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107
Verifying nonlinear real formulas via sums of squares
 Theorem Proving in Higher Order Logics, TPHOLs 2007, volume 4732 of Lect. Notes in Comp. Sci
, 2007
"... Abstract. Techniques based on sums of squares appear promising as a general approach to the universal theory of reals with addition and multiplication, i.e. verifying Boolean combinations of equations and inequalities. A particularly attractive feature is that suitable ‘sum of squares ’ certificates ..."
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Abstract. Techniques based on sums of squares appear promising as a general approach to the universal theory of reals with addition and multiplication, i.e. verifying Boolean combinations of equations and inequalities. A particularly attractive feature is that suitable ‘sum of squares ’ certificates can be found by sophisticated numerical methods such as semidefinite programming, yet the actual verification of the resulting proof is straightforward even in a highly foundational theorem prover. We will describe our experience with an implementation in HOL Light, noting some successes as well as difficulties. We also describe a new approach to the univariate case that can handle some otherwise difficult examples. 1 Verifying nonlinear formulas over the reals Over the real numbers, there are algorithms that can in principle perform quantifier elimination from arbitrary firstorder formulas built up using addition, multiplication and the usual equality and inequality predicates. A classic example of such a quantifier elimination equivalence is the criterion for a quadratic equation to have a real root: ∀a b c. (∃x. ax 2 + bx + c = 0) ⇔ a = 0 ∧ (b = 0 ⇒ c = 0) ∨ a � = 0 ∧ b 2 ≥ 4ac
Elliptic curves with complex multiplication and the conjecture of Birch and SwinnertonDyer
 in Arithmetic Theory of Elliptic Curves
, 1997
"... ..."
Solving the Pell Equation
, 2008
"... We illustrate recent developments in computational number theory by studying their implications for solving the Pell equation. We shall see that, if the solutions to the Pell equation are properly represented, the traditional continued fraction method for solving the equation can be significantly a ..."
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Cited by 28 (0 self)
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We illustrate recent developments in computational number theory by studying their implications for solving the Pell equation. We shall see that, if the solutions to the Pell equation are properly represented, the traditional continued fraction method for solving the equation can be significantly accelerated. The most promising method depends on the use of smooth numbers. As with many algorithms depending on smooth numbers, its run time can presently only conjecturally be established; giving a rigorous analysis is one of the many open problems surrounding the Pell equation.
Efficient Solution of Rational Conics
 Math. Comp
, 1998
"... this paper (section 2), and to Denis Simon for the reference [10]. ..."
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Cited by 26 (0 self)
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this paper (section 2), and to Denis Simon for the reference [10].
The nqueens problem
 American Mathematical Monthly
, 1994
"... which Archimedes devised in epigrams, and which he communicated to students of such matters ..."
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Cited by 22 (1 self)
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which Archimedes devised in epigrams, and which he communicated to students of such matters
Analytic Continuation of Riemann’s Zeta Function and Values at Negative Integers via Euler’s Transformation of Series
 Proc. Amer. Math. Soc
, 1994
"... Abstract. We prove that a series derived using Euler's transformation provides the analytic continuation of ((s) for all complex s ^ 1. At negative integers the series becomes a finite sum whose value is given by an explicit formula for Bernoulli numbers. 1. ..."
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Cited by 21 (2 self)
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Abstract. We prove that a series derived using Euler's transformation provides the analytic continuation of ((s) for all complex s ^ 1. At negative integers the series becomes a finite sum whose value is given by an explicit formula for Bernoulli numbers. 1.
Some old problems and new results about quadratic forms
 Notices Amer. Math. Soc
, 1997
"... It may be a challenging problem to describe the integer solutions to a polynomial equation in several variables. Which integers, for example, are represented by a quadratic polynomial? This ..."
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Cited by 19 (2 self)
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It may be a challenging problem to describe the integer solutions to a polynomial equation in several variables. Which integers, for example, are represented by a quadratic polynomial? This
Archimedes' Cattle Problem
 American Mathematical Monthly
, 1998
"... this paper is to take the Cattle Problem out of the realm of the \astronomical" and put it into manageable form. This is achieved in formulas (12) and (13) which give explicit forms for the solution. For example, the smallest possible value for the total number of cattle satisfying the conditio ..."
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Cited by 12 (4 self)
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this paper is to take the Cattle Problem out of the realm of the \astronomical" and put it into manageable form. This is achieved in formulas (12) and (13) which give explicit forms for the solution. For example, the smallest possible value for the total number of cattle satisfying the conditions of the problem is
EULER’S CONSTANT: EULER’S WORK AND MODERN DEVELOPMENTS
, 2013
"... Abstract. This paper has two parts. The first part surveys Euler’s work on the constant γ =0.57721 ·· · bearing his name, together with some of his related work on the gamma function, values of the zeta function, and divergent series. The second part describes various mathematical developments invol ..."
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Cited by 12 (1 self)
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Abstract. This paper has two parts. The first part surveys Euler’s work on the constant γ =0.57721 ·· · bearing his name, together with some of his related work on the gamma function, values of the zeta function, and divergent series. The second part describes various mathematical developments involving Euler’s constant, as well as another constant, the Euler–Gompertz constant. These developments include connections with arithmetic functions and the Riemann hypothesis, and with sieve methods, random permutations, and random matrix products. It also includes recent results on Diophantine approximation and transcendence related to Euler’s constant. Contents
Explicit 8Descent on Elliptic Curves
, 2005
"... In this thesis I will describe an explicit method for performing an 8descent on elliptic curves. First I will present some basics on descent, in particular I will give a generalization of the definition of ncoverings, which suits the needs of higher descent. Then I will sketch the classical method ..."
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Cited by 11 (0 self)
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In this thesis I will describe an explicit method for performing an 8descent on elliptic curves. First I will present some basics on descent, in particular I will give a generalization of the definition of ncoverings, which suits the needs of higher descent. Then I will sketch the classical method of 2descent, and the two methods that are known for doing a second 2descent, also called 4descent. Next I will locate the starting position for 8descent and supplement the exposition of 4descent by some more detailed geometric information. In Chapter 3, I will describe the construction of the descent map. It is very similar to Cassels ’ method for doing a 4descent, however there are some differences that make our situation more complicated. This descent map can be used to give an explicit description of a subset of the 8Selmer group. However, the set we get is only close to the right one, so I call it the fake Selmer set. The methods for computing the fake Selmer set are described in Chapter 4.