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A Separation Bound for Real Algebraic Expressions [chapter]

Christoph Burnikel, Stefan Funke, Kurt Mehlhorn, Stefan Schirra, Susanne Schmitt
2001 Lecture Notes in Computer Science  
We prove a new separation bound for real algebraic expressions and compare it analytically and experimentally with previous bounds. The bound is used in the sign test of the number type leda real.  ...  We consider the sign computation of real algebraic expressions, a task vital for the implementation of geometric algorithms.  ...  We prove a new separation bound for the following class of real algebraic expressions.  ... 
doi:10.1007/3-540-44676-1_21 fatcat:uqljn7mlsvdttjbtdjutk3o2fq

A Separation Bound for Real Algebraic Expressions

Christoph Burnikel, Stefan Funke, Kurt Mehlhorn, Stefan Schirra, Susanne Schmitt
2007 Algorithmica  
We prove a new separation bound for real algebraic expressions and compare it analytically and experimentally with previous bounds. The bound is used in the sign test of the number type leda real.  ...  We consider the sign computation of real algebraic expressions, a task vital for the implementation of geometric algorithms.  ...  We prove a new separation bound for the following class of real algebraic expressions.  ... 
doi:10.1007/s00453-007-9132-4 fatcat:hc4n4hv3xjezfcu725pvxrttyy

Page 26 of American Mathematical Society. Bulletin of the American Mathematical Society Vol. 60, Issue 1 [page]

1954 American Mathematical Society. Bulletin of the American Mathematical Society  
Let X be a reflexive Banach space and % be a bounded Boolean algebra (B.A.) of projections in X (for some M, ||E|| <M, EC%). B is complete if for every subset {E.}  ...  Consider the polynomial f(z; x) =1+a{2+a;2*+ - - - +a%2", x real. Its discriminant expressed in terms of coefficients is a continuous function of x. From theorem of Schurmalo (e.g. J.  ... 

Counterexamples to Quantifier Elimination for Fewnomial and Exponential Expressions

A. Gabrielov
2007 Moscow Mathematical Journal  
We construct a family of semialgebraic sets of bounded fewnomial complexity, with unbounded fewnomial complexity of their projections to a subspace.  ...  We also construct a set defined by exponential algebraic functions such that its projection cannot be defined by a quantifierfree formula with exponential algebraic functions, even if division is permitted  ...  Although t d is a multivalued function with ramification at t = 0 and t = ∞, we can always choose its branch uniquely since, for bounded complex values of x, solutions to (3.1) are bounded and separated  ... 
doi:10.17323/1609-4514-2007-7-3-453-460 fatcat:2ymzpt5ifbdevpsp5vm5r4eo54

The Polytope-Collision Problem

Shaull Almagor, Joël Ouaknine, James Worrell, Marc Herbstritt
2017 International Colloquium on Automata, Languages and Programming  
Using techniques from transcendental number theory, and separation bounds on algebraic numbers, we are able to solve such instances in PSPACE.  ...  Our proof begins by formulating the problem as the satisfiability of a parametrized family of sentences in the existential first-order theory of real-closed fields.  ...  We can now treat the expression as d 2 expressions of real-algebraic sums of exponentials.  ... 
doi:10.4230/lipics.icalp.2017.24 dblp:conf/icalp/AlmagorOW17 fatcat:tmoworgmavdhxc2ziacj6huroa

Page 4946 of Mathematical Reviews Vol. , Issue 92i [page]

1992 Mathematical Reviews  
In this case the asymptotic formula for the eigenvalues of the pencil L(A) inside ® is found and this formula is expressed in terms of the symbol of L(A) for positive real A. G. V. Rozenblyum (St.  ...  Let # be a separable Hilbert space.  ... 

Page 5337 of Mathematical Reviews Vol. , Issue 93j [page]

1993 Mathematical Reviews  
For example, the interior of the union of all separated orbits has a geometric quotient. Assume now that L = G is a connected reductive algebraic group and that X is irreducible.  ...  Let X be a nonsingular algebraic variety over a real closed field R; here we see the variety as a scheme, and make the assumption that it has some point over R, that is, X(R) 4 @.  ... 

Notes on coarse grainings and functions of observables

Anatolij Dvurečenskij, Pekka Lahti, Sylvia Pulmannová, Kari Ylinen
2005 Reports on mathematical physics  
To this end, conditions for the separability and for the Boolean structure of an observable are given.  ...  Using the Naimark dilation theory we investigate the question under what conditions an observable which is a coarse graining of another observable is a function of it.  ...  Hamhalter, Technical University of Prague, for useful discussions on von Neumann algebras.  ... 
doi:10.1016/s0034-4877(05)80030-8 fatcat:io65xzigjncn3lfewauobb4che

Page 8592 of Mathematical Reviews Vol. , Issue 2004k [page]

2004 Mathematical Reviews  
This improves bounds which can be obtained by using the Thom- Milnor bound and cylindrical algebraic decomposition.  ...  The focus of the paper is the study of local obstructions to the separability of two semi-algebraic subsets A and B of an algebraic set X by a regular function.  ... 

Page 4041 of Mathematical Reviews Vol. , Issue 88h [page]

1988 Mathematical Reviews  
Generators of the algebras are given as closed expressions for matrix elements. The Gel‘fand-Tsetlin basis for U(m) naturally plays a key role.  ...  A subspace Y of a topological space X is said to be a bounded subspace if any continuous real function on XY is bounded on the subspace Y.  ... 

Page 8243 of Mathematical Reviews Vol. , Issue 2002K [page]

2002 Mathematical Reviews  
, in the length of the representation 6 =) b,,4,. i=l For the real forms of those triples this bound is shown to be 306, i.e. the double of the complex case.  ...  The remaining case is the rectangular factor, which as 46L Selfadjoint operator algebras (C*-algebras, von Neumann (W’*-) algebras, etc.) 2002k:46182 a space is simply the space B(H, K) of all bounded  ... 

Page 327 of American Mathematical Society. Bulletin of the American Mathematical Society Vol. 61, Issue 4 [page]

1955 American Mathematical Society. Bulletin of the American Mathematical Society  
Fell: Separable representations of purely infinite algebras. Preliminary report. A representation of an algebra A by bounded operators on a Hilbert space K is called separable if K is separable.  ...  Let A be a purely infinite W*-algebra (for terminology, see Kaplansky, Ann. of Math. vol. 53 (1951) pp. 235-249), and let R be a separable *-representation of A.  ... 

Univariate Real Root Isolation over a Single Logarithmic Extension of Real Algebraic Numbers [chapter]

Adam Strzeboński, Elias P. Tsigaridas
2017 Applications of Computer Algebra  
We present algorithmic, complexity and implementation results for the problem of isolating the real roots of a univariate polynomial B ∈ L[x], where L = Q[lg(α)] and α is a positive real algebraic number  ...  The algorithm approximates the coefficients of B up to a sufficient accuracy and then solves the approximate polynomial. For this we derive worst case (aggregate) separation bounds.  ...  Acknowledgments Both authors would like to thank an anonymous referee for her, or his, very detailed comments that greatly improved the presentation of the results.  ... 
doi:10.1007/978-3-319-56932-1_27 fatcat:pemjhkbm5jf57npi5qxpkxxzra

V*-algebras [article]

Andre Kornell
2016 arXiv   pre-print
More generally, if A is any separable C*-algebra of type I, then V^*(A) is canonically isomorphic to an ℓ^∞-direct sum of type I factors, with one summand for each irreducible representation of A.  ...  We then define a V*-algebra to be a C*-algebra of bounded operators that is closed in the continuum-weak topology.  ...  This axiom implies that the bounded real-valued functions on a complete separable metric space X, can be obtained from the continuous bounded real-valued functions on X by iterating taking limits of monotone  ... 
arXiv:1502.01516v3 fatcat:b4j7p4psujejdkrppp3diectp4

High-Level Filtering for Arrangements of Conic Arcs [chapter]

Ron Wein
2002 Lecture Notes in Computer Science  
In Subsection 5.1.1 we shall give a sketch of a constructive proof for the existence of a separation bound for every algebraic expression.  ...  to a very small separation bound.  ...  bounded by M and two integer roots, where α is a common root.  ... 
doi:10.1007/3-540-45749-6_76 fatcat:moh7447a35f3va36v3agodmtji
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