692 research outputs found
L p -estimates for the heat semigroup on differential forms, and related problems
We consider a complete non-compact Riemannian manifold satisfying the volume
doubling property and a Gaussian upper bound for its heat kernel (on
functions). Let -- k be the Hodge-de Rham Laplacian on
differential k-forms with k 1. By the Bochner decomposition formula --
k = * + R k. Under the assumption that the negative part
R -- k is in an enlarged Kato class, we prove that for all p [1,
], e --t -- k p--p C(t log t) D 4 (1-- 2
p) (for large t). This estimate can be improved if R -- k is strongly
sub-critical. In general, (e --t -- k) t>0 is not
uniformly bounded on L p for any p = 2. We also prove the gradient estimate e
--t p--p Ct -- 1 p , where is the Laplace-Beltrami
operator (acting on functions). Finally we discuss heat kernel bounds on forms
and the Riesz transform on L p for p > 2
Improved bounds for testing Dyck languages
In this paper we consider the problem of deciding membership in Dyck
languages, a fundamental family of context-free languages, comprised of
well-balanced strings of parentheses. In this problem we are given a string of
length in the alphabet of parentheses of types and must decide if it is
well-balanced. We consider this problem in the property testing setting, where
one would like to make the decision while querying as few characters of the
input as possible.
Property testing of strings for Dyck language membership for , with a
number of queries independent of the input size , was provided in [Alon,
Krivelevich, Newman and Szegedy, SICOMP 2001]. Property testing of strings for
Dyck language membership for was first investigated in [Parnas, Ron
and Rubinfeld, RSA 2003]. They showed an upper bound and a lower bound for
distinguishing strings belonging to the language from strings that are far (in
terms of the Hamming distance) from the language, which are respectively (up to
polylogarithmic factors) the power and the power of the input size
.
Here we improve the power of in both bounds. For the upper bound, we
introduce a recursion technique, that together with a refinement of the methods
in the original work provides a test for any power of larger than .
For the lower bound, we introduce a new problem called Truestring Equivalence,
which is easily reducible to the -type Dyck language property testing
problem. For this new problem, we show a lower bound of to the power of
Stable Matching with Evolving Preferences
We consider the problem of stable matching with dynamic preference lists. At
each time step, the preference list of some player may change by swapping
random adjacent members. The goal of a central agency (algorithm) is to
maintain an approximately stable matching (in terms of number of blocking
pairs) at all times. The changes in the preference lists are not reported to
the algorithm, but must instead be probed explicitly by the algorithm. We
design an algorithm that in expectation and with high probability maintains a
matching that has at most blocking pairs.Comment: 13 page
Claw Finding Algorithms Using Quantum Walk
The claw finding problem has been studied in terms of query complexity as one
of the problems closely connected to cryptography. For given two functions, f
and g, as an oracle which have domains of size N and M (N<=M), respectively,
and the same range, the goal of the problem is to find x and y such that
f(x)=g(y). This paper describes an optimal algorithm using quantum walk that
solves this problem. Our algorithm can be generalized to find a claw of k
functions for any constant integer k>1, where the domains of the functions may
have different size.Comment: 12 pages. Introduction revised. A reference added. Weak lower bound
delete
Efficient quantum algorithms for some instances of the non-Abelian hidden subgroup problem
In this paper we show that certain special cases of the hidden subgroup
problem can be solved in polynomial time by a quantum algorithm. These special
cases involve finding hidden normal subgroups of solvable groups and
permutation groups, finding hidden subgroups of groups with small commutator
subgroup and of groups admitting an elementary Abelian normal 2-subgroup of
small index or with cyclic factor group.Comment: 10 page
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