Top PDF STUDY O F T H E G E O ME T R YO F T H ED E R I V E DC A T E G O R Y

STUDY O F T H E G E O ME T R YO F T H ED E R I V E DC A T E G O R Y

STUDY O F T H E G E O ME T R YO F T H ED E R I V E DC A T E G O R Y

The collection of all sheaves of O-modules is a fundamental geometric invariant of an algebraic variety. Traditionally one studies the abelian category of O-modules which is a powerful and robust structure containing complete information about the variety. However, a standard problem with this method of study is that the natural geometric and algebraic constructions (e.g., pushforward, pullback, and tensor product) are not always exact and so do not preserve this abelian category. Derived functors and eventually the derived category were introduced as a means of reconciling this lack of exactness. Beyond creating a conceptual framework for understanding the behavior of non-exact functors, derived categories provide a new important flexibility: they can serve as a replacement of the classical notion of a space and provide a setting for the exploration of noncommutative geometry. This thesis revolves around a fundamental issue that arises during the course of this exploration: how and what kind of geometry is encoded in the derived category of sheaves on an algebraic variety?
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Non-negatively C u r v e d C o h o mo g e n e i t yO n e Ma n i f o l d s

Non-negatively C u r v e d C o h o mo g e n e i t yO n e Ma n i f o l d s

In this section, we begin the study of the invariant metrics on the examples. In general, the family of the G-invariant metrics on M is very large. Though there are relatively more rigidity results in the positively curved metrics, for example, L. Verdiani classified all positively curved cohomogeneity one manifolds in even dimen- sions, see [Ve2] and [Ve3], K.Grove, B.Wilking and W.Ziller obtained a short list of cohomogeneity one manifolds which possibly have invariant positively curved met- rics in [GVWZ] and recently K.Grove, L.Verdiani and W.Ziller have succeeded in constructing positively metric on one of them in [GVZ], the rigidity results in the non-negatively metrics are very few. Recently, B.Wilking considered the transversal Jacobi field and proved some fundamental rigidity theorems in the non-negatively curved metrics, see [Wi2]. His results work for a very general setting.
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RAM I FI C A T I O N O F P R I ME SI NFI E L D SO F MO D U L I O F T H R E E - P O I N T COVERS

RAM I FI C A T I O N O F P R I ME SI NFI E L D SO F MO D U L I O F T H R E E - P O I N T COVERS

• To all my friends in the department—you make me look forward to coming into DRL (seriously, it’s not the beautiful exterior). To Clay and Shea for being wonderful officemates and for their frequent technological help. To Asher, Shuvra, and Scott C. for many useful mathematical conversations. To John for being a great officemate for the last two years. To Wil, Andy, Dragos, Colin, Tobi, Dave F., other Dave F., Linda, Hilaf, Pilar, Armin, Jen, Chris, Martin, Scott M., Paul, Mike, Andrew R., and everyone else who ever came to a Halloween party at 4203 Pine.

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O n I ncl us i o n-D r i v e n L e a r ni ng o f B a y e s i a n Ne t w o r k s ob e r t Ca s t e l or c a s t e l o @

O n I ncl us i o n-D r i v e n L e a r ni ng o f B a y e s i a n Ne t w o r k s ob e r t Ca s t e l or c a s t e l o @

In the context of structure learning, the Bayesian network structure is often identified as the Bayesian network itself because learning the parameters can be done once the structure has been learned. However, another implicit reason, which is important for this paper, is that the Bayesian network structure conveys a model on its own, a conditional independence model. More concretely, as Whittaker (1990, pg. 207), we use the term model to specify an arbitrary family of probability distributions that satisfies a set of CI restrictions in the following way. A probability distribution P is Markov over a graph G if every CI restriction encoded in G is satisfied by P . A graphical Markov model (GMM), denoted by M(G), is the family of probability distributions that are Markov over G (Whittaker, 1990, pg. 13). One also says that G determines the GMM M(G).
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A n Appr o x i ma t i o n Al g o r i t hm f o r B o unde d D e g r e eC l o s e s t P hy l o g e ne t i c 2 nd Ro o tP r o bl e m

A n Appr o x i ma t i o n Al g o r i t hm f o r B o unde d D e g r e eC l o s e s t P hy l o g e ne t i c 2 nd Ro o tP r o bl e m

Let u be the vertex that was chosen randomly in this recursive call (and has degree |C| − 1.) In any optimum ∆-clustering of G, u has at most ∆ − 2 neighbors in the same cluster and therefore has at least |C|−∆+1 adjacent edges crossing the boundary. So u incurs at least |C| − ∆ + 1 disagreements in that optimum clustering. Observe that in both cases any edge causing a disagree- ment is removed. So an edge can incur a disagreement at

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L E -C OUN T I N GQ -R OK T H E OR YANDO T H E R EN E R AL I Z A T I ON S O FC L A SSI CALROK T H E OR Y

L E -C OUN T I N GQ -R OK T H E OR YANDO T H E R EN E R AL I Z A T I ON S O FC L A SSI CALROK T H E OR Y

Recently, the study of q-analogs has become popular in combinatorial research. A q-analog of a combinatorial quantity is an expression in the variable q (often a polynomial) such that when we let q = 1 we get the classical quantity. An additional desired property of a q-analog is that so-called q-versions of theorems known for the original quantity hold. For example, we might have an equation in which some or all of the quantities are q-analogs, and letting q = 1 gives the original known equation.

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An i n t r o d u c t i o n t o Mo d e r nP o r t f o l i o T h e o r y : Ma r k o w i t z , C A P - M APT a n d B l a c k - L i t t e r m a n

An i n t r o d u c t i o n t o Mo d e r nP o r t f o l i o T h e o r y : Ma r k o w i t z , C A P - M APT a n d B l a c k - L i t t e r m a n

Thus, we start with a very steep downward sloping line going through (0, r) and rotate it anti-clockwise while we have an attainable portfolio. The ‘last’ portfolio is called the Optimum Portfolio of Risky Assets (OPRA). We will quantify it by observing that of all such lines going through the attainable region it is the one with maximal gradient. The investor can now place themselves anywhere on this line through an appropriate amount of lending or borrowing and using the remainder (which could be greater than one, if borrowing occurs - this is known as gearing) to buy the OPRA.

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THE B I R A T I O N A L G E O ME T R YO F T R O P I C A LC O MP A C T I FI C A T I O N S

THE B I R A T I O N A L G E O ME T R YO F T R O P I C A LC O MP A C T I FI C A T I O N S

I would like thank my advisor Antonella Grassi for suggesting tropical geometry as a possible dissertation topic (and for her truly commendable patience and willingness to help). Ron Donagi, Tony Pantev, Jonathan Block, and Erik van Erp also helped me find my way mathematically here at Penn. Conversations with Paul Hacking, Jenia Tevelev, Eric Katz, Ilia Zharkov, Eduardo Cattani, and Alicia Dickenstein were beneficial during the preparation of this thesis.

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ARITHM E T I C A N DG E O ME T R YO F T H EO P E NP - A D I C D I S C

ARITHM E T I C A N DG E O ME T R YO F T H EO P E NP - A D I C D I S C

The main result of this thesis (Theorem 4.0.1) says that the special fiber of an abelian branched cover of the open p-adic disc is completely determined by charac- teristic zero fibers. The motivation for such a theorem comes from the global lifting problem for Galois covers of curves: if G is a finite group, k is an algebraically closed field of characteristic p > 0, and f : C → C 0 is a finite G-Galois branched cover of smooth projective k-curves, does there exist a lifting of f to a G-Galois branched cover of smooth projective R-curves, where R is a discrete valuation ring of mixed characteristic with residue field k? This problem has been much studied; see e.g. [9], [13], [8], [14], [16].
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PROGRAM Declarations and Instructions

PROGRAM Declarations and Instructions

PROGRAM Declarations and Instructions T h e K V A L G r o u p b i d s y o u w e l c o m e t o S w e d e n T h e n e x t f e w p a g e s w i l l g i v e y o u t h e m a i n f a c t s a b o u t w h a t[.]

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M o de l so f Addi ng E dg e s be t w e e n t heR o o t a nd D e s ce nda nt si n aC o m pl e t eB i na r yT r e e M i ni mi zi ng T o t a l P a t h L e ng t h

M o de l so f Addi ng E dg e s be t w e e n t heR o o t a nd D e s ce nda nt si n aC o m pl e t eB i na r yT r e e M i ni mi zi ng T o t a l P a t h L e ng t h

This study considered the addition of relations to an orga- nization structure such that the communication of infor- mation between every unit in the organization becomes the most efficient. For each of two models of adding edges between the root and nodes of depth N to a complete bi- nary tree of height H which can describe the basic type of a pyramid organization, we obtained an optimal depth N ∗ which maximizes the total shortening path length.

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T r a ns m i s s i o n Ne t w o r kC o ng e s t i o n i n D e r e g ul a t e d W ho l e s a l eE l e ct r i ci t yM a r k e t

T r a ns m i s s i o n Ne t w o r kC o ng e s t i o n i n D e r e g ul a t e d W ho l e s a l eE l e ct r i ci t yM a r k e t

It can be observed from the above analysis that, with this bidding and a common MCP leads to the network con- gestion for the system under study by overloading of line AC. To relieve the network from congestion, the above system has to be solved under transmission line capac- ity constraints. This will alter the output of generator (supplier) and the catered demand (consumer). Because of network congestion the MCP is not valid for all the nodes of the market and will lead to different locational marginal price (LMP) at each node. The results of con- strained DC power flow are as shown in Table 4.
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Lo c a l i z a t i o n , P a t h i n t e g r a l a n dS u p e r s y m m e t r y

Lo c a l i z a t i o n , P a t h i n t e g r a l a n dS u p e r s y m m e t r y

Localization methods are crucial in mathematical physics. As we have seen, it allow t possible to make certain quantities calculable. It has brought back to the taste of the day some of the algebraic geometry methods as enumerative geometry. Its application to mathematical physics leads to the definition of moduli spaces and, in the best case, to instantons counting. This makes it possible to calculate certain correlation functions, resulting from a path integral. These methods have allowed a better understanding of quantum field theories in physics, they now connect topology, geometry and physic, to the concept of supersymmetry correctly defined from a mathematical point of view.
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ROOT N U M B E R SO F A B E L I A NV A R I E T I E SA N D REPRESENTATIONS O F T H E WE I L - D E L I G N EG R O U P

ROOT N U M B E R SO F A B E L I A NV A R I E T I E SA N D REPRESENTATIONS O F T H E WE I L - D E L I G N EG R O U P

One of the main objects of study in this thesis is the root number W (A, τ ) associ- ated to an abelian variety A of dimension g over a number field F and a continuous irreducible complex finite-dimensional representation τ of Gal(F /F ) with real-valued character. The root number W (A, τ) is a complex number of absolute value 1. As- sume for simplicity that F = Q . Then W (A, τ ) appears in the following conjectural functional equation:

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On t h e e v o l u t i o n a r yf o r mo f t h ec o n s t r a i n t s in e l e c t r o d y n a m i c s

On t h e e v o l u t i o n a r yf o r mo f t h ec o n s t r a i n t s in e l e c t r o d y n a m i c s

and its decomposition in terms of the variables N , b N b I , b γ IJ , is also known throughout Σ. Thus (16) can be solved for λ. Note that (16), is manifestly independent of the choice made for the foliation and flow, and (16) is always a linear hyperbolic equation for λ, with ρ “playing the role of time”. As it was emphasized in the introduction, the global existence of unique smooth solutions (under suitable regularity conditions on the coefficients and source terms) is always guaranteed to such linear first order symmetric hyperbolic equations. This makes it tempting to consider applications of this evolutionary equation for the geometrically distinguished scalar field λ, which has been motivated by the constraints of Maxwell theory.
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D e r i v a t i o n o f t he L o c a l L o r e nt z G a ug e T r a ns f o r m a t i o n o f a D i r a c Spi no r F i e l d i n Q ua nt um E i ns t e i n- C a r t a n T he o r y

D e r i v a t i o n o f t he L o c a l L o r e nt z G a ug e T r a ns f o r m a t i o n o f a D i r a c Spi no r F i e l d i n Q ua nt um E i ns t e i n- C a r t a n T he o r y

I examine the groups which underly classical mechanics, non-relativistic quantum mechanics, special relativity, relativistic quantum mechan- ics, quantum electrodynamics, quantum flavour- dynamics, quantum chromodynamics, and gen- eral relativity. This examination includes the rotations SO(2) and SO(3), the Pauli algebra, the Lorentz transformations, the Dirac algebra, and the U (1), SU (2), and SU(3) gauge trans- formations. I argue that general relativity must be generalized to Einstein-Cartan theory, so that Dirac spinors can be described within the frame- work of gravitation theory.
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´A T H - S Z A B ´O APPLICATIONS O F O Z S V I N V A R I A N T ST O CONTACT G E O M E T R Y

´A T H - S Z A B ´O APPLICATIONS O F O Z S V I N V A R I A N T ST O CONTACT G E O M E T R Y

Lisca, Ozsv´ath, Stipsicz, and Szab´o define a pair of invariants for null-homologous Legendrian and transverse knots in [LOSS08]. These invariants live in the knot Floer homology groups of the ambient space with reversed orientation, and generalize the previously defined invariants of closed contact manifolds, c(Y, ξ). They have been useful in constructing new examples of knot types which are not transversally simple (see [LOSS08, OS08a]).

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An O v e r v i e w o f t h e G e o me t r ya n dC o m b i n a t o r i c s o f t h e Ma c d o n a l d Polynom i a l a n d q - t C a t a l a nN u m b e r

An O v e r v i e w o f t h e G e o me t r ya n dC o m b i n a t o r i c s o f t h e Ma c d o n a l d Polynom i a l a n d q - t C a t a l a nN u m b e r

Now we may look at the definition of the Hilbert scheme. We will focus on the category of finitely generated schemes over Spec k where k is an algebraicly closed field and of characteristic 0. Generally, let X be a scheme over k, define a functor Hilb n X from schemes to sets as follows. On the object level, Hilb n X (T ) is the set of closed subschemes of X × T that are flat and surjective of degree n over T . As for morphisms, suppose we have f : S → T . Then Hilb n X (f) maps subschemes in Hilb n X (T ) to their pull back in X × S.

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A ut o ma t e d T e s tD a t a Ge ne r a t i o n F o rP r o g r a ms H a v i ngA r r a y Of V a r i a bl eL e ng t h And L o o ps W i t h V a r i a bl eN um be r Of I t e r a t i o n

A ut o ma t e d T e s tD a t a Ge ne r a t i o n F o rP r o g r a ms H a v i ngA r r a y Of V a r i a bl eL e ng t h And L o o ps W i t h V a r i a bl eN um be r Of I t e r a t i o n

[11] N. Williams, B. Marre, P. Mouy, and M. Roger, “PathCrawler: Automatic Generation of Path Tests by Combining Static and Dynamic Analysis,” Springer- Verlag Berlin Heidelberg, LNCS 3463, pp. 281-292, 2005. [12] Koushik Sen, Darko Marinov, Gul Agha, “CUTE: A Con- colic Unit Testing Engine for C ,” ACM, pp. 5-9, 09/2005 [13] R. E. Prather, J. P. Myers, “The Path Prefix Software Engineering,” IEEE Trans on Software Engineering, SE- 13(7), pp. 761-766, 07/1987

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