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Local entropy averages and projections of fractal measures
Get PDFWe show that for families of measures on Euclidean space which satisfy an ergodic-theoretic form of "self-similarity" under the operation of re-scaling, the dimension of linear images of the measure behaves in a semi-continuous way. We apply this to prove the following conjecture of Furstenberg: Let m,n be integers which are not powers of the same integer, and let X,Y be closed subsets of the unit interval which are invariant, respectively, under times-m mod 1 and times-n mod 1. Then, for any non-zero t: dim(X+tY)=min{1,dim(X)+dim(Y)}. A similar result holds for invariant measures, and gives a simple proof of the Rudolph-Johnson theorem. Our methods also apply to many other classes of conformal fractals and measures. As another application, we extend and unify Results of Peres, Shmerkin and Nazarov, and of Moreira, concerning projections of products self-similar measures and Gibbs measures on regular Cantor sets. We show that under natural irreducibility assumptions on the maps in the IFS, the image measure has the maximal possible dimension under any linear projection other than the coordinate projections. We also present applications to Bernoulli convolutions and to the images of fractal measures under differentiable maps
The Canonical Generalized Polar Decomposition
Get PDFThe polar decomposition of a square matrix has been generalized by several authors to scalar products on or given by a bilinear or sesquilinear form. Previous work has focused mainly on the case of square matrices, sometimes with the assumption of a Hermitian scalar product. We introduce the canonical generalized polar decomposition , defined for general matrices , where is a partial -isometry and is -selfadjoint with nonzero eigenvalues lying in the open right half-plane, and the nonsingular matrices and define scalar products on and , respectively. We derive conditions under which a unique decomposition exists and show how to compute the decomposition by matrix iterations. Our treatment derives and exploits key properties of -partial isometries and orthosymmetric pairs of scalar products, and also employs an appropriate generalized Moore--Penrose pseudoinverse. We relate commutativity of the factors in the canonical generalized polar decomposition to an appropriate definition of normality. We also consider a related generalized polar decomposition , defined only for square matrices and in which is an automorphism; we analyze its existence and the uniqueness of the selfadjoint factor when is singular
Using the Torso to Compensate for Non-Minimum Phase Behaviour in ZMP Bipedal Walking
No full textIn Zero Moment Point (ZMP) bipedal walking, the conventional method is to use the cart-table model for generating the reference trajectory [1]. However, due to modeling and tracking errors and external disturbances, such as uneven terrain, the generated trajectorymust be adapted by a stabilizer that uses sensory inputs from force and torque sensors placed in the robot’s feet. The problem with the carttable model is that it is non-minimum phase which causes a significant, undesirable undershoot in the ZMP in order to cancel the effect of disturbances. In this paper, a novel scheme is proposed for ZMP feedback stabilization that utilizes the upper body to balance the humanoid robot. This method increases the performance and robustness of walking by reducing the undershoot and maintaining a desired bandwidth. The effectiveness of the proposed scheme is demonstrated using simulation and open problems are discussed
Symplectic group actions and covering spaces
Get PDFFor symplectic group actions which are not Hamiltonian there are two ways to define reduction. Firstly using the cylinder-valued momentum map and secondly lifting the action to any Hamiltonian cover (such as the universal cover), and then performing symplectic reduction in the usual way. We show that provided the action is free and proper, and the Hamiltonian holonomy associated to the action is closed, the natural projection from the latter to the former is a symplectic covering. At the same time we give a classification of all Hamiltonian coverings of a given symplectic group action. The main properties of the lifting of a group action to a cover are studied
A Note on Auto-tuning GEMM for GPUs
Get PDFThe development of high performance dense linear algebra
(DLA) critically depends on highly optimized BLAS, and especially on
the matrix multiplication routine (GEMM). This is especially true for
Graphics Processing Units (GPUs), as evidenced by recently published
results on DLA for GPUs that rely on highly optimized GEMM [13, 11].
However, the current best GEMM performance, e.g. of up to 375 GFlop/s
in single precision and of up to 75 GFlop/s in double precision arithmetic
on NVIDIA's GTX 280, is dicult to achieve. The development involves
extensive GPU knowledge and even backward engineering to understand
some undocumented insides about the architecture that have been of key
importance in the development [12]. In this paper, we describe some GPU
GEMM auto-tuning optimization techniques that allow us to keep up
with changing hardware by rapidly reusing, rather than reinventing, the
existing ideas. Auto-tuning, as we show in this paper, is a very practical
solution where in addition to getting an easy portability, we can often get
substantial speedups even on current GPUs (e.g. up to 27% in certain
cases for both single and double precision GEMMs on the GTX 280)
Why are (the best) women so good at chess? Participation rates and gender differences in intellectual domains
Get PDFA popular explanation for the small number of women at the top level of intellectually demanding activities from chess to science appeals to biological differences in the intellectual abilities of men and women. An alternative explanation is that the extreme values in a large sample are likely to be greater than those in a small one. Although the performance of the 100 best German male chess players is better than that of the 100 best German women, we show that 96 per cent of the observed difference would be expected given the much greater number of men who play chess. There is little left for biological or cultural explanations to account for. In science, where there are many more male than female participants, this statistical sampling explanation, rather than differences in intellectual ability, may also be the main reason why women are under-represented at the top end
On th Roots of Stochastic Matrices
Get PDFIn Markov chain models in finance and healthcare
a transition matrix over a certain time interval is needed
but only a transition matrix over a longer time interval may be available.
The problem arises of determining a stochastic th root of a
stochastic matrix (the given transition matrix).
By exploiting the theory of functions of matrices,
we develop results on the existence and characterization of
matrix th roots,
and in particular on the existence
of stochastic th roots of stochastic matrices.
Our contributions include characterization of when a real matrix has
a real th root,
a classification of th roots of a possibly singular matrix,
a sufficient condition for a th root of a stochastic matrix to have
unit row sums,
and the identification of classes of
stochastic matrices that have stochastic th roots for all .
We also delineate a wide variety of possible configurations
as regards existence,
nature (primary or nonprimary), and number of stochastic roots,
and develop a necessary condition for existence of a stochastic root
in terms of the spectrum of the given matrix
Covers in finitely accessible categories
Get PDFWe show that in a finitely accessible additive category every class
of objects closed under direct limits and pure epimorphic images is covering.
In particular, the classes of
flat objects in a locally finitely presented additive
category and of absolutely pure objects in a locally coherent category are
covering
A stochastic-hybrid Model of DNA replication in mammalian cells
No full textBackground
This study presents a mathematical model which explores the mechanisms of human S phase that link different patterns of synthesis with the temporal S phase programme, which is seen in vivo. A chromosome and genome-wide scale is used in the modelling, which focuses on the behaviour of replication factories relative to the chromatin template. DNA foci that contain ~1Mbp of DNA are the functional units of synthesis and provide targets for the synthetic process, with each focus containing clusters of replicons that are replicated together within a single dedicated replication factory.
Results
The model indicates that random replication factory placement and/or dynamics cannot lead to the observed S phase progression pattern. However, co-ordination of factory activation according to Giemsa staining bands, with bias towards R-bands over G-bands, results in a transition from R- to G-band replication after 3-5hours. Our model predicts that factories move between chromosomes to avoid non-uniform entry into late S-phase. The distribution of replicons has a strong influence on simulations, with grouping of similar sized replicons making replication more efficient.
Conclusions
Mathematical modelling of the mammalian cell S phase provides insight into the organisation of the human genome. For example, the co-ordinated targeting of replication factories to R- and G-bands based on their structural properties and distribution in nuclei can explain the temporal progression of synthesis that is observed in experimental studies. However, further understanding of the S phase programme is likely to require additional spatial information and incorporate a three dimensional approach
Dealing with stochastic reachability
No full textFor stochastic hybrid systems, stochastic reachability is very little supported mainly because of complexity and difficulty of the associated mathematical problems. In this paper, we develop two main directions of studying stochastic reachability as an optimal stopping problem. The first approach studies the hypotheses for the dynamic programming corresponding with the optimal stopping problem for stochastic hybrid systems. In the second approach, we investigate the reachability problem considering approximations of stochastic hybrid systems. The main difficulty arises when we have to prove the convergence of the value functions of the approximating processes to the value function of the initial process. An original proof is provided