202,773 research outputs found
Modular forms of virtually real-arithmetic type I: Mixed mock modular forms yield vector-valued modular forms
The theory of elliptic modular forms has gained significant momentum from the discovery of relaxed yet well-behaved notions of modularity, such as mock modular forms, higher order modular forms, and iterated Eichler-Shimura integrals. Applications beyond number theory range from combinatorics, geometry, and representation theory to string theory and conformal field theory. We unify these relaxed notions in the framework of vector-valued modular forms by introducing a new class of -representations: virtually real-arithmetic types. The key point of the paper is that virtually real-arithmetic types are in general not completely reducible. We obtain a rationality result for Fourier and Taylor coefficients of associated modular forms
Global semantic typing for inductive and coinductive computing
Inductive and coinductive types are commonly construed as ontological
(Church-style) types, denoting canonical data-sets such as natural numbers,
lists, and streams. For various purposes, notably the study of programs in the
context of global semantics, it is preferable to think of types as semantical
properties (Curry-style). Intrinsic theories were introduced in the late 1990s
to provide a purely logical framework for reasoning about programs and their
semantic types. We extend them here to data given by any combination of
inductive and coinductive definitions. This approach is of interest because it
fits tightly with syntactic, semantic, and proof theoretic fundamentals of
formal logic, with potential applications in implicit computational complexity
as well as extraction of programs from proofs. We prove a Canonicity Theorem,
showing that the global definition of program typing, via the usual (Tarskian)
semantics of first-order logic, agrees with their operational semantics in the
intended model. Finally, we show that every intrinsic theory is interpretable
in a conservative extension of first-order arithmetic. This means that
quantification over infinite data objects does not lead, on its own, to
proof-theoretic strength beyond that of Peano Arithmetic. Intrinsic theories
are perfectly amenable to formulas-as-types Curry-Howard morphisms, and were
used to characterize major computational complexity classes Their extensions
described here have similar potential which has already been applied
Instabilities and Non-Reversibility of Molecular Dynamics Trajectories
The theoretical justification of the Hybrid Monte Carlo algorithm depends
upon the molecular dynamics trajectories within it being exactly reversible. If
computations were carried out with exact arithmetic then it would be easy to
ensure such reversibility, but the use of approximate floating point arithmetic
inevitably introduces violations of reversibility. In the absence of evidence
to the contrary, we are usually prepared to accept that such rounding errors
can be made small enough to be innocuous, but in certain circumstances they are
exponentially amplified and lead to blatantly erroneous results. We show that
there are two types of instability of the molecular dynamics trajectories which
lead to this behavior, instabilities due to insufficiently accurate numerical
integration of Hamilton's equations, and intrinsic chaos in the underlying
continuous fictitious time equations of motion themselves. We analyze the
former for free field theory, and show that it is essentially a finite volume
effect. For the latter we propose a hypothesis as to how the Liapunov exponent
describing the chaotic behavior of the fictitious time equations of motion for
an asymptotically free quantum field theory behaves as the system is taken to
its continuum limit, and explain why this means that instabilities in molecular
dynamics trajectories are not a significant problem for Hybrid Monte Carlo
computations. We present data for pure gauge theory and for QCD with
dynamical fermions on small lattices to illustrate and confirm some of our
results.Comment: 28 pages latex with 19 color postscript figures included by eps
An Algebra of Synchronous Scheduling Interfaces
In this paper we propose an algebra of synchronous scheduling interfaces
which combines the expressiveness of Boolean algebra for logical and functional
behaviour with the min-max-plus arithmetic for quantifying the non-functional
aspects of synchronous interfaces. The interface theory arises from a
realisability interpretation of intuitionistic modal logic (also known as
Curry-Howard-Isomorphism or propositions-as-types principle). The resulting
algebra of interface types aims to provide a general setting for specifying
type-directed and compositional analyses of worst-case scheduling bounds. It
covers synchronous control flow under concurrent, multi-processing or
multi-threading execution and permits precise statements about exactness and
coverage of the analyses supporting a variety of abstractions. The paper
illustrates the expressiveness of the algebra by way of some examples taken
from network flow problems, shortest-path, task scheduling and worst-case
reaction times in synchronous programming.Comment: In Proceedings FIT 2010, arXiv:1101.426
- …