The geometry of billiards in ellipses and their poncelet grids

The goal of this paper is an analysis of the geometry of billiards in ellipses, based on properties of confocal central conics. The extended sides of the billiards meet at points which are located on confocal ellipses and hyperbolas. They define the associated Poncelet grid. If a billiard is periodic then it closes for any choice of the initial vertex on the ellipse. This gives rise to a continuous variation of billiards which is called billiard motion though it is neither a Euclidean nor a projective motion. The extension of this motion to the associated Poncelet grid leads to new insights and invariants.

A Note on Hamiltonian Bypasses in Digraphs with Large Degrees

Let D be a 2-strongly connected directed graph of order p ≥ 3. Suppose that d(x) ≥ p for every vertex x ∈ V (D) \ {x0}, where x0 is a vertex of D. In this paper, we show that if D is Hamiltonian or d(x0) > 2(p − 1)/5, then D contains a Hamiltonian path, in which the initial vertex dominates the terminal vertex.

Joint Link Prediction and Network Alignment via Cross-graph Embedding

Link prediction and network alignment are two important problems in social network analysis and other network related applications. Considerable efforts have been devoted to these two problems while often in an independent way to each other. In this paper we argue that these two tasks are relevant and present a joint link prediction and network alignment framework, whereby a novel cross-graph node embedding technique is devised to allow for information propagation. Our approach can either work with a few initial vertex correspondence as seeds, or from scratch. By extensive experiments on public benchmark, we show that link prediction and network alignment can benefit to each other especially for improving the recall for both tasks.

Surprise Probabilities in Markov Chains

In a Markov chain started at a statex, thehitting timeτ(y) is the first time that the chain reaches another statey. We study the probability$\mathbb{P}_x(\tau(y) = t)$that the first visit toyoccurs precisely at a given timet. Informally speaking, the event that a new state is visited at a large timetmay be considered a ‘surprise’. We prove the following three bounds.•In any Markov chain withnstates,$\mathbb{P}_x(\tau(y) = t) \le {n}/{t}$.•In a reversible chain withnstates,$\mathbb{P}_x(\tau(y) = t) \le {\sqrt{2n}}/{t}$ for $t \ge 4n + 4$.•For random walk on a simple graph withn≥ 2 vertices,$\mathbb{P}_x(\tau(y) = t) \le 4e \log(n)/t$.We construct examples showing that these bounds are close to optimal. The main feature of our bounds is that they require very little knowledge of the structure of the Markov chain.To prove the bound for random walk on graphs, we establish the following estimate conjectured by Aldous, Ding and Oveis-Gharan (private communication): for random walk on ann-vertex graph, for every initial vertexx,$$ \sum_y \biggl( \sup_{t \ge 0} p^t(x, y) \biggr) = O(\log n). $$

Rainbow Paths with Prescribed Ends

It was conjectured in [S. Akbari, F. Khaghanpoor, and S. Moazzeni. Colorful paths in vertex coloring of graphs. Preprint] that, if $G$ is a connected graph distinct from $C_7$, then there is a $\chi(G)$-coloring of $G$ in which every vertex $v\in V(G)$ is an initial vertex of a path $P$ with $\chi(G)$ vertices whose colors are different. In [S. Akbari, V. Liaghat, and A. Nikzad. Colorful paths in vertex coloring of graphs. Electron. J. Combin. 18(1): P17, 9pp, 2011] this was proved with $\lfloor\frac{\chi(G)}{2} \rfloor $ vertices instead of $\chi(G)$ vertices. We strengthen this to $\chi(G)-1$ vertices. We also prove that every connected graph with at least one edge has a proper $k$-coloring (for some $k$) such that every vertex of color $i$ has a neighbor of color $i+1$ (mod $k$). $C_5$ shows that $k$ may have to be greater than the chromatic number. However, if the graph is connected, infinite and locally finite, and has finite chromatic number, then the $k$-coloring exists for every $k \geq \chi(G)$. In fact, the $k$-coloring can be chosen such that every vertex is a starting vertex of an infinite path such that the color increases by $1$ (mod $k$) along each edge. The method is based on the circular chromatic number $\chi_c(G)$. In particular, we verify the above conjecture for all connected graphs whose circular chromatic number equals the chromatic number.

PENENTUAN LINTASAN TERPENDEK DARI FMIPA KE REKTORAT DAN FAKULTAS LAIN DI UNSRAT MANADO MENGGUNAKAN ALGORITMA DJIKSTRA

Universitas Sam Ratulangi Manado adalah salah satu perguruan tinggi di Sulawesi Utara yang terdiri atas 11 fakultas dan satu gedung rektorat. Setiap fakultas dan rektorat terhubung dengan fasilitas jalan raya. Secara matematis kondisi seperti ini dapat direpresentasikan sebagai sebuah graf yang bisa diterapkan untuk mencari lintasan terpendek. Pada penelitian ini akan dicari lintasan terpendek dari FMIPA ke rektorat dan fakultas lainnya. Dengan menggunakan algoritma Djikstra, lintasan terpendek dari FMIPA diperoleh dengan memilih minimum lokal atau akses dengan jarak terdekat dari setiap lokasi yang kemudian digabungkan menjadi sebuah kumpulan lintasan dari satu lokasi ke lokasi lainnya dengan jarak terpendek. DETERMINATION OF SHORTEST PATH FROM FMIPA TO RECTORATE AND OTHER FACULTIES AT SAM RATULANGI UNIVERSITY USING DJIKSTRA ALGORITHMABSTRACTSam Ratulangi University is one of the colleges in North Sulawesi consisting of 11 faculties and one rectorate building. Every faculty and rectorate connected by highway facilities. Mathemathically this condition can be represented as an undirected weighted graph that can be applied to find the shortest path. By using the Djikstra algorithm, the shortest paths are obtained by setting the FMIPA as the initial vertex and then select the local minimum or access to the closest distance from each location, then combined the collection of path from one location to another with the shortest distance.

Hierarchy of protein assembly at the vertex ring domain for yeast vacuole docking and fusion

Vacuole tethering, docking, and fusion proteins assemble into a “vertex ring” around the apposed membranes of tethered vacuoles before catalyzing fusion. Inhibitors of the fusion reaction selectively interrupt protein assembly into the vertex ring, establishing a causal assembly hierarchy: (a) The Rab GTPase Ypt7p mediates vacuole tethering and forms the initial vertex ring, independent of t-SNAREs or actin; (b) F-actin disassembly and GTP-bound Ypt7p direct the localization of other fusion factors; (c) The t-SNAREs Vam3p and Vam7p regulate each other's vertex enrichment, but do not affect Ypt7p localization. The v-SNARE Vti1p is enriched at vertices by a distinct pathway that is independent of the t-SNAREs, whereas both t-SNAREs will localize to vertices when trans-pairing of SNAREs is blocked. Thus, trans-SNARE pairing is not required for SNARE vertex enrichment; and (d) The t-SNAREs regulate the vertex enrichment of both G-actin and the Ypt7p effector complex for homotypic fusion and vacuole protein sorting (HOPS). In accord with this hierarchy concept, the HOPS complex, at the end of the vertex assembly hierarchy, is most enriched at those vertices with abundant Ypt7p, which is at the start of the hierarchy. Our findings provide a unique view of the functional relationships between GTPases, SNAREs, and actin in membrane fusion.

Hypothetical evolutions of short-period comets

The hypothetical model of capture I consider is as follows: a comet with an initial conic orbit, meets close to one of its vertices an outer planet and generates one or several little comets (crossing of the Roche limit) with an elliptic orbit. This initial vertex always remains one of the vertices of the captured orbit by the Solar System. Even if there is a discontinuity of all the orbital elements during a very short time of the capture, I assume that the orbital plane is unchanged during the capture, as indeed was the case for Brooks 2 in 1886 (see Belyaev et al., 1986) or will be for Gehrels 3 in 2300 (see Carusi et al., 1985). After this first decisive step, the “new” comets may be subject to some other captures by jovian planets during their evolution with the same scenario. All these hypotheses allow to find particular numerical results for 142 comets whose current orbital elements were found in the Marsden catalog or given by P. Rocher, but I shall only give several examples in the following section.