31,483 research outputs found
A ROOT/IO Based Software Framework for CMS
The implementation of persistency in the Compact Muon Solenoid (CMS) Software
Framework uses the core I/O functionality of ROOT. We will discuss the current
ROOT/IO implementation, its evolution from the prior Objectivity/DB
implementation, and the plans and ongoing work for the conversion to "POOL",
provided by the LHC Computing Grid (LCG) persistency project
Categorification of persistent homology
We redevelop persistent homology (topological persistence) from a categorical
point of view. The main objects of study are diagrams, indexed by the poset of
real numbers, in some target category. The set of such diagrams has an
interleaving distance, which we show generalizes the previously-studied
bottleneck distance. To illustrate the utility of this approach, we greatly
generalize previous stability results for persistence, extended persistence,
and kernel, image and cokernel persistence. We give a natural construction of a
category of interleavings of these diagrams, and show that if the target
category is abelian, so is this category of interleavings.Comment: 27 pages, v3: minor changes, to appear in Discrete & Computational
Geometr
Nilpotent operators and weighted projective lines
We show a surprising link between singularity theory and the invariant
subspace problem of nilpotent operators as recently studied by C. M. Ringel and
M. Schmidmeier, a problem with a longstanding history going back to G.
Birkhoff. The link is established via weighted projective lines and (stable)
categories of vector bundles on those. The setup yields a new approach to
attack the subspace problem. In particular, we deduce the main results of
Ringel and Schmidmeier for nilpotency degree p from properties of the category
of vector bundles on the weighted projective line of weight type (2,3,p),
obtained by Serre construction from the triangle singularity x^2+y^3+z^p. For
p=6 the Ringel-Schmidmeier classification is thus covered by the classification
of vector bundles for tubular type (2,3,6), and then is closely related to
Atiyah's classification of vector bundles on a smooth elliptic curve. Returning
to the general case, we establish that the stable categories associated to
vector bundles or invariant subspaces of nilpotent operators may be naturally
identified as triangulated categories. They satisfy Serre duality and also have
tilting objects whose endomorphism rings play a role in singularity theory. In
fact, we thus obtain a whole sequence of triangulated (fractional) Calabi-Yau
categories, indexed by p, which naturally form an ADE-chain.Comment: More details added. 33 page
Transparent Persistence with Java Data Objects
Flexible and performant Persistency Service is a necessary component of any
HEP Software Framework. The building of a modular, non-intrusive and performant
persistency component have been shown to be very difficult task. In the past,
it was very often necessary to sacrifice modularity to achieve acceptable
performance. This resulted in the strong dependency of the overall Frameworks
on their Persistency subsystems.
Recent development in software technology has made possible to build a
Persistency Service which can be transparently used from other Frameworks. Such
Service doesn't force a strong architectural constraints on the overall
Framework Architecture, while satisfying high performance requirements. Java
Data Object standard (JDO) has been already implemented for almost all major
databases. It provides truly transparent persistency for any Java object (both
internal and external). Objects in other languages can be handled via
transparent proxies. Being only a thin layer on top of a used database, JDO
doesn't introduce any significant performance degradation. Also Aspect-Oriented
Programming (AOP) makes possible to treat persistency as an orthogonal Aspect
of the Application Framework, without polluting it with persistence-specific
concepts.
All these techniques have been developed primarily (or only) for the Java
environment. It is, however, possible to interface them transparently to
Frameworks built in other languages, like for example C++.
Fully functional prototypes of flexible and non-intrusive persistency modules
have been build for several other packages, as for example FreeHEP AIDA and LCG
Pool AttributeSet (package Indicium).Comment: Talk from the 2003 Computing in High Energy and Nuclear Physics
(CHEP03), La Jolla, Ca, USA, March 2003. PSN TUKT00
The Theory of the Interleaving Distance on Multidimensional Persistence Modules
In 2009, Chazal et al. introduced -interleavings of persistence
modules. -interleavings induce a pseudometric on (isomorphism
classes of) persistence modules, the interleaving distance. The definitions of
-interleavings and generalize readily to multidimensional
persistence modules. In this paper, we develop the theory of multidimensional
interleavings, with a view towards applications to topological data analysis.
We present four main results. First, we show that on 1-D persistence modules,
is equal to the bottleneck distance . This result, which first
appeared in an earlier preprint of this paper, has since appeared in several
other places, and is now known as the isometry theorem. Second, we present a
characterization of the -interleaving relation on multidimensional
persistence modules. This expresses transparently the sense in which two
-interleaved modules are algebraically similar. Third, using this
characterization, we show that when we define our persistence modules over a
prime field, satisfies a universality property. This universality result
is the central result of the paper. It says that satisfies a stability
property generalizing one which is known to satisfy, and that in
addition, if is any other pseudometric on multidimensional persistence
modules satisfying the same stability property, then . We also show
that a variant of this universality result holds for , over arbitrary
fields. Finally, we show that restricts to a metric on isomorphism
classes of finitely presented multidimensional persistence modules.Comment: Major revision; exposition improved throughout. To appear in
Foundations of Computational Mathematics. 36 page
Induced Matchings and the Algebraic Stability of Persistence Barcodes
We define a simple, explicit map sending a morphism of
pointwise finite dimensional persistence modules to a matching between the
barcodes of and . Our main result is that, in a precise sense, the
quality of this matching is tightly controlled by the lengths of the longest
intervals in the barcodes of and . As an
immediate corollary, we obtain a new proof of the algebraic stability of
persistence, a fundamental result in the theory of persistent homology. In
contrast to previous proofs, ours shows explicitly how a -interleaving
morphism between two persistence modules induces a -matching between
the barcodes of the two modules. Our main result also specializes to a
structure theorem for submodules and quotients of persistence modules, and
yields a novel "single-morphism" characterization of the interleaving relation
on persistence modules.Comment: Expanded journal version, to appear in Journal of Computational
Geometry. Includes a proof that no definition of induced matching can be
fully functorial (Proposition 5.10), and an extension of our single-morphism
characterization of the interleaving relation to multidimensional persistence
modules (Remark 6.7). Exposition is improved throughout. 11 Figures adde
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