Convergence Rates of Spectral Distribution of Random Inner Product Kernel Matrices
AdvisorRider, Brian (Brian C.)
Committee memberYang, Wei-shih, 1954-
Random Geometric Graph
Random Inner Product Kernel Matrix
Permanent link to this recordhttp://hdl.handle.net/20.500.12613/3132
MetadataShow full item record
AbstractThis dissertation has two parts. In the first part, we focus on random inner product kernel matrices. Under various assumptions, many authors have proved that the limiting empirical spectral distribution (ESD) of such matrices A converges to the Marchenko- Pastur distribution. Here, we establish the corresponding rate of convergence. The strategy is as follows. First, we show that for z = u + iv ∈ C, v > 0, the distance between the Stieltjes transform m_A (z) of ESD of matrix A and Machenko-Pastur distribution m(z) is of order O (log n \ nv). Next, we prove the Kolmogorov distance between ESD of matrix A and Marchenko-Pastur distribution is of order O(3\log n\n). It is the less sharp rate for much more general class of matrices. This uses a Berry-Esseen type bound that has been employed for similar purposes for other families of random matrices. In the second part, random geometric graphs on the unit sphere are considered. Observing that adjacency matrices of these graphs can be thought of as random inner product matrices, we are able to use an idea of Cheng-Singer to establish the limiting for the ESD of these adjacency matrices.
ADA complianceFor Americans with Disabilities Act (ADA) accommodation, including help with reading this content, please contact firstname.lastname@example.org
Showing items related by title, author, creator and subject.
Treatment of Orthodontic White Spots: Etiology of Orthodontic White Spot Lesions and Interventional Fluoride Varnish Treatment: A Randomized Control TrialSciote, James J.; Godel, Jeffrey H.; Tellez Merchán, Marisol (Temple University. Libraries, 2015)Fixed orthodontic appliances harbor plaque and perpetuate the formation of early non-cavitated carious lesions. However, not all patients with poor hygiene develop them. It has been established that fluoride varnish can be used to promote enamel remineralization. The study aimed to assess the efficacy of fluoride varnish in remineralizing early non-cavitated lesions among orthodontic patients. A second goal of this study was to elucidate if BMI and obesity increased susceptibility to development of white spot lesions. A randomized control trial was conducted among 25 patients attending the Orthodontic clinic at Temple University. Patients were ages 11-18 and had fixed orthodontic appliances for a minimum of three months. Eleven were randomly assigned to a test group (Enamel Pro® Varnish fluoride varnish application to white spot lesions every two months) and 14 randomly assigned to a control group (reinforcement of oral hygiene instructions). Data collection was completed every two months over a six-month time period. White spot lesion size was measured using the International Caries Detection and Assessment System (ICDAS). Oral hygiene was assessed using Plaque Index (PI) and S. mutans levels were measured using Stripmutans plaque/salivary tests (Dentocult®). Both the control and experimental group had non-significant decreases in non-cavitated carious lesion count. The control group displayed significant increases in Stripmutans salivary scores (p0.05). PI scores decreased in the control group and increased in the experimental group (p>0.05). There was no correlation between BMI and lesion count in the control or experimental group (p>0.05). A 5% sodium fluoride varnish containing Amorphous Calcium Phosphate (Enamel Pro® Varnish) fluoride varnish application was not efficacious in reducing early non-cavitated carious lesions when compared to reinforcing oral hygiene. There is no correlation with BMI and white spot susceptibility.
Phase Transitions, Magnetism and Surface Adsorptions Assessed by Meta-GGA Functionals and Random Phase ApproximationPerdew, John P.; Perdew, John P.; Ruzsinszky, Adrienn; Wu, Xifan; Yuen, Tan; Tao, Jianmin (Temple University. Libraries, 2014)The meta-GGA functionals and random phase approximation are tested for phase transitions and a strongly correlated transition metal oxide in this dissertation. One of the latest meta-GGA functionals is also employed to study the van der Waals bound system in surface science. Our main purpose is to reveal the performance of new exchange-correlation functionals on various properties and systems. We are also interested in seeking the possible relationship between the performance of a semilocal functional and its exchange enhancement factor. We have studied the structural phase transitions in crystalline Si (insulator to metal), SiO2 (insulator to insulator) and Zr (metal to metal) systems, as a test of exchange energy semilocal functionals on Jacob's ladder. Our results confirm the energy-geometry dilemma of GGAs in three systems. The most sophisticated non-empirical meta-generalized gradient approximations (meta-GGAs) such as TPSS (Tao-Perdew-Staroveov-Scuseria) and revTPSS (revised TPSS) give better lattice constants than PBE, but the phase transition parameters (energy difference and transition pressure) are smaller and less realistic than those from the latter GGA. However, the recent functionals of meta-GGA made simple family (MGGA_MS) behave differently to those previous meta-GGAs, predicting larger and more realistic phase transition parameters. Meanwhile, MGGA_MS also delivers the equilibrium geometry of crystalline materials similar to previous non-empirical meta-GGAs. In contrast to semilocal functionals, the nonlocal functionals such as the range-separated hybrid functional HSE06 (Heyd-Scuseria-Ernzerhof) and non-self consistent random phase approximation (RPA) are not only able to give the accurate equilibrium geometry , but also predict the realistic phase transition parameters for Si and SiO2 systems. The ground state of rutile-type vanadium dioxide (R-VO2) represents a great challenge to the current density functional theory. In this dissertation, we investigated the electronic structures and magnetism of R-VO2 using exchange-correlation functionals of all five rungs on Jacob's ladder. Our calculations show that all semilocal functionals (LSDA, GGAs and meta-GGAs) and hybrid functionals (HSE06) stabilize the spin-polarized states (ferromagnetic and anti-ferromagnetic states) over non-magnetic state, which are completely opposite to experimental observation. Surprisingly, LSDA gives the best energetic descriptions for magnetic and non-magnetic phases of R-VO2 among semilocal functionals and HSE06. Otherwise, RPA calculations are highly dependent on the inputs in the spin polarized case. With PBE inputs, RPA also fails, giving lower energies for spin-polarized states than for the non-magnetic phase. Meanwhile, the results are reversed using LSDA inputs. From the computed equilibrium cell volume, we observe the error cancellation in the exchange-correlation hole of most semilocal functionals in the spin-polarized calculations. LSDA and RPA do not fit to this picture. By analyzing the local magnetic moments of vanadium atoms, it is found that the magnetic property predicted from meta-GGA can be related to its exchange enhancement factor. The physisorption of a molecule on a transition metal surface is also another difficult problem in DFT because of the long-range van der Waals interactions. The recently developed MGGA_MS family of density functionals is able to capture a portion of intermediate range dispersion interactions. Therefore, we employed MGGA_MS2 to study the physisorption of CO2 on Pt (111) surface, and the results are compared to those of PBE, PBE+D2 and optB88-vdW methods. The computed binding curves confirm that that MGGA_MS2 indeed captures the van der Waals interactions near the equilibrium binding distance, and the obtained binding distance is also in good agreement with PBE+D2 and optB88-vdW calculations. By computing the electron density difference map (EDDM), we find that the electron densities of CO2 and Pt (111) surface are strongly polarized in optB88-vdW, creating the dipole moments in two subsystems. Such effect is reduced in MGGA_MS2. For PBE, the polarization of electron density is very weak, but not negligible. The α dependence in the exchange enhancement factor of a meta-GGA is the key to capture the intermediate range van der Waals interactions. In summary, a meta-GGA functional can step out of the famous "energy-geometry dilemma" , predicting good lattice constants and phase transition parameters at the same time. With the proper construction, a meta-GGA can even capture a portion of van der Waals interactions. The RPA is usually more accurate than semilocal functionals for many ground state properties. The strongly correlated systems like R-VO2 are still a big challenge to present-day density functional theory. We will continue to seek more accurate exchange-correlation functionals.
A Nonparametric Test for Deviation from RandomnessIglewicz, Boris; Chitturi, Pallavi; Dong, Yuexiao; Chervoneva, Inna (Temple University. Libraries, 2012)There are many existing tests used to determine if a series consists of a random sample. Often these tests have restrictive distributional assumptions, size distortions, or low power for key useful alternative situations. The interest of this dissertation lies in developing an alternative nonparametric test to determine whether a series consists of a random sample. The proposed test detects deviations from randomness, without a priori distributional assumption, when observations are not independent and identically distributed (i.i.d.), which is suitable for our motivating stock market index data. Departures from i.i.d. are tested by subdividing data into subintervals and then using a conditional probability measure within intervals as a binomial test. This nonparametric test is designed to detect deviations of neighboring observations from randomness when the data set consists of time series observations. Simulation results confirm correct test size for varied distributions and good power for detecting alternative cases. This test is compared to a number of other popular methods and shown to be a competitive alternative. Although the proposed test may be applicable to multiple areas, this dissertation is mostly interested in applications to stock market and regression data. The proposed test is effectively illustrated with the common three stock market index data sets using a newly created transformation, and shown to perform exceptionally well.