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Hilbert’s twentyfourth problem
 American Mathematical Monthly
, 2001
"... 1. INTRODUCTION. For geometers, Hilbert’s influential work on the foundations of geometry is important. For analysts, Hilbert’s theory of integral equations is just as important. But the address “Mathematische Probleme ” [37] that David Hilbert (1862– 1943) delivered at the second International Cong ..."
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1. INTRODUCTION. For geometers, Hilbert’s influential work on the foundations of geometry is important. For analysts, Hilbert’s theory of integral equations is just as important. But the address “Mathematische Probleme ” [37] that David Hilbert (1862– 1943) delivered at the second International Congress of Mathematicians (ICM) in Paris has tremendous importance for all mathematicians. Moreover, a substantial part of
Totally real origami and impossible paper folding
 Amer. Math. Monthly
, 1995
"... Origami is the ancient Japanese art of paper folding. It is possible to fold many intriguing geometrical shapes with paper [M]. In this article, the question we will answer is which shapes are possible to construct and which shapes are impossible to construct using origami. One of the most interesti ..."
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Cited by 7 (0 self)
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Origami is the ancient Japanese art of paper folding. It is possible to fold many intriguing geometrical shapes with paper [M]. In this article, the question we will answer is which shapes are possible to construct and which shapes are impossible to construct using origami. One of the most interesting things we discovered is that it is impossible to construct a cube with twice the volume of a given cube using origami, just as it is impossible to do using a compass and straight edge. As an unexpected surprise, our algebraic characterization of origami is related to David Hilbert’s 17 th problem. Hilbert’s problem is to show that any rational function which is always nonnegative is a sum of squares of rational functions [B]. This problem was solved by Artin in 1926 [Ar]. We would like to thank John Tate for noticing the relationship between our present work and Hilbert’s
Hilbert problems for the geosciences in the 21st century
 NONLINEAR PROCESSES IN GEOPHYSICS (2001) 8: 211222
, 2001
"... The scientific problems posed by the Earth's fluid envelope, and its atmosphere, oceans, and the land surface that interacts with them are central to major socioeconomic and political concerns as we move into the 21st century. It is natural, therefore, that a certain impatience should prevail in at ..."
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The scientific problems posed by the Earth's fluid envelope, and its atmosphere, oceans, and the land surface that interacts with them are central to major socioeconomic and political concerns as we move into the 21st century. It is natural, therefore, that a certain impatience should prevail in attempting to solve these problems. The point of this review paper is that one should proceed with all diligence, but not excessive haste: "festina lente," as the Romans said two thousand years ago, i.e. "hurry in a measured way." The paper traces the necessary progress through the solutions to the ten problems: 1. What is the coarsegrained structure of lowfrequency atmospheric variability, and what is the connection between its episodic and oscillatory description? 2. What can we predict beyond one week, for how long, and by what methods? 3. What are the respective roles of intrinsic ocean variability, coupled oceanatmosphere modes, and atmospheric forcing in seasonaltointerannual variability? 4. What are the implications of the answer to the previous problem for climate prediction on this time scale? 5. How does the oceans' thermohaline circulation change on interdecadal and longer time scales, and what is the role of the atmosphere and sea ice in such changes? 6. What is the role of chemical cycles and biological changes in affecting climate on slow time scales, and how are they affected, in turn, by climate variations? 7. Does the answer to the question above give us some trigger points for climate control? 8. What can we learn about these problems from the atmospheres and oceans of other planets and their satellites? Correspondence to: M. Ghil (ghil@atmos.ucla.edu) 9. Given the answer to the questions so far, what is the role of humans in modifying the clim...
SOME LOWER BOUNDS IN THE B. AND M. SHAPIRO CONJECTURE FOR FLAG VARIETIES
"... Abstract. The B. and M. Shapiro conjecture stated that all solutions of the Schubert Calculus problems associated with real points on the rational normal curve should be real. For Grassmannians, it was proved by Mukhin, Tarasov and Varchenko. For flag varieties, Sottile found a counterexample and su ..."
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Abstract. The B. and M. Shapiro conjecture stated that all solutions of the Schubert Calculus problems associated with real points on the rational normal curve should be real. For Grassmannians, it was proved by Mukhin, Tarasov and Varchenko. For flag varieties, Sottile found a counterexample and suggested that all solutions should be real under certain monotonicity conditions. In this paper, we compute lower bounds on the number of real solutions for some special cases of the B. and M. Shapiro conjecture for flag varieties, when Sottile’s monotonicity conditions are not satisfied. 1.
Are There Absolutely Unsolvable Problems? Gödel’s Dichotomy
 PHILOSOPHIA MATHEMATICA
, 2006
"... This is a critical analysis of the first part of Gödel’s 1951 Gibbs lecture on certain philosophical consequences of the incompleteness theorems. Gödel’s discussion is framed in terms of a distinction between objective mathematics and subjective mathematics, according to which the former consists of ..."
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This is a critical analysis of the first part of Gödel’s 1951 Gibbs lecture on certain philosophical consequences of the incompleteness theorems. Gödel’s discussion is framed in terms of a distinction between objective mathematics and subjective mathematics, according to which the former consists of the truths of mathematics in an absolute sense, and the latter consists of all humanly demonstrable truths. The question is whether these coincide; if they do, no formal axiomatic system (or Turing machine) can comprehend the mathematizing potentialities of human thought, and, if not, there are absolutely unsolvable mathematical problems of diophantine form. Either... the human mind... infinitely surpasses the powers of any finite machine, or else there exist absolutely unsolvable diophantine problems.
The Bernstein Theorem in Higher Dimensions
"... Dedicated to the memory of Guido Stampacchia Abstract. – In this work we have reconsidered the famous paper of Bombieri, De Giorgi and Giusti [4] and, thanks to the software Mathematica Ⓡ we made it possible for anybody to control the difficult computations. 1. ..."
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Dedicated to the memory of Guido Stampacchia Abstract. – In this work we have reconsidered the famous paper of Bombieri, De Giorgi and Giusti [4] and, thanks to the software Mathematica Ⓡ we made it possible for anybody to control the difficult computations. 1.
The HilbertSmith Conjecture by
, 2001
"... In 1900, Hilbert proposed twentythree problems [8]. For an excellent discussion concerning these problems, see the Proceedings of Symposia In Pure Mathematics concerning “Mathematical Developments Arising From Hilbert Problems ” [3]. The abstract by C.T. Yang [22] gives a review of Hilbert’s Fifth ..."
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In 1900, Hilbert proposed twentythree problems [8]. For an excellent discussion concerning these problems, see the Proceedings of Symposia In Pure Mathematics concerning “Mathematical Developments Arising From Hilbert Problems ” [3]. The abstract by C.T. Yang [22] gives a review of Hilbert’s Fifth Problem “How is Lie’s concept of continuous
A Proof Of The HilbertSmith Conjecture by
, 2001
"... The HilbertSmith Conjecture states that if G is a locally compact group which acts effectively on a connected manifold as a topological transformation group, then G is a Lie group. A rather straightforward proof of this conjecture is given. The motivation is work of Cernavskii (“Finitetoone mappi ..."
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The HilbertSmith Conjecture states that if G is a locally compact group which acts effectively on a connected manifold as a topological transformation group, then G is a Lie group. A rather straightforward proof of this conjecture is given. The motivation is work of Cernavskii (“Finitetoone mappings of manifolds”, Trans. of Math. Sk. 65 (107), 1964.) His work is generalized to the orbit map of an effective action of a padic group on compact connected nmanifolds with the aid of some new ideas. There is no attempt to use Smith Theory even though there may be similarities.
HILBERT’S FIRST AND SECOND PROBLEMS AND THE FOUNDATIONS OF MATHEMATICS
, 2004
"... In 1900, David Hilbert gave a seminal lecture in which he spoke about a list of unsolved problems in mathematics that he deemed to be of outstanding importance. The first of these was Cantor’s continuum problem, which has to do with infinite numbers with which Cantor revolutionised set theory. The s ..."
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In 1900, David Hilbert gave a seminal lecture in which he spoke about a list of unsolved problems in mathematics that he deemed to be of outstanding importance. The first of these was Cantor’s continuum problem, which has to do with infinite numbers with which Cantor revolutionised set theory. The smallest infinite number, ℵ0, ‘alephnought, ’ gives the number of positive whole numbers. A set is of this cardinality if it is possible to list its members in an arrangement such that each one is encountered after a finite number (however large) of steps. Cantor’s revolutionary discovery was that the points on a line cannot be so listed, and so the number of points on a line is a strictly higher infinite number (c, ‘the cardinality of the continuum’) than ℵ0. Hilbert’s First Problem asks whether any infinite subset of the real line is of one of these two cardinalities. The axiom that this is indeed the case is known as the Continuum Hypothesis (CH). This problem had unexpected connections with Hilbert’s Second Problem (and even with the Tenth, see the article by M. Davis and the comments on
Vine Copulas as a Way to Describe and Analyze MultiVariate Dependence in Econometrics: Computational Motivation and Comparison with Bayesian Networks and Fuzzy Approaches
"... Abstract. In the last decade, vine copulas emerged as a new efficient techniques for describing and analyzing multivariate dependence in econometrics; see, e.g., [1–3, 7, 9–11, 13, 14, 21]. Our experience has shown, however, that while these techniques have been successfully applied to many practic ..."
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Abstract. In the last decade, vine copulas emerged as a new efficient techniques for describing and analyzing multivariate dependence in econometrics; see, e.g., [1–3, 7, 9–11, 13, 14, 21]. Our experience has shown, however, that while these techniques have been successfully applied to many practical problems of econometrics, there is still a lot of confusion and misunderstanding related to vine copulas. In this paper, we provide a motivation for this new technique from the computational viewpoint. We show that other techniques used to described dependence – Bayesian networks and fuzzy techniques – can be viewed as a particular case of vine copulas. 1 Copulas – A Useful Tool in Econometrics: Motivations and Descriptions Need for studying dependence in econometrics. Many researchers have observed that economics is more complex than physics. In physics, many parameters, many phenomena are independent. As a result, we can observe (and thoroughly