Symbolic dynamics and the stable algebra of matrices

We give an introduction to the "stable algebra of matrices" as related to certain problems in symbolic dynamics. We consider this stable algebra (especially, shift equivalence and strong shift equivalence) for matrices over general rings as well as various specific rings. This algebra is of independent interest and can be followed with little attention to the symbolic dynamics. We include strong connectionsto algebraic K-theory and the inverse spectral problem for nonnegative matrices. We also review key features of the automorphism group of a shift of finite type, and the work of Kim, Roush and Wagoner giving counterexamples to Williams' Shift Equivalence Conjecture.Comment: 121 pages. Main changes from version 1: Author and subject indices were added. Various citations were added, with commentary. Bibliography items are now listed with internal references (i.e., pages of the paper on which they are cited

The many faces of positivity to approximate structured optimization problems

The PhD dissertation proposes tractable linear and semidefinite relaxations for optimization problems that are hard to solve and approximate, such as polynomial or copositive problems. To do this, we exploit the structure and inherent symmetry of these problems. The thesis consists of five essays devoted to distinct problems. First, we consider the kissing number problem. The kissing number is the maximum number of non-overlapping unit spheres that can simultaneously touch another unit sphere, in n-dimensional space. In chapter two we construct a new hierarchy of upper bounds on the kissing number. To implement the hierarchy, in chapter three we propose two generalizations of Schoenberg's theorem on positive definite kernels. In the fourth chapter, we derive new certificates of non-negativity of polynomials on generic sets defined by polynomial equalities and inequalities. These certificates are based on copositive polynomials and allow obtaining new upper and lower bounds for polynomial optimization problems. In chapter five, for any given graph we look for the largest k-colorable subgraph; that is, the induced subgraph that can be colored in k colors such that no two adjacent vertices have the same color. We obtain several new semidefinite programming relaxations to this problem. In the final sixth chapter, we consider the problem of allocating tasks to unrelated parallel selfish machines to minimize the time to complete all the tasks. For this problem, we suggest new upper and lower bounds on the best approximation ratio of a class of truthful task allocation algorithms

Matching the heterotic string on orbifolds and their resolutions

We study the symmetry breaking mechanism under which a 6d orbifold compactification of the 10d heterotic string turns into a smooth Calabi--Yau compactification. This process is naturally required to preserve N=1 supersymmetry in 4d. The cause is the existence of a Fayet--Iliopoulos D--term generated by the anomalous U(1) gauge--symmetry on the orbifold theory. An orbifold is constructed by modding out a symmetry from a toroidal 6d lattice, resulting in an almost everywhere flat variety with the exception of fixed sets under the symmetry action. Those sets are fixed points and fixed tori, and the states localized at them are the so called twisted states. D--flatness leads to a vacuum with non--zero expectation values of twisted scalars. Those scalars play the role of blow--up modes: their vevs deform the local geometry and smooth out the singularities. We study Calabi--Yau manifolds obtained by blowing--up (resolving) the singularities using toric geometry. We analyze the massless spectrum and the anomaly cancellation on the deformed orbifold and in the resolution obtaining a perfect map. On the orbifold we can compute the full particle spectrum and the interactions using the CFT world--sheet description of the heterotic string. To compactify on the resolution, as the metric is unknown, we have to start with the 10d N=1 supergravity and super Yang--Mills effective theory and perform dimensional reduction. In the thesis we first review the 10d heterotic string, the heterotic supergravity, the orbifold and Calabi--Yau compactifications and the toric geometry techniques required for resolving orbifolds. We perform an study of potential orbifold 4d discrete symmetries in factorizable and non--factorizable orbifolds arising from the torus lattice automorphisms. We then come to our focus, which is the orbifold--resolution transition in two compact orbifold models with the Minimal Supersymmetric Standard Model Physics. First, we study the T6/Z7 orbifold and its resolution. This orbifold contains all the ingredients of realistic models. It is simpler because it is prime and has therefore only fixed points and no orbifold brother models. We find the field redefinitions that identify the orbifold and blow--up massless spectrum. A local index theorem is crucial in this process. We study then the Green--Schwarz anomaly cancellation mechanism after dimensional reduction on the resolution and from the massless spectrum and the field redefinitions on the orbifold. We find that both results perfectly agree. This determines the blow--up modes as non--universal axions on the Calabi--Yau manifold. After these encouraging results we study now a more involved case, this is the T6/Z6II orbifold and its resolution. As the orbifold is non--prime there are fixed tori. This makes the identification of the blow--up modes and the search of the field redefinitions more difficult. We overcome that difficulty exploring in a Mini--landscape of phenomenologically promising orbifold models to select a suitable one, and on it we are able to identify the blow--up modes. We find perfect agreement of the massless spectrum, including the orbifold generated mass terms. To culminate, we study in detail the anomaly cancellation mechanism. We find here as in the Z7 case that the blow--up modes play the role of the resolution non--universal axions. Our work interplays between string theory consistency, which expresses itself through 10d anomaly cancellation, and the physics and the geometry of supersymmetric space--time vacua

Nested algebraic Bethe ansaetze for orthogonal and symplectic spin chains

In this thesis the nested algebraic Bethe ansatz technique is applied to various orthogonal and symplectic closed and open spin chain models. Each spin chain considered is regarded as a representation of an underlying quantum group algebra, and expressions for eigenvectors of transfer matrices associated to these models are constructed using the algebra relations, reducing the problem to a set of Bethe equations. The specific models considered are the Ol'shanskii twisted Yangian spin chain, where gl_n bulk symmetry is broken to orthogonal or symplectic symmetry; the MacKay twisted Yangian spin chain, an open spin chain with bulk orthogonal or symplectic symmetry and various boundary types; and the q-deformed orthogonal or symplectic closed spin chain. For the first and third cases, a closed 'trace formula' expression for the eigenvector is also provided

Conformal Field Theory Between Supersymmetry and Indecomposable Structures

This thesis considers conformal field theory in its supersymmetric extension as well as in its relaxation to logarithmic conformal field theory. Compactification of superstring theory on four-dimensional complex manifolds obeying the Calabi-Yau conditions yields the moduli space of N=(4,4) superconformal field theories with central charge c=6 which consists of two continuously connected subspaces. This thesis is concerned with the subspace of K3 compactifications which is not well known yet. In particular, we inspect the intersection point of the Z_2 and Z_4 orbifold subvarieties within the K3 moduli space, explicitly identify the two corresponding points on the subvarieties geometrically, and give an explicit isomorphism of the three conformal field theory models located at that point, a specific Z_2 and a Z_4 orbifold model as well as the Gepner model (2)^4. We also prove the orthogonality of the two subvarieties at the intersection point. This is the starting point for the programme to investigate generic points in K3 moduli space. We use the coordinate identification at the intersection point in order to relate the coordinates of both subvarieties and to explicitly calculate a geometric geodesic between the two subvarieties as well as its generator. A generic point in K3 moduli space can be reached by such a geodesic originating at a known model. We also present advances on the conformal field theoretic side of deformations along such a geodesic using conformal deformation theory. Since a consistent regularisation of the appearing deformation integrals has not been achieved yet, the completion of this programme is still an open problem. Moreover, we regard a relaxation of conformal field theory to logarithmic conformal field theory. The latter allows the indecomposable action of the L_0 Virasoro mode within a representation of the conformal symmetry. In particular, we study general augmented c_{p,q} minimal models which generalise the well-known (augmented) c_{p,1} model series. We calculate logarithmic nullvectors in both types of models. But most importantly, we investigate the low lying Virasoro representation content and fusion algebra of two general augmented c_{p,q} models, the augmented c_{2,3} = 0 model as well as the augmented Yang-Lee model at c_{2,5} = -22/5. These exhibit a much richer structure as the c_{p,1} models with indecomposable representations up to rank 3. We elaborate several of these new rank 3 representations in great detail and uncover astonishing features. Furthermore, we argue that irreducible representations corresponding to the Kac table domain of the proper minimal models cannot be included into the theory. In particular, the true vacuum representation is rather given by a rank 1 indecomposable but not irreducible subrepresentation of a rank 2 representation. We generalise these generic examples to give the representation content and the fusion algebra of general augmented c_{p,q} models as a conjecture. Finally, we open a new connection between logarithmic conformal field theory and quantum spin chains by relating some representations of the augmented c_{2,3} = 0 model to the representation content of a c=0 model which describes an XXZ quantum spin chain

1969-1970 Xavier University College of Business Administration, College of Arts and Sciences, Evening College, Graduate School

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