159,511 research outputs found
On inner product in modular tensor categories. II. Inner product on conformal blocks and affine inner product identities
This is the second part of the paper (the first part is published in Jour. of
AMS, vol.9, 1135--1170, q-alg/9508017). In the first part, we defined for every
modular tensor category (MTC) inner products on the spaces of morphisms and
proved that the inner product on the space \Hom (\bigoplus X_i\otimes X^*_i,
U) is modular invariant. Also, we have shown that in the case of the MTC
arising from the representations of the quantum group U_q \sln at roots of
unity and being a symmetric power of the fundamental representation, this
inner product coincides with so-called Macdonald's inner product on symmetric
polynomials.
In this paper, we apply the same construction to the MTC coming from the
integrable representations of affine Lie algebras. In this case our
construction immediately gives a hermitian form on the spaces of conformal
blocks, and this form is modular invariant (Warning: we cannot prove that it is
positive definite). We show that this form can be rewritten in terms of
asymptotics of KZ equations, and calculate it for , in which case the
formula is a natural affine analogue of Macdonald's inner product identities.
We also formulate as a conjecture similar formula for .Comment: 21 pp., AmsTeX, 3 figures (require epsf
Conclusive inner product modification
The task of changing the overlap between two quantum states can not be
performed by making use of a unitary evolution only. However, by means of a
unitary-reduction process it can be probabilistically modified. Here we study
in detail the problem of mapping two known pure states onto other two states in
such a way that the final inner product between the outcome states is different
from the inner product of the initial states. In this way we design an optimal
non-orthogonal quantum state preparation scheme by starting from an orthonormal
basis. In this scheme the absolute value of the inner product can be reduced
only probabilistically whereas it can be increased deterministically. Our
analysis shows that the phases of the involved inner products play an important
role in the increase of the success probability of the desired process.Comment: 7 pages, 4 figur
Characterization of inner product spaces
We prove that the existence of best coapproximation to any element of the
normed linear space out of any one dimensional subspace and its coincidence
with the best approximation to that element out of that subspace characterizes
a real inner product space of dimension . We conjecture that a finite
dimensional real smooth normed space of dimension is an inner product
space iff given any element on the unit sphere there exists a strongly
orthonormal Hamel basis in the sense of Birkhoff-James containing that element.
This is substantiated by our result on the spaces $(R^n,\|.\|_p).
Inside s-inner product sets and Euclidean designs
A finite set X in the Euclidean space is called an s-inner product set if the
set of the usual inner products of any two distinct points in X has size s.
First, we give a special upper bound for the cardinality of an s-inner product
set on concentric spheres. The upper bound coincides with the known lower bound
for the size of a Euclidean 2s-design. Secondly, we prove the non-existence of
2- or 3-inner product sets on two concentric spheres attaining the upper bound
for any d>1. The efficient property needed to prove the upper bound for an
s-inner product set gives the new concept, inside s-inner product sets. We
characterize the most known tight Euclidean designs as inside s-inner product
sets attaining the upper bound.Comment: 9 pages, no figur
Semi-inner product structures for groupoids
In this paper there are considered some scalar valued groupoid bihomomorphism
structures, being in fact the groupoid counterparts of the inner product notion
originally defined for vectors. These bihomomorphisms, called here the
semi-inner products for groupoids, determine non-negative real valued functions
which fulfill the axioms assumed for a groupoid norm concept [2]
Quantization based Fast Inner Product Search
We propose a quantization based approach for fast approximate Maximum Inner
Product Search (MIPS). Each database vector is quantized in multiple subspaces
via a set of codebooks, learned directly by minimizing the inner product
quantization error. Then, the inner product of a query to a database vector is
approximated as the sum of inner products with the subspace quantizers.
Different from recently proposed LSH approaches to MIPS, the database vectors
and queries do not need to be augmented in a higher dimensional feature space.
We also provide a theoretical analysis of the proposed approach, consisting of
the concentration results under mild assumptions. Furthermore, if a small
sample of example queries is given at the training time, we propose a modified
codebook learning procedure which further improves the accuracy. Experimental
results on a variety of datasets including those arising from deep neural
networks show that the proposed approach significantly outperforms the existing
state-of-the-art
A characterization of inner product spaces
In this paper we present a new criterion on characterization of real inner
product spaces. We conclude that a real normed space is an inner
product space if for some positive integer and all
. Conversely, if is an inner product space,
then the equality above holds for all and all .Comment: 8 Pages, to appear in Kochi J. Math. (Japan
The weak n-inner product space
In this article we study a generalization of the n-inner product which we
name weak n-inner product. As particular case we consider the n-iterated
2-inner product and we give its representation in terms of the standard k-inner
products, k<= n, using the Dodgson's identity for determinants. Finally, we
present several applications, including a brief characterization of a linear
regression model for the random variables in discrete case and a generalization
of the Chebyshev functional using the n-iterated 2-inner product
A generating function for non-standard orthogonal polynomials involving differences: the Meixner case
In this paper we deal with a family of non--standard polynomials orthogonal
with respect to an inner product involving differences. This type of inner
product is the so--called --Sobolev inner product. Concretely, we
consider the case in which both measures appearing in the inner product
correspond to the Pascal distribution (the orthogonal polynomials associated to
this distribution are known as Meixner polynomials). The aim of this work is to
obtain a generating function for the --Meixner--Sobolev orthogonal
polynomials and, by using a limit process, recover a generating function for
Laguerre--Sobolev orthogonal polynomials.Comment: 14 page
Inner products for Convex Bodies
We define a set inner product to be a function on pairs of convex bodies
which is symmetric, Minkowski linear in each dimension, positive definite, and
satisfies the natural analogue of the Cauchy-Schwartz inequality (which is not
implied by the other conditions). We show that any set inner product can be
embedded into an inner product space on the associated support functions,
thereby extending fundamental results of Hormander and Radstrom. The set inner
product provides a geometry on the space of convex bodies. We explore some of
the properties of that geometry, and discuss an application of these ideas to
the reconstruction of ancestral ecological niches in evolutionary biology
- β¦