A covering problem over finite rings

AbstractGiven a finite commutative ring with identity A, define c(A,n,R) as the minimum cardinality of a subset H of An which satisfies the following property: every element in An differs in at most R coordinates from a multiple of an element in H. In this work, we determine the numbers c(Zm,n,0) for all integers m≥2 and n≥1. We also prove the relation c(S×A,n,1)≤c(S,n−1,0)c(A,n,1), where S=Fq or Zq and q is a prime power. As an application, an upper bound is obtained for c(Zpm,n,1), where p is a prime

Bounds on Covering Codes in RT spaces using Ordered Covering Arrays

In this work, constructions of ordered covering arrays are discussed and applied to obtain new upper bounds on covering codes in Rosenbloom-Tsfasman spaces (RT spaces), improving or extending some previous results.Comment: 12 page

Contributions on secretary problems, independent sets of rectangles and related problems

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2011.Cataloged from PDF version of thesis.Includes bibliographical references (p. 187-198).We study three problems arising from different areas of combinatorial optimization. We first study the matroid secretary problem, which is a generalization proposed by Babaioff, Immorlica and Kleinberg of the classical secretary problem. In this problem, the elements of a given matroid are revealed one by one. When an element is revealed, we learn information about its weight and decide to accept it or not, while keeping the accepted set independent in the matroid. The goal is to maximize the expected weight of our solution. We study different variants for this problem depending on how the elements are presented and on how the weights are assigned to the elements. Our main result is the first constant competitive algorithm for the random-assignment random-order model. In this model, a list of hidden nonnegative weights is randomly assigned to the elements of the matroid, which are later presented to us in uniform random order, independent of the assignment. The second problem studied is the jump number problem. Consider a linear extension L of a poset P. A jump is a pair of consecutive elements in L that are not comparable in P. Finding a linear extension minimizing the number of jumps is NP-hard even for chordal bipartite posets. For the class of posets having two directional orthogonal ray comparability graphs, we show that this problem is equivalent to finding a maximum independent set of a well-behaved family of rectangles. Using this, we devise combinatorial and LP-based algorithms for the jump number problem, extending the class of bipartite posets for which this problem is polynomially solvable and improving on the running time of existing algorithms for certain subclasses. The last problem studied is the one of finding nonempty minimizers of a symmetric submodular function over any family of sets closed under inclusion. We give an efficient O(ns)-time algorithm for this task, based on Queyranne's pendant pair technique for minimizing unconstrained symmetric submodular functions. We extend this algorithm to report all inclusion-wise nonempty minimal minimizers under hereditary constraints of slightly more general functions.by José Antonio Soto.Ph.D

Ordered Covering Arrays and Upper Bounds on Covering Codes in NRT spaces

This work shows several direct and recursive constructions of ordered covering arrays using projection, fusion, column augmentation, derivation, concatenation and cartesian product. Upper bounds on covering codes in NRT spaces are also obtained by improving a general upper bound. We explore the connection between ordered covering arrays and covering codes in NRT spaces, which generalize similar results for the Hamming metric. Combining the new upper bounds for covering codes in NRT spaces and ordered covering arrays, we improve upper bounds on covering codes in NRT spaces for larger alphabets. We give tables comparing the new upper bounds for covering codes to existing ones.Comment: 27 page

STRUCTURAL ANALYSIS, PARAGENESIS, AND GEOCHRONOLOGY OF THE ARROW URANIUM DEPOSIT, WESTERN ATHABASCA BASIN, SASKATCHEWAN, CANADA: IMPLICATIONS FOR THE DEVELOPMENT OF THE PATTERSON LAKE CORRIDOR

The Athabasca Basin in northern Saskatchewan hosts the world’s highest-grade uranium deposits, which are commonly spatially associated with structural zones that have undergone episodes of brittle reactivation, alteration, and polyphase fluid movement. The most recent significant discoveries of uranium mineralization in the Athabasca Basin have been associated with a series of geophysical conductors along a NE-SW-trending structural zone, termed the Patterson Lake corridor, in the southwestern portion of the Basin. The Arrow Deposit, which is along this trend and hosted exclusively in the basement rocks below the Athabasca Supergroup sandstones, has an indicated mineral resource of 179.5 Mlbs U3O8 at a grade of 6.88% U3O8, and is the largest undeveloped uranium resource in the Basin. The present study examines the relationships between the ductile framework and brittle reactivation of deep-seated structures, mineral paragenesis, and radiogenic and stable isotope analyses of uranium mineralization at the Arrow Deposit. Hand sample examination, structural analysis from oriented drill core, thin section microscopy, and electron microprobe analysis has been used to generate a detailed paragenesis of the Arrow Deposit, which was used to select mineralized samples for isotopic analysis that were categorized based on cross-cutting relationships, textures, and chemical composition. Paragenetic information was integrated with structural analysis utilizing over 18,000 measurements of foliation, fractures, veins, shears, mylonites, breccias, cataclasites, fault gouges, and plunge and trend of slickenstriae and ductile lineations. Through this study, the structural system at Arrow has been interpreted as a partitioned, sinistral strike-slip dominated, brittle-ductile fault system of complex Riedel-style geometry. The Arrow system developed along sub-vertical, NE-SW-trending heterogeneous high strain zones (named the A1 through A5 shears) along the limb of a regional-scale fold, and further evolved through episodic reactivation events creating small-scale brittle fault linkages oblique to, and connecting the main fault zone, allowing for migration of fluids, alteration of host rocks, and precipitation of uranium. Uranium mineralization at Arrow occurs as botryoidal, cubic, vein, semi-massive, and massive uraninite (UO2), as well as younger alteration phases including uranium-silicates (e.g. coffinite) and uranyl oxy-hydroxide minerals (e.g. uranophane). Regression of the concentrations of substituting elements including Fe, Si, and Ca give an average chemical age of initial uraninite crystallization of approximately 1,425 Ma. In-situ secondary ionization mass spectrometry (SIMS) U-Pb ages obtained in this study (~700, ~1,200, and ~1,300 Ma) are comparable with those obtained from the Shea Creek area and reveal numerous episodes of uranium mineralization, remobilization, and alteration associated with multi-stage deformation during the Proterozoic. The geochronological data on uranium mineralization and post-mineralization alteration and resetting events broadly correspond to major orogenic events that have affected the North American shield. The oldest uraninites (~1,300 to 1,425 Ma) are botryoidal, cubic, and semi-massive occurrences commonly replacing clay minerals and micas. Younger (~1,200 and ~700 Ma) uraninites occur as cubic crystals, semi-massive and massive lenses, and form the matrix of breccias. The youngest uraniferous minerals are the products of alteration and/or remobilization of uraninite through subsequent fluid-flow events. This study demonstrates that the U–Pb isotope systematics of uranium-rich minerals from the Arrow Deposit have been affected by paleo-fluid-flow events that were controlled by regional and global-scale tectonic events. The precision and high spatial resolution of the SIMS method allowed for measurement of δ18O values from ~10 μm spots on uraninites from the Arrow Deposit. The range of δ18O values (-34.5 to -15.2 ‰) are low, and comparable to those obtained from other unconformity-type deposits in the Basin such as Cigar Lake and Shea Creek. The low δ18O values indicate that the uraninite likely underwent recrystallization via interaction with late, relatively low temperature Athabasca fluids with δ18O values in the range of -20 to -16 ‰. The other discoveries along the Patterson Lake corridor (Triple R, Cannon, Bow, Harpoon, Spitfire) have not been studied in detail, and so this study of the structural context of the Arrow Deposit is important as it emphasizes that protracted reactivation of deep-seated NE-SW-trending structures and their subsidiaries was a fundamental control on uranium mineralization in the SW Athabasca Basin. Continued studies integrating mineral paragenesis, geochemistry, and structural geology with geochronological context will aid in understanding the evolution of uranium deposits within the recently established southwestern Athabasca Basin uranium camp

Publications of the Jet Propulsion Laboratory, July 1964 through June 1965

JPL publications bibliography with abstracts - reports on DSIF, Mariner program, Ranger project, Surveyor project, and other space programs, and space science

Covering Radius 1985-1994

We survey important developments in the theory of covering radius during the period 1985-1994. We present lower bounds, constructions and upper bounds, the linear and nonlinear cases, density and asymptotic results, normality, specific classes of codes, covering radius and dual distance, tables, and open problems