Structure in minor-closed classes of matroids

Model theory for universal classes with the amalgamation property: A study in the foundations of model theory and algebra

The matroid structure theory of Geelen, Gerards, and Whittle has led to a hypothesis that a highly connected member of a minor-closed class of matroids representable over a finite field is a mild modification (known as a perturbation) of a frame matroid, the dual of a frame matroid, or a matroid representable over a proper subfield. They introduced the notion of a template to describe these perturbations in more detail. In this dissertation, we determine these templates for various classes and use them to prove results about representability, extremal functions, and excluded minors. Chapter 1 gives a brief introduction to matroids and matroid structure theory. Chapters 2 and 3 analyze this hypothesis of Geelen, Gerards, and Whittle and propose some refined hypotheses. In Chapter 3, we define frame templates and discuss various notions of template equivalence. Chapter 4 gives some details on how templates relate to each other. We define a preorder on the set of frame templates over a finite field, and we determine the minimal nontrivial templates with respect to this preorder. We also study in significant depth a specific type of template that is pertinent to many applications. Chapters 5 and 6 apply the results of Chapters 3 and 4 to several subclasses of the binary matroids and the quaternary matroids---those matroids representable over the fields of two and four elements, respectively. Two of the classes we study in Chapter 5 are the even-cycle matroids and the even-cut matroids. Each of these classes has hundreds of excluded minors. We show that, for highly connected matroids, two or three excluded minors suffice. We also show that Seymour's 1-Flowing Conjecture holds for sufficiently highly connected matroids. In Chapter 6, we completely characterize the highly connected members of the class of golden-mean matroids and several other closely related classes of quaternary matroids. This leads to a determination of the extremal functions for these classes, verifying a conjecture of Archer for matroids of sufficiently large rank.

The purpose of this survey is to provide an introduction to the interesting role which set-theoretic methods have begun to play in the problem of determining the structure and classification of certain classes of abelian groups, in particular, the class of ω1-separable groups, i.e., groups such that every countable subset is contained in a Σ -cyclic summand.

The fascinating ability of myxobacteria to form multicellular spore filled fruiting bodies under starvation conditions was widely studied as a model for cooperative microbial behavior. The potential of a life cycle induced change of secondary metabolism, as a means to discover novel natural products, remains largely underexplored. We therefore studied the model organism Myxococcus xanthus DK1622 under submersed and solid cultivation conditions to find putatively life-cycle related compounds by applying statistical analysis on analytical data. Utilizing the advantageous characteristics of LC-MS, LC-MS/MS, and MALDI-MSI allowed the identification of compounds unambiguously associated with myxobacterial fruiting bodies. Our screening effort resulted in the purification and structure elucidation of a novel compound, the homospermidine lipid, from cultures that had undergone the fruiting process. A combination of molecular networking and targeted LC-MS/MS in conjunction with our in-house metabolomics database subsequently revealed alternative producers of the respective compound as well as a number of compounds belonging to the same structural class. Three further members of this compound class were isolated from an alternative producer and structurally elucidated by NMR. Insights into the biosynthesis of this novel compound class was gained by feeding of isotopically labeled substrates and in silico analysis.

In this work we relate the deterministic complexity of factoring polynomials (over finite fields) to certain combinatorial objects we call m-schemes. We extend the known conditional deterministic subexponential time polynomial factoring algorithm for finite fields to get an underlying m-scheme. We demonstrate how the properties of m-schemes relate to improvements in the deterministic complexity of factoring polynomials over finite fields assuming the generalized Riemann Hypothesis (GRH). In particular, we give the first deterministic polynomial time algorithm (assuming GRH) to find a nontrivial factor of a polynomial of prime degree n where (n-1) is a smooth number.

The subject matter of this paper belongs to the general theory of sets.The objectives here are the examinations of the structure of classes of sets which are closed under various finite and transfinite set operations and those transformations which preserve these operations.The major results of this paper are partial extensions of (a) the well known theory of cr-algebras (e.g.[4]), (b) the work of E. Marczewski [l] on isomorphisms, (c) the work of R. Sikorski [2] on tr-homomorphisms and (d) the work of A .Tarski [3 ] on fields of sets and set functions.Three classes of sets-the algebra, raz-algebra and total algebra-as well as four transformations-the w-homomorphism, weak isomorphism, raz-isomorphism, and total isomorphism-are studied.A class X of sets is an algebra if it is closed under finite set operations, i.e. addition of two sets and complementation.Analogically X is an w-algebra if X is closed under tw-operations, i.e. complementations and addition of not more than m sets, where m is an arbitrary fixed cardinal number.X is a total algebra if it is closed under all operations, i.e. complementation and arbitrary addition.Two classes of sets, X and £, are weakly isomorphic if X and £ considered as partially ordered by the relation of proper inclusion are similar.X and £ are zra-isomorphic if they have the same properties from the point of view of »«-operations; and, finally, X and £ are

A new theory of coherent structure in wall turbulence is presented. The theory is the first to predict packets of hairpin vortices and other structure in turbulence, and their dynamics, based on an analysis of the Navier–Stokes equations, under an assumption of a turbulent mean profile. The assumption of the turbulent mean acts as a restriction on the class of possible structures. It is shown that the coherent structure is a manifestation of essentially low-dimensional flow dynamics, arising from a critical-layer mechanism. Using the decomposition presented in McKeon & Sharma (J. Fluid Mech., vol. 658, 2010, pp. 336–382), complex coherent structure is recreated from minimal superpositions of response modes predicted by the analysis, which take the form of radially varying travelling waves. The leading modes effectively constitute a low-dimensional description of the turbulent flow, which is optimal in the sense of describing the resonant effects around the critical layer and which minimally predicts all types of structure. The approach is general for the full range of scales. By way of example, simple combinations of these modes are offered that predict hairpins and modulated hairpin packets. The example combinations are chosen to represent observed structure, consistent with the nonlinear triadic interaction for wavenumbers that is required for self-interaction of structures. The combination of the three leading response modes at streamwise wavenumbers $6, ~1, ~7$ and spanwise wavenumbers $\pm 6, ~\pm 6, ~\pm 12$, respectively, with phase velocity $2/ 3$, is understood to represent a turbulence ‘kernel’, which, it is proposed, constitutes a self-exciting process analogous to the near-wall cycle. Together, these interactions explain how the mode combinations may self-organize and self-sustain to produce experimentally observed structure. The phase interaction also leads to insight into skewness and correlation results known in the literature. It is also shown that the very large-scale motions act to organize hairpin-like structures such that they co-locate with areas of low streamwise momentum, by a mechanism of locally altering the shear profile. These energetic streamwise structures arise naturally from the resolvent analysis, rather than by a summation of hairpin packets. In addition, these packets are modulated through a ‘beat’ effect. The relationship between Taylor’s hypothesis and coherence is discussed, and both are shown to be the consequence of the localization of the response modes around the critical layer. A pleasing link is made to the classical laminar inviscid theory, whereby the essential mechanism underlying the hairpin vortex is captured by two obliquely interacting Kelvin–Stuart (cat’s eye) vortices. Evidence for the theory is presented based on comparison with observations of structure in turbulent flow reported in the experimental and numerical simulation literature and with exact solutions reported in the transitional literature.

Class consciousness occupied a central role in some classic theories in the social sciences, particularly in sociology. However, these concepts have been mostly overlooked by social psychology. The recent development of the psychology of inequality and social class opens new opportunities to revisit the concept of class consciousness. In this chapter, we provide a conceptual overview of class consciousness highlighting its contemporary relevance, and we discuss how inequality and class consciousness are connected. To explore this connection, we show some preliminary evidence at the social level, showing how class consciousness is lower among rich and egalitarian countries, and at the individual level, showing that for members of the working class, higher levels of class consciousness are connected to lower life satisfaction and worse health-related indicators. We propose a research agenda for social psychology that reconsiders the concept of class consciousness as it might be a fruitful avenue to further explain how certain psychological processes impact the formation and maintenance of social hierarchies.

The asymmetric alkylation of isoflavanones (3-aryl-chroman-4-ones) and protected 3-phenyl-2,3-dihydroquinolin-4(1H)-ones catalyzed by a novel cinchonidine-derived phase transfer catalyst E is reported. This functionalization occurs at the unactivated C3 methine to afford novel products that can easily be functionalized to generate more complex fused ring systems. The process accommodates a variety of isoflavanones and activated electrophiles and installs a stereogenic quaternary center in high yield and with good-to-excellent selectivity. Isoflavanones are a privileged class of natural products with a broad spectrum of biological activities including insecticidal, antimicrobial, antibacterial, estrogenic, antitumor, and anti-HIV activity. (1) Isoflavanones are also precursors for more complex natural products such as pterocarpans and rotenones. (1) Given their therapeutic promise, selective strategies to access new classes of isoflavanones and related structures has high value. (2) The functionalization of the C3 position could promote beneficial interactions with biological targets of interest. Specifically, an alkylation at C3 can rapidly access new members of the general class of biologically active homoisoflavanones. (3).

Recent analyses of classical integrable structures in quantum integrable models solved by various versions of the Bethe ansatz are reviewed. Similarities between elements of quantum and classical theories of integrable systems are discussed. Some key ideas in quantum theory, now standard in the quantum inverse scattering method, are identified with typical constructions in classical soliton theory. Functional relations for quantum transfer matrices become the classical Hirota bilinear difference equation; solving this classical equation gives all the basic results for the spectral properties of quantum systems. Vice versa, typical Bethe ansatz formulas under certain boundary conditions yield solutions of this classical equation. The Baxter T-Q relation and its generalizations arise as auxiliary linear problems for the Hirota equation.

On a theorem of Deshouillers and Freiman

Three basic themes played indispensable roles in Reinhold Niebuhr’s engagements with economic issues: (1) the foundational importance of a theological anthropology for his critical assessment of complex social, economic, and political issues; (2) his focus on two clashing theories of economic order: Adam’s Smith’s doctrine of laissez-faire capitalism and Karl Marx’s revolutionary vision for the collective ownership of property within a centralized state system; and (3) his endorsement of a realistic yet viable alternative to these two classic theories, one that involved pragmatic and incremental steps toward fostering economic justice through the public regulation and oversight of a free-market economy. These pragmatic measures included allocating government resources for vitally important public goods and services at federal, state, and local levels. Though Niebuhr was initially drawn to some form of democratic socialism, he embraced this incremental strategy within the context of President Franklin Delano Roosevelt’s implementation of his New Deal agenda. My analysis of Niebuhr’s contributions concludes with attention to new challenges confronting contemporary quests for economic justice in a post–New Deal era, one marked Soundings, Vol. 95, No. 4, 2012 Copyright © 2012 The Pennsylvania state University, University Park, Pa di sc us sio n diussion

The Möbius matroids

A recent (1971) result due to Wittrick-Williams relating the number of modes of a flexible structure to the zeros of a matrix function is generalized and an independent proof given. It is applied to Timoshenko models of a class of interconnected beamlike lattice structures and an interpretation of the zeros given as pinned modes of the structure. N determining modes of continuum models of flexible structures, particularly when the mode density is high, it is often necessary to make sure that no modes have been missed. Here, a result due to Wittrick and Williams1 (which is at once an algorithm and often referred to as such) may be used to determine the total number of modes in any given interval of frequency. In the words of the authors, it is an algorithm in the sense that it enables one to calculate how many natural frequencies lie below any chosen frequency, without determining them and hence to converge to any required natural frequency to any specified accuracy. References 2-5 are illustrative of the actual use of this algorithm for determining modes of beamlike structures. Our concern in this paper is in the formula itself rather than its use as a computational tool for finding modes of structures. Indeed, it can be regarded as a generalization of Foster's theorem for onedimensional loss-less impedances, familiar in classical electrical circuit theory; see Ref. 6. Also, the proof given by the authors in Ref. 6 leans heavily on a classical theorem of Lord Rayleigh7 and many arguments peculiar to mechanics of structures. Here we formulate the problem in the generality of an abstract wave equation in a Hilbert space and at the same time present an entirely independent mathematical proof. We show, in addition, how it can be applied to Timoshenko models of interconnected beamlike lattice structures.

The matrix theory of framed structures to applicable in principle to frames of arbitrary finite extent, but difficulties of computation preclude its use for analyzing structures with many degrees of freedom. A class of structures is defined that includes some well-known large-span domes. It is shown that for these structures a highly significant decoupling of the stiffness equations is achieved by the use of finite Fourier series with vector coefficients. For certain kinds of loading the theory is extended to treat nonlinear elastic behavior, and computable criteria of elastic buckling are established.