337 research outputs found
Cycles representing the Todd class of a toric variety
In this paper, we describe a way to construct cycles which represent the Todd
class of a toric variety. Given a lattice with an inner product we assign a
rational number m(s) to each rational polyhedral cone s in the lattice, such
that for any toric variety X with fan S, the Todd class of X is the sum over
all cones s in S of m(s)[V(s)]. This constitutes an improved answer to an old
question of Danilov.
In a similar way, beginning with the choice of a complete flag in the
lattice, we obtain the cycle Todd classes constructed by Morelli.
Our construction is based on an intersection product on cycles of a
simplicial toric variety developed by the second-named author. Important
properties of the construction are established by showing a connection to the
canonical representation of the Todd class of a simplicial toric variety as a
product of torus-invariant divisors developed by the first-named author.Comment: 13 pages; version to appear in Journal of the AMS; minor
modifications throughout, corrections to proof of theorem 2; LaTe
On the uselessness of quantum queries
Given a prior probability distribution over a set of possible oracle
functions, we define a number of queries to be useless for determining some
property of the function if the probability that the function has the property
is unchanged after the oracle responds to the queries. A familiar example is
the parity of a uniformly random Boolean-valued function over ,
for which classical queries are useless. We prove that if classical
queries are useless for some oracle problem, then quantum queries are also
useless. For such problems, which include classical threshold secret sharing
schemes, our result also gives a new way to obtain a lower bound on the quantum
query complexity, even in cases where neither the function nor the property to
be determined is Boolean
Single query learning from abelian and non-abelian Hamming distance oracles
We study the problem of identifying an n-bit string using a single quantum
query to an oracle that computes the Hamming distance between the query and
hidden strings. The standard action of the oracle on a response register of
dimension r is by powers of the cycle (1...r), all of which, of course,
commute. We introduce a new model for the action of an oracle--by general
permutations in S_r--and explore how the success probability depends on r and
on the map from Hamming distances to permutations. In particular, we prove that
when r = 2, for even n the success probability is 1 with the right choice of
the map, while for odd n the success probability cannot be 1 for any choice.
Furthermore, for small odd n and r = 3, we demonstrate numerically that the
image of the optimal map generates a non-abelian group of permutations.Comment: 14 page
Cycle-Level Products in Equivariant Cohomology of Toric Varieties
In this paper, we define an action of the group of equivariant Cartier
divisors on a toric variety X on the equivariant cycle groups of X, arising
naturally from a choice of complement map on the underlying lattice. If X is
nonsingular, this gives a lifting of the multiplication in equivariant
cohomology to the level of equivariant cycles. As a consequence, one naturally
obtains an equivariant cycle representative of the equivariant Todd class of
any toric variety. These results extend to equivariant cohomology the results
of Thomas and Pommersheim. In the case of a complement map arising from an
inner product, we show that the equivariant cycle Todd class obtained from our
construction is identical to the result of the inductive, combinatorial
construction of Berline-Vergne. In the case of arbitrary complement maps, we
show that our Todd class formula yields the local Euler-Maclarurin formula
introduced in Garoufalidis-Pommersheim.Comment: 15 pages, to be published in Michigan Mathematical Journal; LaTe
United States v. Gillette: A Tiny Prairie Casenote Opening a Window on the Enveloping Fog Obscuring the Indian Civil Rights Act of 1968
United States v. Gillette is one of the first reported cases on the new post-United States v. Bryant road. As of yet, there is no reliable (legal) GPS to point the way. This tiny prairie casenote is not meant to focus on the answers, but rather clarify the questions and to widen the discussion as the journey continues
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