72,716 research outputs found
Tangle-tree duality: in graphs, matroids and beyond
We apply a recent duality theorem for tangles in abstract separation systems
to derive tangle-type duality theorems for width-parameters in graphs and
matroids. We further derive a duality theorem for the existence of clusters in
large data sets.
Our applications to graphs include new, tangle-type, duality theorems for
tree-width, path-width, and tree-decompositions of small adhesion. Conversely,
we show that carving width is dual to edge-tangles. For matroids we obtain a
duality theorem for tree-width.
Our results can be used to derive short proofs of all the classical duality
theorems for width parameters in graph minor theory, such as path-width,
tree-width, branch-width and rank-width.Comment: arXiv admin note: text overlap with arXiv:1406.379
Product decompositions of quasirandom groups and a Jordan type theorem
We first note that a result of Gowers on product-free sets in groups has an
unexpected consequence: If k is the minimal degree of a representation of the
finite group G, then for every subset B of G with we have
B^3 = G.
We use this to obtain improved versions of recent deep theorems of Helfgott
and of Shalev concerning product decompositions of finite simple groups, with
much simpler proofs.
On the other hand, we prove a version of Jordan's theorem which implies that
if k>1, then G has a proper subgroup of index at most ck^2 for some absolute
constant c, hence a product-free subset of size at least . This
answers a question of Gowers.Comment: 18 pages. In this third version we added an Appendix with a short
proof of Proposition
On the alleged simplicity of impure proof
Roughly, a proof of a theorem, is âpureâ if it draws only on what is âcloseâ or âintrinsicâ to that theorem. Mathematicians employ a variety of terms to identify pure proofs, saying that a pure proof is one that avoids what is âextrinsic,â âextraneous,â âdistant,â âremote,â âalien,â or âforeignâ to the problem or theorem under investigation. In the background of these attributions is the view that there is a distance measure (or a variety of such measures) between mathematical statements and proofs. Mathematicians have paid little attention to specifying such distance measures precisely because in practice certain methods of proof have seemed self- evidently impure by design: think for instance of analytic geometry and analytic number theory. By contrast, mathematicians have paid considerable attention to whether such impurities are a good thing or to be avoided, and some have claimed that they are valuable because generally impure proofs are simpler than pure proofs. This article is an investigation of this claim, formulated more precisely by proof- theoretic means. After assembling evidence from proof theory that may be thought to support this claim, we will argue that on the contrary this evidence does not support the claim
A Universal Approach to Self-Referential Paradoxes, Incompleteness and Fixed Points
Following F. William Lawvere, we show that many self-referential paradoxes,
incompleteness theorems and fixed point theorems fall out of the same simple
scheme. We demonstrate these similarities by showing how this simple scheme
encompasses the semantic paradoxes, and how they arise as diagonal arguments
and fixed point theorems in logic, computability theory, complexity theory and
formal language theory
Existence of periodic solutions of the FitzHugh-Nagumo equations for an explicit range of the small parameter
The FitzHugh-Nagumo model describing propagation of nerve impulses in axon is
given by fast-slow reaction-diffusion equations, with dependence on a parameter
representing the ratio of time scales. It is well known that for all
sufficiently small the system possesses a periodic traveling wave.
With aid of computer-assisted rigorous computations, we prove the existence of
this periodic orbit in the traveling wave equation for an explicit range
. Our approach is based on a novel method of
combination of topological techniques of covering relations and isolating
segments, for which we provide a self-contained theory. We show that the range
of existence is wide enough, so the upper bound can be reached by standard
validated continuation procedures. In particular, for the range we perform a rigorous continuation based on
covering relations and not specifically tailored to the fast-slow setting.
Moreover, we confirm that for the classical interval
Newton-Moore method applied to a sequence of Poincar\'e maps already succeeds.
Techniques described in this paper can be adapted to other fast-slow systems of
similar structure
Learning-Assisted Automated Reasoning with Flyspeck
The considerable mathematical knowledge encoded by the Flyspeck project is
combined with external automated theorem provers (ATPs) and machine-learning
premise selection methods trained on the proofs, producing an AI system capable
of answering a wide range of mathematical queries automatically. The
performance of this architecture is evaluated in a bootstrapping scenario
emulating the development of Flyspeck from axioms to the last theorem, each
time using only the previous theorems and proofs. It is shown that 39% of the
14185 theorems could be proved in a push-button mode (without any high-level
advice and user interaction) in 30 seconds of real time on a fourteen-CPU
workstation. The necessary work involves: (i) an implementation of sound
translations of the HOL Light logic to ATP formalisms: untyped first-order,
polymorphic typed first-order, and typed higher-order, (ii) export of the
dependency information from HOL Light and ATP proofs for the machine learners,
and (iii) choice of suitable representations and methods for learning from
previous proofs, and their integration as advisors with HOL Light. This work is
described and discussed here, and an initial analysis of the body of proofs
that were found fully automatically is provided
"Weak yet strong'' restrictions of Hindman's Finite Sums Theorem
We present a natural restriction of Hindmanâs Finite Sums Theorem that admits a simple combinatorial proof (one that does not also prove the full Finite Sums Theorem) and low computability-theoretic and proof-theoretic upper bounds, yet implies the existence of the Turing Jump, thus realizing the only known lower bound for the full Finite Sums Theorem. This is the first example of this kind. In fact we isolate a rich family of similar restrictions of Hindmanâs Theorem with analogous propertie
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