Identities from representation theory

We give a new Jacobi--Trudi-type formula for characters of finite-dimensional irreducible representations in type $C_n$ using characters of the fundamental representations and non-intersecting lattice paths. We give equivalent determinant formulas for the decomposition multiplicities for tensor powers of the spin representation in type $B_n$ and the exterior representation in type $C_n$. This gives a combinatorial proof of an identity of Katz and equates such a multiplicity with the dimension of an irreducible representation in type $C_n$. By taking certain specializations, we obtain identities for $q$-Catalan triangle numbers, the $q,t$-Catalan number of Stump, $q$-triangle versions of Motzkin and Riordan numbers, and generalizations of Touchard's identity. We use (spin) rigid tableaux and crystal base theory to show some formulas relating Catalan, Motzkin, and Riordan triangle numbers.Comment: 68 pages, 8 figure

Nonlinear Approximations in Filter Design and Wave Propagation

This thesis has two parts. In both parts we use nonlinear approximations to obtain accurate solutions to problems where traditional numerical approaches rapidly become computationally infeasible. The first part describes a systematic method for designing highly accurate and efficient infinite impulse response (IIR) and finite impulse response (FIR) filters given their specifications. In our approach, we first meet the specifications by constructing an IIR filter, without requiring the filter to be causal, and possibly with a large number of poles. We then construct, for any given accuracy, an optimal IIR version of such filter. Finally, also for any given accuracy, we convert the IIR filter to an efficient FIR filter cascade. In this FIR approximation, the non-causal part of the IIR filter only introduces an additional delay. Because our IIR construction does not have to enforce causality, the filters we design are more efficient than filters designed by existing methods. The second part describes a fast algorithm to propagate, for any desired accuracy, a time-harmonic electromagnetic field between two planes separated by free space. The analytic formulation of this problem (circa 1897) requires the evaluation of the Rayleigh-Sommerfeld integral. If the distance between the planes is small, this integral can be accurately evaluated in the Fourier domain; if the distance is large, it can be accurately approximated by asymptotic methods. The computational difficulties arise in the intermediate region where, in order to obtain an accurate solution, it is necessary to apply the oscillatory Rayleigh-Sommerfeld kernel as is. In our approach, we accurately approximate the kernel by a short sum of Gaussians with complex exponents and then efficiently apply the result to input data using the unequally spaced fast Fourier transform. The resulting algorithm has the same computational complexity as methods based on the Fresnel approximation. We demonstrate that while the Fresnel approximation may provide adequate accuracy near the optical axis, the accuracy deteriorates significantly away from the optical axis. In contrast, our method maintains controlled accuracy throughout the entire computational domain

Simulating fermionic systems on classical and quantum computing devices

This thesis presents a theoretical study of topics related to the simulation of quantum mechanical systems on classical and quantum computers. A large part of this work focuses on strongly interacting fermionic systems, more precisely, the behavior of electrons in presence of strong magnetic fields. We show how the energy spectrum of a Hamiltonian describing the fractional quantum Hall effect can be computed on a quantum computer and derive a closed form for the Hamiltonian coefficients in second quantization. We then discuss a mean-field method and a multi-reference state approach that allow for an efficient classical computation and an efficient initial state preparation on a quantum computer. The second part of the thesis presents a detailed description on how long-range interacting fermionic systems can be simulated on classical computers using a variational method, introduce an Ansatz which could potentially simplify numerical simulations and give an explicit quantum circuit that shows how the variational state can be used as an initial state and how it can implemented on a quantum computer. In the last part, two novel protocols are presented that generate a variety of prominent many-body operators from two-body interactions and show how these protocols improve over previous construction schemes for a number of important examples.Diese Arbeit behandelt verschiedene zentrale Probleme theoretischer Natur, welche im Rahmen der Simulation quantenmechanischer Systeme auf klassischen und Quantencomputern auftreten. Ein Großteil dieser Arbeit beschäftigt sich mit stark wechselwirkendenden fermionischen Systemen, genauer gesagt, dem Verhalten von Elektronen innerhalb eines starken Magnetfelds. Es wird dargelegt, wie das Energiespektrum des Quanten- Hall-Effekt-Hamiltonoperators auf einem Quantencomputer berechnet werden kann, und es werden geschlossene Ausdrücke für dessen Hamilton-Koeffizienten in zweiter Quantisierung hergeleitet. Anschließend werden sowohl ein Molekularfeld- als auch ein Multi- Referenz-Ansatz diskutiert, welche eine effiziente Berechnung auf klassischen Rechnern zulassen sowie eine effiziente Implementierung auf Quantencomputern ermöglichen. Der zweite Teil dieser Arbeit erläutert, wie man langreichweitige, wechselwirkende fermionische Systeme mit Hilfe einer neuen Variationsmethode, welche über die Molekularfeldnäherung hinaus geht, auf einem klassischen Computer simulieren kann. Es wird darüber hinaus ein alternativer Ansatz vorgestellt, der Teile dieser Variationsmethode vereinfachen könnte, und gezeigt, wie sich der Ansatz auf einem Quantencomputer realisieren lässt. Im letzten Teil werden zwei neue Methoden vorgestellt, welche es ermöglichen, eine Reihe wichtiger Vielteilchen-Operatoren aus Zweiteilchen-Wechselwirkungen zu erzeugen. Beide Methoden werden durch eine Vielzahl an wichtigen Beispielen veranschaulicht

Structure Constants in N=4\mathcal{N}=4 SYM at Finite Coupling as Worldsheet gg-Function

We develop a novel nonperturbative approach to a class of three-point functions in planar $\mathcal{N}=4$ SYM based on Thermodynamic Bethe Ansatz (TBA). More specifically, we study three-point functions of a non-BPS single-trace operator and two determinant operators dual to maximal Giant Graviton D-branes in AdS$_5\times$S$^{5}$. They correspond to disk one-point functions on the worldsheet and admit a simpler and more powerful integrability description than the standard single-trace three-point functions. We first introduce two new methods to efficiently compute such correlators at weak coupling; one based on large $N$ collective fields, which provides an example of open-closed-open duality discussed by Gopakumar, and the other based on combinatorics. The results so obtained exhibit a simple determinant structure and indicate that the correlator can be interpreted as a generalization of $g$-functions in 2d QFT; an overlap between an integrable boundary state and a state corresponding to the single-trace operator. We then determine the boundary state at finite coupling using the symmetry, the crossing equation and the boundary Yang-Baxter equation. With the resulting boundary state, we derive the ground-state $g$-function based on TBA and conjecture its generalization to other states. This is the first fully nonperturbative proposal for the structure constants of operators of finite length. The results are tested extensively at weak and strong couplings. Finally, we point out that determinant operators can provide better probes of sub-AdS locality than single-trace operators and discuss possible applications.Comment: 224 pages; v3 new results, appendices, and references adde

Modeling single-phase flow and solute transport across scales

textFlow and transport phenomena in the subsurface often span a wide range of length (nanometers to kilometers) and time (nanoseconds to years) scales, and frequently arise in applications of CO₂ sequestration, pollutant transport, and near-well acid stimulation. Reliable field-scale predictions depend on our predictive capacity at each individual scale as well as our ability to accurately propagate information across scales. Pore-scale modeling (coupled with experiments) has assumed an important role in improving our fundamental understanding at the small scale, and is frequently used to inform/guide modeling efforts at larger scales. Among the various methods, there often exists a trade-off between computational efficiency/simplicity and accuracy. While high-resolution methods are very accurate, they are computationally limited to relatively small domains. Since macroscopic properties of a porous medium are statistically representative only when sample sizes are sufficiently large, simple and efficient pore-scale methods are more attractive. In this work, two Eulerian pore-network models for simulating single-phase flow and solute transport are developed. The models focus on capturing two key pore-level mechanisms: a) partial mixing within pores (large void volumes), and b) shear dispersion within throats (narrow constrictions connecting the pores), which are shown to have a substantial impact on transverse and longitudinal dispersion coefficients at the macro scale. The models are verified with high-resolution pore-scale methods and validated against micromodel experiments as well as experimental data from the literature. Studies regarding the significance of different pore-level mixing assumptions (perfect mixing vs. partial mixing) in disordered media, as well as the predictive capacity of network modeling as a whole for ordered media are conducted. A mortar domain decomposition framework is additionally developed, under which efficient and accurate simulations on even larger and highly heterogeneous pore-scale domains are feasible. The mortar methods are verified and parallel scalability is demonstrated. It is shown that they can be used as “hybrid” methods for coupling localized pore-scale inclusions to a surrounding continuum (when insufficient scale separation exists). The framework further permits multi-model simulations within the same computational domain. An application of the methods studying “emergent” behavior during calcite precipitation in the context of geologic CO₂ sequestration is provided.Petroleum and Geosystems Engineerin