15 research outputs found
Yang-Mills measure on the two-dimensional torus as a random distribution
We introduce a space of distributional one-forms on the
torus for which holonomies along axis paths are well-defined and
induce H\"older continuous functions on line segments. We show that there
exists an -valued random variable for which Wilson loop
observables of axis paths coincide in law with the corresponding observables
under the Yang-Mills measure in the sense of L\'evy (2003). It holds
furthermore that embeds into the H\"older-Besov space
for all , so that has the correct
small scale regularity expected from perturbation theory. Our method is based
on a Landau-type gauge applied to lattice approximations.Comment: 37 pages, 4 figures. Minor revisions. To appear in Communications in
Mathematical Physic
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A rough path perspective on renormalization
We develop the algebraic theory of rough path translation. Particular attention is given to the case of branched rough paths, whose underlying algebraic structure (Connes-Kreimer, Grossman-Larson) makes it a useful model case of a regularity structure in the sense of Hairer. Pre-Lie structures are seen to play a fundamental rule which allow a direct understanding of the translated (i.e. renormalized) equation under consideration. This construction is also novel with regard to the algebraic renormalization theory for regularity structures due to Bruned–Hairer–Zambotti (2016), the links with which are discussed in detail. © 2019 The Author(s
Multiscale Systems, Homogenization, and Rough Paths:VAR75 2016: Probability and Analysis in Interacting Physical Systems
In recent years, substantial progress was made towards understanding
convergence of fast-slow deterministic systems to stochastic differential
equations. In contrast to more classical approaches, the assumptions on the
fast flow are very mild. We survey the origins of this theory and then revisit
and improve the analysis of Kelly-Melbourne [Ann. Probab. Volume 44, Number 1
(2016), 479-520], taking into account recent progress in -variation and
c\`adl\`ag rough path theory.Comment: 27 pages. Minor corrections. To appear in Proceedings of the
Conference in Honor of the 75th Birthday of S.R.S. Varadha
A rough path perspective on renormalization
We develop the algebraic theory of rough path translation. Particular
attention is given to the case of branched rough paths, whose underlying
algebraic structure (Connes-Kreimer, Grossman-Larson) makes it a useful model
case of a regularity structure in the sense of Hairer. Pre-Lie structures are
seen to play a fundamental rule which allow a direct understanding of the
translated (i.e. renormalized) equation under consideration. This construction
is also novel with regard to the algebraic renormalization theory for
regularity structures due to Bruned--Hairer--Zambotti (2016), the links with
which are discussed in detail.Comment: Final version to appear in Journal of Functional Analysi
Supersingular isogeny graphs and endomorphism rings:reductions and solutions
In this paper, we study several related computational problems for supersingular elliptic curves, their isogeny graphs, and their endomorphism rings. We prove reductions between the problem of path finding in the -isogeny graph, computing maximal orders isomorphic to the endomorphism ring of a supersingular elliptic curve, and computing the endomorphism ring itself. We also give constructive versions of Deuring’s correspondence, which associates to a maximal order in a certain quaternion algebra an isomorphism class of supersingular elliptic curves. The reductions are based on heuristics regarding the distribution of norms of elements in quaternion algebras. We show that conjugacy classes of maximal orders have a representative of polynomial size, and we define a way to represent endomorphism ring generators in a way that allows for efficient evaluation at points on the curve. We relate these problems to the security of the Charles-Goren-Lauter hash function. We provide a collision attack for special but natural parameters of the hash function and prove that for general parameters its preimage and collision resistance are also equivalent to the endomorphism ring computation problem.SCOPUS: cp.kinfo:eu-repo/semantics/published37th Annual International Conference on the Theory and Applications of Cryptographic Techniques, EUROCRYPT 2018; Tel Aviv; Israel; 29 April 2018 through 3 May 2018ISBN: 978-331978371-0Volume Editors: Nielsen J.B.Rijmen V.Publisher: Springer Verla
Signature moments to characterize laws of stochastic processes
The sequence of moments of a vector-valued random variable can characterize
its law. We study the analogous problem for path-valued random variables, that
is stochastic processes, by using so-called robust signature moments. This
allows us to derive a metric of maximum mean discrepancy type for laws of
stochastic processes and study the topology it induces on the space of laws of
stochastic processes. This metric can be kernelized using the signature kernel
which allows to efficiently compute it. As an application, we provide a
non-parametric two-sample hypothesis test for laws of stochastic processes
Persistence paths and signature features in topological data analysis
We introduce a new feature map for barcodes as they arise in persistent homology computation. The main idea is to first realize each barcode as a path in a convenient vector space, and to then compute its path signature which takes values in the tensor algebra of that vector space. The composition of these two operations - barcode to path, path to tensor series - results in a feature map that has several desirable properties for statistical learning, such as universality and characteristicness, and achieves state-of-the-art results on common classification benchmarks
A stochastic model of chemorepulsion with additive noise and nonlinear sensitivity
We consider a stochastic partial differential equation (SPDE) model for chemorepulsion, with non-linear sensitivity on the one-dimensional torus. By establishing an a priori estimate independent of the initial data, we show that there exists a pathwise unique, global solution to the SPDE. Furthermore, we show that the associated semi-group is Markov and possesses a unique invariant measure, supported on a Hölder–Besov space of positive regularity, which the solution law converges to exponentially fast. The a priori bound also allows us to establish tail estimates on the Lp norm of the invariant measure which are heavier than Gaussian
Persistence Paths and Signature Features in Topological Data Analysis
We introduce a new feature map for barcodes as they arise in persistent homology computation. The main idea is to first realize each barcode as a path in a convenient vector space, and to then compute its path signature which takes values in the tensor algebra of that vector space. The composition of these two operations-barcode to path, path to tensor series-results in a feature map that has several desirable properties for statistical learning, such as universality and characteristicness, and achieves state-of-the-art results on common classification benchmark
Pension reform in the Czech Republic and demographic development
The paper work is focused on the issue of pension reform, which has been recently made in the Czech Republic. The first part of the paper describes the pension systems, their types and differences among them. Further the work is dedicated to progressive development of the pension system. It is focused on the signifficant changes that were made within the so-called Small and Big pension reform and introduces the versions of the pension reforms of particular political parties based on Bezděk report's conclusion. Further the work analyzes the factors, which have impact on the progress of the pension system with focus on future trens of demographic development. In conclusion the paper work evaluates the fullfilment of the goals set by the government and describes the results, which the reform has brought in two years of its functioning