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A Separation Bound for Real Algebraic Expressions
[chapter]

2001
*
Lecture Notes in Computer Science
*

We prove

doi:10.1007/3-540-44676-1_21
fatcat:uqljn7mlsvdttjbtdjutk3o2fq
*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*. ...##
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A Separation Bound for Real Algebraic Expressions

2007
*
Algorithmica
*

We prove

doi:10.1007/s00453-007-9132-4
fatcat:hc4n4hv3xjezfcu725pvxrttyy
*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*. ...##
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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. ...##
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Counterexamples to Quantifier Elimination for Fewnomial and Exponential Expressions

2007
*
Moscow Mathematical Journal
*

We construct

doi:10.17323/1609-4514-2007-7-3-453-460
fatcat:2ymzpt5ifbdevpsp5vm5r4eo54
*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*...##
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The Polytope-Collision Problem

2017
*
International Colloquium on Automata, Languages and Programming
*

Using techniques from transcendental number theory, and

doi:10.4230/lipics.icalp.2017.24
dblp:conf/icalp/AlmagorOW17
fatcat:tmoworgmavdhxc2ziacj6huroa
*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. ...##
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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. ...##
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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 @. ...

##
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Notes on coarse grainings and functions of observables

2005
*
Reports on mathematical physics
*

To this end, conditions

doi:10.1016/s0034-4877(05)80030-8
fatcat:io65xzigjncn3lfewauobb4che
*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*. ...##
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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. ...##
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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. ...##
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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*...##
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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]

2017
*
Applications of Computer Algebra
*

We present algorithmic, complexity and implementation results

doi:10.1007/978-3-319-56932-1_27
fatcat:pemjhkbm5jf57npi5qxpkxxzra
*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. ...##
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V*-algebras
[article]

2016
*
arXiv
*
pre-print

More generally, if

arXiv:1502.01516v3
fatcat:b4j7p4psujejdkrppp3diectp4
*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 ...##
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High-Level Filtering for Arrangements of Conic Arcs
[chapter]

2002
*
Lecture Notes in Computer Science
*

In Subsection 5.1.1 we shall give

doi:10.1007/3-540-45749-6_76
fatcat:moh7447a35f3va36v3agodmtji
*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. ...
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