160,724 research outputs found
Coalgebraic Infinite Traces and Kleisli Simulations
Kleisli simulation is a categorical notion introduced by Hasuo to verify
finite trace inclusion. They allow us to give definitions of forward and
backward simulation for various types of systems. A generic categorical theory
behind Kleisli simulation has been developed and it guarantees the soundness of
those simulations with respect to finite trace semantics. Moreover, those
simulations can be aided by forward partial execution (FPE)---a categorical
transformation of systems previously introduced by the authors.
In this paper, we give Kleisli simulation a theoretical foundation that
assures its soundness also with respect to infinitary traces. There, following
Jacobs' work, infinitary trace semantics is characterized as the "largest
homomorphism." It turns out that soundness of forward simulations is rather
straightforward; that of backward simulation holds too, although it requires
certain additional conditions and its proof is more involved. We also show that
FPE can be successfully employed in the infinitary trace setting to enhance the
applicability of Kleisli simulations as witnesses of trace inclusion. Our
framework is parameterized in the monad for branching as well as in the functor
for linear-time behaviors; for the former we mainly use the powerset monad (for
nondeterminism), the sub-Giry monad (for probability), and the lift monad (for
exception).Comment: 39 pages, 1 figur
Traces on Infinite-Dimensional Brauer Algebras
We describe the central measures for the random walk on graded graphs. Using
this description, we obtain the list of all finite traces on three
infinite-dimensional algebras: on the Brauer algebra, on the partition algebra,
and on the walled Brauer algebra. For the first two algebras, these lists
coincide with the list of all finite traces of the infinite symmetric group.
For the walled Brauer algebra, the list of finite traces coincide with the list
of finite traces of the square of the latter group. We introduce the operation
which corresponds to the graph another graph which called "Pasclization" of the
initial graph and then give the general criteria for coinsidness of the sets of
traces on both graphs.Comment: 9 pages, 20 Re
Traces and Characteristic Classes in Infinite Dimensions
This paper surveys topological results obtained from characteristic classes
built from the two types of traces on the algebra of pseudodifferential
operators of nonpositive order. The main results are the construction of a
universal -polynomial and Chern character that control the -index
theorem for all circle actions on a fixed vector bundle over a manifold, and
, for the diffeomorphism
group of circle bundles with large first Chern class over projective
algebraic Kaehler surfaces.Comment: Parts of Section 2.3 are not correct. This is discussed in T.
McCauley, "S^1-Equivariant Chern-Weil Constructions on Loop Spaces,"
arXiv:1507.0862
On the domain of singular traces
The question whether an operator belongs to the domain of some singular trace
is addressed, together with the dual question whether an operator does not
belong to the domain of some singular trace. We show that the answers are
positive in general, namely for any (compact, infinite rank) positive operator
A we exhibit two singular traces, the first being zero and the second being
infinite on A. However, if we assume that the singular traces are generated by
a "regular" operator, the answers change, namely such traces always vanish on
trace-class, non singularly traceable operators and are always infinite on non
trace-class, non singularly traceable operators. These results are achieved on
a general semifinite factor, and make use of a new characterization of singular
traceability (cf. math.OA/0202108).Comment: 7 pages, LaTeX. Minor corrections, to appear on the International
Journal of Mathematic
All Linear-Time Congruences for Familiar Operators
The detailed behaviour of a system is often represented as a labelled
transition system (LTS) and the abstract behaviour as a stuttering-insensitive
semantic congruence. Numerous congruences have been presented in the
literature. On the other hand, there have not been many results proving the
absence of more congruences. This publication fully analyses the linear-time
(in a well-defined sense) region with respect to action prefix, hiding,
relational renaming, and parallel composition. It contains 40 congruences. They
are built from the alphabet, two kinds of traces, two kinds of divergence
traces, five kinds of failures, and four kinds of infinite traces. In the case
of finite LTSs, infinite traces lose their role and the number of congruences
drops to 20. The publication concentrates on the hardest and most novel part of
the result, that is, proving the absence of more congruences
Traces of Sobolev functions on regular surfaces in infinite dimensions
In a Banach space endowed with a nondegenerate Gaussian measure, we
consider Sobolev spaces of real functions defined in a sublevel set of a Sobolev nondegenerate function . We define
the traces at of the elements of for , as
elements of where is the surface measure of Feyel
and de La Pradelle. The range of the trace operator is contained in
for and even in under
further assumptions. If is a suitable halfspace, the range is characterized
as a sort of fractional Sobolev space at the boundary.
An important consequence of the general theory is an integration by parts
formula for Sobolev functions, which involves their traces at
Irreducible adjoint representations in prime power dimensions, with an application
We construct an infinite family of representations of finite groups with an
irreducible adjoint action and we give an application to the question of
lacunary of Frobenius traces in Galois representations.Comment: To appear in the Journal of the Ramanujan Mathematical Societ
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