9,678 research outputs found
Gradient map of isoparametric polynomial and its application to Ginzburg-Landau system
In this note, we study properties of the gradient map of the isoparametric
polynomial. For a given isoparametric hypersurface in sphere, we calculate
explicitly the gradient map of its isoparametric polynomial which turns out
many interesting phenomenons and applications. We find that it should map not
only the focal submanifolds to focal submanifolds, isoparametric hypersurfaces
to isoparametric hypersurfaces, but also map isoparametric hypersurfaces to
focal submanifolds. In particular, it turns out to be a homogeneous polynomial
automorphism on certain isoparametric hypersurface. As an immediate
consequence, we get the Brouwer degree of the gradient map which was firstly
obtained by Peng and Tang with moving frame method. Following Farina's
construction, another immediate consequence is a counter example of the
Br\'ezis question about the symmetry for the Ginzburg-Landau system in
dimension 6, which gives a partial answer toward the Open problem 2 raised by
Farina.Comment: 10 page
Stochastic phenotype transition of a single cell in an intermediate region of gene-state switching
Multiple phenotypic states often arise in a single cell with different
gene-expression states that undergo transcription regulation with positive
feedback. Recent experiments have shown that at least in E. coli, the gene
state switching can be neither extremely slow nor exceedingly rapid as many
previous theoretical treatments assumed. Rather it is in the intermediate
region which is difficult to handle mathematically.Under this condition, from a
full chemical-master-equation description we derive a model in which the
protein copy-number, for a given gene state, follow a deterministic mean-field
description while the protein synthesis rates fluctuate due to stochastic
gene-state switching. The simplified kinetics yields a nonequilibrium landscape
function, which, similar to the energy function for equilibrium fluctuation,
provides the leading orders of fluctuations around each phenotypic state, as
well as the transition rates between the two phenotypic states. This rate
formula is analogous to Kramers theory for chemical reactions. The resulting
behaviors are significantly different from the two limiting cases studied
previously.Comment: 6 pages,4 figure
An effective likelihood-free approximate computing method with statistical inferential guarantees
Approximate Bayesian computing is a powerful likelihood-free method that has
grown increasingly popular since early applications in population genetics.
However, complications arise in the theoretical justification for Bayesian
inference conducted from this method with a non-sufficient summary statistic.
In this paper, we seek to re-frame approximate Bayesian computing within a
frequentist context and justify its performance by standards set on the
frequency coverage rate. In doing so, we develop a new computational technique
called approximate confidence distribution computing, yielding theoretical
support for the use of non-sufficient summary statistics in likelihood-free
methods. Furthermore, we demonstrate that approximate confidence distribution
computing extends the scope of approximate Bayesian computing to include
data-dependent priors without damaging the inferential integrity. This
data-dependent prior can be viewed as an initial `distribution estimate' of the
target parameter which is updated with the results of the approximate
confidence distribution computing method. A general strategy for constructing
an appropriate data-dependent prior is also discussed and is shown to often
increase the computing speed while maintaining statistical inferential
guarantees. We supplement the theory with simulation studies illustrating the
benefits of the proposed method, namely the potential for broader applications
and the increased computing speed compared to the standard approximate Bayesian
computing methods
Deciphering a novel image cipher based on mixed transformed Logistic maps
Since John von Neumann suggested utilizing Logistic map as a random number
generator in 1947, a great number of encryption schemes based on Logistic map
and/or its variants have been proposed. This paper re-evaluates the security of
an image cipher based on transformed logistic maps and proves that the image
cipher can be deciphered efficiently under two different conditions: 1) two
pairs of known plain-images and the corresponding cipher-images with
computational complexity of ; 2) two pairs of chosen plain-images
and the corresponding cipher-images with computational complexity of ,
where is the number of pixels in the plain-image. In contrast, the required
condition in the previous deciphering method is eighty-seven pairs of chosen
plain-images and the corresponding cipher-images with computational complexity
of . In addition, three other security flaws existing in most
Logistic-map-based ciphers are also reported.Comment: 10 pages, 2 figure
On the lower bound for kissing numbers of -spheres in high dimensions
In this paper, we give some new lower bounds for the kissing number of
-spheres. These results improve the previous work due to Xu (2007). Our
method is based on coding theory.Comment: 15 pages, 4 figures; any comments are welcom
Some sum-product estimates in matrix rings over finite fields
We study some sum-product problems over matrix rings. Firstly, for , we have whenever
. Secondly, if a set in
satisfies for some sufficiently
large , then we have
These improve the results due to The and Vinh (2020), and generalize the
results due to Mohammadi, Pham, and Wang (2021). We also give a new proof for a
recent result due to The and Vinh (2020). Our method is based on spectral graph
theory and linear algebra.Comment: 18 page
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