Embedding Four-directional Paths on Convex Point Sets

A directed path whose edges are assigned labels "up", "down", "right", or "left" is called \emph{four-directional}, and \emph{three-directional} if at most three out of the four labels are used. A \emph{direction-consistent embedding} of an \mbox{$n$-vertex} four-directional path $P$ on a set $S$ of $n$ points in the plane is a straight-line drawing of $P$ where each vertex of $P$ is mapped to a distinct point of $S$ and every edge points to the direction specified by its label. We study planar direction-consistent embeddings of three- and four-directional paths and provide a complete picture of the problem for convex point sets.Comment: 11 pages, full conference version including all proof

Ursinus College Alumni Journal, Spring 1948

Give the steward a break! • President\u27s page • College has full extra-curricular program: Dr. Miller forum speaker; Religious Emphasis Week; Ray Eberle for junior prom • Mrs. Smith resigns alumni secretaryship • College is recipient of paintings and furniture • Congratulations, alumni basketeers! • Women\u27s Club entertains Ursinus senior girls • New associate professor joins college faculty • Nineteen alumni attain doctorates • Still room for students in certain categories • College librarian resigns • A successful season was had by all: Men\u27s basketball; Girls\u27 basketball; Wrestling; Swimming • News about ourselves • Cost of living increase voted faculty members • News around town • Ursinus curriculum undergoes revision • Women\u27s rules liberalizedhttps://digitalcommons.ursinus.edu/alumnijournal/1032/thumbnail.jp

Hamiltonicity of 3-arc graphs

An arc of a graph is an oriented edge and a 3-arc is a 4-tuple $(v,u,x,y)$ of vertices such that both $(v,u,x)$ and $(u,x,y)$ are paths of length two. The 3-arc graph of a graph $G$ is defined to have vertices the arcs of $G$ such that two arcs $uv, xy$ are adjacent if and only if $(v,u,x,y)$ is a 3-arc of $G$. In this paper we prove that any connected 3-arc graph is Hamiltonian, and all iterative 3-arc graphs of any connected graph of minimum degree at least three are Hamiltonian. As a consequence we obtain that if a vertex-transitive graph is isomorphic to the 3-arc graph of a connected arc-transitive graph of degree at least three, then it is Hamiltonian. This confirms the well known conjecture, that all vertex-transitive graphs with finitely many exceptions are Hamiltonian, for a large family of vertex-transitive graphs. We also prove that if a graph with at least four vertices is Hamilton-connected, then so are its iterative 3-arc graphs.Comment: in press Graphs and Combinatorics, 201

Ursinus College Alumni Journal, Summer 1947

Word of appreciation • Old Timers\u27 Day • President\u27s page • Alumni Association tables special committee\u27s report • Student union chosen as war memorial • Women\u27s Club • Dr. Clawson new Ursinus Dean • Board of Directors creates new committee, adds five members • Campus to be used for astronomical observation • Kuhrt Wieneke named coach of football • Three resign from faculty • Fourteen members added to faculty • Dr. Distler addresses graduating class • Sports: Men\u27s basketball; baseball; men\u27s tennis; track; women\u27s swimming; women\u27s basketball; women\u27s tennis • Local alumni associations meet • Work progressing on alumni register • Summer assembly • News about ourselves • Necrology • News around townhttps://digitalcommons.ursinus.edu/alumnijournal/1031/thumbnail.jp

Ursinus College Alumni Journal, Winter 1947

An auspicious beginning for a worthwhile project • President\u27s page • Committees plan new position at college • Status of the war memorial campaign • 964 students enrolled at Ursinus • Miss Moll resumes duties at Ursinus • General Arnold Founders\u27 Day speaker • Three faculty promotions, one appointment announced • New gymnasium nearing completion • Questionnaires outstanding from 900 alumni • Sports: football, soccer, hockey • The shape of things to come? • The attack on illiteracy in British Guiana • News around town • News about ourselves • Faculty members complete laboratory manual • Necrologyhttps://digitalcommons.ursinus.edu/alumnijournal/1029/thumbnail.jp

Shift invariant preduals of ℓ₁(ℤ)

The Banach space ℓ<sub>1</sub>(ℤ) admits many non-isomorphic preduals, for example, C(K) for any compact countable space K, along with many more exotic Banach spaces. In this paper, we impose an extra condition: the predual must make the bilateral shift on ℓ<sub>1</sub>(ℤ) weak<sup>*</sup>-continuous. This is equivalent to making the natural convolution multiplication on ℓ<sub>1</sub>(ℤ) separately weak*-continuous and so turning ℓ<sub>1</sub>(ℤ) into a dual Banach algebra. We call such preduals <i>shift-invariant</i>. It is known that the only shift-invariant predual arising from the standard duality between C<sub>0</sub>(K) (for countable locally compact K) and ℓ<sub>1</sub>(ℤ) is c<sub>0</sub>(ℤ). We provide an explicit construction of an uncountable family of distinct preduals which do make the bilateral shift weak<sup>*</sup>-continuous. Using Szlenk index arguments, we show that merely as Banach spaces, these are all isomorphic to c<sub>0</sub>. We then build some theory to study such preduals, showing that they arise from certain semigroup compactifications of ℤ. This allows us to produce a large number of other examples, including non-isometric preduals, and preduals which are not Banach space isomorphic to c<sub>0</sub>

A comparison of variational and Markov chain Monte Carlo methods for inference in partially observed stochastic dynamic systems

In recent work we have developed a novel variational inference method for partially observed systems governed by stochastic differential equations. In this paper we provide a comparison of the Variational Gaussian Process Smoother with an exact solution computed using a Hybrid Monte Carlo approach to path sampling, applied to a stochastic double well potential model. It is demonstrated that the variational smoother provides us a very accurate estimate of mean path while conditional variance is slightly underestimated. We conclude with some remarks as to the advantages and disadvantages of the variational smoother. © 2008 Springer Science + Business Media LLC