12,393 research outputs found
The triviality of the 61-stem in the stable homotopy groups of spheres
We prove that the 2-primary is zero. As a consequence, the
Kervaire invariant element is contained in the strictly defined
4-fold Toda bracket .
Our result has a geometric corollary: the 61-sphere has a unique smooth
structure and it is the last odd dimensional case - the only ones are and .
Our proof is a computation of homotopy groups of spheres. A major part of
this paper is to prove an Adams differential . We prove this
differential by introducing a new technique based on the algebraic and
geometric Kahn-Priddy theorems. The success of this technique suggests a
theoretical way to prove Adams differentials in the sphere spectrum inductively
by use of differentials in truncated projective spectra.Comment: 67 pages, minor changes, accepted versio
Enhanced -infinity obstruction theory
We extend the Bousfield-Kan spectral sequence for the computation of the
homotopy groups of the space of minimal A-infinity algebra structures on a
graded projective module. We use the new part to define obstructions to the
extension of truncated minimal A-infinity algebra structures. We also consider
the Bousfield-Kan spectral sequence for the moduli space of A-infinity
algebras. We compute up to the second page, terms and differentials, of these
spectral sequences in terms of Hochschild cohomology.Comment: 42 pages, color figure
Motivic Brown-Peterson invariants of the rationals
Fix the base field Q of rational numbers and let BP denote the family of
motivic truncated Brown-Peterson spectra over Q. We employ a "local-to-global"
philosophy in order to compute the motivic Adams spectral sequence converging
to the bi-graded homotopy groups of BP. Along the way, we provide a new
computation of the homotopy groups of BP over the 2-adic rationals, prove a
motivic Hasse principle for the spectra BP, and deduce several classical and
recent theorems about the K-theory of particular fields.Comment: 32 pages, 6 figures; Introduction and exposition improved, typos
corrected, now published in Geometry & Topolog
Automatic Search of Truncated Impossible Differentials for Word-Oriented Block Ciphers (Full Version)
Impossible differential cryptanalysis is a powerful technique to recover the secret key of block ciphers by
exploiting the fact that in block ciphers specific input and output
differences are not compatible.
This paper introduces a novel tool to search truncated impossible differentials for
word-oriented block ciphers with bijective Sboxes. Our tool generalizes the earlier
-method and the UID-method. It allows to reduce
the gap between the best impossible differentials found by these methods and the best known
differentials found by ad hoc methods that rely on cryptanalytic insights.
The time and space complexities of our tool in judging an -round truncated impossible differential are about and respectively,
where is the number of words in the plaintext and , are constants depending on the machine and the block cipher.
In order to demonstrate the strength of our tool, we show that it does not only allow to automatically rediscover the
longest truncated impossible differentials of many word-oriented block ciphers, but also finds new
results. It independently rediscovers all 72 known truncated impossible differentials on 9-round CLEFIA.
In addition, finds new truncated impossible differentials for AES, ARIA, Camellia without
FL and FL layers, E2, LBlock, MIBS and Piccolo.
Although our tool does
not improve the lengths of impossible differentials for existing block ciphers, it helps to
close the gap between the best known results of previous tools and those of manual cryptanalysis
Double complexes and vanishing of Novikov cohomology
We consider a non-standard totalisation functor to produce a cochain complex
from a given double complex: instead of sums or products, totalisation is
defined via truncated products of modules. We give an elementary proof of the
fact that a double complex with exact rows (resp, columns) yields an acyclic
cochain complex under totalisation using right (resp, left) truncated products.
As an application we consider the algebraic mapping torus T(h) of a self map h
of a cochain complex C. We show that if C consists of finitely presented
modules then T(h) has trivial negative Novikov cohomology; if in addition h is
a quasi-isomorphism, then T(h) has trivial positive Novikov cohomology as well.
As a consequence we obtain a new proof that a finitely dominated cochain
complex over a Laurent polynomial ring has trivial Novikov cohomology.Comment: 6 pages; diagrams typeset with Paul taylors "diagrams" macro package;
v2: 7 pages, expanded introduction, minor changes in exposition; v3: minor
changes to abstract, typos correcte
Adaptive Higher-order Spectral Estimators
Many applications involve estimation of a signal matrix from a noisy data
matrix. In such cases, it has been observed that estimators that shrink or
truncate the singular values of the data matrix perform well when the signal
matrix has approximately low rank. In this article, we generalize this approach
to the estimation of a tensor of parameters from noisy tensor data. We develop
new classes of estimators that shrink or threshold the mode-specific singular
values from the higher-order singular value decomposition. These classes of
estimators are indexed by tuning parameters, which we adaptively choose from
the data by minimizing Stein's unbiased risk estimate. In particular, this
procedure provides a way to estimate the multilinear rank of the underlying
signal tensor. Using simulation studies under a variety of conditions, we show
that our estimators perform well when the mean tensor has approximately low
multilinear rank, and perform competitively when the signal tensor does not
have approximately low multilinear rank. We illustrate the use of these methods
in an application to multivariate relational data.Comment: 29 pages, 3 figure
Hypergeometric Properties of Genus 3 Generalized Legendre Curves
Inspired by a result of Manin, we study the relationship between certain
period integrals and the trace of Frobenius of genus 3 generalized Legendre
curves. We show that both of these properties can be computed in terms of
"matching" classical and finite field hypergeometric functions, a phenomenon
that has also been observed in elliptic curves and many higher dimensional
varieties.Comment: 13 page
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