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[math/0408099] Tropical Mathematics
These are the notes for the Clay Mathematics Institute Senior Scholar Lecture which was delivered by Bernd Sturmfels in Park City, Utah, on July 22, 2004. The topic of this lecture is the ``tropical approach'' in mathematics, which has gotten a lot of attention recently in combinatorics, algebraic geometry and related fields. It offers an an elementary introduction to this subject, touching upon Arithmetic, Polynomials, Curves, Phylogenetics and Linear Spaces. Each section ends with a suggestion for further research. The bibliography contains numerousreferences for further reading in this field.
group-theory  mathematics  representation  combinatorics  rather-interesting  to-write-about 
22 hours ago by Vaguery
[1804.03032] k-NN Graph Construction: a Generic Online Approach
Nearest neighbor search and k-nearest neighbor graph construction are two fundamental issues arise from many disciplines such as information retrieval, data-mining, machine learning and computer vision. Despite continuous efforts have been taken in the last several decades, these two issues remain challenging. They become more and more imminent given the big data emerges in various fields and has been expanded significantly over the years. In this paper, a simple but effective solution both for k-nearest neighbor search and k-nearest neighbor graph construction is presented. Namely, these two issues are addressed jointly. On one hand, the k-nearest neighbor graph construction is treated as a nearest neighbor search task. Each data sample along with its k-nearest neighbors are joined into the k-nearest neighbor graph by sequentially performing the nearest neighbor search on the graph under construction. On the other hand, the built k-nearest neighbor graph is used to support k-nearest neighbor search. Since the graph is built online, dynamic updating of the graph, which is not desirable from most of the existing solutions, is supported. Moreover, this solution is feasible for various distance measures. Its effectiveness both as a k-nearest neighbor construction and k-nearest neighbor search approach is verified across various datasets in different scales, various dimensions and under different metrics.
graph-theory  algorithms  to-understand  rather-interesting  computational-complexity  data-structures 
2 days ago by Vaguery
[1804.10962] Stress anisotropy in shear-jammed packings of frictionless disks
We perform computational studies of repulsive, frictionless disks to investigate the development of stress anisotropy in mechanically stable (MS) packings. We focus on two protocols for generating MS packings: 1) isotropic compression and 2) applied simple or pure shear strain γ at fixed packing fraction ϕ. MS packings of frictionless disks occur as geometric families (i.e. parabolic segments with positive curvature) in the ϕ-γ plane. MS packings from protocol 1 populate parabolic segments with both signs of the slope, dϕ/dγ>0 and dϕ/dγ<0. In contrast, MS packings from protocol 2 populate segments with dϕ/dγ<0 only. For both simple and pure shear, we derive a relationship between the stress anisotropy and dilatancy dϕ/dγ obeyed by MS packings along geometrical families. We show that for MS packings prepared using isotropic compression, the stress anisotropy distribution is Gaussian centered at zero with a standard deviation that decreases with increasing system size. For shear jammed MS packings, the stress anisotropy distribution is a convolution of Weibull distributions that depend on strain, which has a nonzero average and standard deviation in the large-system limit. We also develop a framework to calculate the stress anisotropy distribution for packings generated via protocol 2 in terms of the stress anisotropy distribution for packings generated via protocol 1. These results emphasize that for repulsive frictionless disks, different packing-generation protocols give rise to different MS packing probabilities, which lead to differences in macroscopic properties of MS packings.
physics!  sandpiles  materials-science  simulation  rather-interesting  condensed-matter  phase-transitions  looking-to-see 
2 days ago by Vaguery
Coder-Physicists Are Simulating the Universe to Unlock Its Secrets | Quanta Magazine
These small, faint galaxies have always presented problems. The “missing satellite problem,” for instance, is the expectation, based on standard cold dark matter models, that hundreds of satellite galaxies should orbit every spiral galaxy. But the Milky Way has just dozens. This has caused some physicists to contemplate more complicated models of dark matter. However, when Hopkins and colleagues incorporated realistic superbubbles into their simulations, they saw many of those excess satellite galaxies go away. Hopkins has also found potential resolutions to two other problems, called “cusp-core” and “too-big-to-fail,” that have troubled the cold dark matter paradigm.
simulation  looking-to-see  astronomy  rather-interesting  to-write-about  the-mangle-in-practice  (totally) 
3 days ago by Vaguery
Tiling with TriCurves
There are a number of ways one can define a tricurve, the one used here is to start with an arc of some angle, replicate two identical curves ard rotate each about some angle about the ends of the arc. The Tricurve is the enclosed area.
plane-geometry  tiling  rather-interesting  define-your-terms  representation  to-write-about  mathematical-recreations 
8 days ago by Vaguery
Tiling with One Arc-Sided Shape | Math ∞ Blog
A flat puzzle (tiling) with dozens or hundreds of identical pieces may sound a little dull and predictable. But what is the most interesting shape we can use, to get the most unusual designs and the most variety? To make it more visually interesting, let’s say we want a shape with no straight edges—only curves. The following guidelines should help us get started.
plane-geometry  representation  tiling  rather-interesting  mathematical-recreations  to-write-about 
8 days ago by Vaguery
The propagation of error in classical geometry constructions | Joel David Hamkins
I’d like to discuss the issue of error and error propagation in the constructions of classical geometry. How does error propagate in these constructions? How sensitive are the familiar classical constructions to small errors in the use of the straightedge or compass?
plane-geometry  con  structions  robustness  rather-interesting  to-write-about  nudge-targets  consider:multiobjective-selection 
9 days ago by Vaguery
[1801.08003] Threadable Curves
We define a plane curve to be threadable if it can rigidly pass through a point-hole in a line L without otherwise touching L. Threadable curves are in a sense generalizations of monotone curves. We have two main results. The first is a linear-time algorithm for deciding whether a polygonal curve is threadable---O(n) for a curve of n vertices---and if threadable, finding a sequence of rigid motions to thread it through a hole. We also sketch an argument that shows that the threadability of algebraic curves can be decided in time polynomial in the degree of the curve. The second main result is an O(n polylog n)-time algorithm for deciding whether a 3D polygonal curve can thread through hole in a plane in R^3, and if so, providing a description of the rigid motions that achieve the threading.
computational-geometry  geometry  rather-interesting  definition  nudge-targets  consider:feature-discovery  to-write-about  consider:algorithms 
9 days ago by Vaguery
[1808.02841] On divergent Series
This is the translation of Leonhard Euler's paper "De Seriebus divergentibus" written in Latin into English. Leonhard Euler defines and discusses divergent series. He is especially interested in the example 1!−2!+3!−etc. and uses different methods to sum it. He finds a value of about 0.59....
mathematics  history  translation  series  to-write-about  rather-interesting 
9 days ago by Vaguery
[1808.07006] Observations on continued fractions
This is a translation of Euler's Latin paper "De fractionibus continuis observationes" into English. In this paper Euler describes his theory of continued fractions. He teaches, how to transform series into continued fractions, solves the Riccati-Differential equation by means of continued fractions and finds many other interesting formulas and results (e.g, the continued fraction for the quotient of two hypergeometric series usually attributed to Gau{\ss})
continued-fractions  translation  mathematics  history  rather-interesting  to-write-about 
9 days ago by Vaguery
[1810.00173] On solids whose (entire) surface can be unfolded onto a plane
This is the English translation of Leonhard Euler's Latin paper "De solidis quorum superficiem in planum explicare licet". Euler explains several methods to obtain equations for developable surfaces. Therefore, this paper might be interesting for anyone studying the history of Differential Geometry.
geometry  history  mathematics  translation  rather-interesting 
9 days ago by Vaguery
DRM-free Bookshops
A regularly updated list of online shops that sell e-books without DRM.
books  shopping  DRM  copyright  rather-interesting  publishing 
12 days ago by Vaguery
THE EXTENDED EVOLUTIONARY SYNTHESIS | Proceedings of the Royal Society of London B: Biological Sciences
Scientific activities take place within the structured sets of ideas and assumptions that define a field and its practices. The conceptual framework of evolutionary biology emerged with the Modern Synthesis in the early twentieth century and has since expanded into a highly successful research program to explore the processes of diversification and adaptation. Nonetheless, the ability of that framework satisfactorily to accommodate the rapid advances in developmental biology, genomics and ecology has been questioned. We review some of these arguments, focusing on literatures (evo-devo, developmental plasticity, inclusive inheritance and niche construction) whose implications for evolution can be interpreted in two ways—one that preserves the internal structure of contemporary evolutionary theory and one that points towards an alternative conceptual framework. The latter, which we label the ‘extended evolutionary synthesis' (EES), retains the fundaments of evolutionary theory, but differs in its emphasis on the role of constructive processes in development and evolution, and reciprocal portrayals of causation. In the EES, developmental processes, operating through developmental bias, inclusive inheritance and niche construction, share responsibility for the direction and rate of evolution, the origin of character variation and organism–environment complementarity. We spell out the structure, core assumptions and novel predictions of the EES, and show how it can be deployed to stimulate and advance research in those fields that study or use evolutionary biology.
evolutionary-biology  academic-culture  models-and-modes  theoretical-biology  define-your-terms  rather-interesting  to-write-about 
15 days ago by Vaguery
Same-different problems strain convolutional neural networks | the morning paper
Digging deeper, when learning did occur in SD, increasing item size never strained performance. But increasing the overall image size, or increasing the number of items did. (Gray bars in the above figures indicate the number of trials in which learning failed). The results suggest that straining is not simply a direct outcome of an increase in image variability. Using CNNs with more than twice the number of kernels (wide), or twice as many layers (deep) did not change the observed trend.
neural-networks  representation  problem-solving  rather-interesting  ontology  generalization  to-write-about  nudge-targets  consider:feature-discovery 
17 days ago by Vaguery
Treating The Prodrome | Slate Star Codex
I think what they are saying is that, as the world becomes even more random and confusing, the brain very slowly adjusts its highest level parameters. It concludes, on a level much deeper than consciousness, that the world does not make sense, that it’s not really useful to act because it’s impossible to predict the consequences of actions, and that it’s not worth drawing on prior knowledge because anything could happen at any time. It gets a sort of learned helplessness about cognition, where since it never works it’s not even worth trying. The onslaught of random evidence slowly twists the highest-level beliefs into whatever form best explains random evidence (usually: that there’s a conspiracy to do random things), and twists the fundamental parameters into a form where they expect evidence to be mostly random and aren’t going to really care about it one way or the other.
psychology  rather-interesting  consider:what-are-we-doing-to-our-machines?  emergence  to-write-about 
17 days ago by Vaguery
[1712.08175] Conversion of Love waves in a forest of trees
We inspect the propagation of shear polarized surface waves akin to Love waves through a forest of trees of same height atop a guiding layer on a soil substrate. We discover that the foliage of trees { brings a radical change in} the nature of the dispersion relation of these surface waves, which behave like spoof plasmons in the limit of a vanishing guiding layer, and like Love waves in the limit of trees with a vanishing height. When we consider a forest with trees of increasing or decreasing height, this hybrid "Spoof Love" wave is either reflected backwards or converted into a downward propagating bulk wave. An asymptotic analysis shows the forest behaves like an anisotropic wedge with effective boundary conditions.
materials-science  simulation  nonlinear-dynamics  rather-interesting  side-effects  consider:pragmatics-of-a-tree  to-write-about 
26 days ago by Vaguery
A non-spatial account of place and grid cells based on clustering models of concept learning | bioRxiv
One view is that conceptual knowledge is organized as a "cognitive map" in the brain, using the circuitry in the medial temporal lobe (MTL) that supports spatial navigation. In contrast, we find that a domain-general learning algorithm explains key findings in both spatial and conceptual domains. When the clustering model is applied to spatial navigation tasks, so called place and grid cells emerge because of the relatively uniform sampling of possible inputs in these tasks. The same mechanism applied to conceptual tasks, where the overall space can be higher-dimensional and sampling sparser, leads to representations more aligned with human conceptual knowledge. Although the types of memory supported by the MTL are superficially dissimilar, the information processing steps appear shared.
models-and-modes  emergence  data-analysis  rather-interesting  to-write-about  consider:the-mangle 
26 days ago by Vaguery

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