Thesis (Ph. D.)--Princeton University, 1927. / Caption title: On types of knotted curves by J.W. Alexander and G.B. Briggs.

Thesis (Ph. D.)--Columbia University. / "Books of reference": p. 1-8.

Inaug.--Diss.--Göttingen. / Lebenslauf.

Thesis (M.S.)--Brigham Young University. Dept. of Mechanical Engineering, 2005. / Includes bibliographical references (p. 73-75).

Thesis (Ph.D.)--Columbia University. / "Books of reference": p. 1-8.

In this thesis, we explore the role of planarity in mean curvature flow in higher codimension and investigate its implications for singularity formation in a certain class of flows. In Chapter 1, we show that the blow-ups of compact 𝑛-dimensional solutions to mean curvature flow in ℝⁿ initially satisfying the pinching condition |𝐴|² < c |𝐻|² for a suitable constant c = c(𝑛) must be codimension one. We do this by establishing a new a priori estimate via a maximum principle argument. In Chapter 2, we consider ancient solutions to the mean curvature flow in ℝⁿ⁺¹ (𝑛 ≥ 3) that are weakly convex, uniformly two-convex, and satisfy derivative estimates |∇𝐴| ≤ 𝛾1 |𝐻|², |∇² 𝐴| \leq 𝛾2 |𝐻|³. We show that such solutions are noncollapsed. The proof is an adaptation of the foundational work of Huisken and Sinestrari on the flow of two-convex hypersurfaces. As an application, in arbitrary codimension, we classify the singularity models of compact 𝑛-dimensional (𝑛 ≥ 5) solutions to the mean curvature flow in ℝⁿ that satisfy the pinching condition |𝐴|² < c |𝐻|² for c = min {1/𝑛-2, 3(𝑛+1)/2𝑛(𝑛+2)}. Using recent work of Brendle and Choi, together with the estimate of Chapter 1, we conclude that any blow-up model at the first singular time must be a codimension one shrinking sphere, shrinking cylinder, or translating bowl soliton. Finally, in Chapters 3 and 4, we prove a canonical neighborhood theorem for the mean curvature flow of compact 𝑛-dimensional submanifolds in ℝⁿ (𝑛 ≥ 5) satisfying a pinching condition |𝐴|² < c |𝐻|² for $c = min {1/𝑛-2, 3(𝑛+1)/2𝑛(𝑛+2)}. We first discuss, in some detail, a well-known compactness theorem of the mean curvature flow. Then, adapting an argument of Perelman and using the conclusions of Chapter 2, we characterize regions of high curvature in the pinched solutions of the mean curvature flow under consideration.

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.

A polyhedral graph G is called PCC if every vertex of G has strictly positive combinatorial curvature and the graph is not a prism or antiprism. In this thesis it is shown that the maximum order of a 3-regular PCC graph is 132 and the 3-regular PCC graphs which match that bound are enumerated. A new PCC graph with two 39-faces and 208 vertices is constructed, matching the number of vertices of the largest PCC graphs discovered by Nicholson and Sneddon. A conjecture that there are no PCC graphs with faces of size larger than 39 is made, along with a proof that if there are no faces of size larger than 122, then there is an upper bound of 244 on the order of PCC graphs. / Graduate

A series of experiments were carried out to measure the effects of convex surface curvature on film cooling. In the first series of experiments cooling holes were positioned along the vane such that their non-dimensional curvature parameter, 2r/d, was matched. Single row of holes with the same diameter were placed at high and moderate curvature position along a turbine vane resulting in 2r/d = 28 and 40, accordingly. A third row of holes was installed on the vane at the same location as the moderate curvature row with a larger hole diameter, resulting in 2r/d = 28, matching the high curvature row. Adiabatic temperature measurements were then carried out for blowing ratios of M = 0.30 to 1.60 tested at a density ratio of DR = 1.20. The results indicated that there was some scaling of performance present with matching 2r/d, but there was not an exact matching of performance. The second series of experiments focused on the effects of a changing surface curvature downstream of injection. Two row of holes were positioned along the vane surface such that the local radius of curvature and hole diameters were equivalent, with one row positioned upstream of the maximum curvature point and the other downstream of the maximum curvature point. Adiabatic temperature measurements were carried out for blowing ratios of M = 0.30 to 1.60 and tested at a density ratio of DR = 1.20. The results show that the change in curvature downstream plays a significant role in the performance of film cooling and that the local surface curvature is insufficient in capturing its effects. Additional experiments were carried out to measure the effects of the approaching boundary layer influence on film cooling as well as the effect of injection angle at a weakly convex surface. / text