67,599 research outputs found
Noetherian approximation of algebraic spaces and stacks
We show that every scheme/algebraic space/stack that is quasi-compact with
quasi-finite diagonal can be approximated by a noetherian scheme/algebraic
space/stack. More generally, we show that any stack which is etale-locally a
global quotient stack can be approximated. Examples of applications are
generalizations of Chevalley's, Serre's and Zariski's theorems and Chow's lemma
to the non-noetherian setting. We also show that every quasi-compact algebraic
stack with quasi-finite diagonal has a finite generically flat cover by a
scheme.Comment: 39 pages; complete overhaul of paper; generalized results and
simplified proofs (no groupoid-calculations); added more applications and
appendices with standard results on constructible properties and limits for
stacks; generalized Thm C (no finite presentation hypothesis); some minor
changes in 2,1-2.8, 8.2, 8.8 and 8.9; final versio
Some applications of the ultrapower theorem to the theory of compacta
The ultrapower theorem of Keisler-Shelah allows such model-theoretic notions
as elementary equivalence, elementary embedding and existential embedding to be
couched in the language of categories (limits, morphism diagrams). This in turn
allows analogs of these (and related) notions to be transported into unusual
settings, chiefly those of Banach spaces and of compacta. Our interest here is
the enrichment of the theory of compacta, especially the theory of continua,
brought about by the immigration of model-theoretic ideas and techniques
Maps from Riemannian manifolds into non-degenerate Euclidean cones
Let be a connected, non-compact -dimensional Riemannian manifold. In
this paper we consider smooth maps with images
inside a non-degenerate cone. Under quite general assumptions on , we
provide a lower bound for the width of the cone in terms of the energy and the
tension of and a metric parameter. As a side product, we recover some
well known results concerning harmonic maps, minimal immersions and K\"ahler
submanifolds. In case is an isometric immersion, we also show that, if
is sufficiently well-behaved and has non-positive sectional curvature,
cannot be contained into a non-degenerate cone of
.Comment: 19 pages, to appea
Market Completion with Derivative Securities
Let be a -martingale representing the price of a primitive
asset in an incomplete market framework. We present easily verifiable
conditions on model coefficients which guarantee the completeness of the market
in which in addition to the primitive asset one may also trade a derivative
contract . Both and are defined in terms of the solution
to a -dimensional stochastic differential equation: and
. From a purely mathematical point
of view we prove that every local martingale under can be
represented as a stochastic integral with respect to the
-martingale . Notably, in contrast to recent
results on the endogenous completeness of equilibria markets, our conditions
allow the Jacobian matrix of to be singular everywhere on
. Hence they cover, as a special case, the prominent example of a
stochastic volatility model being completed with a European call (or put)
option
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