Viterbi Sequences and Polytopes

A Viterbi path of length n of a discrete Markov chain is a sequence of n+1 states that has the greatest probability of ocurring in the Markov chain. We divide the space of all Markov chains into Viterbi regions in which two Markov chains are in the same region if they have the same set of Viterbi paths. The Viterbi paths of regions of positive measure are called Viterbi sequences. Our main results are (1) each Viterbi sequence can be divided into a prefix, periodic interior, and suffix, and (2) as n increases to infinity (and the number of states remains fixed), the number of Viterbi regions remains bounded. The Viterbi regions correspond to the vertices of a Newton polytope of a polynomial whose terms are the probabilities of sequences of length n. We characterize Viterbi sequences and polytopes for two- and three-state Markov chains.Comment: 15 pages, 2 figures, to appear in Journal of Symbolic Computatio

Applications of Graphical Condensation for Enumerating Matchings and Tilings

A technique called graphical condensation is used to prove various combinatorial identities among numbers of (perfect) matchings of planar bipartite graphs and tilings of regions. Graphical condensation involves superimposing matchings of a graph onto matchings of a smaller subgraph, and then re-partitioning the united matching (actually a multigraph) into matchings of two other subgraphs, in one of two possible ways. This technique can be used to enumerate perfect matchings of a wide variety of bipartite planar graphs. Applications include domino tilings of Aztec diamonds and rectangles, diabolo tilings of fortresses, plane partitions, and transpose complement plane partitions.Comment: 25 pages; 21 figures Corrected typos; Updated references; Some text revised, but content essentially the sam

The Rising Incidence Of Small Endocrine Cancers In The United States: Effects On Surgical Therapy In An Age Of Imaging

The increasing utilization of imaging technology has led to the diagnosis of cancers earlier in their clinical course. When small tumor size is coupled with relatively indolent histology, excellent oncologic outcomes require the risks of surgery to be carefully considered. However, characteristics and outcomes of small cancers of the thyroid and endocrine pancreas remain poorly defined, and evidence to guide their management is sparse. Patients with tall cell (mTCV) and diffuse sclerosing (mDSV) variants of papillary thyroid microcarcinoma (mPTC), follicular (mFTC) and Hurthle cell microcarcinoma (mHCC), parathyroid carcinoma (PC) and pancreatic neuroendocrine tumors (PNETs) ≤ 2 cm in size were selected from the National Cancer Institute\u27s Surveillance, Epidemiology, and End Results (SEER) database, 1988-2009. Data regarding incidence, characteristics, and outcomes were extracted and analyzed with χ2 tests, ANOVA, the Kaplan Meier method, log-rank tests, and Cox proportional hazards. 97 mTCV, 90 mDSV, 371 mFTC, 193 mHCC, and 263 PNETs ≤ 2 cm were identified. The incidence of mTCV, mDSV, and mFTC remained stable throughout the study period, while the incidences of mHCC and PNETs ≤ 2 cm increased by 400% and 710% over the study period, respectively. Although survival was similar, mTCV and mDSV were associated with higher rates of extrathyroidal extension and nodal metastasis in comparison to classic mPTC. mFHCC had over eight times the rate of distant metastases compared to mPTC and was associated with compromised 10-year disease specific survival (95.4 vs. 99.3%, P\u3c0.001). Rates of extrapancreatic extension, nodal metastasis, and distant metastasis in PNETs ≤ 2 cm were 17.9%, 27.3%, and 9.1%, respectively. The incidence of many endocrine cancers is increasing, presumably due to increased detection. All histologies studied were capable of exhibiting aggressive behavior despite small tumor size. Further studies that specifically examine the risks and benefits of surgical therapy in small tumors may clarify future surgical decision making

Image segmentation using fuzzy LVQ clustering networks

In this note we formulate image segmentation as a clustering problem. Feature vectors extracted from a raw image are clustered into subregions, thereby segmenting the image. A fuzzy generalization of a Kohonen learning vector quantization (LVQ) which integrates the Fuzzy c-Means (FCM) model with the learning rate and updating strategies of the LVQ is used for this task. This network, which segments images in an unsupervised manner, is thus related to the FCM optimization problem. Numerical examples on photographic and magnetic resonance images are given to illustrate this approach to image segmentation

Ununfoldable Polyhedra with Convex Faces

Unfolding a convex polyhedron into a simple planar polygon is a well-studied problem. In this paper, we study the limits of unfoldability by studying nonconvex polyhedra with the same combinatorial structure as convex polyhedra. In particular, we give two examples of polyhedra, one with 24 convex faces and one with 36 triangular faces, that cannot be unfolded by cutting along edges. We further show that such a polyhedron can indeed be unfolded if cuts are allowed to cross faces. Finally, we prove that ``open'' polyhedra with triangular faces may not be unfoldable no matter how they are cut.Comment: 14 pages, 9 figures, LaTeX 2e. To appear in Computational Geometry: Theory and Applications. Major revision with two new authors, solving the open problem about triangular face

Language of physics, language of math: Disciplinary culture and dynamic epistemology

Mathematics is a critical part of much scientific research. Physics in particular weaves math extensively into its instruction beginning in high school. Despite much research on the learning of both physics and math, the problem of how to effectively include math in physics in a way that reaches most students remains unsolved. In this paper, we suggest that a fundamental issue has received insufficient exploration: the fact that in science, we don't just use math, we make meaning with it in a different way than mathematicians do. In this reflective essay, we explore math as a language and consider the language of math in physics through the lens of cognitive linguistics. We begin by offering a number of examples that show how the use of math in physics differs from the use of math as typically found in math classes. We then explore basic concepts in cognitive semantics to show how humans make meaning with language in general. The critical elements are the roles of embodied cognition and interpretation in context. Then we show how a theoretical framework commonly used in physics education research, resources, is coherent with and extends the ideas of cognitive semantics by connecting embodiment to phenomenological primitives and contextual interpretation to the dynamics of meaning making with conceptual resources, epistemological resources, and affect. We present these ideas with illustrative case studies of students working on physics problems with math and demonstrate the dynamical nature of student reasoning with math in physics. We conclude with some thoughts about the implications for instruction.Comment: 27 pages, 9 figure