1,201 research outputs found
On the sum-of-squares degree of symmetric quadratic functions
We study how well functions over the boolean hypercube of the form
can be approximated by sums of squares of low-degree
polynomials, obtaining good bounds for the case of approximation in
-norm as well as in -norm. We describe three
complexity-theoretic applications: (1) a proof that the recent breakthrough
lower bound of Lee, Raghavendra, and Steurer on the positive semidefinite
extension complexity of the correlation and TSP polytopes cannot be improved
further by showing better sum-of-squares degree lower bounds on
-approximation of ; (2) a proof that Grigoriev's lower bound on
the degree of Positivstellensatz refutations for the knapsack problem is
optimal, answering an open question from his work; (3) bounds on the query
complexity of quantum algorithms whose expected output approximates such
functions.Comment: 33 pages. Second version fixes some typos and adds reference
Idempotent generated algebras and Boolean powers of commutative rings
A Boolean power S of a commutative ring R has the structure of a commutative
R-algebra, and with respect to this structure, each element of S can be written
uniquely as an R-linear combination of orthogonal idempotents so that the sum
of the idempotents is 1 and their coefficients are distinct. In order to
formalize this decomposition property, we introduce the concept of a Specker
R-algebra, and we prove that the Boolean powers of R are up to isomorphism
precisely the Specker R-algebras. We also show that these algebras are
characterized in terms of a functorial construction having roots in the work of
Bergman and Rota. When R is indecomposable, we prove that S is a Specker
R-algebra iff S is a projective R-module, thus strengthening a theorem of
Bergman, and when R is a domain, we show that S is a Specker R-algebra iff S is
a torsion-free R-module. For an indecomposable R, we prove that the category of
Specker R-algebras is equivalent to the category of Boolean algebras, and hence
is dually equivalent to the category of Stone spaces. In addition, when R is a
domain, we show that the category of Baer Specker R-algebras is equivalent to
the category of complete Boolean algebras, and hence is dually equivalent to
the category of extremally disconnected compact Hausdorff spaces. For a totally
ordered R, we prove that there is a unique partial order on a Specker R-algebra
S for which it is an f-algebra over R, and show that S is equivalent to the
R-algebra of piecewise constant continuous functions from a Stone space X to R
equipped with the interval topology.Comment: 18 page
Hilbert Modules - Square Roots of Positive Maps
We reflect on the notions of positivity and square roots. We review many
examples which underline our thesis that square roots of positive maps related
to *-algebras are Hilbert modules. As a result of our considerations we discuss
requirements a notion of positivity on a *-algebra should fulfill and derive
some basic consequences.Comment: 24 page
A computer algebra user interface manifesto
Many computer algebra systems have more than 1000 built-in functions, making
expertise difficult. Using mock dialog boxes, this article describes a proposed
interactive general-purpose wizard for organizing optional transformations and
allowing easy fine grain control over the form of the result even by amateurs.
This wizard integrates ideas including:
* flexible subexpression selection;
* complete control over the ordering of variables and commutative operands,
with well-chosen defaults;
* interleaving the choice of successively less main variables with applicable
function choices to provide detailed control without incurring a combinatorial
number of applicable alternatives at any one level;
* quick applicability tests to reduce the listing of inapplicable
transformations;
* using an organizing principle to order the alternatives in a helpful
manner;
* labeling quickly-computed alternatives in dialog boxes with a preview of
their results,
* using ellipsis elisions if necessary or helpful;
* allowing the user to retreat from a sequence of choices to explore other
branches of the tree of alternatives or to return quickly to branches already
visited;
* allowing the user to accumulate more than one of the alternative forms;
* integrating direct manipulation into the wizard; and
* supporting not only the usual input-result pair mode, but also the useful
alternative derivational and in situ replacement modes in a unified window.Comment: 38 pages, 12 figures, to be published in Communications in Computer
Algebr
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