We introduce a symbolic computational approach to detecting all permutation and parity symmetries in any general evolution equation, and to generating associated invariant polynomials, from given monomials, under the action of these symmetries. Traditionally, discrete point symmetries of differential equations are systemically found by solving complicated nonlinear systems of partial differential equations; in the presence of Lie symmetries, the process can be simplified further. Here, we show how to find parity- and permutation-type discrete symmetries purely based on algebraic calculations. Furthermore, we show that such symmetries always form groups, thereby allowing for the generation of new group-invariant conserved quantities from known conserved quantities. This work also contains an implementation of the said results in Mathematica. In addition, it includes, as a motivation for this work, an investigation of the connection between variational symmetries, described by local Lie groups, and conserved quantities in Hamiltonian systems.

Thesis advisor: Ying Ran / Recently there has been a surging interest in the topological phases of matter, including the symmetry-protected topological phases, symmetry-enriched topological phases, and topological semimetals. This thesis is aiming at finding new ways of searching and probing these topological phases of matter in order to deepen our understanding of them. The body of the thesis consists of three parts. In the first part, we study the search of filling-enforced topological phases of matter in materials. It shows the existence of symmetry-protected topological phases enforced by special electron fillings or fractional spin per unit-cell. This is an extension of the famous Lieb-Schultz-Mattis theorem. The original LSM theorem states that the symmetric gapped ground state of the system must exhibit topological order when there's fractional spin or fractional electron filling per unit-cell. However, the LSM theorem can be circumvented when commensurate magnetic flux is present in the system, which enlarge the unit-cells to accommodate integer numbers of electrons. We utilize this point to prove that the ground state of the system must be a symmetry-protected topological phase when magnetic translation symmetry is satisfied, which we coin the name “generalized LSM theorem”. The theorem is proved using two different methods. The first proof is to use the tensor network representation of the ground state wave-function. The second proof consists of a physical argument based on the idea of entanglement pumping. As a byproduct of this theorem, a large class of decorated quantum dimer models are introduced, which satisfy the condition of the generalized LSM theorem and exhibit SPT phases as their ground states. In part II, we switch to the nonlinear response study of Weyl semimetals. Weyl semimetals (WSM) have been discovered in time-reversal symmetric materials, featuring monopoles of Berry’s curvature in momentum space. WSM have been distinguished between Type-I and II where the velocity tilting of the cone in the later ensures a finite area Fermi surface.To date it has not been clear whether the two types results in any qualitatively new phenomena. In this part we focus on the shift-current response ($\sigma_{shift}(\omega)$), a second order optical effect generating photocurrents. We find that up to an order unity constant, $\sigma_{shift}(\omega)\sim \frac{e^3}{h^2}\frac{1}{\omega}$ in Type-II WSM, diverging in the low frequency $\omega\rightarrow 0$ limit. This is in stark contrast to the vanishing behavior ($\sigma_{shift}(\omega)\propto \omega$) in Type-I WSM. In addition, in both Type-I and Type-II WSM, a nonzero chemical potential $\mu$ relative to nodes leads to a large peak of shift-current response with a width $\sim |\mu|/\hbar$ and a height $\sim \frac{e^3}{h}\frac{1}{|\mu|}$, the latter diverging in the low doping limit. We show that the origin of these divergences is the singular Berry’s connections and the Pauli-blocking mechanism. Similar results hold for the real part of the second harmonic generation, a closely related nonlinear optical response. In part III, we propose a new kind of thermo-optical experiment: the nonreciprocal directional dichroism induced by a temperature gradient. The nonreciprocal directional dichroism effect, which measures the difference in the optical absorption coefficient between counterpropagating lights, occurs only in systems lacking inversion symmetry. The introduction of temperature-gradient in an inversion-symmetric system will also yield nonreciprocal directional dichroism effect. This effect is then applied to quantum magnetism, where conventional experimental techniques have difficulty detecting magnetic mobile excitations such as magnons or spinons exclusively due to the interference of phonons and local magnetic impurities. A model calculation is presented to further demonstrate this phenomenon. / Thesis (PhD) — Boston College, 2021. / Submitted to: Boston College. Graduate School of Arts and Sciences. / Discipline: Physics.

In this thesis, we study the structure and symmetry of singularity models of mean curvature flow. In chapter 1, we prove the quantitative long range curvature estimate and related results. The famous structure theorem of White asserts that in convex 𝛼-noncollapsed ancient solutions to the mean curvature flow, rescaled curvature is bounded in terms of rescaled distance. We improve this result and show that rescaled curvature is bounded by a quadratic function of rescaled distance using Ecker-Huisken's interior estimate. This method together with an induction on scale argument similar to the work of Brendle-Huisken can push the result to high curvature regions. We show that for a mean convex flow and any 𝑅 > 0, the rescaled curvature is bounded by 𝑪(𝑅+1)² in a parabolic neighborhood of rescaled size 𝑅 in the high curvature regions. We will then describe how this can be applied to give an alternative proof to a simplified version of White's structure theorem. In chapter 2, we discuss the symmetry structure of translators. We show that with mild assumptions, every convex, noncollapsed translator in ℝ⁴ has 𝑆𝑂(2) symmetry. In higher dimensions, we can prove an analogous result with a curvature assumption. With mild assumptions, we show that every convex, uniformly 3-convex, noncollapsed translator in ℝⁿ+¹ has 𝑆𝑂(n-1) symmetry.

No description available.

This thesis examines the relationship between Leibniz’s Principle of Identity of Indiscernibles and symmetry. In his 1717 correspondence with Samuel Clarke, Leibniz argued that “There is no such thing as a pair of individuals that are indiscernible from each other” (Leibniz 16). In other words, any objects sharing all their properties are in fact one and the same object. This is Leibniz’s Principle of Identity of Indiscernibles (the “PII”). The principle and its converse Leibniz’s Law express a conditional relationship between the identity of an object and its properties. Our investigation will use applications of Leibniz’s principles from the history of philosophy to examine this relationship and we’ll find that imperfect applications result in either perfect qualitative identity between multiple objects (“multiple indiscernibles”), imperfect qualitative identity between multiple objects (“incongruent counterparts”), or, finally, a relative identity between two facets of one object (a “singular discernible”). My project will also trace the historical thread leading from Leibniz to the development of symmetry groups in mathematics. Leibniz’s principles are embedded in science’s ability to distinguish the objective from the subjective, owing to their usefulness discerning an object’s intrinsic properties (properties belonging to the object itself) from extrinsic properties (properties based in relations the object is in with other objects). Symmetry is the relativity of identity, and the PII is an exploratory instrument illuminating this relationship: it injects structure into investigations of identity, but also affords the opportunity to capture pre-existing convictions about identity a thinker brings to the application.

Evaluating side-to-side symmetry in the lower extremity has been significant in assessing injury risk and the success of rehabilitation programs. Considering limb dominance in the lower limbs is also important as limb dominance could influence symmetry measures. There is a need to assess symmetry, particularly in healthy populations, in tasks other than walking and running and establish how the dominant limb can impact symmetry. By evaluating symmetry in healthy adults, how the limbs function with respect to one another can be determined. Therefore, the first purpose of this study was to investigate the impact of lower limb dominance on walking and sitting-to-standing. Data was collected from 49 healthy older adults, aged 50-89 years old. Using loadsol® sensors (Novel, St. Paul, MN, USA), plantar loading data such as peak impact force and loading rate was calculated. Participants completed one sit-to-stand trial and three 10-meter walking trials, as these serve as prime examples of daily activities. The secondary purpose of this study was to assess the impact of lower limb dominance on athletic tasks like running and agility. The pedar-X® pressure insoles (Novel, St. Paul, MN, USA) were used to collect plantar loading data such as peak force, contact area, and contact time, from 10 athletes. Participants completed five t-drill trials and five agility ladder drill trials. The acceleration phase of the t-drill served as standard running. A mixed effects model was used to test if differences existed in various plantar loading outcome measures based on limb dominance. Non-parametric tests were used for non-normally distributed data. The statistical analysis determined that no significant differences existed between the dominant limb and non-dominant limb for the 10-meter walking trials peak impact force (p=0.245) or average loading rate (p=0.943). During the sit-to-stand trial, no significant differences existed in peak impact force (p=0.317) or average loading rate (p=0.943). For the agility ladder drill, the maximum force (p=0.427), contact area (p=0.517), or contact time (p=0.734) showed no significant differences. In the T-drill, the maximum force (p=0.385), contact area (p=0.571), or contact time (p=0.571) had no significant differences. These drive the conclusion that limb dominance does not need to be considered when assessing side-to-side symmetry. / Master of Science / Understanding how the left and right lower limbs of the body compare is important to preventing injuries and measuring if rehabilitation interventions are beneficial. A factor in that is knowing how the dominant limb can affect how the lower limbs compare to one another. Through symmetry, especially in healthy adults, a greater comprehension for over limb functionality can be better understood. There is need to assess the lower limb symmetry in healthy populations in tasks aside from walking and running as well as establish how the dominant limb is impacting that symmetry. The first purpose of this study was to observe how lower limb dominance affects walking and standing from a seated position. Data was collected from 49 healthy older adults, aged 50-89 years old. Insoles were placed in participants' shoes to collect plantar loading data. Each participant did two tasks: one trial of the sit-to-stand and three trials of 10-meter walking. The second purpose of this study was to observe how lower limb dominance affects athletic tasks such as running and agility. Loading insoles were used to collect data from 10 current or previous athletes. Each participant did five t-drill trials and five agility ladder trials. Statistical analyses established no significant differences were shown between the dominant and non-dominant limbs peak impact force for the 10-meter walking trials (p=0.245) nor for the average loading rate (p=0.943). For the sit-to-stand trial, no significant differences were seen in peak impact force (p=0.317) or average loading rate (p=0.943). In the agility ladder drill, no significant differences were shown for the maximum force (p=0.427), contact area (p=0.517), or contact time (p=0.734). In the agility ladder drill, no significant differences existed for the maximum force (p=0.385), contact area (p=0.571), or contact time (p=0.571). These findings suggested that the dominant limb does not impact lower limb comparisons.

Human locomotion is a topic that has been studied for many years in biomechanics. To perform athletic tasks or everyday tasks, balance and symmetry is needed. Symmetry is the perfect balance and correspondence of the body or parts of the body. This concept has often been used to evaluate the normality of movements. Limb symmetry, specifically, is the equal actions of the lower limbs during movement. This is needed to perform tasks safely and efficiently without injury. Gait and movement symmetry has been used to predict lower limb injury risk for many populations and improve performance for athletes. It has also been used in assessment for rehabilitation processes and return to sport processes following injury or surgery. For many years, healthy gait was considered to be symmetrical for simplification purposes. However, many studies have contradicted that conclusion showing that even for has asymmetrical patterns. Deficits in symmetry can reduce quality of life for some individuals and can have detrimental health effects. Many measures have been used to assess symmetry in various tasks that have important implications on gait patterns. Another component of gait and movement that affects performance and injury risk is limb stiffness. Limb stiffness is the body's resistance to deformation when moments and forces are applied to it. The body has been shown to be modeled as a spring mass system that can restore and reuse energy. This is associated with the stretch shortening cycle during cyclic movements, such as running and walking. Limb stiffness is also associated with musculoskeletal loading that impacts performance and injury. Therefore, optimizing limb stiffness is important to improve utilization of elastic energy for athletic performance and reduce injuries associated with high and low limb stiffness values. Imbalances in limb stiffness have been shown to increase injury risk during walking and other tasks. Studying these imbalances using symmetry indices could give insight into the injury risk associated with this metric. In addition, limb stiffness in humans has been shown to change with the type of contact surface. This is associated with compensation methods used by humans when contacting different surfaces. Studying the relationship between limb stiffness symmetry and different surfaces during walking is important to observe how humans adjust and how it impacts injury risk. The purpose of this research was to assess the impact that surface stiffness has on limb stiffness symmetry during walking in healthy adults. To assess limb stiffness differences when transitioning to different surface stiffnesses anteriorly and posteriorly, the Normalized Symmetry Index (NSI) was determined for the two transition conditions and the control. The results showed that limb stiffness NSI was significant between the conditions (p=0.012). More specifically, a difference was seen between the stiff to compliant transition and the control (p=0.020) and the compliant to stiff transition and the control (p=0.032). These results show that humans do compensate when transitioning onto different surfaces. This is essential for understanding how humans adjust during real world walking and what patterns are used to maintain stability. To assess limb stiffness symmetry, when surface stiffness is different between limbs, the limb stiffness NSI was compared between two conditions. This included the side-to-side stiffness difference condition and the control condition. The results revealed that surface stiffness was not significant between conditions (p=0.244). Based on these results, limb stiffness symmetry is not significantly impacted when the surface stiffness is different between limbs. This contradicts prior studies that observed changes limb stiffness and symmetry depending on the surface stiffness. This may be due to overcompensation or the ability of the healthy adult population to quickly adjust to the surface stiffness changes before the measurements were taken. Simulating uneven surfaces is important to understand how humans compensate to maintain stability on surfaces in real world walking and for imbalances due to disorders. Further research is needed to study the changes in limb stiffness symmetry on different surfaces during walking to improve injury prevention methods. / Master of Science / Humans perform many daily tasks and athletic tasks that have been observed in human movement analysis. To perform these tasks safely and efficiently, many factors must be considered. One of the important factors in performing tasks is symmetry. Symmetry is the perfect balance between parts of the body, such as the lower limbs during walking or gait. Gait in healthy adults was considered to be symmetrical for simplification purposes. However, studies have revealed that gait asymmetry is present in the healthy adult population during walking and other movements. Gait symmetry has been used to assess normality of gait patterns in healthy individuals and in clinical populations. Asymmetrical gait patterns can lead to injury and have detrimental effects on health. Therefore, limb symmetry has been an important metric to assess lower limb injury risk and improving injury prevention methods to correct asymmetrical patterns in healthy adults and other populations. Another aspect of human movement that impacts injury is limb stiffness. Limb stiffness is the body's resistance to deformation under applied forces. High limb stiffness values have been associated with bony injuries due to increased loading. However, low stiffness values have been associated with soft tissue injuries. Therefore, regulating limb stiffness is important to reduce injuries in the long term. The type of contact surface during walking and other tasks has been shown to change limb stiffness values. Humans often encounter changes to surfaces when walking. For example, hikers who encounter uneven terrain or everyday walking on uneven pavement. Uneven surfaces have been shown to require more energy and work to move forwards during walking. Therefore, simulating uneven surfaces in the real world is important to understand how humans compensate on different surfaces. This could be important for understanding how limb stiffness imbalances on different surfaces affect injury. To quantify these imbalances, the metric of limb stiffness symmetry will be used. Limb stiffness imbalances due to surface stiffness are essential to assess how humans adapt to instability during real world walking. Therefore, this study aims to determine how humans adjust when transitioning to different surface stiffnesses and when surface stiffness is different between limbs. To determine how humans adjust when transitioning to different surfaces of different stiffnesses, the limb stiffness symmetry was calculated using the Normalized Symmetry Index (NSI). This was calculated for three different surface stiffness conditions, consisting of a stiff to compliant transition, a compliant to stiff transition, and the control condition. The results showed that there was a significant difference between the NSI values of the three conditions. However, there was no difference between the two transition conditions. This indicated that there was no difference between the transition order. Based on the results, limb stiffness symmetry does change when transitioning to different surface stiffness conditions. This agrees with previous literature that suggests that surface stiffness has an impact on limb stiffness. This information is beneficial to understand the patterns humans use to compensate to maintain stability. To determine how limb stiffness symmetry is impacted when surface stiffness is different between limbs, the limb stiffness NSI was calculated for two surface conditions. This included the side-to-side condition and the control condition. The results showed that there was no statistical difference between the limb stiffness NSI values of the two conditions. This shows that limb stiffness symmetry doesn't change when the surface stiffness is different between limbs, which disagrees with previous literature. Overall, this information is important to understand how humans compensate when transitioning on different surfaces or walking on uneven surfaces. This is important to understand how stability is maintained despite imbalances for improvement of injury prevention methods.

No description available.

This thesis is devoted to the study of gauged linear sigma models (GLSMs) and mirror symmetry. The first chapter of this thesis aims to introduce some basics of GLSMs and mirror symmetry. The second chapter contains the author's contributions to new exact results for GLSMs obtained by applying supersymmetric localization. The first part of that chapter concerns supermanifolds. We use supersymmetric localization to show that A-twisted GLSM correlation functions for certain supermanifolds are equivalent to corresponding Atwisted GLSM correlation functions for hypersurfaces. The second part of that chapter defines associated Cartan theories for non-abelian GLSMs by studying partition functions as well as elliptic genera. The third part of that chapter focuses on N=(0,2) GLSMs. For those deformed from N=(2,2) GLSMs, we consider A/2-twisted theories and formulate the genuszero correlation functions in terms of Jeffrey-Kirwan-Grothendieck residues on Coulomb branches, which generalize the Jeffrey-Kirwan residue prescription relevant for the N=(2,2) locus. We reproduce known results for abelian GLSMs, and can systematically calculate more examples with new formulas that render the quantum sheaf cohomology relations and other properties manifest. We also include unpublished results for counting deformation parameters. The third chapter is about mirror symmetry. In the first part of the third chapter, we propose an extension of the Hori-Vafa mirrror construction [25] from abelian (2,2) GLSMs they considered to non-abelian (2,2) GLSMs with connected gauge groups, a potential solution to an old problem. We formally show that topological correlation functions of B-twisted mirror LGs match those of A-twisted gauge theories. In this thesis, we study two examples, Grassmannians and two-step flag manifolds, verifying in each case that the mirror correctly reproduces details ranging from the number of vacua and correlations functions to quantum cohomology relations. In the last part of the third chapter, we propose an extension of the Hori-Vafa construction [25] of (2,2) GLSM mirrors to (0,2) theories obtained from (2,2) theories by special tangent bundle deformations. Our ansatz can systematically produce the (0,2) mirrors of toric varieties and the results are consistent with existing examples which were produced by laborious guesswork. The last chapter briefly discusses some directions that the author would like to pursue in the future. / Doctor of Philosophy / In this thesis, I summarize my work on gauged linear sigma models (GLSMs) and mirror symmetry. We begin by using supersymmetric localization to show that A-twisted GLSM correlation functions for certain supermanifolds are equivalent to corresponding A-twisted GLSM correlation functions for hypersurfaces. We also define associated Cartan theories for non-abelian GLSMs. We then consider N =(0,2) GLSMs. For those deformed from N =(2,2) GLSMs, we consider A/2-twisted theories and formulate the genus-zero correlation functions on Coulomb branches. We reproduce known results for abelian GLSMs, and can systematically compute more examples with new formulas that render the quantum sheaf cohomology relations and other properties are manifest. We also include unpublished results for counting deformation parameters. We then turn to mirror symmetry, a duality between seemingly-different two-dimensional quantum field theories. We propose an extension of the Hori-Vafa mirror construction [25] from abelian (2,2) GLSMs to non-abelian (2,2) GLSMs with connected gauge groups, a potential solution to an old problem. In this thesis, we study two examples, Grassmannians and two-step flag manifolds, verifying in each case that the mirror correctly reproduces details ranging from the number of vacua and correlations functions to quantum cohomology relations. We then propose an extension of the HoriVafa construction [25] of (2,2) GLSM mirrors to (0,2) theories obtained from (2,2) theories by special tangent bundle deformations. Our ansatz can systematically produce the (0,2) mirrors of toric varieties and the results are consistent with existing examples. We conclude with a discussion of directions that we would like to pursue in the future.

Trunk-mounted monitoring equipment like GPSports SPIHPU units are designed to use global positioning (GPS), accelerometer and heart rate monitoring to evaluate the physical demands of an activity. A medical staff might also consider markers such as running symmetry in evaluation of injury occurrence and rehabilitation. A running symmetry is a ratio of the synchronization of the right and left lower limbs during the gait cycle. An asymmetry due to, a pathology or musculoskeletal injury, results in abnormal loading on the foot that may be detected by trunk-mounted accelerometry. The aim of this study is to evaluate the ability of SPIHPU units to detect running asymmetry. Subjects wore the HPISPU units (100Hz, 16g tri-axial accelerometer, 50Hz magnetometer) while engaged in various running activities. In the first study, artificially inducing a leg length discrepancy led to a difference between running symmetry scores. This discrepancy was confirmed using individual accelerometers attached to the lower leg near the foot. Next, varying running speed did not result in differences in running symmetry. However, the SPIHPU units did detect a running asymmetry between fatigued and non-fatigued conditions. Finally, two case studies showed that the units could identify asymmetry immediately after a lower leg injury and during rehabilitation of anterior cruciate ligament reconstruction surgery. The results of this study show that the HPUSPI units can be reliably used to monitor running symmetry and to detect asymmetrical gait patterns. / Master of Science