meager set (the union of a countable number of nowhere-dense sets). 2) Countable ordinals are the natural numbers themselves and their natural extension: taking the limit over all natural numbers we get ω, adding +1 to it consecutively and taking the limit yet again we get ω + ω = ω · 2, and so on. Each time, the limit is taken over a countable sequence. 3) This means that a natural statement about measurability (or other simple subjects) cannot be proved or disproved in the natural set theory of ZFC, which seems to contain all the mathematics used in physics, biology, computer science, and engineering. Vassily Lyubetsky, Vladimir Kanovei Editors x mathematics Article Models of Set Theory in which Nonconstructible Reals First Appear at a Given
Load more

Recommended publications
  • A Note on Indicator-Functions
    proceedings of the american mathematical society Volume 39, Number 1, June 1973 A NOTE ON INDICATOR-FUNCTIONS J. MYHILL Abstract. A system has the existence-property for abstracts (existence property for numbers, disjunction-property) if when- ever V(1x)A(x), M(t) for some abstract (t) (M(n) for some numeral n; if whenever VAVB, VAor hß. (3x)A(x), A,Bare closed). We show that the existence-property for numbers and the disjunc- tion property are never provable in the system itself; more strongly, the (classically) recursive functions that encode these properties are not provably recursive functions of the system. It is however possible for a system (e.g., ZF+K=L) to prove the existence- property for abstracts for itself. In [1], I presented an intuitionistic form Z of Zermelo-Frankel set- theory (without choice and with weakened regularity) and proved for it the disfunction-property (if VAvB (closed), then YA or YB), the existence- property (if \-(3x)A(x) (closed), then h4(t) for a (closed) comprehension term t) and the existence-property for numerals (if Y(3x G m)A(x) (closed), then YA(n) for a numeral n). In the appendix to [1], I enquired if these results could be proved constructively; in particular if we could find primitive recursively from the proof of AwB whether it was A or B that was provable, and likewise in the other two cases. Discussion of this question is facilitated by introducing the notion of indicator-functions in the sense of the following Definition. Let T be a consistent theory which contains Heyting arithmetic (possibly by relativization of quantifiers).
    [Show full text]
  • Soft Set Theory First Results
    An Inlum~md computers & mathematics PERGAMON Computers and Mathematics with Applications 37 (1999) 19-31 Soft Set Theory First Results D. MOLODTSOV Computing Center of the R~msian Academy of Sciences 40 Vavilova Street, Moscow 117967, Russia Abstract--The soft set theory offers a general mathematical tool for dealing with uncertain, fuzzy, not clearly defined objects. The main purpose of this paper is to introduce the basic notions of the theory of soft sets, to present the first results of the theory, and to discuss some problems of the future. (~) 1999 Elsevier Science Ltd. All rights reserved. Keywords--Soft set, Soft function, Soft game, Soft equilibrium, Soft integral, Internal regular- ization. 1. INTRODUCTION To solve complicated problems in economics, engineering, and environment, we cannot success- fully use classical methods because of various uncertainties typical for those problems. There are three theories: theory of probability, theory of fuzzy sets, and the interval mathematics which we can consider as mathematical tools for dealing with uncertainties. But all these theories have their own difficulties. Theory of probabilities can deal only with stochastically stable phenomena. Without going into mathematical details, we can say, e.g., that for a stochastically stable phenomenon there should exist a limit of the sample mean #n in a long series of trials. The sample mean #n is defined by 1 n IZn = -- ~ Xi, n i=l where x~ is equal to 1 if the phenomenon occurs in the trial, and x~ is equal to 0 if the phenomenon does not occur. To test the existence of the limit, we must perform a large number of trials.
    [Show full text]
  • Data Envelopment Analysis with Fuzzy Parameters: an Interactive Approach
    International Journal of Operations Research and Information Systems, 2(3), 39-53, July-September 2011 39 Data Envelopment Analysis with Fuzzy Parameters: An Interactive Approach Adel Hatami-Marbini, Universite Catholique de Louvain, Belgium Saber Saati, Islamic Azad University, Iran Madjid Tavana, La Salle University, USA ABSTRACT Data envelopment analysis (DEA) is a methodology for measuring the relative efficiencies of a set of deci- sion making units (DMUs) that use multiple inputs to produce multiple outputs. In the conventional DEA, all the data assume the form of specific numerical values. However, the observed values of the input and output data in real-life problems are sometimes imprecise or vague. Previous methods have not considered the preferences of the decision makers (DMs) in the evaluation process. This paper proposes an interactive evaluation process for measuring the relative efficiencies of a set of DMUs in fuzzy DEA with consideration of the DMs’ preferences. The authors construct a linear programming (LP) model with fuzzy parameters and calculate the fuzzy efficiency of the DMUs for different α levels. Then, the DM identifies his or her most preferred fuzzy goal for each DMU under consideration. A modified Yager index is used to develop a ranking order of the DMUs. This study allows the DMs to use their preferences or value judgments when evaluating the performance of the DMUs. Keywords: Data Envelopment Analysis, Efficiency Evaluation, Fuzzy Mathematical Programming, Interactive Solution, Preference Modeling INTRODUCTION efficiency. A DMU is considered efficient when no other DMU can produce more outputs using The changing economic conditions have chal- an equal or lesser amount of inputs.
    [Show full text]
  • Fuzzy Integer Linear Programming with Fuzzy Decision Variables
    Applied Mathematical Sciences, Vol. 4, 2010, no. 70, 3493 - 3502 Fuzzy Integer Linear Programming with Fuzzy Decision Variables C. Sudhagar1 Department of Mathematics and Applied Sciences Middle East College of Information Technology, Muscat, Oman [email protected] K. Ganesan Department of Mathematics, SRM University, Chennai, India gansan [email protected] Abstract In this paper a new method for dealing with Fuzzy Integer Linear Programming Problems (FILPP) has been proposed. FILPP with fuzzy variables model was taken for solution. This solution method is based on the fuzzy ranking method. The proposed method can serve deci- sion makers by providing the reasonable range of values for the fuzzy variable, which is comparatively better than the currently available solu- tions. Numerical examples demonstrate the effectiveness and accuracy of the proposed method. Mathematics Subject Classification: 65K05, 90C10, 90C70, 90C90 Keywords: Fuzzy numbers, Ranking, Fuzzy integer linear programming 1 Introduction Linear Programming Problems(LPP) have an outstanding relevance in the field of Decision making, Artificial intelligence, Control theory, Management sciences, Job placement interventions etc. In many practical applications the available information in the system under consideration are not precise. In such situations, it is more appropriate to use the fuzzy LPP. 1Corresponding author 3494 C. Sudhagar and K. Ganesan The concept of fuzzy linear programming problems was first introduced by Tanaka et al.,[13, 12] After his work, several kinds of fuzzy linear programming problems have appeared in the literature and different methods have been proposed to solve such problems. Numerous methods for comparison of fuzzy numbers have been suggested in the literature. In Campos Verdegay paper[2] linear programming problems with fuzzy constraints and fuzzy coefficients in both matrix and right hand side of the constraint set are considered.
    [Show full text]
  • What Is Fuzzy Probability Theory?
    Foundations of Physics, Vol.30,No. 10, 2000 What Is Fuzzy Probability Theory? S. Gudder1 Received March 4, 1998; revised July 6, 2000 The article begins with a discussion of sets and fuzzy sets. It is observed that iden- tifying a set with its indicator function makes it clear that a fuzzy set is a direct and natural generalization of a set. Making this identification also provides sim- plified proofs of various relationships between sets. Connectives for fuzzy sets that generalize those for sets are defined. The fundamentals of ordinary probability theory are reviewed and these ideas are used to motivate fuzzy probability theory. Observables (fuzzy random variables) and their distributions are defined. Some applications of fuzzy probability theory to quantum mechanics and computer science are briefly considered. 1. INTRODUCTION What do we mean by fuzzy probability theory? Isn't probability theory already fuzzy? That is, probability theory does not give precise answers but only probabilities. The imprecision in probability theory comes from our incomplete knowledge of the system but the random variables (measure- ments) still have precise values. For example, when we flip a coin we have only a partial knowledge about the physical structure of the coin and the initial conditions of the flip. If our knowledge about the coin were com- plete, we could predict exactly whether the coin lands heads or tails. However, we still assume that after the coin lands, we can tell precisely whether it is heads or tails. In fuzzy probability theory, we also have an imprecision in our measurements, and random variables must be replaced by fuzzy random variables and events by fuzzy events.
    [Show full text]
  • Connes on the Role of Hyperreals in Mathematics
    Found Sci DOI 10.1007/s10699-012-9316-5 Tools, Objects, and Chimeras: Connes on the Role of Hyperreals in Mathematics Vladimir Kanovei · Mikhail G. Katz · Thomas Mormann © Springer Science+Business Media Dordrecht 2012 Abstract We examine some of Connes’ criticisms of Robinson’s infinitesimals starting in 1995. Connes sought to exploit the Solovay model S as ammunition against non-standard analysis, but the model tends to boomerang, undercutting Connes’ own earlier work in func- tional analysis. Connes described the hyperreals as both a “virtual theory” and a “chimera”, yet acknowledged that his argument relies on the transfer principle. We analyze Connes’ “dart-throwing” thought experiment, but reach an opposite conclusion. In S, all definable sets of reals are Lebesgue measurable, suggesting that Connes views a theory as being “vir- tual” if it is not definable in a suitable model of ZFC. If so, Connes’ claim that a theory of the hyperreals is “virtual” is refuted by the existence of a definable model of the hyperreal field due to Kanovei and Shelah. Free ultrafilters aren’t definable, yet Connes exploited such ultrafilters both in his own earlier work on the classification of factors in the 1970s and 80s, and in Noncommutative Geometry, raising the question whether the latter may not be vulnera- ble to Connes’ criticism of virtuality. We analyze the philosophical underpinnings of Connes’ argument based on Gödel’s incompleteness theorem, and detect an apparent circularity in Connes’ logic. We document the reliance on non-constructive foundational material, and specifically on the Dixmier trace − (featured on the front cover of Connes’ magnum opus) V.
    [Show full text]
  • Be Measurable Spaces. a Func- 1 1 Tion F : E G Is Measurable If F − (A) E Whenever a G
    2. Measurable functions and random variables 2.1. Measurable functions. Let (E; E) and (G; G) be measurable spaces. A func- 1 1 tion f : E G is measurable if f − (A) E whenever A G. Here f − (A) denotes the inverse!image of A by f 2 2 1 f − (A) = x E : f(x) A : f 2 2 g Usually G = R or G = [ ; ], in which case G is always taken to be the Borel σ-algebra. If E is a topological−∞ 1space and E = B(E), then a measurable function on E is called a Borel function. For any function f : E G, the inverse image preserves set operations ! 1 1 1 1 f − A = f − (A ); f − (G A) = E f − (A): i i n n i ! i [ [ 1 1 Therefore, the set f − (A) : A G is a σ-algebra on E and A G : f − (A) E is f 2 g 1 f ⊆ 2 g a σ-algebra on G. In particular, if G = σ(A) and f − (A) E whenever A A, then 1 2 2 A : f − (A) E is a σ-algebra containing A and hence G, so f is measurable. In the casef G = R, 2the gBorel σ-algebra is generated by intervals of the form ( ; y]; y R, so, to show that f : E R is Borel measurable, it suffices to show −∞that x 2E : f(x) y E for all y. ! f 2 ≤ g 2 1 If E is any topological space and f : E R is continuous, then f − (U) is open in E and hence measurable, whenever U is op!en in R; the open sets U generate B, so any continuous function is measurable.
    [Show full text]
  • Expectation and Functions of Random Variables
    POL 571: Expectation and Functions of Random Variables Kosuke Imai Department of Politics, Princeton University March 10, 2006 1 Expectation and Independence To gain further insights about the behavior of random variables, we first consider their expectation, which is also called mean value or expected value. The definition of expectation follows our intuition. Definition 1 Let X be a random variable and g be any function. 1. If X is discrete, then the expectation of g(X) is defined as, then X E[g(X)] = g(x)f(x), x∈X where f is the probability mass function of X and X is the support of X. 2. If X is continuous, then the expectation of g(X) is defined as, Z ∞ E[g(X)] = g(x)f(x) dx, −∞ where f is the probability density function of X. If E(X) = −∞ or E(X) = ∞ (i.e., E(|X|) = ∞), then we say the expectation E(X) does not exist. One sometimes write EX to emphasize that the expectation is taken with respect to a particular random variable X. For a continuous random variable, the expectation is sometimes written as, Z x E[g(X)] = g(x) d F (x). −∞ where F (x) is the distribution function of X. The expectation operator has inherits its properties from those of summation and integral. In particular, the following theorem shows that expectation preserves the inequality and is a linear operator. Theorem 1 (Expectation) Let X and Y be random variables with finite expectations. 1. If g(x) ≥ h(x) for all x ∈ R, then E[g(X)] ≥ E[h(X)].
    [Show full text]
  • More on Discrete Random Variables and Their Expectations
    MASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.436J/15.085J Fall 2018 Lecture 6 MORE ON DISCRETE RANDOM VARIABLES AND THEIR EXPECTATIONS Contents 1. Comments on expected values 2. Expected values of some common random variables 3. Covariance and correlation 4. Indicator variables and the inclusion-exclusion formula 5. Conditional expectations 1COMMENTSON EXPECTED VALUES (a) Recall that E is well defined unless both sums and [X] x:x<0 xpX (x) E x:x>0 xpX (x) are infinite. Furthermore, [X] is well-defined and finite if and only if both sums are finite. This is the same as requiring! that ! E[|X|]= |x|pX (x) < ∞. x " Random variables that satisfy this condition are called integrable. (b) Noter that fo any random variable X, E[X2] is always well-defined (whether 2 finite or infinite), because all the terms in the sum x x pX (x) are nonneg- ative. If we have E[X2] < ∞,we say that X is square integrable. ! (c) Using the inequality |x| ≤ 1+x 2 ,wehave E [|X|] ≤ 1+E[X2],which shows that a square integrable random variable is always integrable. Simi- larly, for every positive integer r,ifE [|X|r] is finite then it is also finite for every l<r(fill details). 1 Exercise 1. Recall that the r-the central moment of a random variable X is E[(X − E[X])r].Showthat if the r -th central moment of an almost surely non-negative random variable X is finite, then its l-th central moment is also finite for every l<r. (d) Because of the formula var(X)=E[X2] − (E[X])2,wesee that: (i) if X is square integrable, the variance is finite; (ii) if X is integrable, but not square integrable, the variance is infinite; (iii) if X is not integrable, the variance is undefined.
    [Show full text]
  • Decision Analysis in the UK Energy Supply Chain Risk Management: Tools Development and Application
    Decision Analysis in the UK Energy Supply Chain Risk Management: Tools Development and Application by Amin Vafadarnikjoo Registration number: 100166891 Thesis submitted to The University of East Anglia for the Degree of Doctor of Philosophy (PhD) August 2020 Norwich Business School “This copy of the thesis has been supplied on condition that anyone who consults it is understood to recognise that its copyright rests with the author and that use of any information derived therefrom must be in accordance with current UK Copyright Law. In addition, any quotation or extract must include full attribution.” 1 Abstract Large infrastructures like electricity supply networks are widely presumed to be crucial for the functioning of societies as they create conditions for essential economic activities. There has always been a continuing concern and complexity around risks in the field of energy security and particularly power grids within energy supply chain. Drawing on this complexity and a need for useful tools, this research contributes to developing and utilising proper decision-making tools (i.e. methods and models) to deal with the risk identification and mitigation in the UK energy supply chain as a compound networked system. This thesis is comprised of four study phases (Figure I.A.). It is aimed at developing decision-making tools for risk identification, risk interdependency analysis, risk prioritisation, and long-term risk mitigation strategy recommendations. The application of the tools has focused on the UK power supply chain. The five new tools which are introduced and applied in this thesis are: (1) Proposed Expert Selection Model (ESM) and its application under hesitant fuzzy environment (i.e.
    [Show full text]
  • POST-OLYMPIAD PROBLEMS (1) Evaluate ∫ Cos2k(X)Dx Using Combinatorics. (2) (Putnam) Suppose That F : R → R Is Differentiable
    POST-OLYMPIAD PROBLEMS MARK SELLKE R 2π 2k (1) Evaluate 0 cos (x)dx using combinatorics. (2) (Putnam) Suppose that f : R ! R is differentiable, and that for all q 2 Q we have f 0(q) = f(q). Need f(x) = cex for some c? 1 1 (3) Show that the random sum ±1 ± 2 ± 3 ::: defines a conditionally convergent series with probability 1. 2 (4) (Jacob Tsimerman) Does there exist a smooth function f : R ! R with a unique critical point at 0, such that 0 is a local minimum but not a global minimum? (5) Prove that if f 2 C1([0; 1]) and for each x 2 [0; 1] there is n = n(x) with f (n) = 0, then f is a polynomial. 2 2 (6) (Stanford Math Qual 2018) Prove that the operator χ[a0;b0]Fχ[a1;b1] from L ! L is always compact. Here if S is a set, χS is multiplication by the indicator function 1S. And F is the Fourier transform. (7) Let n ≥ 4 be a positive integer. Show there are no non-trivial solutions in entire functions to f(z)n + g(z)n = h(z)n. (8) Let x 2 Sn be a random point on the unit n-sphere for large n. Give first-order asymptotics for the largest coordinate of x. (9) Show that almost sure convergence is not equivalent to convergence in any topology on random variables. P1 xn (10) (Noam Elkies on MO) Show that n=0 (n!)α is positive for any α 2 (0; 1) and any real x.
    [Show full text]
  • 10.0 Lesson Plan
    10.0 Lesson Plan 1 Answer Questions • Robust Estimators • Maximum Likelihood Estimators • 10.1 Robust Estimators Previously, we claimed to like estimators that are unbiased, have minimum variance, and/or have minimum mean squared error. Typically, one cannot achieve all of these properties with the same estimator. 2 An estimator may have good properties for one distribution, but not for n another. We saw that n−1 Z, for Z the sample maximum, was excellent in estimating θ for a Unif(0,θ) distribution. But it would not be excellent for estimating θ when, say, the density function looks like a triangle supported on [0,θ]. A robust estimator is one that works well across many families of distributions. In particular, it works well when there may be outliers in the data. The 10% trimmed mean is a robust estimator of the population mean. It discards the 5% largest and 5% smallest observations, and averages the rest. (Obviously, one could trim by some fraction other than 10%, but this is a commonly-used value.) Surveyors distinguish errors from blunders. Errors are measurement jitter 3 attributable to chance effects, and are approximately Gaussian. Blunders occur when the guy with the theodolite is standing on the wrong hill. A trimmed mean throws out the blunders and averages the good data. If all the data are good, one has lost some sample size. But in exchange, you are protected from the corrosive effect of outliers. 10.2 Strategies for Finding Estimators Some economists focus on the method of moments. This is a terrible procedure—its only virtue is that it is easy.
    [Show full text]