Doob-Martin compactification of a Markov chain for growing random words sequentially

We consider a Markov chain that iteratively generates a sequence of random finite words in such a way that the $n^{\mathrm{th}}$ word is uniformly distributed over the set of words of length $2n$ in which $n$ letters are $a$ and $n$ letters are $b$: at each step an $a$ and a $b$ are shuffled in uniformly at random among the letters of the current word. We obtain a concrete characterization of the Doob-Martin boundary of this Markov chain. Writing $N(u)$ for the number of letters $a$ (equivalently, $b$) in the finite word $u$, we show that a sequence $(u_n)_{n \in \mathbb{N}}$ of finite words converges to a point in the boundary if, for an arbitrary word $v$, there is convergence as $n$ tends to infinity of the probability that the selection of $N(v)$ letters $a$ and $N(v)$ letters $b$ uniformly at random from $u_n$ and maintaining their relative order results in $v$. We exhibit a bijective correspondence between the points in the boundary and ergodic random total orders on the set $\{a_1, b_1, a_2, b_2, \ldots \}$ that have distributions which are separately invariant under finite permutations of the indices of the $a'$s and those of the $b'$s. We establish a further bijective correspondence between the set of such random total orders and the set of pairs $(\mu,\nu)$ of diffuse probability measures on $[0,1]$ such that $\frac{1}{2}(\mu+\nu)$ is Lebesgue measure: the restriction of the random total order to $\{a_1, b_1, \ldots, a_n, b_n\}$ is obtained by taking $X_1, \ldots, X_n$ (resp. $Y_1, \ldots, Y_n$) i.i.d. with common distribution $\mu$ (resp. $\nu$), letting $(Z_1, \ldots, Z_{2n})$ be $\{X_1, Y_1, \ldots, X_n, Y_n\}$ in increasing order, and declaring that the $k^{\mathrm{th}}$ smallest element in the restricted total order is $a_i$ (resp. $b_j$) if $Z_k = X_i$ (resp. $Z_k = Y_j$).Comment: 24 pages, revised to deal with reviewer's comment

DFacTo: Distributed Factorization of Tensors

We present a technique for significantly speeding up Alternating Least Squares (ALS) and Gradient Descent (GD), two widely used algorithms for tensor factorization. By exploiting properties of the Khatri-Rao product, we show how to efficiently address a computationally challenging sub-step of both algorithms. Our algorithm, DFacTo, only requires two sparse matrix-vector products and is easy to parallelize. DFacTo is not only scalable but also on average 4 to 10 times faster than competing algorithms on a variety of datasets. For instance, DFacTo only takes 480 seconds on 4 machines to perform one iteration of the ALS algorithm and 1,143 seconds to perform one iteration of the GD algorithm on a 6.5 million x 2.5 million x 1.5 million dimensional tensor with 1.2 billion non-zero entries.Comment: Under review for NIPS 201

Impact of the recent results by the CMS and ATLAS Collaborations at the CERN Large Hadron Collider on an effective Minimal Supersymmetric extension of the Standard Model

We discuss the impact for light neutralinos in an effective Minimal Supersymmetric extension of the Standard Model of the recent results presented by the CMS and ATLAS Collaborations at the CERN Large Hadron Collider for a search of supersymmetry in proton-proton collisions at a center-of-mass energy of 7 TeV with an integrated luminosity of 35 inverse pb. We find that, in the specific case of light neutralinos, efficiencies for the specific signature searched by ATLAS (jets+missing transverse energy and an isolated lepton) imply a lower sensitivity compared to CMS (which searches for jets +missing transverse energy). Focusing on the CMS bound, if squark soft masses of the three families are assumed to be degenerate, the combination of the ensuing constraint on squark and gluino masses with the experimental limit on the b to s + gamma decay imply a lower bound on the neutralino mass that can reach the value of 11.9 GeV, depending on the gluino mass. On the other hand, when the universality condition among squark soft parameters is relaxed, the lower bound on the neutralino mass is not constrained by the CMS measurement and then remains at the value 7.5 GeV derived in previous papers.Comment: 6 pages, 6 figures, typeset with ReVTeX4. A version of the paper with full resolution figures can be found at http://www.to.infn.it/~scopel/cms_mssm2.pd

Correct Effective Potential of Supersymmetric Yang-Mills Theory on M^4\times S^1

We study an ${\cal N}=1$ supersymmetric Yang-Mills theory defined on $M^4\times S^1$. The vacuum expectation values for adjoint scalar field in vector multiplet, though important, has been overlooked in evaluating one-loop effective potential of the theory. We correctly take the vacuum expectation values into account in addition to the Wilson line phases to give an expression for the effective potential, and gauge symmetry breaking is discussed. In evaluating the potential, we employ the Scherk-Schwarz mechanism and introduce bare mass for gaugino in order to break supersymmetry. We also obtain masses for the scalars, the adjoint scalar, and the component gauge field for the $S^1$ direction in case of the SU(2) gauge group. We observe that large supersymmetry breaking gives larger mass for the scalar. This analysis is easily applied to the $M^4\times S^1/Z_2$ case.Comment: 12 pages, 1 figur

Male Partners' Involvement Towards Prenatal Screening and Diagnostic Testing for Down Syndrome

Introduction: Now, male partners' involvement in prenatal screening and diagnostic testing for Down syndrome is becoming increasingly recognized as well to ensure that parents are well informed of the risks and benefits of screening. The aim of study was to understand the degree of male partners' involvement during pregnancy in Singapore population. Methods: A cross-sectional survey of male partners' attending prenatal counseling was performed. The instrument used to measure the level of involvement is a self-assessment questionnaire that identifies the role of male partners with a Likert scale. Descriptive statistics was used to analyze data gained. Result: A total of 107 participants completed the questionnaire. Sixty-seven percent of male partners were found to have a highlevel of involvement while 32.7% was found to have a medium level of involvement. Most of them stated that women can pursue prenatal testing without their permission. Male partners found it more important for them to accompany their spouse to amniocentesis or CVS than to the Down syndrome screening test. When participants were asked about how much information about Down syndrome they sought prior to the appointment, how much discussion they had with their spouse about Down syndrome testing, and about whether they or their spouse should be the first person to receive test results, most stated that they were undecided. Conclusion: These results revealed that male partners were very well involved in the Down syndrome testing during pregnancy and future studies should assess possible underlying factors that influence male partners' involvement