Divisors on graphs, Connected flags, and Syzygies

We study the binomial and monomial ideals arising from linear equivalence of divisors on graphs from the point of view of Gr\"obner theory. We give an explicit description of a minimal Gr\"obner bases for each higher syzygy module. In each case the given minimal Gr\"obner bases is also a minimal generating set. The Betti numbers of the binomial ideal and its natural initial ideal coincide and they correspond to the number of 'connected flags' in the graph. In particular the Betti numbers are independent of the characteristic of the base field. For complete graphs the problem was previously studied by Postnikov and Shapiro and by Manjunath and Sturmfels. The case of a general graph was stated as an open problem.Comment: to appear in International Mathematics Research Notices (IMRN

Biochemical and immunochemical analysis of the arrangement of connexin43 in rat heart gap junction membranes

A 43 × 10^3 M_r protein (designated connexin43 or Cx43) is a major constituent of heart gap junctions. The understanding of its arrangement in junctional membranes has been extended by means of site-directed antibodies raised against synthetic peptides of Cx43. These represent part of the first extracellular loop (EL-46), the cytoplasmic loop (CL-100), the second extracellular loop (EL-186) and carboxy-terminal sequences (CT-237 and CT-360). All of the antibodies raised reacted with their respective peptides and the Cx43 protein on Western blots. By immunoelectron microscopy two of the antibodies (CL-100 and CT-360) were shown to label the cytoplasmic surface of isolated gap junction membranes. Immunofluorescent labeling at locations of neonatal cardiac myocyte-myocyte apposition required an alkali/urea treatment when the EL-46 and EL-186 antibodies were used. Immunoblot analysis of endoproteinase Lys-C-digested gap junctions revealed that the Cx43 protein passed through the lipid bilayer four times. Alkaline phosphatase digestion of isolated junctions was used to show that the CT-360 antibody recognized many phosphorylated forms of Cx43. Our results unequivocally confirm models of the organization of Cx43 that were based on a more limited set of data and a priori considerations of the sequence

Chip-firing based methods in the Riemann--Roch theory of directed graphs

Baker and Norine proved a Riemann--Roch theorem for divisors on undirected graphs. The notions of graph divisor theory are in duality with the notions of the chip-firing game of Bj\"orner, Lov\'asz and Shor. We use this connection to prove Riemann--Roch-type results on directed graphs. We give a simple proof for a Riemann--Roch inequality on Eulerian directed graphs, improving a result of Amini and Manjunath. We also study possibilities and impossibilities of Riemann--Roch-type equalities in strongly connected digraphs and give examples. We intend to make the connections of this theory to graph theoretic notions more explicit via using the chip-firing framework.Comment: 22 pages, 4 figure

Weakly Supervised Localization using Deep Feature Maps

Object localization is an important computer vision problem with a variety of applications. The lack of large scale object-level annotations and the relative abundance of image-level labels makes a compelling case for weak supervision in the object localization task. Deep Convolutional Neural Networks are a class of state-of-the-art methods for the related problem of object recognition. In this paper, we describe a novel object localization algorithm which uses classification networks trained on only image labels. This weakly supervised method leverages local spatial and semantic patterns captured in the convolutional layers of classification networks. We propose an efficient beam search based approach to detect and localize multiple objects in images. The proposed method significantly outperforms the state-of-the-art in standard object localization data-sets with a 8 point increase in mAP scores

Load Balancing via Random Local Search in Closed and Open systems

In this paper, we analyze the performance of random load resampling and migration strategies in parallel server systems. Clients initially attach to an arbitrary server, but may switch server independently at random instants of time in an attempt to improve their service rate. This approach to load balancing contrasts with traditional approaches where clients make smart server selections upon arrival (e.g., Join-the-Shortest-Queue policy and variants thereof). Load resampling is particularly relevant in scenarios where clients cannot predict the load of a server before being actually attached to it. An important example is in wireless spectrum sharing where clients try to share a set of frequency bands in a distributed manner.Comment: Accepted to Sigmetrics 201

Tropicalization of Canonical Curves: the Planar Case

We study a topological version of the tropical lifting problem for canonical curves. This leads us to a tropical analogue of the notion of graph curves that we refer to as tropical graph curves. We study the analogous tropical lifting problem for graph curves and use this as a tool to show that every three regular, three edge connected planar graph of a given genus can be realized as the tropicalization of a canonical curve of the same genus.Comment: 26 pages, 14 Figures. This Is a revised version of our preprint "Tropical Graph Curves". We have completely rewritten the introduction, revised the body of the paper including several proofs and added a new section "Conclusion and Future Work

Embeddings and immersions of tropical curves

We construct immersions of trivalent abstract tropical curves in the Euclidean plane and embeddings of all abstract tropical curves in higher dimensional Euclidean space. Since not all curves have an embedding in the plane, we define the tropical crossing number of an abstract tropical curve to be the minimum number of self-intersections, counted with multiplicity, over all its immersions in the plane. We show that the tropical crossing number is at most quadratic in the number of edges and this bound is sharp. For curves of genus up to two, we systematically compute the crossing number. Finally, we use our immersed tropical curves to construct totally faithful nodal algebraic curves via lifting results of Mikhalkin and Shustin.Comment: 23 pages, 14 figures, final submitted versio