Renormalization is a powerful tool showing up in different contexts of mathematics and physics. In the context of circle diffeomorphisms, the renormalization operator acts like a microscope and allows to study the dynamics of a circle diffeomorphism on a small scale. The convergence of renormalization leads to a proof of the so-called rigidity theorem, which classifies the dynamics of circle diffeomorphisms geometrically: the conjugacy between $C^3$ circle diffeomorphism with Diophantine rotation number and the corresponding rotation is $C^1$. In this thesis, we define the renormalization of circle diffeomorphisms and study its dynamics. In particular, we prove that the renormalization of orientation preserving $C^3$ circle diffeomorphisms with irrational rotation number of bounded type converges to rotations at exponential speed. We also introduce the necessary relevant concepts such as rotation number, distortion and non-linearity and discuss some of their properties. This thesis is a summary and supplement to the book One-Dimensional Dynamics: from Poincaré to Renormalization. / Omnormalisering är en kraftfull teknik som dyker upp i olika sammanhang inom matematik och fysik. I samband med cirkeldiffeomorfier är omnormaliseringsoperatorn ett dynamiskt system, som fungerar som ett mikroskop och gör att vi kan studera dynamiken hos en cirkeldiffeomorfi på en liten skala. Omnormaliseringens konvergens leder till ett bevis för det så kallade rigiditetssatsen, som klassificerar dynamiken hos cirkeldiffeomorfier geometriskt: konjugatet mellan $C^3$ cirkeldiffeomorfi med diofantiska rotationstal och den motsvarande rotationen är $C^1$. I denna avhandling definierar vi omnormaliseringen av cirkeldiffeomorfier och studerar dess dynamik. I synnerhet bevisar vi att omnormaliseringen av orienteringsbevarande $C^3$ cirkeldiffeomorfier med irrationellt rotationstal av begränsad typ konvergerar till rotationer med exponentiell hastighet. Vi introducerar också nödvändiga och relevanta begrepp så som rotationstal, distorsion och icke-linjäritet och diskuterar några av deras egenskaper. Denna avhandling är en sammanfattning och ett komplement till boken One- Dimensional Dynamics: from Poincaré to Renormalization.

Ph. D.

Accurate theoretical predictions of optical rotation are of substantial utility to the chemical community enabling the determination of absolute configuration without the need for poten- tially lengthy total synthesis. The requirements for robust calculation of gas-phase optical rotation are well understood, but too expensive for routine use. In an effort to reduce this cost we have examined the performance of the LPol and ORP basis sets, created for use in density functional theory calculations of optical rotation, finding that at the coupled cluster level of theory they perform the same or better than comparably sized general basis sets that are often used. We have also examined the performance of a perturbational approach to inclusion of explicit solvent molecules in an effort to extend the calculation of response properties from the gas phase to the condensed phase. This N-body approach performs admirably for interaction energies and even dipole moments but breaks down for optical rotation, exhibiting large basis set superposition errors and requiring higher-order terms in the expansion to provide reasonable accuracy. In addition, we have begun the process of implementing a gauge invariant version of coupled cluster response properties to address the fundamentally unphysical lack of gauge invariance in coupled cluster optical rotations. Correcting this problem, which arises from the non- variational nature of the coupled cluster wavefunction, involves reformulating the response amplitude and function expressions and solving for all necessary amplitudes simultaneously. / Ph. D.

Master of Science

In a large part of the soil fertility studies that have been attempted in the past, a great number of the results obtained were based almost entirely on crop yields. These yields were used as the final criterion upon which to draw conclusions as to whether a given system of soil management was a profitable one. At the Virginia Agricultural Experiment Station, Blacksburg, Virginia, a series of rotation and continuous cropping experiments have been undergoing a specified treatment for a period of 24 years. Conclusions as to the expediency of the practices followed have been approached from a purely crop yield basis. This method is serving its purpose in showing exactly how much the land is producing at the present time under a given soil management practice. It is believed, however, that a soil management program goes further than this. A practice based on crop production alone may prove, over a long period of time, to be a poor one. Perhaps just as important are the conditions arising within the soil relating to these practices. Whether or not the system evolved is one of soil building rather than one of soil depletion, is a question which necessarily must be considered before any recommendation as to the soil management practices to be followed, may be attempted. / Master of Science

Master of Science

The effectiveness of exercise has been increasingly studied as exercise has been popular for the improvement of physical performance and rehabilitation of lumbar spine. A variety of exercises have been used to reduce back pain or spinal degeneration. However, there are no studies to determine effects of exercise on lumbar spine kinematics, including lumbar-pelvic coordination and instantaneous axis of rotation. The current study aimed to examine these lumbar spine kinematical changes due to exercise and therapy. We hypothesized that exercise and therapy will affect the changes of lumbar spine kinematics. Lumbar-Pelvic motions were recorded from 86 healthy subjects while performing lifting and lowering tasks of 10% and 25% of body weight. The influence of exercise was quantified from coefficients of curve-fitting for pelvic and lumbar angles. There was a significant difference (p<0.05) for the range of lumbar motion (distribution, D) between the control group and the cardiovascular exercise group after 12-week program. However, there was no significance for lumbar-pelvic coordination, C. A second study was performed to investigate the changes of instantaneous axis of rotation (IAR) at which trunk angle reached 25º. Results indicated that a superior-inferior location of IAR was significantly (p<0.05) modified by the cardiovascular exercise after 12 weeks, but there was no significant effectiveness of the physical therapy exercise. Finding of lumbar spine kinematics during lifting and lowering a weight which are the most popular manual handling activities may provide great understanding of the exercise effectiveness. Future studies are recommended to assess whether the changes of lumbar spine kinematics lead to the decrease instances of lumbar spine injuries or low back pain. / Master of Science

Real-time face recognition has recently become available for the government and industry due to developments in face recognition algorithms, human head detection algorithms, and faster/low cost computers. Despite these advances, however, there are still some critical issues that affect the performance of real-time face recognition software. This paper addresses the problem of off-centered and out-of-pose faces in pictures, particularly in regard to the eigenface method for face recognition. We first demonstrate how the representation of faces by the eigenface method, and ultimately the performance of the software depend on the location of the eyes in the pictures. The eigenface method for face recognition is described: specifically, the creation of a face basis using the singular value decomposition, the reduction of dimension, and the unique representation of faces in the basis. Two different approaches for aligning the eyes in images are presented. The first considers the rotation of images using the orthogonal Procrustes Problem. The second approach looks at locating features in images using energy-minimizing active contours. We then conclude with a simple and fast algorithm for locating faces in images. Future research is also discussed. / Master of Science

An investigation of the amount of exchangeable potassium found in the surface and subsoils of a 33-year rotation experiment at Blacksburg, Virginia, was made. The rotation is located on Dunmore silt loam, Which is rated as a relatively fertile soil in Virginia. There are indications that sheet erosion has removed considerable amounts of the surface soil from the steeper slopes of the first nine treatments, and it is evident that an accumulation of this material has occurred on the four lower plots. It is believed by the writer, and others familiar with these plots, that.this erosion occurred prior to the beginning of the rotation in 1909. Composite soil samples were taken from each of the 8 treatments in each of the 4 series of this rotation and resulted in a total of thirty-two fertilized subplot samples, and a corresponding thirty-two unfertilized subplot samples were collected and analyzed for exchangeable potassium. Similar analyses were ma.de on the thirty-two subsoils of both the fertilized and the unfertilized treatments. The resulting data from the surface and subsoils were then statistically analyzed and the following conclusion drawn: 1. The applications of both muriate of potash and manure significantly increased the amount of exchangeable potassium found in both the surface and subsoils. 2. The surface soils are significantly higher in exchangeable potassium than the subsoils. 3. Exchangeable potassium may be leached from the surface horizon to the subsoil when applied in a complete fertilizer in excess of plant needs. 4. Manure applications which contained considerably less potassium than the potassium added to the soils in the muriate applications, resulted in larger amounts of exchangeable potassium in the surface soils, a 30 per cent increase in crop yields, and significantly less exchangeable potassium leached to the subsoils. 5. The difference in exchangeable potassium between the fertilized subplots receiving muriate of potash. and the unfertilized subplots. resulted in an increase in the amount of potassium found in the surface soils which was equivalent to 996 pounds of 50 per cent potash fertilizer, and this figure would represent an amount equivalent to 15 per cent of the total muriate added to the surface soils. 6. The difference in exchangeable potassium found in the subsoil between the fertilized and unfertilized plots represents an equivalent of approximately 9 per cent of the total potash added. 7. Significantly larger amounts of exchangeable potassium were leached to the subsoil from plots, receiving muriate of potash than soils which received manurial treatments. Greater utilization of the applied potassium is indicated by the manurial applications. 8. A chemical analysis should be made of the manure which is applied to this experiment in order to obtain a more comprehensive understanding of the yields, and physical and chemical properties of these soils. / M.S.

Mental rotation skills predict later achievement in STEM (Wai, Lubinski, & Benbow, 2009). Prior research shows that children’s mental rotation skills improve after training (Hawes, Gilligan-Lee, & Mix, 2022; Uttal et al., 2013). However, most studies have used dynamic stimuli where children see objects rotating. We hypothesized that reading books that practice mental rotation with only static images could improve children’ mental rotation skills. We preregistered a pretest-training-posttest design with 4- and 5-year-olds (N = 80). Children completed a mental rotation assessment at pretest, four to six reading days with an experimenter over two weeks in one of two randomly-assigned conditions (Mental Rotation Book Condition versus Control Book Condition), and a mental rotation assessment at posttest. The Mental Rotation Books involved mental rotation practice while the Control Books did not. Consistent with our hypothesis, condition was a significant predictor of posttest mental rotation accuracy, controlling for age, verbal ability, and pretest mental rotation accuracy. Children in the Mental Rotation Book condition significantly improved from pretest (M = .59, SD = .24) to posttest (M = .75, SD = .21), while the control group did not. However, condition was not a significant predictor of posttest mental transformation skills, math achievement, or spatial vocabulary, controlling for age, verbal ability, and respective pretest scores. Book-reading may be a scalable method for improving mental rotation skills in early childhood and warrants further intervention studies using book-reading at home or in schools to improve spatial skills. / Psychology