Lq estimates for rearrangements of Fourier series

This work is concerned with estimating the upper envelopes S* of the absolute values of the partial sums of rearranged trigonometric sums. A.M. Garsia [Annals of Math. 79 (1964), 634-9] gave an estimate for the L₂ norms of the S*, averaged over all rearrangements of the original (finite) sum. This estimate enabled him to prove that the Fourier series of any function in L₂ can be rearranged so that it converges a.e. The main result of this thesis is a similar estimate of the L_q norms of the S*, for all even integers q. This holds for finite linear combinations of functions which satisfy a condition which is a generalization of orthonormality in the L₂ case. This estimate for finite sums is extended to Fourier series of L_q functions; it is shown that there are functions to which the Men’shov-Paley Theorem does not apply, but whose Fourier series can nevertheless be rearranged so that the S* of the rearranged series is in L_q.

An arithmetical theorem for partially ordered sets

The simplest multiplicative systems in which arithmetical ideas can be defined are semigroups. For such systems irreducible (prime) elements can be introduced and conditions under which the fundamental theorem of arithmetic holds have been investigated (Clifford (3)). After identifying associates, the elements of the semigroup form a partially ordered set with respect to the ordinary division relation. This suggests the possibility of an analogous arithmetical result for abstract partially ordered sets. Although nothing corresponding to product exists in a partially ordered set, there is a notion similar to g.c.d. This is the meet operation, defined as greatest lower bound. Thus irreducible elements, namely those elements not expressible as meets of proper divisors can be introduced. The assumption of the ascending chain condition then implies that each element is representable as a reduced meet of irreducibles. The central problem of this thesis is to determine conditions on the structure of the partially ordered set in order that each element have a unique such representation.

Part I contains preliminary results and introduces the principal tools of the investigation. In the second part, basic properties of the lattice of ideals and the connection between its structure and the irreducible decompositions of elements are developed. The proofs of these results are identical with the corresponding ones for the lattice case (Dilworth (2)). The last part contains those results whose proofs are peculiar to partially ordered sets and also contains the proof of the main theorem.

A new high-order Fourier continuation-based elasticity solver for complex three-dimensional geometries

This thesis presents a new approach for the numerical solution of three-dimensional problems in elastodynamics. The new methodology, which is based on a recently introduced Fourier continuation (FC) algorithm for the solution of Partial Differential Equations on the basis of accurate Fourier expansions of possibly non-periodic functions, enables fast, high-order solutions of the time-dependent elastic wave equation in a nearly dispersionless manner, and it requires use of CFL constraints that scale only linearly with spatial discretizations. A new FC operator is introduced to treat Neumann and traction boundary conditions, and a block-decomposed (sub-patch) overset strategy is presented for implementation of general, complex geometries in distributed-memory parallel computing environments. Our treatment of the elastic wave equation, which is formulated as a complex system of variable-coefficient PDEs that includes possibly heterogeneous and spatially varying material constants, represents the first fully-realized three-dimensional extension of FC-based solvers to date. Challenges for three-dimensional elastodynamics simulations such as treatment of corners and edges in three-dimensional geometries, the existence of variable coefficients arising from physical configurations and/or use of curvilinear coordinate systems and treatment of boundary conditions, are all addressed. The broad applicability of our new FC elasticity solver is demonstrated through application to realistic problems concerning seismic wave motion on three-dimensional topographies as well as applications to non-destructive evaluation where, for the first time, we present three-dimensional simulations for comparison to experimental studies of guided-wave scattering by through-thickness holes in thin plates.

Crystal Coulomb energies: I. Organic donor-acceptor complexes--a review. II. Classical Ewald calculations of the Coulomb binding energy of four donor-acceptor crystals and Wurster's blue perchlorate. III. A rapid-convergence quantum-mechanical formalism for crystal electronic energies

Chapter I

Theories for organic donor-acceptor (DA) complexes in solution and in the solid state are reviewed, and compared with the available experimental data. As shown by McConnell et al. (Proc. Natl. Acad. Sci. U.S., 53, 46-50 (1965)), the DA crystals fall into two classes, the holoionic class with a fully or almost fully ionic ground state, and the nonionic class with little or no ionic character. If the total lattice binding energy 2ε₁ (per DA pair) gained in ionizing a DA lattice exceeds the cost 2ε_o of ionizing each DA pair, ε₁ + ε_o less than 0, then the lattice is holoionic. The charge-transfer (CT) band in crystals and in solution can be explained, following Mulliken, by a second-order mixing of states, or by any theory that makes the CT transition strongly allowed, and yet due to a small change in the ground state of the non-interacting components D and A (or D⁺ and A^-). The magnetic properties of the DA crystals are discussed.

Chapter II

A computer program, EWALD, was written to calculate by the Ewald fast-convergence method the crystal Coulomb binding energy E_C due to classical monopole-monopole interactions for crystals of any symmetry. The precision of E_C values obtained is high: the uncertainties, estimated by the effect on E_C of changing the Ewald convergence parameter η, ranged from ± 0.00002 eV to ± 0.01 eV in the worst case. The charge distribution for organic ions was idealized as fractional point charges localized at the crystallographic atomic positions: these charges were chosen from available theoretical and experimental estimates. The uncertainty in E_C due to different charge distribution models is typically ± 0.1 eV (± 3%): thus, even the simple Hückel model can give decent results.

E_C for Wurster's Blue Perchl orate is -4.1 eV/molecule: the crystal is stable under the binding provided by direct Coulomb interactions. E_C for N-Methylphenazinium Tetracyanoquino- dimethanide is 0.1 eV: exchange Coulomb interactions, which cannot be estimated classically, must provide the necessary binding.

EWALD was also used to test the McConnell classification of DA crystals. For the holoionic (1:1)-(N,N,N',N'-Tetramethyl-para- phenylenediamine: 7,7,8,8-Tetracyanoquinodimethan) E_C = -4.0 eV while 2ε_o = 4.6₅ eV: clearly, exchange forces must provide the balance. For the holoionic (1:1)-(N,N,N',N'-Tetramethyl-para- phenylenediamine:para-Chloranil) E_C = -4.4 eV, while 2ε_o = 5.0 eV: again EC falls short of 2ε₁. As a Gedankenexperiment, two nonionic crystals were assumed to be ionized: for (1:1)-(Hexamethyl- benzene:para-Chloranil) E_C = -4.5 eV, 2ε_o = 6.6 eV; for (1:1)- (Napthalene:Tetracyanoethylene) E_C = -4.3 eV, 2ε_o = 6.5 eV. Thus, exchange energies in these nonionic crystals must not exceed 1 eV.

Chapter III

A rapid-convergence quantum-mechanical formalism is derived to calculate the electronic energy of an arbitrary molecular (or molecular-ion) crystal: this provides estimates of crystal binding energies which include the exchange Coulomb inter- actions. Previously obtained LCAO-MO wavefunctions for the isolated molecule(s) ("unit cell spin-orbitals") provide the starting-point. Bloch's theorem is used to construct "crystal spin-orbitals". Overlap between the unit cell orbitals localized in different unit cells is neglected, or is eliminated by Löwdin orthogonalization. Then simple formulas for the total kinetic energy Q^(XT)_λ, nuclear attraction [λ/λ]^XT, direct Coulomb [λλ/λ'λ']^XT and exchange Coulomb [λλ'/λ'λ]^XT integrals are obtained, and direct-space brute-force expansions in atomic wavefunctions are given. Fourier series are obtained for [λ/λ]^XT, [λλ/λ'λ']^XT, and [λλ/λ'λ]^XT with the help of the convolution theorem; the Fourier coefficients require the evaluation of Silverstone's two-center Fourier transform integrals. If the short-range interactions are calculated by brute-force integrations in direct space, and the long-range effects are summed in Fourier space, then rapid convergence is possible for [λ/λ]^XT, [λλ/λ'λ']^XT and [λλ'/λ'λ]^XT. This is achieved, as in the Ewald method, by modifying each atomic wavefunction by a "Gaussian convergence acceleration factor", and evaluating separately in direct and in Fourier space appropriate portions of [λ/λ]^XT, etc., where some of the portions contain the Gaussian factor.

The power of quantum Fourier sampling

How powerful are Quantum Computers? Despite the prevailing belief that Quantum Computers are more powerful than their classical counterparts, this remains a conjecture backed by little formal evidence. Shor's famous factoring algorithm [Shor97] gives an example of a problem that can be solved efficiently on a quantum computer with no known efficient classical algorithm. Factoring, however, is unlikely to be NP-Hard, meaning that few unexpected formal consequences would arise, should such a classical algorithm be discovered. Could it then be the case that any quantum algorithm can be simulated efficiently classically? Likewise, could it be the case that Quantum Computers can quickly solve problems much harder than factoring? If so, where does this power come from, and what classical computational resources do we need to solve the hardest problems for which there exist efficient quantum algorithms?

We make progress toward understanding these questions through studying the relationship between classical nondeterminism and quantum computing. In particular, is there a problem that can be solved efficiently on a Quantum Computer that cannot be efficiently solved using nondeterminism? In this thesis we address this problem from the perspective of sampling problems. Namely, we give evidence that approximately sampling the Quantum Fourier Transform of an efficiently computable function, while easy quantumly, is hard for any classical machine in the Polynomial Time Hierarchy. In particular, we prove the existence of a class of distributions that can be sampled efficiently by a Quantum Computer, that likely cannot be approximately sampled in randomized polynomial time with an oracle for the Polynomial Time Hierarchy.

Our work complements and generalizes the evidence given in Aaronson and Arkhipov's work [AA2013] where a different distribution with the same computational properties was given. Our result is more general than theirs, but requires a more powerful quantum sampler.

Chains of Non-regular de Branges Spaces

We consider canonical systems with singular left endpoints, and discuss the concept of a scalar spectral measure and the corresponding generalized Fourier transform associated with a canonical system with a singular left endpoint. We use the framework of de Branges’ theory of Hilbert spaces of entire functions to study the correspondence between chains of non-regular de Branges spaces, canonical systems with singular left endpoints, and spectral measures.

We find sufficient integrability conditions on a Hamiltonian H which ensure the existence of a chain of de Branges functions in the first generalized Pólya class with Hamiltonian H. This result generalizes de Branges’ Theorem 41, which showed the sufficiency of stronger integrability conditions on H for the existence of a chain in the Pólya class. We show the conditions that de Branges came up with are also necessary. In the case of Krein’s strings, namely when the Hamiltonian is diagonal, we show our proposed conditions are also necessary.

We also investigate the asymptotic conditions on chains of de Branges functions as t approaches its left endpoint. We show there is a one-to-one correspondence between chains of de Branges functions satisfying certain asymptotic conditions and chains in the Pólya class. In the case of Krein’s strings, we also establish the one-to-one correspondence between chains satisfying certain asymptotic conditions and chains in the generalized Pólya class.