148 research outputs found
Eliciting implicit assumptions of proofs in the MIZAR Mathematical Library by property omission
When formalizing proofs with interactive theorem provers, it often happens
that extra background knowledge (declarative or procedural) about mathematical
concepts is employed without the formalizer explicitly invoking it, to help the
formalizer focus on the relevant details of the proof. In the contexts of
producing and studying a formalized mathematical argument, such mechanisms are
clearly valuable. But we may not always wish to suppress background knowledge.
For certain purposes, it is important to know, as far as possible, precisely
what background knowledge was implicitly employed in a formal proof. In this
note we describe an experiment conducted on the MIZAR Mathematical Library of
formal mathematical proofs to elicit one such class of implicitly employed
background knowledge: properties of functions and relations (e.g.,
commutativity, asymmetry, etc.).Comment: 11 pages, 3 tables. Preliminary version presented at the 3rd Workshop
on Modules and Libraries for Proof Assistants (MLPA-11), affiliated with the
2nd Conference on Interactive Theorem Proving (ITP-2011), Nijmegen, the
Netherland
Vortex structure in p-wave superconductors
We study vortices in p-wave superconductors in a Ginzburg-Landau setting. The
state of the superconductor is described by a pair of complex wave functions,
and the p-wave symmetric energy functional couples these in both the kinetic
(gradient) and potential energy terms, giving rise to systems of partial
differential equations which are nonlinear and coupled in their second
derivative terms. We prove the existence of energy minimizing solutions in
bounded domains , and consider the existence and
qualitative properties (such as the asymptotic behavior) of equivariant
solutions defined in all of . The coupling of the equations at
highest order changes the nature of the solutions, and many of the usual
properties of classical Ginzburg-Landau vortices either do not hold for the
p-wave solutions or are not immediately evident
Positive solutions to indefinite Neumann problems when the weight has positive average
We deal with positive solutions for the Neumann boundary value problem
associated with the scalar second order ODE where is positive on and is an indefinite weight. Complementary to previous
investigations in the case , we provide existence results
for a suitable class of weights having (small) positive mean, when
at infinity. Our proof relies on a shooting argument for a suitable equivalent
planar system of the type with
a continuous function defined on the whole real line.Comment: 17 pages, 3 figure
Large Formal Wikis: Issues and Solutions
We present several steps towards large formal mathematical wikis. The Coq
proof assistant together with the CoRN repository are added to the pool of
systems handled by the general wiki system described in
\cite{DBLP:conf/aisc/UrbanARG10}. A smart re-verification scheme for the large
formal libraries in the wiki is suggested for Mizar/MML and Coq/CoRN, based on
recently developed precise tracking of mathematical dependencies. We propose to
use features of state-of-the-art filesystems to allow real-time cloning and
sandboxing of the entire libraries, allowing also to extend the wiki to a true
multi-user collaborative area. A number of related issues are discussed.Comment: To appear in The Conference of Intelligent Computer Mathematics: CICM
201
Ginzburg-Landau model with small pinning domains
We consider a Ginzburg-Landau type energy with a piecewise constant pinning
term in the potential . The function is different from
1 only on finitely many disjoint domains, called the {\it pinning domains}.
These pinning domains model small impurities in a homogeneous superconductor
and shrink to single points in the limit ; here, \v is the inverse of
the Ginzburg-Landau parameter. We study the energy minimization in a smooth
simply connected domain with Dirichlet boundary
condition on \d \O, with topological degree {\rm deg}_{\d \O} (g) = d
>0. Our main result is that, for small \v, minimizers have distinct
zeros (vortices) which are inside the pinning domains and they have a degree
equal to 1. The question of finding the locations of the pinning domains with
vortices is reduced to a discrete minimization problem for a finite-dimensional
functional of renormalized energy. We also find the position of the vortices
inside the pinning domains and show that, asymptotically, this position is
determined by {\it local renormalized energy} which does not depend on the
external boundary conditions.Comment: 39 page
Premise Selection for Mathematics by Corpus Analysis and Kernel Methods
Smart premise selection is essential when using automated reasoning as a tool
for large-theory formal proof development. A good method for premise selection
in complex mathematical libraries is the application of machine learning to
large corpora of proofs. This work develops learning-based premise selection in
two ways. First, a newly available minimal dependency analysis of existing
high-level formal mathematical proofs is used to build a large knowledge base
of proof dependencies, providing precise data for ATP-based re-verification and
for training premise selection algorithms. Second, a new machine learning
algorithm for premise selection based on kernel methods is proposed and
implemented. To evaluate the impact of both techniques, a benchmark consisting
of 2078 large-theory mathematical problems is constructed,extending the older
MPTP Challenge benchmark. The combined effect of the techniques results in a
50% improvement on the benchmark over the Vampire/SInE state-of-the-art system
for automated reasoning in large theories.Comment: 26 page
The magnetic neutron scattering resonance of high-T_c superconductors in external magnetic fields: an SO(5) study
The magnetic resonance at 41 meV observed in neutron scattering studies of Y
Ba_2 Cu_3 O_7 holds a key position in the understanding of high-T_c
superconductivity. Within the SO(5) model for superconductivity and
antiferromagnetism, we have calculated the effect of an applied magnetic field
on the neutron scattering cross-section of the magnetic resonance. In the
presence of vortices, the neutron scattering cross-section shows clear
signatures of not only the fluctuations in the superconducting order parameter
\psi, but also the modulation of the phase of \psi due to vortices. In
reciprocal space we find that i) the scattering amplitude is zero at
(pi/a,pi/a), ii) the resonance peak is split into a ring with radius pi/d
centered at (pi/a,pi/a), d being the vortex lattice constant, and consequently,
iii) the splitting pi/d scales with the magnetic field as sqrt{B}.Comment: 4 pages including 3 eps-figures - minor changes and one reference
added. Accepted for publication in Phys. Rev.
Evolutionary Analysis of Cystatins of Early-Emerging Metazoans Reveals a Novel Subtype in Parasitic Cnidarians
© 2021 by the authors.The evolutionary aspects of cystatins are greatly underexplored in early-emerging metazoans. Thus, we surveyed the gene organization, protein architecture, and phylogeny of cystatin homologues mined from 110 genomes and the transcriptomes of 58 basal metazoan species, encompassing free-living and parasite taxa of Porifera, Placozoa, Cnidaria (including Myxozoa), and Ctenophora. We found that the cystatin gene repertoire significantly differs among phyla, with stefins present in most of the investigated lineages but with type 2 cystatins missing in several basal metazoan groups. Similar to liver and intestinal flukes, myxozoan parasites possess atypical stefins with chimeric structure that combine motifs of classical stefins and type 2 cystatins. Other early metazoan taxa regardless of lifestyle have only the classical representation of cystatins and lack multi-domain ones. Our comprehensive phylogenetic analyses revealed that stefins and type 2 cystatins clustered into taxonomically defined clades with multiple independent paralogous groups, which probably arose due to gene duplications. The stefin clade split between the subclades of classical stefins and the atypical stefins of myxozoans and flukes. Atypical stefins represent key evolutionary innovations of the two parasite groups for which their origin might have been linked with ancestral gene chimerization, obligate parasitism, life cycle complexity, genome reduction, and host immunity.This research was funded by the Ministry of Education, Youth, and Sports of the Czech Republic, grant number LTAUSA17201; by the European Commission under the H2020 ProgrammeâParaFishControl, grant number 634429; by the Czech Science Foundation, grant number 19-28399X (to A. S. Holzer, G. Alama-Bermejo, and J. KyslĂk) and 21-16565S and by the Czech Academy of Sciences and Hungarian Academy of Sciences, grant number MTA 19-07. This publication reflects the views of the authors only; the European Commission cannot be held responsible for any use which may be made of the information contained therein
Vortex state in a doped Mott insulator
We analyze the recent vortex core spectroscopy experiments in cuprate
superconductor and discuss what can be learned from them about the nature of
the ground state in these compounds. We argue that the data are inconsistent
with the assumption of a simple metallic ground state and exhibit
characteristics of a doped Mott insulator. A theory for a vortex core in such a
doped Mott insulator is developed based on the U(1) gauge field slave boson
model and is shown to exhibit properties qualitatively consistent with the
experimental data.Comment: 11 pages REVTeX, 3 ps figures; version to appear in PR
About curvature, conformal metrics and warped products
We consider the curvature of a family of warped products of two
pseduo-Riemannian manifolds and furnished with metrics of
the form and, in particular, of the type , where are smooth
functions and is a real parameter. We obtain suitable expressions for the
Ricci tensor and scalar curvature of such products that allow us to establish
results about the existence of Einstein or constant scalar curvature structures
in these categories. If is Riemannian, the latter question involves
nonlinear elliptic partial differential equations with concave-convex
nonlinearities and singular partial differential equations of the
Lichnerowicz-York type among others.Comment: 32 pages, 3 figure
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