5,224 research outputs found
The signature of a splice
Get PDFWe study the behavior of the signature of colored links [Flo05, CF08] under
the splice operation. We extend the construction to colored links in integral
homology spheres and show that the signature is almost additive, with a
correction term independent of the links. We interpret this correction term as
the signature of a generalized Hopf link and give a simple closed formula to
compute it.Comment: Updated version. Sign corrected in Theorems 2.2 and 2.10 of the
previous version. Also Corollary 2.6 was corrected and an Example added. 24
pages, 5 figures. To appear in IMR
Slopes and signatures of links
Full text linkWe define the slope of a colored link in an integral homology sphere,
associated to admissible characters on the link group. Away from a certain
singular locus, the slope is a rational function which can be regarded as a
multivariate generalization of the Kojima--Yamasaki -function. It is the
ratio of two Conway potentials, provided that the latter makes sense;
otherwise, it is a new invariant. The slope is responsible for an extra
correction term in the signature formula for the splice of two links, in the
previously open exceptional case where both characters are admissible. Using a
similar construction for a special class of tangles, we formulate generalized
skein relations for the signature
Inception-v4, Inception-ResNet and the Impact of Residual Connections on Learning
Full text linkVery deep convolutional networks have been central to the largest advances in
image recognition performance in recent years. One example is the Inception
architecture that has been shown to achieve very good performance at relatively
low computational cost. Recently, the introduction of residual connections in
conjunction with a more traditional architecture has yielded state-of-the-art
performance in the 2015 ILSVRC challenge; its performance was similar to the
latest generation Inception-v3 network. This raises the question of whether
there are any benefit in combining the Inception architecture with residual
connections. Here we give clear empirical evidence that training with residual
connections accelerates the training of Inception networks significantly. There
is also some evidence of residual Inception networks outperforming similarly
expensive Inception networks without residual connections by a thin margin. We
also present several new streamlined architectures for both residual and
non-residual Inception networks. These variations improve the single-frame
recognition performance on the ILSVRC 2012 classification task significantly.
We further demonstrate how proper activation scaling stabilizes the training of
very wide residual Inception networks. With an ensemble of three residual and
one Inception-v4, we achieve 3.08 percent top-5 error on the test set of the
ImageNet classification (CLS) challeng
On Primitivity of Sets of Matrices
Full text linkA nonnegative matrix is called primitive if is positive for some
integer . A generalization of this concept to finite sets of matrices is
as follows: a set of matrices is
primitive if is positive for some indices
. The concept of primitive sets of matrices comes up in a
number of problems within the study of discrete-time switched systems. In this
paper, we analyze the computational complexity of deciding if a given set of
matrices is primitive and we derive bounds on the length of the shortest
positive product.
We show that while primitivity is algorithmically decidable, unless it
is not possible to decide primitivity of a matrix set in polynomial time.
Moreover, we show that the length of the shortest positive sequence can be
superpolynomial in the dimension of the matrices. On the other hand, defining
to be the set of matrices with no zero rows or columns, we give
a simple combinatorial proof of a previously-known characterization of
primitivity for matrices in which can be tested in polynomial
time. This latter observation is related to the well-known 1964 conjecture of
Cerny on synchronizing automata; in fact, any bound on the minimal length of a
synchronizing word for synchronizing automata immediately translates into a
bound on the length of the shortest positive product of a primitive set of
matrices in . In particular, any primitive set of
matrices in has a positive product of length
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