Khovanov Homology and Calculus of Functors

A central question in knot theory is the classification of knots. Given two knots, how can we determine if they are different or the same? To answer this question, we develop and study knot invariants which are properties of knots that remain unchanged under isotopy. Khovanov homology is a powerful knot invariant that is able to distinguish many knots. However, because it is constructed in a combinatorial and algebraic manner, Khovanov homology lacks any geometric or topological motivation. Since Khovanov homology encodes information about topological objects, it would be ideal if it could be interpreted from a topological perspective. One way to approach this is through the lens of manifold calculus of functors and more specifically, the Taylor tower for spaces of long knots. Recent developments have shown that the Jones polynomial, another knot invariant, is encoded in the Taylor tower for knots. Since the Jones polynomial can be extracted from Khovanov homology, it is natural to ask if the Taylor tower can provide a space level realization of Khovanov homology. This paper provides an introduction to Khovanov homology and calculus of functors and offers conjectures that relate the two notions

Rational torsion on optimal curves and rank-one quadratic twists

AbstractWhen an elliptic curve E′/Q of square-free conductor N has a rational point of odd prime order l∤N, Dummigan (2005) in [Du] explicitly constructed a rational point of order l on the optimal curve E, isogenous over Q to E′, under some conditions. In this paper, we show that his construction also works unconditionally. And applying it to Heegner points of elliptic curves, we find a family of elliptic curves E′/Q such that a positive proportion of quadratic twists of E′ has (analytic) rank 1. This family includes the infinite family of elliptic curves of the same property in Byeon, Jeon, and Kim (2009) [B-J-K]

Yaw angle effect on the aerodynamic performance of hatchback vehicle fitted with wing spoiler

Research on spoiler available to date was mainly done to optimize the performance of spoiler in non-zero yaw condition. However, the effect of spoiler is most needed during cornering to ensure the stability of the vehicle. Therefore, this study aims to inspect the effect of yaw angles change on the aerodynamic performance of the NACA 0018 wing spoiler and the subsequent influence on the flow characteristics of the hatchback vehicle. Computational Fluid Dynamics (CFD) has been applied to model the flow. Comparison between numerically obtained results and experimental data was done to validate the CFD method. The findings show that both the drag coefficient, Cd, and lift coefficient, Cl have increased with increasing yaw angle. However, the spoiler has performed in favor of reducing the Cd and Cl even with increasing yaw angle. The averaged proportion contributions from the spoiler to the overall Cd and Cl are 2.7% and 4.1%, respectively. The other body parts that have contributed to the Cd and Cl reductions were the base and slant, and the roof

ON THE JACOBIAN OF A FAMILY OF HYPERELLIPTIC CURVES

In this paper, we study the algebraic rank and the analytic rank of the Jacobian of hyperelliptic curves y² = x⁵ + m² for integers m. Namely, we first provide a condition on m that gives a bound of the size of Selmer group and then we provide a condition on m that makes L-functions non-vanishing. As a consequence, we construct a Jacobian that satisfies the rank part of the Birch–Swinnerton-Dyer conjecture

Counterexamples in the Theory of Ulrich Modules

The theory of Ulrich modules has many powerful and broad applications ranging from the original purpose of giving a criterion for when a local Cohen-Macaulay ring is Gorenstein to new methods of finding Chow forms of a variety to longstanding open conjectures in multiplicity theory. For example, the existence of Ulrich modules and Ulrich-like objects has been the main approach to Lech's conjecture, which has been open for over 60 years. However, existence results have been very difficult to establish and for over thirty years, it was unknown whether (complete) local domains always have Ulrich modules. Recently, Ma introduced the weaker notion of (weakly) lim Ulrich sequences and showed that their existence for (complete) local domains implies Lech's conjecture. Ma then asks if (weakly) lim Ulrich sequences always exist for complete local domains. In this thesis, we answer the question of existence for both Ulrich modules and weakly lim Ulrich sequences in the negative by constructing (complete) local domains that do not have any Ulrich modules or weakly lim Ulrich sequences. A key insight in our proofs is the classification of MCM modules over a ring $R$ via the $S_2$-ification of $R$. Moreover, for local domains of dimension 2, we show that the existence of weakly lim Ulrich sequences implies the existence of lim Ulrich sequences. Finally, our counterexamples are not standard-graded or Cohen--Macaulay. As such, we construct candidate counterexample rings that are standard-graded and/or Cohen--Macaulay from our original counterexamples.PHDMathematicsUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/169885/1/fyhee_1.pd

Experimental study of yaw angle effect on aerodynamic performance of road vehicle with rear spoiler

A spoiler is an external structure added to the trailing edge of the roof of a vehicle to increase the downward force, and hence, improves its traction. To date, the performance of spoilers had mainly been optimized for the straight-ahead driving condition. However, during cornering, good traction is critical to ensure that sufficient centripetal force is generated for the vehicle (o drive through the curve without slip. Hence, this research aims to investigate the effects of yaw angle change corresponding to cornering on the flow characteristics of various typical spoiler configurations, and the subsequent influence on the aerodynamic performance of vehicles. The vehicle type tested is the hatchback vehicle while the rear spoiler configuration employed are those typically found in real-life applications, namely, strip type, standing wing type, and the combination of the strip and wing (i.e. combo type). Ahmed model which had been scaled down to 75% of the original model was adopted to simulate the idealized hatchback type vehicle. The rear spoilers are being mounted on the roof end of the vehicle model. The experimentations are conducted in a low-speed wind tunnel at the Reynolds number of 2.7 x 106. The yaw angle tested ranges from 0° to 14°, at 4° increment. At zero yaw angle, the C1 of the vehicle model fitted with either of the three spoiler types is lower as compared to the vehicle model without a rear spoiler. Although the Srip-type spoiler provides the same advantage when comes to the Cd. performance, however, the Wing-type and Combo-type spoilers show an increase in Cd. Under the influence of the yaw angle, both the Cd and C1 of the vehicle model increase disregard the use of the rear spoiler. When without a spoiler, there exists a critical yaw angle value where the Cd and C1 of the vehicle model will surge by as much as 29% and 91%, respectively. However, the application of the three spoiler types could prevent such an undesirable tendency. In addition, at low yaw angle range, the vehicle model fitted with the strip-type spoiler shows a reduction in Cd, whereas the cases fitted with the wing-type and combo-type spoilers show a higher Cd. However, all the cases fitted with a spoiler produce a lower C1 as compared to the without spoilers case. Overall, the use of the of the rear spoilers could improve the driving stability and safety of the vehicle. However, the wing type and combo type could cause an increase in the aerodynamic drag of the vehicle operated at zero yaw angle or low yaw angle range

Yaw Angle Effect On The Aerodynamic Performance Of Hatchback Vehicle Fitted With Wing Spoiler

Research on spoiler available to date was mainly done to optimize the performance of spoiler in non-zero yaw condition. However, the effect of spoiler is most needed during cornering to ensure the stability of the vehicle. Therefore, this study aims to inspect the effect of yaw angles change on the aerodynamic performance of the NACA 0018 wing spoiler and the subsequent influence on the flow characteristics of the hatchback vehicle. Computational Fluid Dynamics (CFD) has been applied to model the flow. Comparison between numerically obtained results and experimental data was done to validate the CFD method. The findings show that both the drag coefficient, Cd, and lift coefficient, Cl have increased with increasing yaw angle. However, the spoiler has performed in favor of reducing the Cd and Cl even with increasing yaw angle. The averaged proportion contributions from the spoiler to the overall Cd and Cl are 2.7% and 4.1%, respectively. The other body parts that have contributed to the Cd and Cl reductions were the base and slant, and the roof

Histopathological retrospective study of canine renal disease in Korea, 2003~2008

Renal disease includes conditions affecting the glomeruli, tubules, interstitium, pelvis, and vasculature. Diseases of the kidney include glomerular diseases, diseases of the tubules and interstitium, diseases of renal pelvis, and developmental abnormalities. Renal tissue samples (n = 70) submitted to the Department of Veterinary Pathology of Konkuk University from 2003 to 2008 were included in this study. Tissue histopathology was performed using light microscopy with hematoxylin and eosin stains. Masson's trichrome, Congo Red, and Warthin starry silver staining were applied in several individual cases. Glomerular diseases (22.9%), tubulointerstitial diseases (8.6%), neoplastic diseases (8.6%), conditions secondary to urinary obstruction (24.3%), and other diseases (35.7%) were identified. Glomerulonephritis (GN) cases were classified as acute proliferative GN (5.7%), membranous GN (4.3%), membranoproliferative GN (4.3%), focal segmental GN (2.9%), and other GN (4.2%). The proportion of canine GN cases presently identified was not as high as the proportions identified in human studies. Conversely, urinary obstruction and end-stage renal disease cases were relatively higher in dogs than in human populations