Consider (1) -yn1+ q1y1 = (λr11 + µr12)y1 on [0, 1] y’1(0) = cot α1 and = y’1(1) = a1λ + b1 y1(0) y1(1) c1λ+d1 (2) - yn2 + q2y2 = (λr21 + µr22)y2 on [0, 1] y’2(0) = cot α2 and = y’2(1) = a2µ + b2 y2(0) y2(1) c2µ + d2 subject to certain definiteness conditions; where qi and rij are continuous real valued functions on [0, 1], the angle αi is in [0, π) and ai, bi, ci, di are real numbers with δi = aidi − bici > 0 and ci = 0 for I, j = 1,2. Under the Uniform Left Definite condition we have proved an asymptotic theorem and an oscillation theorem. Analysis of (1) and (2) subject to the Uniform Ellipticity condition focus on the location of eigenvalues, perturbation theory and the local analysis of eigenvalues. We also gave a bound for the number of nonreal eigenvalues. We also have studied the system T1(x1) = (λA11 + µA12)(x1) and T2(x2) = (λA21 + µA22)(x2) where Aij (j =1, 2) and Ti are linear operators acting on finite dimensional Hilbert spaces Hi (i = 1, 2). For a pair of commutative operators Γ = (Γ0, Γ1) constructed from Aij and Ti on the Hilbert space tensor product H1 ⊗ H2, we can associate a natural Koszul complex namely Dºr-(λ,μ) D1 r-(λ,μ) 0 H H ø H H 0 We have constructed a basis for the Koszul quotient space N(D1Г−(λ,µ))/R(D0Г−( λ,µ)) in terms of the root basis of (Г0, Г1). (For equations pl refer the PDF file)

x, 68 p. : ill. (some col.) / Motivated by the Adams spectral sequence for computing stable homotopy groups, Priddy defined a class of algebras called Koszul algebras with nice homological properties. Many important algebras arising naturally in mathematics are Koszul, and the Koszul property is often tied to important structure in the settings which produced the algebras. However, the strong defining conditions for a Koszul algebra imply that such algebras must be quadratic. A very natural generalization of Koszul algebras called K 2 algebras was recently introduced by Cassidy and Shelton. Unlike other generalizations of the Koszul property, the class of K 2 algebras is closed under many standard operations in ring theory. The class of K 2 algebras includes Artin-Schelter regular algebras of global dimension 4 on three linear generators as well as graded complete intersections. Our work comprises two distinct projects. Each project was motivated by an aspect of the theory of Koszul algebras which we regard as sufficiently powerful or fundamental to warrant an interpretation for K 2 algebras. A very useful theorem due to Backelin and Fröberg states that if A is a Koszul algebra and I is a quadratic ideal of A which is Koszul as a left A -module, then the factor algebra A/I is a Koszul algebra. We prove that if A is Koszul algebra and A I is a K 2 module, then A/I is a K 2 algebra provided A/I acts trivially on Ext A ( A/I,k ). As an application of our theorem, we show that the class of sequentially Cohen-Macaulay Stanley-Reisner rings are K 2 algebras and we give examples that suggest the class of K 2 Stanley-Reisner rings is actually much larger. Another important recent development in ring theory is the use of A ∞ -algebras. One can characterize Koszul algebras as those graded algebras whose Yoneda algebra admits only trivial A ∞ -structure. We show that, in contrast to the situation for Koszul algebras, vanishing of higher A ∞ -structure on the Yoneda algebra of a K 2 algebra need not be determined in any obvious way by the degrees of defining relations. We also demonstrate that obvious patterns of vanishing among higher multiplications cannot detect the K 2 property. This dissertation includes previously unpublished co-authored material. / Committee in charge: Dr. Brad Shelton, Chair; Dr. Victor Ostrik, Member; Dr. Nicholas Proudfoot, Member; Dr. Arkady Vaintrob, Member; Dr. David Boush, Outside Member

Made available in DSpace on 2015-05-15T11:46:19Z (GMT). No. of bitstreams: 1 arquivototal.pdf: 1046176 bytes, checksum: b07998b12889ca7bedcf163b7e83cd18 (MD5) Previous issue date: 2010-12-20 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES / O complexo de Koszul é uma ferramenta de vital importância na Álgebra Comutativa. Ele nos permitirá definir alguns invariantes que nos dão informações refinadas acerca de um determinado módulo. Entre eles podemos ressaltar a profundidade e a multiplicidade de tal módulo em relação à um ideal. A primeira mede o comprimento da maior M-sequência formada por elementos do anel e a segunda nos dà informações assintóticas acerca do comprimento de módulos quocientes

This thesis consists of four papers on the topics of free loop spaces, Koszul duality and A∞-algebras.  In Paper I we consider a definition of differential operators for noncommutative algebras. This definition is inspired by the connections between differential operators of commutative algebras, L∞-algebras and BV-algebras. We show that the definition is reasonable by establishing results that are analoguous to results in the commutative case. As a by-product of this definition we also obtain definitions for noncommutative versions of Gerstenhaber and BV-algebras.  In Paper II we calculate the free loop space homology of (n-1)-connected manifolds of dimension of at least 3n-2. The Chas-Sullivan loop product and the loop bracket are calculated. Over a field of characteristic zero the BV-operator is determined as well. Explicit expressions for the Betti numbers are also established, showing that they grow exponentially.  In Paper III we restrict our coefficients to a field of characteristic 2. We study the Dyer-Lashof operations that exist on free loop space homology in this case. Explicit calculations are carried out for manifolds that are connected sums of products of spheres.  In Paper IV we extend the Koszul duality methods used in Paper II by incorporating A∞-algebras and A∞-coalgebras. This extension of Koszul duality enables us to compute free loop space homology of manifolds that are not necessarily formal and coformal. As an example we carry out the computations for a non-formal simply connected 7-manifold. / Denna avhandling består av fyra artiklar inom ämnena fria öglerum, Koszuldualitet och A∞-algebror. I Artikel I behandlar vi en definition av differentialoperatorer för ickekommutativa algebror. Denna definition är inspirerad av kopplingar mellan differentialoperatorer för kommutativa algebror, L∞-algebror och BV-algebror. Vi visar att definitionen är rimlig genom att etablera resultat som är analoga med resultat i det kommutativa fallet. Som en biprodukt får vi också definitioner för ickekommutativa varianter av Gerstenhaber och BV-algebror. I Artikel II beräknar vi den fria öglerumshomologin av (n-1)-sammanhängande mångfalder av dimension minst 3n-2. Chas-Sullivans ögleprodukt och öglehake beräknas. Över en kropp av karakteristik noll beräknas även BV-operatorn. Explicita uttryck för Bettitalen fastställs också, vilka visar att de växer exponentiellt. I Artikel III begränsar vi koefficienterna till en kropp av karakteristik 2. Vi studerar Dyer- Lashofoperationer som existerar på den fria öglerumshomologin i detta fall. Explicita beräkningar görs för mångfalder som är sammanhängande summor av produkter av sfärer. I Artikel IV utvidgar vi Koszuldualitetmetoden som används i Artikel II genom att inkorporera A∞-algebror och A∞-koalgebror. Denna utvidgning av Koszuldualitet gör det möjligt att beräkna fri öglerumshomologi för mångfalder som inte nödvändigtvis är formella och koformella. Som ett exempel utför vi beräkningar för en ickeformell enkelt sammanhängande 7-mångfald. / <p>At the time of the doctoral defense, the following papers were unpublished and had a status as follows: Paper 3: Manuscript. Paper 4: Manuscript.</p>

Graded ideals over the polynomial ring are studied deeply with a huge of methods and results. Over the exterior algebra, there are not much known about the structures of minimal graded resolutions, Gröbner fans of graded ideals or the Koszul property of algebras defined by graded ideals. We study componentwise linearity, linear resolutions of graded ideals as well as universally, initially and strongly Koszul properties of graded algebras defined by a graded ideals over the exterior algebra. After that, we apply our results to Orlik-Solomon ideals of hyperplane arrangements and show in which way the exterior algebra is useful in the study of related combinatorial objects.

Toric face rings are a generalization of Stanley-Reisner rings and affine monoid rings. New problems and results are obtained by a systematic study of toric face rings, shedding new lights to the understanding of Stanley-Reisner rings and affine monoid rings. We study algebra retracts of Stanley-Reisner rings, in particular, classify all the $\mathbb{Z}$-graded algebra retracts. We consider the Koszul property of toric face rings via Betti numbers and properties of the defining ideal. The last chapter is devoted to local cohomology of seminormal toric face rings and applications to singularities of toric face rings in positive characteristics.

This thesis studies derived equivalences between total spaces of vector bundles and dg-quivers. A dg-quiver is a graded quiver whose path algebra is a dg-algebra. A quiver with superpotential is a dg-quiver whose differential is determined by a "function" Φ. It is known that the bounded derived category of representations of quivers with superpotential with finite dimensional cohomology is a Calabi- Yau triangulated category. Hence quivers with superpotential can be viewed as noncommutative Calabi- Yau manifolds. One might then ask if there are derived equivalences between Calabi-Yau manifolds and quivers with superpotential. In this thesis, we answer this question and, generalizing Bridgeland [15], give a recipe on how to construct such derived equivalences.

Les algèbres Artin-Schelter régulières sont des analogues non-commutatifs d'algèbres de polynomes. En dimension globale 3, ces algèbres graduées sont homogènes et ont des relations de degré 2 ou 3. Dans cette thèse, nous nous intéressons à certaines algèbres Artin-Schelter régulières de dimension globale 3, à relations cubiques. Nous commencons par calculer l'homologie de Hochschild des algèbres Artin-Schelter régulières de dimension globale 3, cubiques de type A à coefficients génériques. Soit $A$ une telle algèbre. Nous suivons la méthode employée par M. Van den Bergh (K-Theory 8 (1994) 213-230) dans le cas quadratique, en considérant cette algèbre comme déformation d'une algèbre de polynomes, avec crochet de Poisson remarquable. Nous calculons alors l'homologie de Poisson et nous montrons que la suite spectrale de Brylinski associée dégénère. Pour cela, nous utilisons le fait que cette algèbre est de Koszul au sens généralisé défini par R. Berger (J. Algebra 239 (2001) 705-734) et nous donnons un nouveau quasi-isomorphisme entre la résolution de Koszul de $A$ par des $A$-$A$-bimodules et la bar-résolution de $A$. Nous déduisons la cohomologie de de Rham, l'homologie cyclique et l'homologie cyclique périodique de l'homologie de Hochschild de $A$, en utilisant des résultats classiques. La propriété de Koszul généralisée nous permet d'écrire un quasi-isomorphisme explicite entre le complexe qui calcule la cohomologie de Hochschild de $A$ et le complexe qui calcule l'homologie de Hochschild de $A$, obtenant ainsi une dualité de Poincaré. Nous déduisons alors la cohomologie de Hochschild de $A$ de l'homologie de Hochschild de $A$. Nous déterminons le centre de $A$, ce qui n'était pas connu. Nous terminons par divers compléments. En particulier, nous explicitons une injection de la résolution de Koszul par des $A$-$A$-bimodules vers la bar-résolution de $A$, valable pour toute algèbre de Koszul généralisée $A$.

Nous généralisons la dualité de Koszul des algèbres et des opérades aux PROPs. Alors que les opérades sont des objets algébriques qui représentent les opérations à plusieurs entrées et une seule sortie sur les différents types d'algèbres, les PROPs modélisent les opérations à plusieurs entrées et plusieurs sorties agissant sur des structures algébriques telles que les bigèbres et les bigèbres de Lie. Nous introduisons un nouveau produit monoidal qui décrit les compositions entre ces opérations et nous restreignons notre étude à la partie connexe de chaque PROP, que nous appelons "propérade", par analogie avec les opérades. Nous généralisons aux propéades les différents objets homologiques associés aux algèbres et aux opérades comme les bar et cobar constructions, les modules et les propérades quasi-libres. Pour une propérade (resp. un PROP) donnée, nous construisons une copropérade (resp. un coPROP) dual ainsi qu'un complexe de Koszul dont l'acyclicité est un critère qui permet de déterminer si la cobar construction fournit une résolution quasi-libre, appelée modèle minimal, de la propérade (resp. du PROP) de départ. Pour démontrer ce théorème, nous introduisons une graduation supplémentaire qui provient ici des différents foncteurs analytiques engendrés par le produit monoidal. Cette théorie nous permet de définir des notions de "bigèbres" à homotopie près, sur un PROP de Koszul. Cette notion est l'équivalente au niveau des "bigèbres" de celle d'algèbre à homotpie près, qui est très importante en topologie algèbrique.

<p>We investigate the spectrum of certain Banach algebras. Properties</p><p>like generators of maximal ideals and generalized Shilov boundaries are studied. In particular we show that if the ∂-equation has solutions in the algebra of bounded functions or continuous functions up to the boundary of a domain D ⊂⊂ Cⁿ then every maximal ideal over D is generated by the coordinate functions. This implies that the fibres over D in the spectrum are trivial and that the projection on Cn of the n − 1 order generalized Shilov boundary is contained in the boundary of D.</p><p>For a domain D ⊂⊂ Cⁿ where the boundary of the Nebenhülle coincide</p><p>with the smooth strictly pseudoconvex boundary points of D we show that there always exist points p ∈ D such that D has the Gleason property at p.</p><p>If the boundary of an open set U is smooth we show that there exist points in</p><p>U such that the maximal ideals over those points are generated by the coordinate functions.</p><p>An example is given of a Riemann domain, Ω, spread over Cⁿ where the fibers over a point p ∈ Ω consist of m > n elements but the maximal ideal over p is generated by n functions.</p>