19,471 research outputs found
More Than 1700 Years of Word Equations
Geometry and Diophantine equations have been ever-present in mathematics.
Diophantus of Alexandria was born in the 3rd century (as far as we know), but a
systematic mathematical study of word equations began only in the 20th century.
So, the title of the present article does not seem to be justified at all.
However, a linear Diophantine equation can be viewed as a special case of a
system of word equations over a unary alphabet, and, more importantly, a word
equation can be viewed as a special case of a Diophantine equation. Hence, the
problem WordEquations: "Is a given word equation solvable?" is intimately
related to Hilbert's 10th problem on the solvability of Diophantine equations.
This became clear to the Russian school of mathematics at the latest in the mid
1960s, after which a systematic study of that relation began.
Here, we review some recent developments which led to an amazingly simple
decision procedure for WordEquations, and to the description of the set of all
solutions as an EDT0L language.Comment: The paper will appear as an invited address in the LNCS proceedings
of CAI 2015, Stuttgart, Germany, September 1 - 4, 201
Neutral evolution and turnover over centuries of English word popularity
Here we test Neutral models against the evolution of English word frequency
and vocabulary at the population scale, as recorded in annual word frequencies
from three centuries of English language books. Against these data, we test
both static and dynamic predictions of two neutral models, including the
relation between corpus size and vocabulary size, frequency distributions, and
turnover within those frequency distributions. Although a commonly used Neutral
model fails to replicate all these emergent properties at once, we find that
modified two-stage Neutral model does replicate the static and dynamic
properties of the corpus data. This two-stage model is meant to represent a
relatively small corpus (population) of English books, analogous to a `canon',
sampled by an exponentially increasing corpus of books in the wider population
of authors. More broadly, this mode -- a smaller neutral model within a larger
neutral model -- could represent more broadly those situations where mass
attention is focused on a small subset of the cultural variants.Comment: 12 pages, 5 figures, 1 tabl
Multi-timescale Solar Cycles and the Possible Implications
Based on analysis of the annual averaged relative sunspot number (ASN) during
1700 -- 2009, 3 kinds of solar cycles are confirmed: the well-known 11-yr cycle
(Schwabe cycle), 103-yr secular cycle (numbered as G1, G2, G3, and G4,
respectively since 1700); and 51.5-yr Cycle. From similarities, an
extrapolation of forthcoming solar cycles is made, and found that the solar
cycle 24 will be a relative long and weak Schwabe cycle, which may reach to its
apex around 2012-2014 in the vale between G3 and G4. Additionally, most Schwabe
cycles are asymmetric with rapidly rising-phases and slowly decay-phases. The
comparisons between ASN and the annual flare numbers with different GOES
classes (C-class, M-class, X-class, and super-flare, here super-flare is
defined as X10.0) and the annal averaged radio flux at frequency of 2.84
GHz indicate that solar flares have a tendency: the more powerful of the flare,
the later it takes place after the onset of the Schwabe cycle, and most
powerful flares take place in the decay phase of Schwabe cycle. Some
discussions on the origin of solar cycles are presented.Comment: 8 pages, 4 figure
Phenomenology of current-induced skyrmion motion in antiferromagnets
We study current-driven skyrmion motion in uniaxial thin film
antiferromagnets in the presence of the Dzyaloshinskii-Moriya interactions and
in an external magnetic field. We phenomenologically include relaxation and
current-induced torques due to both spin-orbit coupling and spatially
inhomogeneous magnetic textures in the equation for the N\'eel vector of the
antiferromagnet. Using the collective coordinate approach we apply the theory
to a two-dimensional antiferromagnetic skyrmion and estimate the skyrmion
velocity under an applied DC electric current.Comment: 14 pages, 3 figures, 1 tabl
From Malthus to Ohlin: Trade, Growth and Distribution Since 1500
A recent endogenous growth literature has focused on the transition from a Malthusian world where real wages were linked to factor endowments, to one where modern growth has broken that link. In this paper we present evidence on another, related phenomenon: the dramatic reversal in distributional trends -- from a steep secular fall to a steep secular rise in wage-land rent ratios -- which occurred some time early in the 19th century. What explains this reversal? While it may seem logical to locate the causes in the Industrial Revolutionary forces emphasized by endogenous growth theorists, we provide evidence that something else mattered just as much: the opening up of the European economy to international trade.
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