The secretary problem with items arriving according to a random permutation avoiding a pattern of length three

In the classical secretary problem, $n$ ranked items arrive one by one, and each item's rank relative to its predecessors is noted. The observer must select or reject each item as it arrives, with the object of selecting the item of highest rank. For $M_n\in\{0,1,\cdots, n-1\}$, let $\mathcal{S}(n,M_n)$ denote the strategy whereby the observer rejects the first $M_n$ items, and then selects the first later-arriving item whose rank is higher than that of any of the first $M_n$ items (if such an item exists). If the ranked items arrive in a uniformly random order, it is well-known that the limiting optimal probability of success is $\frac1e$, which occurs if $M_n\sim\frac ne$. It has been shown that when the ranked items arrive according to certain non-uniform distributions on the set of permutations, $\frac1e$ serves as a lower bound for the optimal probability. There is a fundamental reason for this phenomenon. We consider certain distributions for which that reason does not apply. We begin by noting a cooked-up class of distributions for which $\mathcal{S}(n,M)$ yields the lowest possible probability of success -- namely $\frac1n$, for all $M$. We then consider the uniform distribution over all permutations avoiding a particular pattern of length three. In the case of the pattern 231 or 132, for any choice of $M_n$, the strategy $\mathcal{S}(n,M_n)$ yields the very same probability of success; namely $\frac{n+1}{2(2n-1)}$, which gives a limiting probability of $\frac14$. For the pattern 213, the optimal strategy is obtained for $M\in\{0,1\}$, also yielding a limiting probability of $\frac14$. For the pattern 123, the optimal strategy is obtained for $M=1$, yielding a limiting probability of $\frac34$. For the other two patterns, 312 and 321, an optimal strategy will yield a limiting probability of at least $\frac7{16}$

Modeling the Behavior of Novice Young Drivers Using Data from In- Vehicle Data Recorders

Novice young drivers suffer from increased crash risk that translates into over-representation in road injuries. A better understanding of the driving behavior of novice young drivers and of their determinants is needed to tackle this problem. To this extent, this study analyzes the behavior of novice young drivers within a Graduated Driver Licensing (GDL) program. Data on driving behavior of novice drivers and their parents is collected using in-vehicle data recorders, which calculate compound risk indices as measures of the risk taking behavior of the various drivers. Data is used to estimate a negative binomial model to identify the major factors that affect the driving behavior of the young drivers. Estimation results suggest that the risk taking behavior of young drivers is influenced by that of their parents and decreases with higher levels of supervised driving and stricter monitoring by the parents

Obtaining research biopsies during pediatric colonoscopy: Safety and adverse events

AIM: To investigate the safety profile of acquiring additional intestinal biopsies for research purposes in children undergoing a medically indicated colonoscopy. METHODS: A retrospective review of 122 pediatric patients who underwent colonoscopy over a 9 mo time period was completed. 38/122 participants consented to a research study in which 4 additional biopsies were obtained, in addition to routine biopsies. The outcomes after colonoscopy were measured in the research participants, and compared to 84 control participants who did not consent for the study. Groups were compared with regard to number of biopsies obtained, underlying diagnosis, and both serious and minor adverse outcomes. Data was collected including: age, gender, race, indication, diagnosis, number of biopsies obtained per case and post procedure adverse events. Medical records were reviewed and a questionnaire was completed by each of the ten gastroenterologists who performed procedures during the study. Physicians were asked about individual patient outcomes to ensure that all adverse events, such as perforation, excessive bleeding, infection, and minor gastrointestinal outcomes, were captured and included. RESULTS: The research group had more biopsies obtained (mean = 13.58 +/- 4.21) compared to controls (mean = 9.33 +/- 4.40),

Volatility of Linear and Nonlinear Time Series

Previous studies indicate that nonlinear properties of Gaussian time series with long-range correlations, $u_i$, can be detected and quantified by studying the correlations in the magnitude series $|u_i|$, i.e., the ``volatility''. However, the origin for this empirical observation still remains unclear, and the exact relation between the correlations in $u_i$ and the correlations in $|u_i|$ is still unknown. Here we find analytical relations between the scaling exponent of linear series $u_i$ and its magnitude series $|u_i|$. Moreover, we find that nonlinear time series exhibit stronger (or the same) correlations in the magnitude time series compared to linear time series with the same two-point correlations. Based on these results we propose a simple model that generates multifractal time series by explicitly inserting long range correlations in the magnitude series; the nonlinear multifractal time series is generated by multiplying a long-range correlated time series (that represents the magnitude series) with uncorrelated time series [that represents the sign series $sgn(u_i)$]. Our results of magnitude series correlations may help to identify linear and nonlinear processes in experimental records.Comment: 7 pages, 5 figure

Mass growth and mergers: direct observations of the luminosity function of LRG satellite galaxies out to z=0.7 from SDSS and BOSS images

We present a statistical study of the luminosity functions of galaxies surrounding luminous red galaxies (LRGs) at average redshifts =0.34 and =0.65. The luminosity functions are derived by extracting source photometry around more than 40,000 LRGs and subtracting foreground and background contamination using randomly selected control fields. We show that at both studied redshifts the average luminosity functions of the LRGs and their satellite galaxies are poorly fitted by a Schechter function due to a luminosity gap between the centrals and their most luminous satellites. We utilize a two-component fit of a Schechter function plus a log-normal distribution to demonstrate that LRGs are typically brighter than their most luminous satellite by roughly 1.3 magnitudes. This luminosity gap implies that interactions within LRG environments are typically restricted to minor mergers with mass ratios of 1:4 or lower. The luminosity functions further imply that roughly 35% of the mass in the environment is locked in the LRG itself, supporting the idea that mass growth through major mergers within the environment is unlikely. Lastly, we show that the luminosity gap may be at least partially explained by the selection of LRGs as the gap can be reproduced by sparsely sampling a Schechter function. In that case LRGs may represent only a small fraction of central galaxies in similar mass halos.Comment: ApJ accepted versio

Searching for a Gravitational Heating Signature in Nearby Luminous Ellipticals

We present a new deep optical study of a luminosity limited sample of nearby elliptical galaxies, attempting to observe the effects of gravitational interactions on the ISM of these objects. This study is motivated by recent observations of M86, a nearby elliptical galaxy that shows possible evidence for gas heating through a recent gravitational interaction. The complete sample includes luminous ellipticals in clusters, groups and the field. For each of the galaxies we objectively derive a tidal parameter which measures the deviation of the stellar body from a smooth, relaxed model and find that 73% of them show tidal disturbance signatures in their stellar bodies. This is the first time that such an analysis is done on a statistically complete sample and it confirms that elliptical galaxies continue to grow and evolve through gravitational interactions even in the local Universe. Our study of ellipticals in a wide range of interaction stages, along with available ISM data will attempt to shed light on this possibly alternative mechanism for maintaining the observed ISM temperatures of elliptical galaxies.Comment: To appear in proceedings of The Monster's Fiery Breath: Feedback in Galaxies, Groups, and Clusters (AIP conference series

Light-Heavy Symmetry: Geometric Mass Hierarchy for Three Families

The Universal Seesaw pattern coupled with a Light$\leftrightarrow$Heavy symmetry principle leads to the Diophantine equation $\displaystyle N = \sum_{i=1}^Nn_i$, where $n_i\geq 0$ and distinct. Its unique non-trivial solution $(3=0+1+2)$ gives rise to the geometric mass hierarchy $m_W$, $m_W\epsilon$, $m_W\epsilon^2$ for $N=3$ fermion families. This is realized in a model where the hybrid (yet Up$\leftrightarrow$Down symmetric) quark mass relations $m_d m_t \approx m_c^2\leftrightarrow m_u m_b \approx m_s^2$ play a crucial role in expressing the CKM mixings in terms of simple mass ratios, notably $\sin\theta_C \approx {m_c\over m_b}$.Comment: 12 pages, no figures, Revtex fil

Rhizomatic affective spaces and the therapeutic potential of music in prison: a qualitative meta-synthesis

Research literature supports that music programmes in prisons can have a therapeutic effect in prisoners’ lives that could promote personal development and assist the process towards desistance. The authors use a meta-synthetic approach to examine 12 qualitative articles published worldwide to explore the therapeutic potential of such programmes. The findings suggest that music programmes in prison are perceived by participating prisoners as a liberating process which encourages participation and allows for non-coercive personal development. The therapeutic potential of music programmes is located in the combination of the benefits emanating from the effect and practice of music and the creation of mental, spatial and temporal zones of free expression and those that derive from the egalitarian and non-authoritative approach employed by the facilitators. These findings are discussed along with aspects involved in the provision of offender treatment as well as factors that affect treatment response and engagement