2,791 research outputs found
Domain Representable Spaces Defined by Strictly Positive Induction
Recursive domain equations have natural solutions. In particular there are
domains defined by strictly positive induction. The class of countably based
domains gives a computability theory for possibly non-countably based
topological spaces. A space is a topological space characterized by
its strong representability over domains. In this paper, we study strictly
positive inductive definitions for spaces by means of domain
representations, i.e. we show that there exists a canonical fixed point of
every strictly positive operation on spaces.Comment: 48 pages. Accepted for publication in Logical Methods in Computer
Scienc
The principle of pointfree continuity
In the setting of constructive pointfree topology, we introduce a notion of
continuous operation between pointfree topologies and the corresponding
principle of pointfree continuity. An operation between points of pointfree
topologies is continuous if it is induced by a relation between the bases of
the topologies; this gives a rigorous condition for Brouwer's continuity
principle to hold. The principle of pointfree continuity for pointfree
topologies and says that any relation which induces
a continuous operation between points is a morphism from to
. The principle holds under the assumption of bi-spatiality of
. When is the formal Baire space or the formal unit
interval and is the formal topology of natural numbers, the
principle is equivalent to spatiality of the formal Baire space and formal unit
interval, respectively. Some of the well-known connections between spatiality,
bar induction, and compactness of the unit interval are recast in terms of our
principle of continuity.
We adopt the Minimalist Foundation as our constructive foundation, and
positive topology as the notion of pointfree topology. This allows us to
distinguish ideal objects from constructive ones, and in particular, to
interpret choice sequences as points of the formal Baire space
Uniformly rigid spaces
We define a new category of non-archimedean analytic spaces over a complete
discretely valued field, which we call uniformly rigid. It extends the category
of rigid spaces, and it can be described in terms of bounded functions on
products of open and closed polydiscs. We relate uniformly rigid spaces to
their associated classical rigid spaces, and we transfer various constructions
and results from rigid geometry to the uniformly rigid setting. In particular,
we prove an analog of Kiehl's patching theorem for coherent ideals, and we
define the uniformly rigid generic fiber of a formal scheme of formally finite
type. This uniformly rigid generic fiber is more intimately linked to its model
than the classical rigid generic fiber obtained via Berthelot's construction.Comment: 46 pages; typos corrected, terminology changed ("semi-affinoid
pre-subdomains" -> "representable subsets", "semi-affinoid subspaces" ->
"open semi-affinoid subspaces"), proof of Cor. 2.15 (formerly 2.14)
rewritten, included proof that the urig G-top is finer than the Zar-top
(Prop. 2.39), added proofs in Section 4, arguments in some proofs explained
in greater detail, to appear in AN
Families of p-divisible groups with constant Newton polygon
A p-divisible group over a base scheme in characteristic p in general does
not admit a slope filtration. Let X be a p-divisible group with constant Newton
polygon over a normal noetherian scheme S; we prove that there exists an
isogeny from X to Y such that Y admits a slope filtration. In case S is regular
this was proved by N. Katz for dim(S) = 1 and by T. Zink for dim(S) > 0. We
give an example of a p-divisible group over a non-normal base which does not
admit an isogeny to a p-divisible group with a slope filtration.Comment: To be published in Documenta Mathematic
Integral representations and Liouville theorems for solutions of periodic elliptic equations
The paper contains integral representations for certain classes of
exponentially growing solutions of second order periodic elliptic equations.
These representations are the analogs of those previously obtained by S. Agmon,
S. Helgason, and other authors for solutions of the Helmholtz equation. When
one restricts the class of solutions further, requiring their growth to be
polynomial, one arrives to Liouville type theorems, which describe the
structure and dimension of the spaces of such solutions. The Liouville type
theorems previously proved by M. Avellaneda and F.-H. Lin, and J. Moser and M.
Struwe for periodic second order elliptic equations in divergence form are
significantly extended. Relations of these theorems with the analytic structure
of the Fermi and Bloch surfaces are explained.Comment: 48 page
Applying quantitative semantics to higher-order quantum computing
Finding a denotational semantics for higher order quantum computation is a
long-standing problem in the semantics of quantum programming languages. Most
past approaches to this problem fell short in one way or another, either
limiting the language to an unusably small finitary fragment, or giving up
important features of quantum physics such as entanglement. In this paper, we
propose a denotational semantics for a quantum lambda calculus with recursion
and an infinite data type, using constructions from quantitative semantics of
linear logic
A necessary and sufficient condition for induced model structures
A common technique for producing a new model category structure is to lift
the fibrations and weak equivalences of an existing model structure along a
right adjoint. Formally dual but technically much harder is to lift the
cofibrations and weak equivalences along a left adjoint. For either technique
to define a valid model category, there is a well-known necessary "acyclicity"
condition. We show that for a broad class of "accessible model structures" - a
generalization introduced here of the well-known combinatorial model structures
- this necessary condition is also sufficient in both the right-induced and
left-induced contexts, and the resulting model category is again accessible. We
develop new and old techniques for proving the acyclity condition and apply
these observations to construct several new model structures, in particular on
categories of differential graded bialgebras, of differential graded comodule
algebras, and of comodules over corings in both the differential graded and the
spectral setting. We observe moreover that (generalized) Reedy model category
structures can also be understood as model categories of "bialgebras" in the
sense considered here.Comment: 49 pages; final journal version to appear in the Journal of Topolog
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