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Robust Geometric Computation
, 1997
"... Nonrobustness refers to qualitative or catastrophic failures in geometric algorithms arising from numerical errors. Section... ..."
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Cited by 80 (14 self)
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Nonrobustness refers to qualitative or catastrophic failures in geometric algorithms arising from numerical errors. Section...
Closed forms: what they are and why we care
, 2010
"... The term “closed form” is one of those mathematical notions that is commonplace, yet virtually devoid of rigor. And, there is disagreement even on the intuitive side; for example, most everyone would say that π + log 2 is a closed form, but some of us would think that the Euler constant γ is not cl ..."
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Cited by 15 (7 self)
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The term “closed form” is one of those mathematical notions that is commonplace, yet virtually devoid of rigor. And, there is disagreement even on the intuitive side; for example, most everyone would say that π + log 2 is a closed form, but some of us would think that the Euler constant γ is not closed. Like others before us, we shall try to supply some missing rigor to the notion of closed forms and also to give examples from modern research where the question of closure looms both important and elusive.
A closedform solution might be given by a tree. the valuation of quadratic polynomials. Submitted for publication
, 2015
"... Abstract. The padic valuation of an integer x is the largest power of the prime p that divides x. It is denoted by νp(x). This work describes properties of the valuation ν2(n2 + a), with a ∈ N. A distinction of the behavior of these valuations for a ≡ 7 mod 8 or not is presented. ..."
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Cited by 1 (1 self)
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Abstract. The padic valuation of an integer x is the largest power of the prime p that divides x. It is denoted by νp(x). This work describes properties of the valuation ν2(n2 + a), with a ∈ N. A distinction of the behavior of these valuations for a ≡ 7 mod 8 or not is presented.
Near Integral Points of Sets Definable in O Minimal Structures ∗
, 2004
"... Modifying the proof of a theorem of Wilkie, it is shown that if a one dimnsional set S is definable in an O minimal expansion of the ordered field of the reals, and if it is regularly exponentially near to many integral points, then there is an unbounded set, which is R definable without parameters, ..."
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Modifying the proof of a theorem of Wilkie, it is shown that if a one dimnsional set S is definable in an O minimal expansion of the ordered field of the reals, and if it is regularly exponentially near to many integral points, then there is an unbounded set, which is R definable without parameters, and which is exponentially near to S. 1
Algebraic properties of the Lambert W . . .
"... It is shown that the Lambert W function cannot be expressed in terms of the elementary, Liouvillian, functions. The proof is based on a theorem due to Rosenlicht. A related function, the Wright ω function is similarly shown to be not Liouvillian. ..."
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It is shown that the Lambert W function cannot be expressed in terms of the elementary, Liouvillian, functions. The proof is based on a theorem due to Rosenlicht. A related function, the Wright ω function is similarly shown to be not Liouvillian.
3Department of Mathematical Sciences
, 2004
"... The Lambert W function [5, 9] is a multivalued function defined as the solution of W (x)eW (x) = x, (1) one of the simplest possible nonalgebraic equations. The Wright ω function [4] also satisfies a simple transcendental equation (away from its discontinuities): ω(x) + lnω(x) = x. (2) These are ..."
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The Lambert W function [5, 9] is a multivalued function defined as the solution of W (x)eW (x) = x, (1) one of the simplest possible nonalgebraic equations. The Wright ω function [4] also satisfies a simple transcendental equation (away from its discontinuities): ω(x) + lnω(x) = x. (2) These are, of course, “implicitly elementary ” functions in the sense discussed in [7]. One can ask whether there are explicit formulations of those functions in terms of known functions, or are they genuinely new functions. A common class of “wellknown ” functions are the Liouvillian functions. Definition 1 Let (k, ′ ) be a differential field of characteristic 0. A differential extension (K, ′ ) of k is called Liouvillian over k if there are θ1,..., θn ∈ K such that K = C(x, θ1,..., θn) and for all i, at least one of the following holds: 1. θi is algebraic over k(θ1,..., θi−1); 1
Closed Forms: What they are and why they matter
, 2010
"... The term “closed form” is one of those mathematical notions that is commonplace, yet virtually devoid of rigor. And, there is disagreement even on the intuitive side; for example, most everyone would say that π + log 2 is a closed form, but some of us would think that the Euler constant γ is not clo ..."
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The term “closed form” is one of those mathematical notions that is commonplace, yet virtually devoid of rigor. And, there is disagreement even on the intuitive side; for example, most everyone would say that π + log 2 is a closed form, but some of us would think that the Euler constant γ is not closed. Like others before us, we shall try to supply some missing rigor to the notion of closed forms and also to give examples from modern research where the question of closure looms both important and elusive.
CLOSED FORMS: WHAT THEY ARE AND WHY THEY MATTER
, 2010
"... The term “closed form” is one of those mathematical notions that is commonplace, yet virtually devoid of rigor. And, there is disagreement even on the intuitive side; for example, most everyone would say that (π + log 2) is a closed form, but some of us would think that the Euler constant γ is not ..."
Abstract
 Add to MetaCart
(Show Context)
The term “closed form” is one of those mathematical notions that is commonplace, yet virtually devoid of rigor. And, there is disagreement even on the intuitive side; for example, most everyone would say that (π + log 2) is a closed form, but some of us would think that the Euler constant γ is not closed. Like others before us, we shall try to supply some missing rigor to the notion of closed forms and also to give examples from modern research where the question of closure looms both important and elusive.