A Hermite interpolatory subdivision scheme for C2C^2-quintics on the Powell-Sabin 12-split

In order to construct a $C^1$-quadratic spline over an arbitrary triangulation, one can split each triangle into 12 subtriangles, resulting in a finer triangulation known as the Powell-Sabin 12-split. It has been shown previously that the corresponding spline surface can be plotted quickly by means of a Hermite subdivision scheme. In this paper we introduce a nodal macro-element on the 12-split for the space of quintic splines that are locally $C^3$ and globally $C^2$. For quickly evaluating any such spline, a Hermite subdivision scheme is derived, implemented, and tested in the computer algebra system Sage. Using the available first derivatives for Phong shading, visually appealing plots can be generated after just a couple of refinements.Comment: 17 pages, 7 figure

Subdivision schemes for curve design and image analysis

Subdivision schemes are able to produce functions, which are smooth up to pixel accuracy, in a few steps through an iterative process. They take as input a coarse control polygon and iteratively generate new points using some algebraic or geometric rules. Therefore, they are a powerful tool for creating and displaying functions, in particular in computer graphics, computer-aided design, and signal analysis. A lot of research on univariate subdivision schemes is concerned with the convergence and the smoothness of the limit curve, especially for schemes where the new points are a linear combination of points from the previous iteration. Much less is known for non-linear schemes: in many cases there are only ad hoc proofs or numerical evidence about the regularity of these schemes. For schemes that use a geometric construction, it could be interesting to study the continuity of geometric entities. Dyn and Hormann propose sufficient conditions such that the subdivision process converges and the limit curve is tangent continuous. These conditions can be satisfied by any interpolatory scheme and they depend only on edge lengths and angles. The goal of my work is to generalize these conditions and to find a sufficient constraint, which guarantees that a generic interpolatory subdivision scheme gives limit curves with continuous curvature. To require the continuity of the curvature it seems natural to come up with a condition that depends on the difference of curvatures of neighbouring circles. The proof of the proposed condition is not completed, but we give a numerical evidence of it. A key feature of subdivision schemes is that they can be used in different fields of approximation theory. Due to their well-known relation with multiresolution analysis they can be exploited also in image analysis. In fact, subdivision schemes allow for an efficient computation of the wavelet transform using the filterbank. One current issue in signal processing is the analysis of anisotropic signals. Shearlet transforms allow to do it using the concept of multiple subdivision schemes. One drawback, however, is the big number of filters needed for analysing the signal given. The number of filters is related to the determinant of the expanding matrix considered. Therefore, a part of my work is devoted to find expanding matrices that give a smaller number of filters compared to the shearlet case. We present a family of anisotropic matrices for any dimension d with smaller determinant than shearlets. At the same time, these matrices allow for the definition of a valid directional transform and associated multiple subdivision schemes

Point sets containing their triangle centers

Let S be a set of at least five points in the plane, not all on a line. Suppose that for any three points $${a,b,c\in S}$$ the nine-point center of triangle abc also belongs to S. We show that S must be dense in the plane. We also consider several problems about partitioning the plane into two sets containing their triangle center

Geometric Inference in Bayesian Hierarchical Models with Applications to Topic Modeling

Unstructured data is available in abundance with the rapidly growing size of digital information. Labeling such data is expensive and impractical, making unsupervised learning an increasingly important field. Big data collections often have rich latent structure that statistical modeler is challenged to uncover. Bayesian hierarchical modeling is a particularly suitable approach for complex latent patterns. Graphical model formalism has been prominent in developing various procedures for inference in Bayesian models, however the corresponding computational limits often fall behind the demands of the modern data sizes. In this thesis we develop new approaches for scalable approximate Bayesian inference. In particular, our approaches are driven by the analysis of latent geometric structures induced by the models. Our specific contributions include the following. We develop full geometric recipe of the Latent Dirichlet Allocation topic model. Next, we study several approaches for exploiting the latent geometry to first arrive at a fast weighted clustering procedure augmented with geometric corrections for topic inference, and then a nonparametric approach based on the analysis of the concentration of mass and angular geometry of the topic simplex, a convex polytope constructed by taking the convex hull of vertices representing the latent topics. Estimates produced by our methods are shown to be statistically consistent under some conditions. Finally, we develop a series of models for temporal dynamics of the latent geometric structures where inference can be performed in online and distributed fashion. All our algorithms are evaluated with extensive experiments on simulated and real datasets, culminating at a method several orders of magnitude faster than existing state-of-the-art topic modeling approaches, as demonstrated by experiments working with several million documents in a dozen minutes.PHDStatisticsUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttps://deepblue.lib.umich.edu/bitstream/2027.42/146051/1/moonfolk_1.pd

COURSE PROPOSAL FOR A MATHEMATICAL MODELING COURSE IN A HIGH SCHOOL CURRICULUM

In the winter of 2015, I will be piloting a course on mathematical modeling at Hawken School, an independent high school in Gates Mills, OH. As I develop all elements of this course, such as lesson plans, assessments, and rubrics, I will be mindful of factors such as the newly adopted Common Core mathematics standards, the variety of student backgrounds in such a course, and how various mathematical societies and organizations such as SIAM, MAA, and COMAP can help in implementing it. However, there is one basic question driving my interest in and design of this course: “when am I ever going to need this?” This quote uttered by many high school math students sounds like nails on a chalkboard to teachers who cannot imagine how anyone can take for granted the great beauty in math. While every math student may not appreciate it as an art in its own right, I do believe that every student should be able to appreciate math for its infinite applications. Because of this, I, like many other math teachers, always try to do my best to incorporate many applications of math to a variety of fields in my courses. In doing so, I hope to impress upon them what is probably the most important aspect of mathematics education: the development of critical thinking. However, due to constraints, such as time, adhering to state content standards, etc., such application problems often get pushed to the side in the average math course. This is why the thoughtful addition of a semester long course in mathematical modeling would be an excellent addition to the high school curriculum

Diszkrét és kombinatórikus geometriai kutatások = Topics in discrete and combinatorial geometry

A most lezárult OTKA grant, 8 résztvevő diszkrét geometriai kutatását támogatta. Itt a témák ilusztrálására kiemelünk néhányat az elért 72 publikációból. 1. Jelentős eredmények születtek (8 cikk) gráfok síkba rajzolhatóságáról, például az úgynevezett metszési számról. 2. Többek között sikerült igazolni Katchalski és Lewis 20 éves sejtését, mely szerint diszjunkt egységkörökből álló rendszereknél ha bármely három körnek van közös metsző egyenese akkor van olyan egyenes, amely legfeljebb 2 kör kivételével valamennyit metsz. 3. Littlewood (1964) problémájaként ismert volt az a kérdés, hogy hány henger érintheti kölcsönösen egymást? Viszonylag alacsony felső korlátot találtunk és egy régóta ismert elhelyzés valótlanságát igazoltuk. 4. Többszörös fedések egyszerű fedésekre való szétbontását vizsgáltuk és értünk el lényeges előrelépést. 5. A Borsuk-féle darabolási problémanak azt a variánsát vizsgáltuk, amelyben a darabolást u. n. hengeres darabolásra korlátozták. 6. Bebizonyítottuk, hogy ''nem nagyon elnyúlt'' ellipszisek esetében a sík legritkább fedésének meghatározásánál el lehet tekinteni az u.n. nem-keresztezési feltételtől. 7. A sejtetthez nagyon közeli korlátot találtunk arra a problémára, hogy az n-dimenziós térben legfeljebb hány homotetikus konvex test helyezhető el úgy, hogy bármely kettő érintse egymást. | Discrete geometry in Hungary flourished since the sixties as a result of the work of László Fejes Tóth. The supported research of 8 participant also belongs to this area. Here we illustrate the achieved 72 publications by mentioning a few results. 1. Important theorems (8 papers) were proved concerning graph drawing. 2. Among others, a 20 year old problem of Katchalsky was proved, stating that in a packing of congruent circles, if any three has a common transversal, then there is a line, which avoids at most two of the circles. 3. Concerning a conjecture of Littlewood we found a small upper bound for the number of infinite cylinders which mutually touch each other. 4. We studied decomposability of multiple coverings into single coverings. 5. We studied that variant of the famous Borsuk problem where the partitions are restricted to cylindrical partitions. 6. We proved that in case of ellipses which are not ''too long'' at determining the thinnest covering one can omit the usually needed noncrossing condition. 7. A bound close to the conjectured bound was found concerning the number of n-dimensional homothetic convex solids which mutually touch each other