Numerical solutions of elliptic inverse problems via the equation error method

To estimate a parameter in an elliptic boundary value problem, the method of equation error chooses the value that minimizes the error in the PDE and boundary condition (the solution of the BVP having been replaced by a measurement). The estimated parameter converges to the exact value as the measured data converge to the exact value, provided Tikhonov regularization is used to control the instability inherent in the problem. The error in the estimated solution can be bounded in an appropriate quotient norm; estimates can be derived for both the underlying (infinite-dimensional) problem and a finite-element discretization that can be implemented in a practical algorithm. Numerical experiments demonstrate the efficacy and limitations of the method.

Extracellular matrix of the charophycean green algae

A comprehensive knowledge of cell wallstructure and function throughout the plant kingdom is essential to understanding cell wall evolution. The fundamental understanding of the charophycean green algal cell wall is broadening. The similarities and differences that exist between land plant and algal cell walls provide opportunities to understand plant evolution. A variety of polymers previously associated with higher plants were discovered in the charophycean green algae (CGA), including homogalacturonans, cross-linking glycans, arabinogalactan protein, β-glucans, and cellulose. The cellulose content of CGA cell walls ranged from 6% to 43%, with the higher valuescomparable to that found in the primary cell wall of land plants (20-30%). (1,3)β-glucans were found in the unicellular Chlorokybus atmophyticus, Penium margaritaceum, and Cosmarium turpini, the unbranched filamentous Klebsormidium flaccidum, and the multicellular Chara corallina. The discovery of homogalacturonan in Penium margaritaceum representsthe first confirmation of land plant-type pectinsin desmids and the second rigorous characterization of a pectin polymer from the charophycean algae. Homogalacturonan was also indicated from the basal species Chlorokybus atmophyticus and Klebsormidium flaccidum. There is evidence of branched pectins in Cosmarium turpini and linkage analysis suggests the presence of type I rhamnogalacturonan (RGI). Cross-linking β-glucans are associated with cellulose microfibrils during land plant cell growth, and were found in the cell wall of CGA. The evidence of mixed-linkage glucan (MLG) in the 11 charophytesis both suprising and significant given that MLG was once thought to be specific to some grasses. The organization and structure of Cosmarium turpini and Chara corallina MLG was found to be similar to that of Equisetumspp., whereas the basal species of the CGA, Chlorokybus atmophyticus and Klebsormidium flaccidum, have unique organization of alternating of 3- and 4-linkages. The significance of this result on the evolution of the MLG synthetic pathway has yet to be determined. The extracellular matrix (ECM) of Chlorokybus atmophyticus, Klebsormidium flaccidum, and Spirogyra spp. exhibits significant biochemical diversity, ranging from distinct “land plant” polymers to polysaccharides unique to these algae. The neutral sugar composition of Chlorokybus atmophyticus hot water extract and Spirogyra extracellular polymeric substance (EPS), combined with antibody labeling results, revealed the distinct possibility of an arabinogalactan protein in these organisms. Polysaccharide analysis of Zygnematales (desmid) EPS, indicated a probable range of different EPS backbones and substitution patterns upon the core portions of the molecules. Desmid EPS is predominately composed of a complex matrix of branched, uronic acid containing polysaccharides with ester sulfate substitutions and, as such, has an almost infinite capacity for various hydrogen bonding, hydrophobic interaction and ionic cross-bridging motifs, which characterize their unique function in biofilms. My observations support the hypothesis that members of the CGA represent the phylogenetic line that gave rise to vascular plants and that the primary cell wall of vascular plants many have evolved directly from structures typical of the cell wall of filamentous green algae found in the charophycean green algae.

Applications of finite geometries to designs and codes

This dissertation concerns the intersection of three areas of discrete mathematics: finite geometries, design theory, and coding theory. The central theme is the power of finite geometry designs, which are constructed from the points and t-dimensional subspaces of a projective or affine geometry. We use these designs to construct and analyze combinatorial objects which inherit their best properties from these geometric structures. A central question in the study of finite geometry designs is Hamada’s conjecture, which proposes that finite geometry designs are the unique designs with minimum p-rank among all designs with the same parameters. In this dissertation, we will examine several questions related to Hamada’s conjecture, including the existence of counterexamples. We will also study the applicability of certain decoding methods to known counterexamples. We begin by constructing an infinite family of counterexamples to Hamada’s conjecture. These designs are the first infinite class of counterexamples for the affine case of Hamada’s conjecture. We further demonstrate how these designs, along with the projective polarity designs of Jungnickel and Tonchev, admit majority-logic decoding schemes. The codes obtained from these polarity designs attain error-correcting performance which is, in certain cases, equal to that of the finite geometry designs from which they are derived. This further demonstrates the highly geometric structure maintained by these designs. Finite geometries also help us construct several types of quantum error-correcting codes. We use relatives of finite geometry designs to construct infinite families of q-ary quantum stabilizer codes. We also construct entanglement-assisted quantum error-correcting codes (EAQECCs) which admit a particularly efficient and effective error-correcting scheme, while also providing the first general method for constructing these quantum codes with known parameters and desirable properties. Finite geometry designs are used to give exceptional examples of these codes.

Problems in the classical calculus of variations

This dissertation concerns convergence analysis for nonparametric problems in the calculus of variations and sufficient conditions for weak local minimizer of a functional for both nonparametric and parametric problems. Newton's method in infinite-dimensional space is proved to be well-defined and converges quadratically to a weak local minimizer of a functional subject to certain boundary conditions. Sufficient conditions for global converges are proposed and a well-defined algorithm based on those conditions is presented and proved to converge. Finite element discretization is employed to achieve an implementable line-search-based quasi-Newton algorithm and a proof of convergence of the discretization of the algorithm is included. This work also proposes sufficient conditions for weak local minimizer without using the language of conjugate points. The form of new conditions is consistent with the ones in finite-dimensional case. It is believed that the new form of sufficient conditions will lead to simpler approaches to verify an extremal as local minimizer for well-known problems in calculus of variations.

OUT-OF-STEP DETECTION BASED ON ZUBOV’S APPROXIMATION BOUNDARY METHOD

Disturbances in power systems may lead to electromagnetic transient oscillations due to mismatch of mechanical input power and electrical output power. Out-of-step conditions in power system are common after the disturbances where the continuous oscillations do not damp out and the system becomes unstable. Existing out-of-step detection methods are system specific as extensive off-line studies are required for setting of relays. Most of the existing algorithms also require network reduction techniques to apply in multi-machine power systems. To overcome these issues, this research applies Phasor Measurement Unit (PMU) data and Zubov’s approximation stability boundary method, which is a modification of Lyapunov’s direct method, to develop a novel out-of-step detection algorithm. The proposed out-of-step detection algorithm is tested in a Single Machine Infinite Bus system, IEEE 3-machine 9-bus, and IEEE 10-machine 39-bus systems. Simulation results show that the proposed algorithm is capable of detecting out-of-step conditions in multi-machine power systems without using network reduction techniques and a comparative study with an existing blinder method demonstrate that the decision times are faster. The simulation case studies also demonstrate that the proposed algorithm does not depend on power system parameters, hence it avoids the need of extensive off-line system studies as needed in other algorithms.

THREE HUNDRED YEARS OF THE ST. PETERSBURG PARADOX

The St. Petersburg Paradox was first presented by Nicholas Bernoulli in 1713. It is related to a gambling game whose mathematical expected payoff is infinite, but no reasonable person would pay more than $25 to play it. In the history, a number of ideas in different areas have been developed to solve this paradox, and this report will mainly focus on mathematical perspective of this paradox. Different ideas and papers will be reviewed, including both classical ones of 18th and 19th century and some latest developments. Each model will be evaluated by simulation using Mathematica.