On the Mumford-Tate conjecture for Abelian varieties with reduction conditions

In this thesis we study Galois representations corresponding to abelian varieties with certain reduction conditions. We show that these conditions force the image of the representations to be "big," so that the Mumford-Tate conjecture (:= MT) holds. We also prove that the set of abelian varieties satisfying these conditions is dense in a corresponding moduli space.

The main results of the thesis are the following two theorems.

Theorem A: Let A be an absolutely simple abelian variety, End° (A) = k : imaginary quadratic field, g = dim(A). Assume either dim(A) ≤ 4, or A has bad reduction at some prime ϕ, with the dimension of the toric part of the reduction equal to 2r, and gcd(r,g) = 1, and (r,g) ≠ (15,56) or (m -1, m(m+1)/2). Then MT holds.

Theorem B: Let M be the moduli space of abelian varieties with fixed polarization, level structure and a k-action. It is defined over a number field F. The subset of M(Q) corresponding to absolutely simple abelian varieties with a prescribed stable reduction at a large enough prime ϕ of F is dense in M(C) in the complex topology. In particular, the set of simple abelian varieties having bad reductions with fixed dimension of the toric parts is dense.

Besides this we also established the following results:

(1) MT holds for some other classes of abelian varieties with similar reduction conditions. For example, if A is an abelian variety with End° (A) = Q and the dimension of the toric part of its reduction is prime to dim( A), then MT holds.

(2) MT holds for Ribet-type abelian varieties.

(3) The Hodge and the Tate conjectures are equivalent for abelian 4-folds.

(4) MT holds for abelian 4-folds of type II, III, IV (Theorem 5.0(2)) and some 4-folds of type I.

(5) For some abelian varieties either MT or the Hodge conjecture holds.

Symmetric representation of an integral domain over a subdomain

Let F(θ) be a separable extension of degree n of a field F. Let Δ and D be integral domains with quotient fields F(θ) and F respectively. Assume that Δ ᴝ D. A mapping φ of Δ into the n x n D matrices is called a Δ/D rep if (i) it is a ring isomorphism and (ii) it maps d onto dI_n whenever d ϵ D. If the matrices are also symmetric, φ is a Δ/D symrep.

Every Δ/D rep can be extended uniquely to an F(θ)/F rep. This extension is completely determined by the image of θ. Two Δ/D reps are called equivalent if the images of θ differ by a D unimodular similarity. There is a one-to-one correspondence between classes of Δ/D reps and classes of Δ ideals having an n element basis over D.

The condition that a given Δ/D rep class contain a Δ/D symrep can be phrased in various ways. Using these formulations it is possible to (i) bound the number of symreps in a given class, (ii) count the number of symreps if F is finite, (iii) establish the existence of an F(θ)/F symrep when n is odd, F is an algebraic number field, and F(θ) is totally real if F is formally real (for n = 3 see Sapiro, “Characteristic polynomials of symmetric matrices” Sibirsk. Mat. Ž. 3 (1962) pp. 280-291), and (iv) study the case D = Z, the integers (see Taussky, “On matrix classes corresponding to an ideal and its inverse” Illinois J. Math. 1 (1957) pp. 108-113 and Faddeev, “On the characteristic equations of rational symmetric matrices” Dokl. Akad. Nauk SSSR 58 (1947) pp. 753-754).

The case D = Z and n = 2 is studied in detail. Let Δ’ be an integral domain also having quotient field F(θ) and such that Δ’ ᴝ Δ. Let φ be a Δ/Z symrep. A method is given for finding a Δ’/Z symrep ʘ such that the Δ’ ideal class corresponding to the class of ʘ is an extension to Δ’ of the Δ ideal class corresponding to the class of φ. The problem of finding all Δ/Z symreps equivalent to a given one is studied.

Photographic investigations in molecular spectroscopy

The determination of the energy levels and the probabilities of transition between them, by the formal analysis of observed electronic, vibrational, and rotational band structures, forms the direct goal of all investigations of molecular spectra, but the significance of such data lies in the possibility of relating them theoretically to more concrete properties of molecules and the radiation field. From the well developed electronic spectra of diatomic molecules, it has been possible, with the aid of the non-relativistic quantum mechanics, to obtain accurate moments of inertia, molecular potential functions, electronic structures, and detailed information concerning the coupling of spin and orbital angular monenta with the angular momentum of nuclear rotation. The silicon fluori1e molecule has been investigated in this laboratory, and is found to emit bands whose vibrational and rotational structures can be analyzed in this detailed fashion.

Like silicon fluoride, however, the great majority of diatomic molecules are formed only under the unusual conditions of electrical discharge, or in high temperature furnaces, so that although their spectra are of great theoretical interest, the chemist is eager to proceed to a study of polyatomic molecules, in the hope that their more practically interesting structures might also be determined with the accuracy and assurance which characterize the spectroscopic determinations of the constants of diatomic molecules. Some progress has been made in the determination of molecule potential functions from the vibrational term values deduced from Raman and infrared spectra, but in no case can the calculations be carried out with great generality, since the number of known term values is always small compared with the total number of potential constants in even so restricted a potential function as the simple quadratic type. For the determination of nuclear configurations and bond distances, however, a knowledge of the rotational terms is required. The spectra of about twelve of the simpler polyatomic molecules have been subjected to rotational analyses, and a number of bond distances are known with considerable accuracy, yet the number of molecules whose rotational fine structure has been resolved even with the most powerful instruments is small. Consequently, it was felt desirable to investigate the spectra of a number of other promising polyatomic molecules, with the purpose of carrying out complete rotational analyses of all resolvable bands, and ascertaining the value of the unresolved band envelopes in determining the structures of such molecules, in the cases in which resolution is no longer possible. Although many of the compounds investigated absorbed too feebly to be photographed under high dispersion with the present infrared sensitizations, the location and relative intensities of their bands, determined by low dispersion measurements, will be reported in the hope that these compounds may be reinvestigated in the future with improved techniques.

Nonlinear optics and wavelength translation via cavity-optomechanics

The field of cavity-optomechanics explores the interaction of light with sound in an ever increasing array of devices. This interaction allows the mechanical system to be both sensed and controlled by the optical system, opening up a wide variety of experiments including the cooling of the mechanical resonator to its quantum mechanical ground state and the squeezing of the optical field upon interaction with the mechanical resonator, to name two.

In this work we explore two very different systems with different types of optomechanical coupling. The first system consists of two microdisk optical resonators stacked on top of each other and separated by a very small slot. The interaction of the disks causes their optical resonance frequencies to be extremely sensitive to the gap between the disks. By careful control of the gap between the disks, the optomechanical coupling can be made to be quadratic to first order which is uncommon in optomechanical systems. With this quadratic coupling the light field is now sensitive to the energy of the mechanical resonator and can directly control the potential energy trapping the mechanical motion. This ability to directly control the spring constant without modifying the energy of the mechanical system, unlike in linear optomechanical coupling, is explored.

Next, the bulk of this thesis deals with a high mechanical frequency optomechanical crystal which is used to coherently convert photons between different frequencies. This is accomplished via the engineered linear optomechanical coupling in these devices. Both classical and quantum systems utilize the interaction of light and matter across a wide range of energies. These systems are often not naturally compatible with one another and require a means of converting photons of dissimilar wavelengths to combine and exploit their different strengths. Here we theoretically propose and experimentally demonstrate coherent wavelength conversion of optical photons using photon-phonon translation in a cavity-optomechanical system. For an engineered silicon optomechanical crystal nanocavity supporting a 4 GHz localized phonon mode, optical signals in a 1.5 MHz bandwidth are coherently converted over a 11.2 THz frequency span between one cavity mode at wavelength 1460 nm and a second cavity mode at 1545 nm with a 93% internal (2% external) peak efficiency. The thermal and quantum limiting noise involved in the conversion process is also analyzed and, in terms of an equivalent photon number signal level, are found to correspond to an internal noise level of only 6 and 4 times 10x^-3 quanta, respectively.

We begin by developing the requisite theoretical background to describe the system. A significant amount of time is then spent describing the fabrication of these silicon nanobeams, with an emphasis on understanding the specifics and motivation. The experimental demonstration of wavelength conversion is then described and analyzed. It is determined that the method of getting photons into the cavity and collected from the cavity is a fundamental limiting factor in the overall efficiency. Finally, a new coupling scheme is designed, fabricated, and tested that provides a means of coupling greater than 90% of photons into and out of the cavity, addressing one of the largest obstacles with the initial wavelength conversion experiment.

The subvarieties of a field

Number systems which satisfy part but not all of the postulates for a field are called subvarieties of a field. The purpose of this paper is the determination of as great as possible a number of such varieties by suitable definitions of the class of elements and of the two operations involved.

Two postulate systems are considered. The first gives rise to 284 varieties, instances of all of which are given for infinite classes of elements, and of all except three for finite classes.

Of the 8192 combinations of postulates arising from the second system, not more than 1146 can be consistent. Instances are given of 1054 of these. As the postulates of this system are not independent, no conclusion has been reached regarding the remaining cases.

Efficient methods for empirical tests of behavioral economics theories in laboratory and field experiments

In the quest for a descriptive theory of decision-making, the rational actor model in economics imposes rather unrealistic expectations and abilities on human decision makers. The further we move from idealized scenarios, such as perfectly competitive markets, and ambitiously extend the reach of the theory to describe everyday decision making situations, the less sense these assumptions make. Behavioural economics has instead proposed models based on assumptions that are more psychologically realistic, with the aim of gaining more precision and descriptive power. Increased psychological realism, however, comes at the cost of a greater number of parameters and model complexity. Now there are a plethora of models, based on different assumptions, applicable in differing contextual settings, and selecting the right model to use tends to be an ad-hoc process. In this thesis, we develop optimal experimental design methods and evaluate different behavioral theories against evidence from lab and field experiments.

We look at evidence from controlled laboratory experiments. Subjects are presented with choices between monetary gambles or lotteries. Different decision-making theories evaluate the choices differently and would make distinct predictions about the subjects' choices. Theories whose predictions are inconsistent with the actual choices can be systematically eliminated. Behavioural theories can have multiple parameters requiring complex experimental designs with a very large number of possible choice tests. This imposes computational and economic constraints on using classical experimental design methods. We develop a methodology of adaptive tests: Bayesian Rapid Optimal Adaptive Designs (BROAD) that sequentially chooses the "most informative" test at each stage, and based on the response updates its posterior beliefs over the theories, which informs the next most informative test to run. BROAD utilizes the Equivalent Class Edge Cutting (EC²) criteria to select tests. We prove that the EC² criteria is adaptively submodular, which allows us to prove theoretical guarantees against the Bayes-optimal testing sequence even in the presence of noisy responses. In simulated ground-truth experiments, we find that the EC² criteria recovers the true hypotheses with significantly fewer tests than more widely used criteria such as Information Gain and Generalized Binary Search. We show, theoretically as well as experimentally, that surprisingly these popular criteria can perform poorly in the presence of noise, or subject errors. Furthermore, we use the adaptive submodular property of EC² to implement an accelerated greedy version of BROAD which leads to orders of magnitude speedup over other methods.

We use BROAD to perform two experiments. First, we compare the main classes of theories for decision-making under risk, namely: expected value, prospect theory, constant relative risk aversion (CRRA) and moments models. Subjects are given an initial endowment, and sequentially presented choices between two lotteries, with the possibility of losses. The lotteries are selected using BROAD, and 57 subjects from Caltech and UCLA are incentivized by randomly realizing one of the lotteries chosen. Aggregate posterior probabilities over the theories show limited evidence in favour of CRRA and moments' models. Classifying the subjects into types showed that most subjects are described by prospect theory, followed by expected value. Adaptive experimental design raises the possibility that subjects could engage in strategic manipulation, i.e. subjects could mask their true preferences and choose differently in order to obtain more favourable tests in later rounds thereby increasing their payoffs. We pay close attention to this problem; strategic manipulation is ruled out since it is infeasible in practice, and also since we do not find any signatures of it in our data.

In the second experiment, we compare the main theories of time preference: exponential discounting, hyperbolic discounting, "present bias" models: quasi-hyperbolic (α, β) discounting and fixed cost discounting, and generalized-hyperbolic discounting. 40 subjects from UCLA were given choices between 2 options: a smaller but more immediate payoff versus a larger but later payoff. We found very limited evidence for present bias models and hyperbolic discounting, and most subjects were classified as generalized hyperbolic discounting types, followed by exponential discounting.

In these models the passage of time is linear. We instead consider a psychological model where the perception of time is subjective. We prove that when the biological (subjective) time is positively dependent, it gives rise to hyperbolic discounting and temporal choice inconsistency.

We also test the predictions of behavioral theories in the "wild". We pay attention to prospect theory, which emerged as the dominant theory in our lab experiments of risky choice. Loss aversion and reference dependence predicts that consumers will behave in a uniquely distinct way than the standard rational model predicts. Specifically, loss aversion predicts that when an item is being offered at a discount, the demand for it will be greater than that explained by its price elasticity. Even more importantly, when the item is no longer discounted, demand for its close substitute would increase excessively. We tested this prediction using a discrete choice model with loss-averse utility function on data from a large eCommerce retailer. Not only did we identify loss aversion, but we also found that the effect decreased with consumers' experience. We outline the policy implications that consumer loss aversion entails, and strategies for competitive pricing.

In future work, BROAD can be widely applicable for testing different behavioural models, e.g. in social preference and game theory, and in different contextual settings. Additional measurements beyond choice data, including biological measurements such as skin conductance, can be used to more rapidly eliminate hypothesis and speed up model comparison. Discrete choice models also provide a framework for testing behavioural models with field data, and encourage combined lab-field experiments.

Velocity resolved - scalar modeled simulations of high Schmidt number turbulent transport

The objective of this thesis is to develop a framework to conduct velocity resolved - scalar modeled (VR-SM) simulations, which will enable accurate simulations at higher Reynolds and Schmidt (Sc) numbers than are currently feasible. The framework established will serve as a first step to enable future simulation studies for practical applications. To achieve this goal, in-depth analyses of the physical, numerical, and modeling aspects related to Sc>>1 are presented, specifically when modeling in the viscous-convective subrange. Transport characteristics are scrutinized by examining scalar-velocity Fourier mode interactions in Direct Numerical Simulation (DNS) datasets and suggest that scalar modes in the viscous-convective subrange do not directly affect large-scale transport for high Sc. Further observations confirm that discretization errors inherent in numerical schemes can be sufficiently large to wipe out any meaningful contribution from subfilter models. This provides strong incentive to develop more effective numerical schemes to support high Sc simulations. To lower numerical dissipation while maintaining physically and mathematically appropriate scalar bounds during the convection step, a novel method of enforcing bounds is formulated, specifically for use with cubic Hermite polynomials. Boundedness of the scalar being transported is effected by applying derivative limiting techniques, and physically plausible single sub-cell extrema are allowed to exist to help minimize numerical dissipation. The proposed bounding algorithm results in significant performance gain in DNS of turbulent mixing layers and of homogeneous isotropic turbulence. Next, the combined physical/mathematical behavior of the subfilter scalar-flux vector is analyzed in homogeneous isotropic turbulence, by examining vector orientation in the strain-rate eigenframe. The results indicate no discernible dependence on the modeled scalar field, and lead to the identification of the tensor-diffusivity model as a good representation of the subfilter flux. Velocity resolved - scalar modeled simulations of homogeneous isotropic turbulence are conducted to confirm the behavior theorized in these a priori analyses, and suggest that the tensor-diffusivity model is ideal for use in the viscous-convective subrange. Simulations of a turbulent mixing layer are also discussed, with the partial objective of analyzing Schmidt number dependence of a variety of scalar statistics. Large-scale statistics are confirmed to be relatively independent of the Schmidt number for Sc>>1, which is explained by the dominance of subfilter dissipation over resolved molecular dissipation in the simulations. Overall, the VR-SM framework presented is quite effective in predicting large-scale transport characteristics of high Schmidt number scalars, however, it is determined that prediction of subfilter quantities would entail additional modeling intended specifically for this purpose. The VR-SM simulations presented in this thesis provide us with the opportunity to overlap with experimental studies, while at the same time creating an assortment of baseline datasets for future validation of LES models, thereby satisfying the objectives outlined for this work.

A kinetic theory description for external spherical flows with arbitrary Knudsen number by a moment method

The Maxwell integral equations of transfer are applied to a series of problems involving flows of arbitrary density gases about spheres. As suggested by Lees a two sided Maxwellian-like weighting function containing a number of free parameters is utilized and a sufficient number of partial differential moment equations is used to determine these parameters. Maxwell's inverse fifth-power force law is used to simplify the evaluation of the collision integrals appearing in the moment equations. All flow quantities are then determined by integration of the weighting function which results from the solution of the differential moment system. Three problems are treated: the heat-flux from a slightly heated sphere at rest in an infinite gas; the velocity field and drag of a slowly moving sphere in an unbounded space; the velocity field and drag torque on a slowly rotating sphere. Solutions to the third problem are found to both first and second-order in surface Mach number with the secondary centrifugal fan motion being of particular interest. Singular aspects of the moment method are encountered in the last two problems and an asymptotic study of these difficulties leads to a formal criterion for a "well posed" moment system. The previously unanswered question of just how many moments must be used in a specific problem is now clarified to a great extent.