Part-products of SS-restricted integer compositions

If $S$ is a cofinite set of positive integers, an "$S$-restricted composition of $n$" is a sequence of elements of $S$, denoted $\vec{\lambda}=(\lambda_1,\lambda_2,...)$, whose sum is $n$. For uniform random $S$-restricted compositions, the random variable ${\bf B}(\vec{\lambda})=\prod_i \lambda_i$ is asymptotically lognormal. The proof is based upon a combinatorial technique for decomposing a composition into a sequence of smaller compositions.Comment: 18 page

Locally Restricted Compositions IV. Nearly Free Large Parts and Gap-Freeness

We define the notion of asymptotically free for locally restricted compositions, which means roughly that large parts can often be replaced by any larger parts. Two well-known examples are Carlitz and alternating compositions. We show that large parts have asymptotically geometric distributions. This leads to asymptotically independent Poisson variables for numbers of various large parts. Based on this we obtain asymptotic formulas for the probability of being gap free and for the expected values of the largest part, number of distinct parts and number of parts of multiplicity k, all accurate to o(1).Comment: 28 page

The largest missing value in a composition of an integer

AbstractIn this paper we find, asymptotically, the mean and variance for the largest missing value (part size) in a composition of an integer n. We go on to show that the probability that the largest missing value and the largest part of a composition differ by one is relatively high and we find the mean for the average largest value in compositions that have this property. The average largest value of compositions with at least one non-zero missing value is also found, and used to calculate how many distinct values exceed the largest missing value on average

Structure, energetics, and mechanical stability of Fe-Cu bcc alloys from first-principles calculations

Atomic volumes, magnetic moments, mixing energies, and the elastic properties of bcc Fe1–xCux solid solutions are studied by ab initio calculations based on the cluster expansion framework. For the calculation of concentration-dependent elastic moduli in disordered solid solutions, we introduce a generalization of the cluster expansion technique that is designed to handle tensorial quantities in high-symmetry phases. Calculated mixing energies, atomic volumes, and magnetic moments are found to be in good agreement with available measurements for metastable alloys prepared through nonequilibrium processing techniques. Additionally, the predicted variations of the bulk modulus and shear moduli C44 and C[prime] with respect to copper concentration are calculated for the disordered bcc phase. While the bulk modulus and C44 are positive for all concentrations, C[prime] is predicted to be positive only for Cu concentration less than 50 atomic %, and negative otherwise. Our results thus indicate that the mechanical instability of bcc Cu persists over a wide range of compositions. The implications of the present results are discussed in relation to the observed metastability of bcc Fe-Cu alloys, and the strengthening mechanism of nanoscale bcc precipitates in an alpha-Fe matrix

Guaranteed Rank Minimization via Singular Value Projection

Minimizing the rank of a matrix subject to affine constraints is a fundamental problem with many important applications in machine learning and statistics. In this paper we propose a simple and fast algorithm SVP (Singular Value Projection) for rank minimization with affine constraints (ARMP) and show that SVP recovers the minimum rank solution for affine constraints that satisfy the "restricted isometry property" and show robustness of our method to noise. Our results improve upon a recent breakthrough by Recht, Fazel and Parillo (RFP07) and Lee and Bresler (LB09) in three significant ways: 1) our method (SVP) is significantly simpler to analyze and easier to implement, 2) we give recovery guarantees under strictly weaker isometry assumptions 3) we give geometric convergence guarantees for SVP even in presense of noise and, as demonstrated empirically, SVP is significantly faster on real-world and synthetic problems. In addition, we address the practically important problem of low-rank matrix completion (MCP), which can be seen as a special case of ARMP. We empirically demonstrate that our algorithm recovers low-rank incoherent matrices from an almost optimal number of uniformly sampled entries. We make partial progress towards proving exact recovery and provide some intuition for the strong performance of SVP applied to matrix completion by showing a more restricted isometry property. Our algorithm outperforms existing methods, such as those of \cite{RFP07,CR08,CT09,CCS08,KOM09,LB09}, for ARMP and the matrix-completion problem by an order of magnitude and is also significantly more robust to noise.Comment: An earlier version of this paper was submitted to NIPS-2009 on June 5, 200

Vissim model development of a mid-block pedestrian crossing between signalized intersections

Pedestrian mid-block crossings through arterial roadways have significant impacts on vehicular level of service and safety. Past research has been conducted to simulate these crossings. However, a practical, realistic model and methodology have not been established for analysis and planning of mid-block pedestrian crossings. The objective of this study was to create a practical traffic microsimulation methodology for mid-block pedestrian crossings through a case study using a versatile transportation software package. VISSIM v3.70 was selected because of its ability to model individual vehicle and pedestrian interactions. The developed methodology provided a framework for model development at any pedestrian mid-block crossing through a case study. Input parameters for the case study model included an innovative field study that was designed using a Geometric distribution to estimate the proportion of drivers yielding to pedestrians at the crosswalk. Segment travel times and crosswalk approach vehicle queue lengths were used for calibration and validation, respectively. Level of service analyses were performed using the final model

Locally Restricted Compositions IV. Nearly Free Large Parts and Gap-Freeness

We define the notion of $t$-free for locally restricted compositions, which means roughly that if such a composition contains a part $c_i$ and nearby parts are at least $t$ smaller, then $c_i$ can be replaced by any larger part. Two well-known examples are Carlitz and alternating compositions. We show that large parts have asymptotically geometric distributions. This leads to asymptotically independent Poisson variables for numbers of various large parts. Based on this we obtain asymptotic formulas for the probability of being gap free and for the expected values of the largest part and number distinct parts, all accurate to $o(1)$

How to Couple from the Past Using a Read-Once Source of Randomness

We give a new method for generating perfectly random samples from the stationary distribution of a Markov chain. The method is related to coupling from the past (CFTP), but only runs the Markov chain forwards in time, and never restarts it at previous times in the past. The method is also related to an idea known as PASTA (Poisson arrivals see time averages) in the operations research literature. Because the new algorithm can be run using a read-once stream of randomness, we call it read-once CFTP. The memory and time requirements of read-once CFTP are on par with the requirements of the usual form of CFTP, and for a variety of applications the requirements may be noticeably less. Some perfect sampling algorithms for point processes are based on an extension of CFTP known as coupling into and from the past; for completeness, we give a read-once version of coupling into and from the past, but it remains unpractical. For these point process applications, we give an alternative coupling method with which read-once CFTP may be efficiently used.Comment: 28 pages, 2 figure

Gamma-GQM Time Headway Model: Endogenous Effects in Rural Two-lane Two-way Roads

AbstractStudy of vehicle time headway distributions is essential in many traffic engineering applications, such as capacity and level of service analysis and, in recent years, in the fields of vehicle generation in traffic micro-simulation models and driving simulation applications. This paper presents results from an experimental analysis of vehicle time headway distributions on two-lane two-way rural roads. Analysis focused on estimating a well-known model, the gamma-generalized queuing model (gamma-GQM). A trendless analysis of observed time headways was also carried out. The endogenous traffic parameters considered as affecting time headway distributions were flow rate and flow composition (percentage of heavy vehicles). Exogenous conditions, such as weather and geometric futures, were common to all time periods and cross-sections analysed. Gamma-GQM pdf appears to be very suitable for representing real headway distributions in all the analysed situations; it fits real-time headway distributions well, despite flow rate range and traffic composition (range of percentage of heavy vehicles)