Trees with an On-Line Degree Ramsey Number of Four

On-line Ramsey theory studies a graph-building game between two players. The player called Builder builds edges one at a time, and the player called Painter paints each new edge red or blue after it is built. The graph constructed is called the background graph. Builder's goal is to cause the background graph to contain a monochromatic copy of a given goal graph, and Painter's goal is to prevent this. In the S[subscript k]-game variant of the typical game, the background graph is constrained to have maximum degree no greater than k. The on-line degree Ramsey number [˚over R][subscript Δ](G) of a graph G is the minimum k such that Builder wins an S[subscript k]-game in which G is the goal graph. Butterfield et al. previously determined all graphs G satisfying [˚ over R][subscript Δ](G)≤3. We provide a complete classification of trees T satisfying [˚ over R][subscript Δ](T)=4.National Science Foundation (U.S.) (Grant DMS-0754106)United States. National Security Agency (Grant H98230-06-1-0013

Towards an integrated understanding of neural networks

Thesis: Ph. D., Massachusetts Institute of Technology, Department of Mathematics, 2018.Cataloged from PDF version of thesis.Includes bibliographical references (pages 123-136).Neural networks underpin both biological intelligence and modern Al systems, yet there is relatively little theory for how the observed behavior of these networks arises. Even the connectivity of neurons within the brain remains largely unknown, and popular deep learning algorithms lack theoretical justification or reliability guarantees. This thesis aims towards a more rigorous understanding of neural networks. We characterize and, where possible, prove essential properties of neural algorithms: expressivity, learning, and robustness. We show how observed emergent behavior can arise from network dynamics, and we develop algorithms for learning more about the network structure of the brain.by David Rolnick.Ph. D

Looking for evidence of noncompetitive behavior in Minnesota's banking industry

Banks and banking - Minnesota

Neural Networks as Paths through the Space of Representations

Deep neural networks implement a sequence of layer-by-layer operations that are each relatively easy to understand, but the resulting overall computation is generally difficult to understand. We consider a simple hypothesis for interpreting the layer-by-layer construction of useful representations: perhaps the role of each layer is to reformat information to reduce the "distance" to the desired outputs. With this framework, the layer-wise computation implemented by a deep neural network can be viewed as a path through a high-dimensional representation space. We formalize this intuitive idea of a "path" by leveraging recent advances in *metric* representational similarity. We extend existing representational distance methods by computing geodesics, angles, and projections of representations, going beyond mere layer distances. We then demonstrate these tools by visualizing and comparing the paths taken by ResNet and VGG architectures on CIFAR-10. We conclude by sketching additional ways that this kind of representational geometry can be used to understand and interpret network training, and to describe novel kinds of similarities between different models.Comment: 10 pages, submitted to ICLR 202

Nuclear dependence of the transverse single-spin asymmetry in the production of charged hadrons at forward rapidity in polarized p+pp+p, p+p+Al, and p+p+Au collisions at sNN=200\sqrt{s_{_{NN}}}=200 GeV

We report on the nuclear dependence of transverse single-spin asymmetries (TSSAs) in the production of positively-charged hadrons in polarized $p^{\uparrow}+p$, $p^{\uparrow}+$Al and $p^{\uparrow}+$Au collisions at $\sqrt{s_{_{NN}}}=200$ GeV. The measurements have been performed at forward rapidity ($1.4<\eta<2.4$) over the range of $1.8<p_{T}<7.0$ GeV$/c$ and $0.1<x_{F}<0.2$. We observed a positive asymmetry $A_{N}$ for positively-charged hadrons in \polpp collisions, and a significantly reduced asymmetry in $p^{\uparrow}$+$A$ collisions. These results reveal a nuclear dependence of charged hadron $A_N$ in a regime where perturbative techniques are relevant. These results provide new opportunities to use \polpA collisions as a tool to investigate the rich phenomena behind TSSAs in hadronic collisions and to use TSSA as a new handle in studying small-system collisions.Comment: 303 authors from 66 institutions, 9 pages, 2 figures, 1 table. v1 is version accepted for publication in Physical Review Letters. Plain text data tables for the points plotted in figures for this and previous PHENIX publications are (or will be) publicly available at http://www.phenix.bnl.gov/papers.htm

Measurements of double-helicity asymmetries in inclusive J/ψJ/\psi production in longitudinally polarized p+pp+p collisions at s=510\sqrt{s}=510 GeV

We report the double helicity asymmetry, $A_{LL}^{J/\psi}$, in inclusive $J/\psi$ production at forward rapidity as a function of transverse momentum $p_T$ and rapidity $|y|$. The data analyzed were taken during $\sqrt{s}=510$ GeV longitudinally polarized $p$$+$$p$ collisions at the Relativistic Heavy Ion Collider (RHIC) in the 2013 run using the PHENIX detector. At this collision energy, $J/\psi$ particles are predominantly produced through gluon-gluon scatterings, thus $A_{LL}^{J/\psi}$ is sensitive to the gluon polarization inside the proton. We measured $A_{LL}^{J/\psi}$ by detecting the decay daughter muon pairs $\mu^+ \mu^-$ within the PHENIX muon spectrometers in the rapidity range $1.2<|y|<2.2$. In this kinematic range, we measured the $A_{LL}^{J/\psi}$ to be $0.012 \pm 0.010$~(stat)~$\pm$~$0.003$(syst). The $A_{LL}^{J/\psi}$ can be expressed to be proportional to the product of the gluon polarization distributions at two distinct ranges of Bjorken $x$: one at moderate range $x \approx 0.05$ where recent RHIC data of jet and $\pi^0$ double helicity spin asymmetries have shown evidence for significant gluon polarization, and the other one covering the poorly known small-$x$ region $x \approx 2\times 10^{-3}$. Thus our new results could be used to further constrain the gluon polarization for $x< 0.05$.Comment: 335 authors, 10 pages, 4 figures, 3 tables, 2013 data. Version accepted for publication by Phys. Rev. D. Plain text data tables for the points plotted in figures for this and previous PHENIX publications are (or will be) publicly available at http://www.phenix.bnl.gov/papers.htm