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Some Concrete Aspects Of Hilbert's 17th Problem
 In Contemporary Mathematics
, 1996
"... This paper is dedicated to the memory of Raphael M. Robinson and Olga TausskyTodd. 1. Introduction ..."
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Cited by 94 (4 self)
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This paper is dedicated to the memory of Raphael M. Robinson and Olga TausskyTodd. 1. Introduction
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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Cited by 19 (2 self)
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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
SUMS OF SQUARES OVER TOTALLY REAL FIELDS ARE RATIONAL SUMS OF SQUARES
"... Abstract. Let K be a totally real number field with Galois closure L. We prove that if f ∈ Q[x1,..., xn] is a sum of m squares in K[x1,..., xn], then f is a sum of 4m · 2 [L:Q]+1([L: Q] + 1 2 squares in Q[x1,..., xn]. Moreover, our argument is constructive and generalizes to the case of commutative ..."
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Cited by 2 (0 self)
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Abstract. Let K be a totally real number field with Galois closure L. We prove that if f ∈ Q[x1,..., xn] is a sum of m squares in K[x1,..., xn], then f is a sum of 4m · 2 [L:Q]+1([L: Q] + 1 2 squares in Q[x1,..., xn]. Moreover, our argument is constructive and generalizes to the case of commutative Kalgebras. This result gives a partial resolution to a question of Sturmfels on the algebraic degree of certain semidefinite programing problems. 1.
ON THE ABSENCE OF UNIFORM DENOMINATORS IN HILBERT’S 17TH PROBLEM
, 2003
"... Abstract. Hilbert showed that for most (n, m) there exist psd forms p(x1,..., xn) of degree m which cannot be written as a sum of squares of forms. His 17th problem asked whether, in this case, there exists a form h so that h 2 p is a sum of squares of forms; that is, p is a sum of squares of ration ..."
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Cited by 1 (0 self)
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Abstract. Hilbert showed that for most (n, m) there exist psd forms p(x1,..., xn) of degree m which cannot be written as a sum of squares of forms. His 17th problem asked whether, in this case, there exists a form h so that h 2 p is a sum of squares of forms; that is, p is a sum of squares of rational functions with denominator h. We show that, for every such (n, m) there does not exist a single form h which serves in this way as a denominator for every psd p(x1,..., xn) of degree m. 1.
Some aspects of the algebraic theory of quadratic forms
, 2009
"... (Notes for lectures at AWS 2009) ..."
LOSSLESS AND DISSIPATIVE . . .
 SIAM J. CONTROL OPTIM
, 2002
"... This paper deals with linear shiftinvariant distributed systems. By this we mean systems described by constant coefficient linear partial differential equations. We define dissipativity with respect to a quadratic differential form, i.e., a quadratic functional in the system variables and their par ..."
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This paper deals with linear shiftinvariant distributed systems. By this we mean systems described by constant coefficient linear partial differential equations. We define dissipativity with respect to a quadratic differential form, i.e., a quadratic functional in the system variables and their partial derivatives. The main result states the equivalence of dissipativity and the existence of a storage function or a dissipation rate. The proof of this result involves the construction of the dissipation rate. We show that this problem can be reduced to Hilbert's 17th problem on the representation of a nonnegative rational function as a sum of squares of rational functions.
unknown title
"... I work on a wide range of problems that arise from other areas of mathematics and the physical sciences. Currently, I am focused on using mathematical and computational tools to solve basic problems in theoretical neuroscience, and in this regard, I have begun collaborations with scientists at the R ..."
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I work on a wide range of problems that arise from other areas of mathematics and the physical sciences. Currently, I am focused on using mathematical and computational tools to solve basic problems in theoretical neuroscience, and in this regard, I have begun collaborations with scientists at the Redwood Center for Theoretical Neuroscience and mathematicians at U.C. Berkeley. I am also interested in theoretical questions involving semidefinite programming, optimization, and computational algebra. The following is a description of several interrelated lines of research in which I will actively participate in the coming years. The first three sections contain very brief discussions of topics related to theoretical neuroscience that I have only begun exploring in recent months. The final sections describe more theoretical studies that I have been investigating in recent years and therefore contain more detailed descriptions. 1. Sparse coding and compressed sensing Sparse coding refers to the process of representing a real vector input (such as an image) as a sparse linear combination of an overcomplete set of vectors (called a sparse basis). Here, overcomplete refers to the fact that there are many more vectors in the sparse basis
Representations as Sums of Squares
, 2009
"... This is a translation of a paper [5] I wrote in 1971, and may help for Parimala’s course. Evidently completely outdated, but still may be useful. I changed some notation so as to be compatible with the course. ..."
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This is a translation of a paper [5] I wrote in 1971, and may help for Parimala’s course. Evidently completely outdated, but still may be useful. I changed some notation so as to be compatible with the course.