Here we present an error estimation method developed for the numerical solutions of interface problems. The goal is to approximate the error of a given numerical approximation to the true solution. As opposed to classic a priori error estimates which may seek to bound the norm of the error by some expression involving a discretization parameter, the focus of this work will be to produce numerical values of the errors associated with a particular solution. The spatial distribution of these errors is also available. This type of estimate is useful in settings where simulations are used as replacements for experiments. In such settings it is important to be able to understand, and, if possible, quantify such errors We begin by describing the Method of Nearby Problems, [1, 2, 3], which is motivated by defect correction methods [4]. The general idea of the Method of Nearby Problems is to construct a problem for which an analytic solution is known and, at the same time, is a small perturbation of the original problem of interest. With this idea in mind, we move forward to describe a modification of the original Method of Nearby Problems that we will study here Although our ultimate goal is to apply the error estimates to numerical solutions of interface problems, we first examine their effectiveness on continuous elliptic problems. We will construct these estimates and study their convergence to the true error as the mesh is refined. We also provide analysis of the error estimate for a one-dimensional example The next step will be to examine the application and performance of the estimates to numerical solutions of interface problems. Our model problem will be an elliptic boundary value problem in which the coefficients are discontinuous across an internal boundary. We will obtain numerical solutions of these problems using the Immersed Interface Method [5, 6]. The Immersed Interface Method solutions preserve the discontinuities and irregularities of the solution at the interface We will next examine solutions obtained using regularization methods. Motivated by the Immersed Boundary Method [7], we regularize the interface problem and then use numerical methods which apply to smooth problems. Here we must analyze both the error due to regularization, as well as discretization error. We will apply this methodology to the discontinuous boundary value problem and provide an analysis of the regularization error for a representative one-dimensional problem Finally, we discuss future directions such as generalizing the method to include Immersed Interface solutions of problems with singular sources. We also discuss requirements of error estimates for transient problems / acase@tulane.edu

Field observations of faulting and associated deformation are used here to reconstruct the structural and kinematic evolution of northern Owens Valley, California. This work consists of three stand-alone research contributions (Chapters Three, Four, and Five). Chapter Three presents a model for the structural evolution of northern Owens Valley; focusing on the origin and evolution of the 'Coyote Warp', as well as the relationship between normal shear along the Sierran Nevada range-front and dextral shear along the Owens Valley fault zone. This model relies on the theoretical relationship between fault spacing, fault dip, and seismogenic thickness in order to make predictions of crustal-scale conjugate normal fault intersection. Application of this model suggests that the structural evolution of northern Owens Valley can be explained in the context of a failed conjugate system, whereby fault intersection within the seismogenic crust results in the locking of one of the graben faults, and subsequent asymmetric range uplift and adjacent basin subsidence. Chapter Four presents a geologically based extensional slip rate history for the central portion of northern Owens Valley. Results suggest that the rate of extensional strain increased significantly since Middle Pleistocene time. These results are in agreement with similar observations of extension within and around northern Owens Valley, and correspond to a decrease in nearby rates of dextral shear over the same time interval. These observations are explained by a counter-clockwise rotation in the orientation of regional shear since Middle Pleistocene time. Furthermore, results from this study contribute to a geologically based extensional slip budget that is in agreement with geodetic based estimates of present-day strain accumulation. In Chapter Five, observations of fault length from several normal fault populations are used to examine the mechanisms that control the distribution of strain within the Eastern California Shear Zone. Results suggest that boundary fault spacing within shear induced fault networks plays a significant role in the redistribution of slip by placing geometric limitations on intermediary cross-cutting normal faults. Such redistribution is expected to occur over a timescale that is related to the lifespan of these constrained faults / acase@tulane.edu

This thesis is focused on developing a numerical method for solving a problem where fluid flow governed by Stokes equations is coupled with flow through a porous medium which is governed by Darcy's equations. For closed smooth boundaries, flow in each region is represented as boundary integrals with densities to be determined to satisfy the boundary conditions. With the solution in this form the dimension of the problem reduces but the integrands are singular. The proposed numerical method is based on regularizing the integrands and discretizing the integrals by a quadrature. The numerical error is discussed and some convergence results are shown for Stokes flow in 3D. A way of reducing the error is discussed. Next, the fluid quantities in two dimensions are reformulated as single and double layer potentials and a solution method with higher accuracy is proposed. It is based on regularizing the kernels and subtracting the highest regularization error term. The method is applied to several test cases of Stokes and Darcy flows, and an increase of the convergence rate from first to second is observed. In the coupled case, the two solutions are coupled through appropriate interface conditions, which in this case do not require the velocity to be continuous. An example of a coupled problem is presented A fast summation technique for impulse methods in three dimensions is presented. Impulse methods provide a representation of Euler flows in terms of impulse variables. The fundamental solution of Darcy flow is a potential dipole, thus this is a robust method for computing Darcy velocity for dense grids / acase@tulane.edu

In standard numerical solvers for the Navier-Stokes equations, a Poisson problem must be solved on a uniform grid at each time step. Typically, the pressure or some related quantity is determined from this problem, and, in order to achieve high accuracy, careful treatment must be taking when solving the problem. This is especially true if an elastic structure is submerged in the flow because the pressure may be discontinous or have discontinous gradients at the structure. In applications in which such a structure is closed, methods have been developed that lead to the correct behavior of the solution. Jump conditions, or other information about the behavior at the structure is used to modify a standard method in order to capture the correct behavior. However, in applications with open immersed structures, such as swimming organisms, the literature does not contain much information about proper behavior near the structure This dissertation works towards filling these gaps in the literature. As a model of the situation, functions which are harmonic in the planar compliment of an arc, and with Dirichlet conditions on the arc are considered. For the cases of line segments and arcs of circles, specific properties of the behavior of such functions are determined using recent work by Jiang and Rokhlin The numerical method presented is a modified finite difference scheme with additional treatment at grid nodes near the boundary. Such treatment is necessary since the gradients of the solution are discontinuous across the boundary and unbounded at the endpoints Several examples are given. The examples show that applying a standard finite difference method does not lead to proper convergence. This fact is linked to the singular endpoint behavior of the solution. However, other examples show that the method developed in this dissertation has the same order of accuracy as the finite difference method has when applied to approximate smooth functions / acase@tulane.edu

In 2006, David Bellamy proved that if X is a pseudo-circle in the complex plane which separates 0 from infinity, and if n epsilon Z+, then the preimage of X under z zn is also a pseudo-circle [1]. He ended his paper with two questions. The first question asks whether the preimage under z zn of a hereditarily indecomposable continuum which is irreducible with respect to separating 0 from infinity is necessarily hereditarily indecomposable. The second questions asks whether the preimage under z zn of a continuum which properly contains a pseudo-circle can ever be hereditarily indecomposable. In this paper, the author provides affirmative answers to both questions. In addition, the author explores the behavior of other properties of continua, when taking their preimage under z zn, and gives various examples of interesting continua which can be constructed using this technique / acase@tulane.edu

Bulk samples collected from the Upper Cambrian (Sunwaptan) Hellnmaria Member of the Notch Peak Formation in Utah and the Upper Cambrian--Ordovician (Sunwaptan--Skullrockian) Wilberns and Tanyard Formations in Texas have yielded faunas of linguliform brachiopods that include many new taxa. Two new genera and eight new species are described. A new genus is proposed for species originally assigned to Angulotreta Palmer, 1954. In Utah, the stratigraphic ranges of several previously described species are extended. The two collections allow biostratigraphic correlation within Laurentia as well as globally. New zonation based on linguliform brachiopods is proposed for the Sunwaptan and lower Skullrockian stages of North America The Hellnmaria Member was deposited in a deep subtidal environment with possible access to the open ocean. The Utah fauna has affinities to coeval faunas in Kazakhstan and includes Zhanatella rotunda Koneva, 1986. This species was described previously from Kazakhstan as well as the Montagne Noire region of France and has potential for global biostratigraphy In Texas, collections from the Taenicephalus and lower Idahoia trilobite zones yielded a fauna with little affinity to brachiopods at a similar stratigraphic position in Utah but with strong affinity to a fauna from the same interval in Wyoming. Upper Sunwaptan and lowest Skullrockian strata yielded a fauna with close affinities to the Utah fauna. There is complete turnover of linguliform fauna in coincidence with two trilobite extinction events A lingulate specimen from the Hellnmaria Member exhibits a repair scar on its larval shell. Because lingulate brachiopod larvae are planktotrophic, the individual is interpreted to have been part of the pelagic realm at injury. The injury consists of a visible break and several areas of damage to the exterior of the shell, all consistent with damage done deliberately by a predator. This implies that durophagous predation was a part of the planktic realm during the Cambrian, and that the planktic realm was more modern in its structure than previously realized. This suggests that brachiopods may have been under predation pressure as larvae as well as adults, and this pressure may have played a role in their evolution and extinction / acase@tulane.edu

This work presents some instances of Experimental Mathematics in Number Theory. The arithmetical properties of an arctangent sum is explored, in particular, its connections to different mathematical objects are shown. One of this connections is the link between the sum and a sequence of type tn=Pnt n-1, 0.0.1 where P(x) is a polynomial with integer coefficients. These type of sequences arise in different types of problems like the integration of rational functions and the evaluation of infinite sums The asymptotic behavior of the p-adic valuation for sequences of type (0.0.1) is described. In particular, the connection between the zeros of the polynomials P(x) in the finite field Z/pZ and the growth of the p-adic valuation is presented Finally, in the last chapter a relation between Dirichlet Series and the evaluation of a class of logarithmic integrals is studied / acase@tulane.edu

Cooperative assembly of inorganic species/surfactant can lead to highly ordered composite mesostrucrures and mesoporous silica of various mesophases (e.g. cubic, hexagonal, lamellar structure). Oriented composite mesostructure and mesoporous silica have received much attention due to its favorable orientations in broad applications. Synthesis of the oriented composite mesostructures and mesoporous silica has been accomplished with assistance of either external field (e.g. magnetic field, mechanical field) or by controlling nature of interface between composite mesostructure and substrate. It is still a challenge to control number and length of oriented hexagonal tubules or mesopores within hexagonal composite mesostructure or mesoporous silica. This dissertation demonstrates capillary growth of composite mesostructure within nanoscale channels to obtain oriented composite mesostructure and corresponding mesoporous nanowires with various composition and well-controlled nanowire diameter. The dissertation also discusses confined effect on the self-assembly process and generalizes a confined assembly approach to synthesize nanowires of a variety of mesoporous mateials. The highly ordered oriented mesoporous materials in nanoscale pore channels are desirable for many of possibility to be realized in applications. The dissertation also has demonstrated novel oriented mesostructured nanowires via electrodeposition using oriented mesoporous silica nanowires as template / acase@tulane.edu

The research project highlighted in this thesis utilizes a systematic synthetic protocol for synthesis of multimetallic dithiolene complexes. Monometallic dithiolene complexes are widely studied and easy to synthesize, whereas, multimetallic dithiolene complexes are rare. The synthetic protocols available in the literature for trimetallic complexes are not well developed and often lack control over specific product formation. Therefore we devised a convergent approach for the synthesis of multimetallic dithiolene complexes with a high degree of control over product formation Recently, we have used 1,3-dithiole-2-one (R2C2S 2C=O) and 1,2-di-n-alkyltin complexs (R2S 2C2SnR'2) as protected forms of dithiolene ligands for synthesis of mono-, di- and trimetallic dithiolene complexes. Dimetallic compounds [(P-P)M(S2C6H2S2)M(P-P)] (M = Ni, Pd; (P-P) = chelating bis(phosphine)) are prepared from O=CS 2C6H2S2C=O or nBu2SnS2C6H2S 2SnnBu2, which are protected forms of 1,2,4,5-benzenetetrathiolate (btt). Selective monodeprotections of O=CS2C6H2S2C=O or nBu 2SnS2C6H2S2Sn nBu2 lead to [(P-P)Ni(S2C6H 2S2C=O)] or [(P-P)Ni(S2C6H2S 2SnnBu2)]; the former is used to prepare trimetallic compounds [(dcpe)Ni(S2C6H 2S2)M(S2C6H2S2)Ni(dcpe)] (M = Ni or Pt; dcpe = 1,2-bis(dicyclohexylphosphino)ethane), in which, 1,2,4,5-benzenetetrathiolate acts as the connector. Dimetallic compounds of type [(R2C 2S2)M((PPh2)2C6H2(PPh 2)2)M(S2C2R2)] (M = Ni, Pd, Pt; R = CH3, CH3O-p-C6H 4) are prepared from R2C2S2SnR' 2 and Cl2M((PPh2)2C6H 2(PPh2)2)MCl2, in which 1,2,4,5-tetrakis(diphenylphosphino)benzene (tppb) acts as the connector. Both classes of dimetallic compounds (btt- and tppb-bridged) are redox active and display two oxidation processes, of which the first is generally reversible. The tppb-bridged dimetallic complexes are characterized by two simultaneous 1e- oxidations, whereas the btt-bridged dimetallic complexes are characterized by single 1e- oxidation processes. Electrochemical studies reveal that 1,2,4,5-benzenetetrathiolate is the redox-active moiety in the btt-linked compounds, while in the tppb-linked compounds, 1,2,4,5-tetrakis(diphenylphosphino)benzene ligand is effectively an insulator between redox active metallodithiolene fragments Structural identification of [(dcpe)Ni(S2C6H 2S2)Ni(dcpe)][BArF24] reveals appreciable shortening and lengthening of C--S and C--C bond distances, respectively, within the tetrathioarene fragment compared to charge-neutral [(dcpe)Ni(S2C 6H2S2)Ni(dcpe)], indicating this to be the redox active moiety. Near IR spectroscopy upon solution-generated cations ([(dcpe)Ni(S 2C6H2S2)Ni(dcpe)]+ and [(R2C2S2)M((PPh2)2C 6H2(PPh2)2)M(S2C2R 2)]+ M = Ni, Pd, Pt; R = CH3O-p-C 6H4) and upon the neutral trinickel complex reveals multiple intense absorptions in the 700-1400 nm region The reaction of P4S10 with acyloins, RC(O)CH(OH)R, in refluxing dioxane, followed by the addition of alkylating agents, forms dithiolene thiophosphoryl thiolate compounds, (R2C2S 2)P(S)(SR´), which are readily isolated and purified. Deprotection of ((H3CO-p-C6H4)2 C2S2)P(S)(SMe) in MeO-/MeOH, followed by addition of NiCl2˙6H2O and then I2, produces square planar [Ni(S2C2(C6H4- p-OCH3)2)2] in 93% yield / acase@tulane.edu

We used mild reaction conditions in a base driven process to isolate new lantho-carboxylates while exploring the effects of deprotonation and functional group character on the development of crystalline materials. To this end several functionalized carboxylic acids were reacted with selected lanthanide (III) salts in aqueous media under relatively mild conditions. 23 unique crystal structures were reported with new crystalline compounds involving lantho-orotates, quinates, hydroxypicolinates and hydroxynicotinates along with examples of hetero-ligand lanthanide complexes. Titration studies gave insight into the solution dynamics of the mixed lanthanide salt/weak acid systems establishing an experimentally observable deprotonation order for these species in water and a possible explanation to some observed behaviors in these systems and crystalline products. Exploratory research broke away from single-acid component reactions in favor of a more complex addition scheme which employed Eu(OH)3 as a starting material along with two types of weak acid. These reactions helped set the stage for future research prospects in the area of hetero-ligand lanthanide chemistry / acase@tulane.edu