1,193,206 research outputs found
On the dual nature of partial theta functions and Appell-Lerch sums
In recent work, Hickerson and the author demonstrated that it is useful to
think of Appell--Lerch sums as partial theta functions. This notion can be used
to relate identities involving partial theta functions with identities
involving Appell--Lerch sums. In this sense, Appell--Lerch sums and partial
theta functions appear to be dual to each other. This duality theory is not
unlike that found by Andrews between various sets of identities of
Rogers-Ramanujan type with respect to Baxter's solution to the hard hexagon
model of statistical mechanics. As an application we construct bilateral
-series with mixed mock modular behaviour.Comment: To be published in Advances in Mathematic
Negative index Jacobi forms and quantum modular forms
In this paper, we consider the Fourier coefficients of a special class of
meromorphic Jaocbi forms of negative index. Much recent work has been done on
such coefficients in the case of Jacobi forms of positive index, but almost
nothing is known for Jacobi forms of negative index. In this paper we show,
from two different perspectives, that their Fourier coefficients have a simple
decomposition in terms of partial theta functions. The first perspective uses
the language of Lie super algebras, and the second applies the theory of
elliptic functions. In particular, we find a new infinite family of rank-crank
type PDEs generalizing the famous example of Atkin and Garvan. We then describe
the modularity properties of these coefficients, showing that they are "mixed
partial theta functions", along the way determining a new class of quantum
modular partial theta functions which is of independent interest.Comment: 29 pages, minor correction
Termination Casts: A Flexible Approach to Termination with General Recursion
This paper proposes a type-and-effect system called Teqt, which distinguishes
terminating terms and total functions from possibly diverging terms and partial
functions, for a lambda calculus with general recursion and equality types. The
central idea is to include a primitive type-form "Terminates t", expressing
that term t is terminating; and then allow terms t to be coerced from possibly
diverging to total, using a proof of Terminates t. We call such coercions
termination casts, and show how to implement terminating recursion using them.
For the meta-theory of the system, we describe a translation from Teqt to a
logical theory of termination for general recursive, simply typed functions.
Every typing judgment of Teqt is translated to a theorem expressing the
appropriate termination property of the computational part of the Teqt term.Comment: In Proceedings PAR 2010, arXiv:1012.455
General Recursion via Coinductive Types
A fertile field of research in theoretical computer science investigates the
representation of general recursive functions in intensional type theories.
Among the most successful approaches are: the use of wellfounded relations,
implementation of operational semantics, formalization of domain theory, and
inductive definition of domain predicates. Here, a different solution is
proposed: exploiting coinductive types to model infinite computations. To every
type A we associate a type of partial elements Partial(A), coinductively
generated by two constructors: the first, return(a) just returns an element
a:A; the second, step(x), adds a computation step to a recursive element
x:Partial(A). We show how this simple device is sufficient to formalize all
recursive functions between two given types. It allows the definition of fixed
points of finitary, that is, continuous, operators. We will compare this
approach to different ones from the literature. Finally, we mention that the
formalization, with appropriate structural maps, defines a strong monad.Comment: 28 page
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