Exceptional collections and D-branes probing toric singularities

We demonstrate that a strongly exceptional collection on a singular toric surface can be used to derive the gauge theory on a stack of D3-branes probing the Calabi-Yau singularity caused by the surface shrinking to zero size. A strongly exceptional collection, i.e., an ordered set of sheaves satisfying special mapping properties, gives a convenient basis of D-branes. We find such collections and analyze the gauge theories for weighted projective spaces, and many of the Y^{p,q} and L^{p,q,r} spaces. In particular, we prove the strong exceptionality for all p in the Y^{p,p-1} case, and similarly for the Y^{p,p-2r} case.Comment: 49 pages, 6 figures; v2 refs added; v3 published versio

Monoids with tests and the algebra of possibly non-halting programs

We study the algebraic theory of computable functions, which can be viewed as arising from possibly non-halting computer programs or algorithms, acting on some state space, equipped with operations of composition, if-then-else and while-do defined in terms of a Boolean algebra of conditions. It has previously been shown that there is no finite axiomatisation of algebras of partial functions under these operations alone, and this holds even if one restricts attention to transformations (representing halting programs) rather than partial functions, and omits while-do from the signature. In the halting case, there is a natural “fix”, which is to allow composition of halting programs with conditions, and then the resulting algebras admit a finite axiomatisation. In the current setting such compositions are not possible, but by extending the notion of if-then-else, we are able to give finite axiomatisations of the resulting algebras of (partial) functions, with while-do in the signature if the state space is assumed finite. The axiomatisations are extended to consider the partial predicate of equality. All algebras considered turn out to be enrichments of the notion of a (one-sided) restriction semigrou

Rings of small rank over a Dedekind domain and their ideals

In 2001, M. Bhargava stunned the mathematical world by extending Gauss's 200-year-old group law on integral binary quadratic forms, now familiar as the ideal class group of a quadratic ring, to yield group laws on a vast assortment of analogous objects. His method yields parametrizations of rings of degree up to 5 over the integers, as well as aspects of their ideal structure, and can be employed to yield statistical information about such rings and the associated number fields. In this paper, we extend a selection of Bhargava's most striking parametrizations to cases where the base ring is not Z but an arbitrary Dedekind domain R. We find that, once the ideal classes of R are properly included, we readily get bijections parametrizing quadratic, cubic, and quartic rings, as well as an analogue of the 2x2x2 cube law reinterpreting Gauss composition for which Bhargava is famous. We expect that our results will shed light on the analytic distribution of extensions of degree up to 4 of a fixed number field and their ideal structure.Comment: 39 pages, 1 figure. Harvard College senior thesis, edite

The new economic geography versus urban economics : an evaluation using local wage rates in Great Britain

This paper tests two competing models, one deriving from new economic geography theory (NEG) emphasising varying market potential, the other with a basis in urban economics theory (UE) in which the main emphasis is on producer service linkages. Using wage rate variations across small regions of Great Britain, the paper finds that, taking commuting into account, it is UE theory rather than NEG theory that has explanatory power. However since the two hypotheses are non-nested, the evaluation of the competing hypotheses is difficult and therefore the conclusions are provisional. Nevertheless this paper provides evidence that we should be cautious about the ability of NEG to work at all levels of spatial resolution, and re-emphasises the need to focus on supply-side variations in producer services inputs and labour efficiency variations, including the role of commuting, in local economic analysis.

Interacting Hopf Algebras

We introduce the theory IH of interacting Hopf algebras, parametrised over a principal ideal domain R. The axioms of IH are derived using Lack's approach to composing PROPs: they feature two Hopf algebra and two Frobenius algebra structures on four different monoid-comonoid pairs. This construction is instrumental in showing that IH is isomorphic to the PROP of linear relations (i.e. subspaces) over the field of fractions of R

Fast, uniform, and compact scalar multiplication for elliptic curves and genus 2 Jacobians with applications to signature schemes

We give a general framework for uniform, constant-time one-and two-dimensional scalar multiplication algorithms for elliptic curves and Jacobians of genus 2 curves that operate by projecting to the x-line or Kummer surface, where we can exploit faster and more uniform pseudomultiplication, before recovering the proper "signed" output back on the curve or Jacobian. This extends the work of L{\'o}pez and Dahab, Okeya and Sakurai, and Brier and Joye to genus 2, and also to two-dimensional scalar multiplication. Our results show that many existing fast pseudomultiplication implementations (hitherto limited to applications in Diffie--Hellman key exchange) can be wrapped with simple and efficient pre-and post-computations to yield competitive full scalar multiplication algorithms, ready for use in more general discrete logarithm-based cryptosystems, including signature schemes. This is especially interesting for genus 2, where Kummer surfaces can outperform comparable elliptic curve systems. As an example, we construct an instance of the Schnorr signature scheme driven by Kummer surface arithmetic

Operator bases, SS-matrices, and their partition functions

Relativistic quantum systems that admit scattering experiments are quantitatively described by effective field theories, where $S$-matrix kinematics and symmetry considerations are encoded in the operator spectrum of the EFT. In this paper we use the $S$-matrix to derive the structure of the EFT operator basis, providing complementary descriptions in (i) position space utilizing the conformal algebra and cohomology and (ii) momentum space via an algebraic formulation in terms of a ring of momenta with kinematics implemented as an ideal. These frameworks systematically handle redundancies associated with equations of motion (on-shell) and integration by parts (momentum conservation). We introduce a partition function, termed the Hilbert series, to enumerate the operator basis--correspondingly, the $S$-matrix--and derive a matrix integral expression to compute the Hilbert series. The expression is general, easily applied in any spacetime dimension, with arbitrary field content and (linearly realized) symmetries. In addition to counting, we discuss construction of the basis. Simple algorithms follow from the algebraic formulation in momentum space. We explicitly compute the basis for operators involving up to $n=5$ scalar fields. This construction universally applies to fields with spin, since the operator basis for scalars encodes the momentum dependence of $n$-point amplitudes. We discuss in detail the operator basis for non-linearly realized symmetries. In the presence of massless particles, there is freedom to impose additional structure on the $S$-matrix in the form of soft limits. The most na\"ive implementation for massless scalars leads to the operator basis for pions, which we confirm using the standard CCWZ formulation for non-linear realizations.Comment: 75 pages plus appendice

Chaos from Symmetry: Navier Stokes equations, Beltrami fields and the Universal Classifying Crystallographic Group

In this report-article, the general setup to classify and construct Arnold-Beltrami Flows on three-dimensional torii, previously introduced by one of us, is further pursued. The idea of a Universal Classifying Group (UCG) is improved. In particular, we construct for the first time such group for the hexagonal lattice. Mastering the cubic and hexagonal instances, we can cover all cases. We upgrade Beltrami flows to a special type of periodic solutions of the NS equations, presenting the relation between the classification of these flows with the classification of contact structures. The recent developments in contact and symplectic geometry, considering singular contact structures, in the framework of b-manifolds, is also reviewed and we show that the choice of the critical surface for the b-deformation seems to be strongly related to its group-theoretical structures. This opens directions of investigation towards a classification of the critical surfaces or boundaries in terms of the UCG and subgroups. Furthermore, as a result of this research programme a complete set of MATHEMATICA Codes (for the cubic and hexagonal cases) have been produced that are able to construct Beltrami Flows with an arbitrarily large number of parameters and analyze their hidden symmetry structures. Indeed the main goal is the systematic organization of the parameter space into group irreps. The two Codes are a further result, being the unavoidable basis for further investigations. The presented exact solutions illustrate the new conceptions and ideas here discussed. The main message is: the more symmetric is the Beltrami Flow, the highest the probability of an on-set of chaotic trajectories. In various applications we need chaos on small scales and a more orderly motion on larger ones. Merging elementary chaotic solutions with large directional ordered flows is the target for future research.Comment: LaTeX source, 152 pages and 69 figures. arXiv admin note: substantial text overlap with arXiv:1501.0460