Class Notes for Math 901/902: Abstract Algebra, Instructor Tom Marley

Topics include: Free groups and presentations; Automorphism groups; Semidirect products; Classification of groups of small order; Normal series: composition, derived, and solvable series; Algebraic field extensions, splitting fields, algebraic closures; Separable algebraic extensions, the Primitive Element Theorem; Inseparability, purely inseparable extensions; Finite fields; Cyclotomic field extensions; Galois theory; Norm and trace maps of an algebraic field extension; Solvability by radicals, Galois' theorem; Transcendence degree; Rings and modules: Examples and basic properties; Exact sequences, split short exact sequences; Free modules, projective modules; Localization of (commutative) rings and modules; The prime spectrum of a ring; Nakayama's lemma; Basic category theory; The Hom functors; Tensor products, adjointness; Left/right Noetherian and Artinian modules; Composition series, the Jordan-Holder Theorem; Semisimple rings; The Artin-Wedderburn Theorem; The Density Theorem; The Jacobson radical; Artinian rings; von Neumann regular rings; Wedderburn's theorem on finite division rings; Group representations, character theory; Integral ring extensions; Burnside's paqb Theorem; Injective modules.

Dynamic Traffic Grooming Algorithms for Reconfigurable SONET Over WDM Networks

The emergence of wavelength-division multiplexing (WDM) technology provides the capability for increasing the bandwidth of synchronous optical network (SONET) rings by grooming low-speed traffic streams onto different high-speed wavelength channels. Since the cost of SONET add–drop multiplexers (SADM) at each node dominates the total cost of these networks, how to assign the wavelength, groom the traffic, and bypass the traffic through the intermediate nodes has received a lot of attention from researchers recently. Moreover, the traffic pattern of the optical network changes from time to time. How to develop dynamic reconfiguration algorithms for traffic grooming is an important issue. In this paper, two cases (best fit and full fit) for handling reconfigurable SONET over WDM networks are proposed. For each approach, an integer linear programming model and heuristic algorithms (TS-1 and TS-2, based on the tabu search method) are given. The results demonstrate that the TS-1 algorithm can yield better solutions but has a greater running time than the greedy algorithm for the best fit case. For the full fit case, the tabu search heuristic yields competitive results compared with an earlier simulated annealing based method and it is more stable for the dynamic case.

Spatial methods for plot-based sampling of wildlife populations

Classical sampling methods can be used to estimate the mean of a finite or infinite population. Block kriging also estimates the mean, but of an infinite population in a continuous spatial domain. In this paper, I consider a finite population version of block kriging (FPBK) for plot-based sampling. The data are assumed to come from a spatial stochastic process. Minimizing mean-squared-prediction errors yields best linear unbiased predictions that are a finite population version of block kriging. FPBK has versions comparable to simple random sampling and stratified sampling, and includes the general linear model. This method has been tested for several years for moose surveys in Alaska, and an example is given where results are compared to stratified random sampling. In general, assuming a spatial model gives three main advantages over classical sampling: (1) FPBK is usually more precise than simple or stratified random sampling, (2) FPBK allows small area estimation, and (3) FPBK allows nonrandom sampling designs.

Vanishing of Ext and Tor over complete intersections

Let (R,m) be a local complete intersection, that is, a local ring whose m-adic completion is the quotient of a complete regular local ring by a regular sequence. Let M and N be finitely generated R-modules. This dissertation concerns the vanishing of Tor(M, N) and Ext(M, N). In this context, M satisfies Serre's condition (S_{n}) if and only if M is an nth syzygy. The complexity of M is the least nonnegative integer r such that the nth Betti number of M is bounded by a polynomial of degree r-1 for all sufficiently large n. We use this notion of Serre's condition and complexity to study the vanishing of Tor_{i}(M, N). In particular, building on results of C. Huneke, D. Jorgensen and R. Wiegand [32], and H. Dao [21], we obtain new results showing that good depth properties on the R-modules M, N and MtensorN force the vanishing of Tor_{i}(M, N) for all i>0. We give examples showing that our results are sharp. We also show that if R is a one-dimensional domain and M and MtensorHom(M,R) are torsion-free, then M is free if and only if M has complexity at most one. If R is a hypersurface and Ext^{i}(M, N) has finite length for all i>>0, then the Herbrand difference [18] is defined as length(Ext^{2n}(M, N))-(Ext^{2n-1}(M, N)) for some (equivalently, every) sufficiently large integer n. In joint work with Hailong Dao, we generalize and study the Herbrand difference. Using the Grothendieck group of finitely generated R-modules, we also examined the number of consecutive vanishing of Ext^{i}(M, N) needed to ensure that Ext^{i}(M, N) = 0 for all i>>0. Our results recover and improve on most of the known bounds in the literature, especially when R has dimension two.

Applications of Linear Programming to Coding Theory

Maximum-likelihood decoding is often the optimal decoding rule one can use, but it is very costly to implement in a general setting. Much effort has therefore been dedicated to find efficient decoding algorithms that either achieve or approximate the error-correcting performance of the maximum-likelihood decoder. This dissertation examines two approaches to this problem. In 2003 Feldman and his collaborators defined the linear programming decoder, which operates by solving a linear programming relaxation of the maximum-likelihood decoding problem. As with many modern decoding algorithms, is possible for the linear programming decoder to output vectors that do not correspond to codewords; such vectors are known as pseudocodewords. In this work, we completely classify the set of linear programming pseudocodewords for the family of cycle codes. For the case of the binary symmetric channel, another approximation of maximum-likelihood decoding was introduced by Omura in 1972. This decoder employs an iterative algorithm whose behavior closely mimics that of the simplex algorithm. We generalize Omura's decoder to operate on any binary-input memoryless channel, thus obtaining a soft-decision decoding algorithm. Further, we prove that the probability of the generalized algorithm returning the maximum-likelihood codeword approaches 1 as the number of iterations goes to infinity.

Avian Use of Harvested Crop Fields in North Dakota During Spring Migration

Broad-spectrum herbicide applications and improved harvesting efficiency of crops have reduced the availability of weed seeds and waste grains for game and nongame wildlife. Over the last decade, corn and soybean plantings have steadily increased in the Prairie Pothole Region (PPR) of North Dakota, while sunflower plantings have declined. The PPR is an important corridor for migratory birds, and changes in food availabilities at stopover habitats may affect how food resources are used. In early spring 2003 and 2004, we compared bird use of harvested fields of sunflower, soybeans, small grains, and corn in the PPR of North Dakota. Across both years and all crop types, we observed 20,400 birds comprising 29 species. Flocks of Lapland Longspurs (Calcarius lapponicus) and Horned Larks (Eremophila alpestris) and flocks of Red-winged Blackbirds (Agelaius phoeniceus) made up 60% and 15%, respectively, of the bird counts. We found that species richness and bird densities were higher in harvested sunflower fields and cornfields than in harvested small-grain and soybean fields, with soybean fields harboring the fewest species and lowest bird density. Blackbird densities tended to be lower in fields tilled after fall harvest than in fields not tilled. These results suggest that some granivorous bird populations in the Northern Great Plains could be positively affected by planting of row crops with postharvest vertical structure (e.g., sunflower, corn) and use of no-till land management practices.

Parental Ethnotheories of Child Development: Looking Beyond Independence and Individualism in American Belief Systems

Over the past several decades, the topic of child development in a cultural context has received a great deal of theoretical and empirical investigation. Investigators from the fields of indigenous and cultural psychology have argued that childhood is socially and historically constructed, rather than a universal process with a standard sequence of developmental stages or descriptions. As a result, many psychologists have become doubtful that any stage theory of cognitive or socialemotional development can be found to be valid for all times and places. In placing more theoretical emphasis on contextual processes, they define culture as a complex system of common symbolic action patterns (or scripts) built up through everyday human social interaction by means of which individuals create common meanings and in terms of which they organize experience. Researchers understand culture to be organized and coherent, but not homogenous or static, and realize that the complex dynamic system of culture constantly undergoes transformation as participants (adults and children) negotiate and re-negotiate meanings through social interaction. These negotiations and transactions give rise to unceasing heterogeneity and variability in how different individuals and groups of individuals interpret values and meanings. However, while many psychologists—both inside and outside the fields of indigenous and cultural psychology–are now willing to give up the idea of a universal path of child development and a universal story of parenting, they have not necessarily foreclosed on the possibility of discovering and describing some universal processes that underlie socialization and development-in-context. The roots of such universalities would lie in the biological aspects of child development, in the evolutionary processes of adaptation, and in the unique symbolic and problem-solving capacities of the human organism as a culture-bearing species. For instance, according to functionalist psychological anthropologists, shared (cultural) processes surround the developing child and promote in the long view the survival of families and groups if they are to demonstrate continuity in the face of ecological change and resource competition, (e.g. Edwards & Whiting, 2004; Gallimore, Goldenberg, & Weisner, 1993; LeVine, Dixon, LeVine, Richman, Leiderman, Keefer, & Brazelton, 1994; LeVine, Miller, & West, 1988; Weisner, 1996, 2002; Whiting & Edwards, 1988; Whiting & Whiting, 1980). As LeVine and colleagues (1994) state: A population tends to share an environment, symbol systems for encoding it, and organizations and codes of conduct for adapting to it (emphasis added). It is through the enactment of these population-specific codes of conduct in locally organized practices that human adaptation occurs. Human adaptation, in other words, is largely attributable to the operation of specific social organizations (e.g. families, communities, empires) following culturally prescribed scripts (normative models) in subsistence, reproduction, and other domains [communication and social regulation]. (p. 12) It follows, then, that in seeking to understand child development in a cultural context, psychologists need to support collaborative and interdisciplinary developmental science that crosses international borders. Such research can advance cross-cultural psychology, cultural psychology, and indigenous psychology, understood as three sub-disciplines composed of scientists who frequently communicate and debate with one another and mutually inform one another’s research programs. For example, to turn to parental belief systems, the particular topic of this chapter, it is clear that collaborative international studies are needed to support the goal of crosscultural psychologists for findings that go beyond simply describing cultural differences in parental beliefs. Comparative researchers need to shed light on whether parental beliefs are (or are not) systematically related to differences in child outcomes; and they need meta-analyses and reviews to explore between- and within-culture variations in parental beliefs, with a focus on issues of social change (Saraswathi, 2000). Likewise, collaborative research programs can foster the goals of indigenous psychology and cultural psychology and lay out valid descriptions of individual development in their particular cultural contexts and the processes, principles, and critical concepts needed for defining, analyzing, and predicting outcomes of child development-in-context. The project described in this chapter is based on an approach that integrates elements of comparative methodology to serve the aim of describing particular scenarios of child development in unique contexts. The research team of cultural insiders and outsiders allows for a look at American belief systems based on a dialogue of multiple perspectives.