Asymmetric binary covering codes

An asymmetric binary covering code of length n and radius R is a subset C of the n-cube Q_n such that every vector x in Q_n can be obtained from some vector c in C by changing at most R 1's of c to 0's, where R is as small as possible. K^+(n,R) is defined as the smallest size of such a code. We show K^+(n,R) is of order 2^n/n^R for constant R, using an asymmetric sphere-covering bound and probabilistic methods. We show K^+(n,n-R')=R'+1 for constant coradius R' iff n>=R'(R'+1)/2. These two results are extended to near-constant R and R', respectively. Various bounds on K^+ are given in terms of the total number of 0's or 1's in a minimal code. The dimension of a minimal asymmetric linear binary code ([n,R]^+ code) is determined to be min(0,n-R). We conclude by discussing open problems and techniques to compute explicit values for K^+, giving a table of best known bounds.Comment: 16 page

Two-batch liar games on a general bounded channel

We consider an extension of the 2-person R\'enyi-Ulam liar game in which lies are governed by a channel $C$, a set of allowable lie strings of maximum length $k$. Carole selects $x\in[n]$, and Paul makes $t$-ary queries to uniquely determine $x$. In each of $q$ rounds, Paul weakly partitions $[n]=A_0\cup >... \cup A_{t-1}$ and asks for $a$ such that $x\in A_a$. Carole responds with some $b$, and if $a\neq b$, then $x$ accumulates a lie $(a,b)$. Carole's string of lies for $x$ must be in the channel $C$. Paul wins if he determines $x$ within $q$ rounds. We further restrict Paul to ask his questions in two off-line batches. We show that for a range of sizes of the second batch, the maximum size of the search space $[n]$ for which Paul can guarantee finding the distinguished element is $\sim t^{q+k}/(E_k(C)\binom{q}{k})$ as $q\to\infty$, where $E_k(C)$ is the number of lie strings in $C$ of maximum length $k$. This generalizes previous work of Dumitriu and Spencer, and of Ahlswede, Cicalese, and Deppe. We extend Paul's strategy to solve also the pathological liar variant, in a unified manner which gives the existence of asymptotically perfect two-batch adaptive codes for the channel $C$.Comment: 26 page

Orbit determination of highly elliptical Earth orbiters using VLBI and delta VLBI measurements

The feasibility of using very long baseline interferometric (VLBI) data acquired by the deep space network to navigate highly elliptical Earth orbiting satellites was shown. The navigation accuracy improvements achievable with VLBI and delta VLBI data types are determined for comparison with the Doppler capability. The sensitivity of the VLBI navigation accuracy to the baseline orientation relative to the orbit plane and the effects of major error sources such as gravitational harmonics and atmospheric are examined. It is found that VLBI measurements perform as well as strategies using conventional Doppler, while substantially reducing the required antenna support

Large Deviation Principles and Complete Equivalence and Nonequivalence Results for Pure and Mixed Ensembles

We consider a general class of statistical mechanical models of coherent structures in turbulence, which includes models of two-dimensional fluid motion, quasi-geostrophic flows, and dispersive waves. First, large deviation principles are proved for the canonical ensemble and the microcanonical ensemble. For each ensemble the set of equilibrium macrostates is defined as the set on which the corresponding rate function attains its minimum of 0. We then present complete equivalence and nonequivalence results at the level of equilibrium macrostates for the two ensembles.Comment: 57 page

The Large Deviation Principle for Coarse-Grained Processes

The large deviation principle is proved for a class of $L^2$-valued processes that arise from the coarse-graining of a random field. Coarse-grained processes of this kind form the basis of the analysis of local mean-field models in statistical mechanics by exploiting the long-range nature of the interaction function defining such models. In particular, the large deviation principle is used in a companion paper to derive the variational principles that characterize equilibrium macrostates in statistical models of two-dimensional and quasi-geostrophic turbulence. Such macrostates correspond to large-scale, long-lived flow structures, the description of which is the goal of the statistical equilibrium theory of turbulence. The large deviation bounds for the coarse-grained process under consideration are shown to hold with respect to the strong $L^2$ topology, while the associated rate function is proved to have compact level sets with respect to the weak topology. This compactness property is nevertheless sufficient to establish the existence of equilibrium macrostates for both the microcanonical and canonical ensembles.Comment: 19 page

Innovative Opportunities for Elementary and Middle School Teachers to Maintain Currency in Mathematics and Science: A Community College-School System Partnership

Since 1992 the Manassas Campus of Northern Virginia Community College – in response to requests from local school systems – has developed four innovative methods of assisting elementary, secondary and middle school teachers to enhance their content knowledge in science and mathematics, as well as integrate curriculum units for classroom presentation. These methods are based on the assumptions that: - While teachers at this level have fundamental understanding of math and science, if they wish to incorporate new concepts or technologies from these fields, graduate level content courses are generally beyond their background level. - Community College faculty can often provide a bridge that connects advanced content in science and mathematics with the applications that can be adapted to elementary/middle school curriculum. - Presenting content to a mixed audience of teachers from K-8 allows teachers to see how content can be adapted to grade levels above and below. - Content delivery methods must be interactive and must be responsive to the multiple demands on these teachers’ time. This requires flexibility in scheduling and course requirements

MOSFIRE Spectroscopy of Quiescent Galaxies at 1.5 < z < 2.5. II - Star Formation Histories and Galaxy Quenching

We investigate the stellar populations for a sample of 24 quiescent galaxies at 1.5 < z < 2.5 using deep rest-frame optical spectra obtained with Keck MOSFIRE. By fitting templates simultaneously to the spectroscopic and photometric data, and exploring a variety of star formation histories, we obtain robust measurements of median stellar ages and residual levels of star formation. After subtracting the stellar templates, the stacked spectrum reveals the Halpha and [NII] emission lines, providing an upper limit on the ongoing star formation rate of 0.9 +/- 0.1 Msun/yr. By combining the MOSFIRE data to our sample of Keck LRIS spectra at lower redshift, we analyze in a consistent manner the quiescent population at 1 < z < 2.5. We find a tight relation (with a scatter of 0.13 dex) between the stellar age and the rest-frame U-V and V-J colors, which can be used to estimate the age of quiescent galaxies given their colors. Applying this age--color relation to large, photometric samples, we are able to model the number density evolution for quiescent galaxies of various ages. We find evidence for two distinct quenching paths: a fast quenching that produces compact post-starburst systems, and a slow quenching of larger galaxies. Fast quenching accounts for about a fifth of the growth of the red sequence at z~1.4, and half at z~2.2. We conclude that fast quenching is triggered by dramatic events such as gas-rich mergers, while slow quenching is likely caused by a different physical mechanism.Comment: 28 pages, 15 figures, accepted in Ap