10,126 research outputs found
On modular decompositions of system signatures
Considering a semicoherent system made up of components having i.i.d.
continuous lifetimes, Samaniego defined its structural signature as the
-tuple whose -th coordinate is the probability that the -th component
failure causes the system to fail. This -tuple, which depends only on the
structure of the system and not on the distribution of the component lifetimes,
is a very useful tool in the theoretical analysis of coherent systems.
It was shown in two independent recent papers how the structural signature of
a system partitioned into two disjoint modules can be computed from the
signatures of these modules. In this work we consider the general case of a
system partitioned into an arbitrary number of disjoint modules organized in an
arbitrary way and we provide a general formula for the signature of the system
in terms of the signatures of the modules.
The concept of signature was recently extended to the general case of
semicoherent systems whose components may have dependent lifetimes. The same
definition for the -tuple gives rise to the probability signature, which may
depend on both the structure of the system and the probability distribution of
the component lifetimes. In this general setting, we show how under a natural
condition on the distribution of the lifetimes, the probability signature of
the system can be expressed in terms of the probability signatures of the
modules. We finally discuss a few situations where this condition holds in the
non-i.i.d. and nonexchangeable cases and provide some applications of the main
results
The algebra of parallel endomorphisms of a germ of pseudo-Riemannian metric
On a (pseudo-)Riemannian manifold (M,g), some fields of endomorphisms i.e.
sections of End(TM) may be parallel for g. They form an associative algebra A,
which is also the commutant of the holonomy group of g. As any associative
algebra, A is the sum of its radical and of a semi-simple algebra S. We show in
arXiv:1402.6642 that S may be of eight different types, including the generic
type S=R.Id, and the K\"ahler and hyperk\"ahler types where S is respectively
isomorphic to the complex field C or to the quaternions H. We show here that
for any self adjoint nilpotent element N of the commutant of such an S in
End(TM), the set of germs of metrics such that A contains S and {N} is
non-empty. We parametrise it. Generically, the holonomy algebra of those
metrics is the full commutant of in O(g). Apart from some
"degenerate" cases, the algebra A is then , where (N) is the
ideal spanned by N. To prove it, we introduce an analogy with complex
Differential Calculus, the ring R[X]/(X^n) replacing the field C. This
describes totally the local situation when the radical of A is principal and
consists of self adjoint elements. We add a glimpse on the case where this
radical is not principal.Comment: 47 pages. This version is only a part of the first version of this
preprint. The other part is now published separately, see arXiv:1402.6642.
Here some typos are corrected; some statements, remarks and tables are made
more concis
Unfolding Mixed-Symmetry Fields in AdS and the BMV Conjecture: I. General Formalism
We present some generalities of unfolded on-shell dynamics that are useful in
analysing the BMV conjecture for mixed-symmetry fields in constantly curved
backgrounds. In particular we classify the Lorentz-covariant Harish-Chandra
modules generated from primary Weyl tensors of arbitrary mass and shape, and in
backgrounds with general values of the cosmological constant. We also discuss
the unfolded notion of local degrees of freedom in theories with and without
gravity and with and without massive deformation parameters, using the language
of Weyl zero-form modules and their duals.Comment: Corrected typos, references added, two figures, some remarks and two
subsections added for clarit
Initial Semantics for Reduction Rules
We give an algebraic characterization of the syntax and operational semantics
of a class of simply-typed languages, such as the language PCF: we characterize
simply-typed syntax with variable binding and equipped with reduction rules via
a universal property, namely as the initial object of some category of models.
For this purpose, we employ techniques developed in two previous works: in the
first work we model syntactic translations between languages over different
sets of types as initial morphisms in a category of models. In the second work
we characterize untyped syntax with reduction rules as initial object in a
category of models. In the present work, we combine the techniques used earlier
in order to characterize simply-typed syntax with reduction rules as initial
object in a category. The universal property yields an operator which allows to
specify translations---that are semantically faithful by construction---between
languages over possibly different sets of types.
As an example, we upgrade a translation from PCF to the untyped lambda
calculus, given in previous work, to account for reduction in the source and
target. Specifically, we specify a reduction semantics in the source and target
language through suitable rules. By equipping the untyped lambda calculus with
the structure of a model of PCF, initiality yields a translation from PCF to
the lambda calculus, that is faithful with respect to the reduction semantics
specified by the rules.
This paper is an extended version of an article published in the proceedings
of WoLLIC 2012.Comment: Extended version of arXiv:1206.4547, proves a variant of a result of
PhD thesis arXiv:1206.455
Noncommutative knot theory
The classical abelian invariants of a knot are the Alexander module, which is
the first homology group of the the unique infinite cyclic covering space of
S^3-K, considered as a module over the (commutative) Laurent polynomial ring,
and the Blanchfield linking pairing defined on this module. From the
perspective of the knot group, G, these invariants reflect the structure of
G^(1)/G^(2) as a module over G/G^(1) (here G^(n) is the n-th term of the
derived series of G). Hence any phenomenon associated to G^(2) is invisible to
abelian invariants. This paper begins the systematic study of invariants
associated to solvable covering spaces of knot exteriors, in particular the
study of what we call the n-th higher-order Alexander module, G^(n+1)/G^(n+2),
considered as a Z[G/G^(n+1)$-module. We show that these modules share almost
all of the properties of the classical Alexander module. They are torsion
modules with higher-order Alexander polynomials whose degrees give lower bounds
for the knot genus. The modules have presentation matrices derived either from
a group presentation or from a Seifert surface. They admit higher-order linking
forms exhibiting self-duality. There are applications to estimating knot genus
and to detecting fibered, prime and alternating knots. There are also
surprising applications to detecting symplectic structures on 4-manifolds.
These modules are similar to but different from those considered by the author,
Kent Orr and Peter Teichner and are special cases of the modules considered
subsequently by Shelly Harvey for arbitrary 3-manifolds.Comment: Published by Algebraic and Geometric Topology at
http://www.maths.warwick.ac.uk/agt/AGTVol4/agt-4-19.abs.htm
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