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O-minimal residue fields of o-minimal fields
Let R be an o-minimal field with a proper convex subring V. We axiomatize the
class of all structures (R,V) such that k_ind, the corresponding residue field
with structure induced from R via the residue map, is o-minimal. More
precisely, in previous work it was shown that certain first order conditions on
(R,V) are sufficient for the o-minimality of k_ind. Here we prove that these
conditions are also necessary
Dp-minimal valued fields
We show that dp-minimal valued fields are henselian and that a dp-minimal
field admitting a definable type V topology is either real closed,
algebraically closed or admits a non-trivial definable henselian valuation. We
give classifications of dp-minimal ordered abelian groups and dp-minimal
ordered fields without additional structure
Minimal Superspace Vector Fields for 5D Minimal Supersymmetry
We investigate a minimal superspace description for 5D superconformal Killing
vectors. The vielbein appropriate for AdS symmetry is discussed within the
confines of this minimal supergeometry.Comment: 10 page
Writing representations over minimal fields
The chief aim of this paper is to describe a procedure which, given a
-dimensional absolutely irreducible matrix representation of a finite group
over a finite field , produces an equivalent representation such
that all matrix entries lie in a subfield of which is
as small as possible. The algorithm relies on a matrix version of Hilbert's
Theorem 90, and is probabilistic with expected running time when is bounded. Using similar
methods we then describe an algorithm which takes as input a prime number and a
power-conjugate presentation for a finite soluble group, and as output produces
a full set of absolutely irreducible representations of the group over fields
whose characteristic is the specified prime, each representation being written
over its minimal field.Comment: 9 page
Presburger sets and p-minimal fields
We prove a cell decomposition theorem for Presburger sets and introduce a
dimension theory for Z-groups with the Presburger structure. Using the cell
decomposition theorem we obtain a full classification of Presburger sets up to
definable bijection. We also exhibit a tight connection between the definable
sets in an arbitrary p-minimal field and Presburger sets in its value group. We
give a negative result about expansions of Presburger structures and prove
uniform elimination of imaginaries for Presburger structures within the
Presburger language.Comment: to appear in the Journal of Symbolic Logi
Correction to "minimal unit vector fields"
The paper "Minimal unit vector fields" by O. Gil-Medrano and E.
Llinares-Fuster \cite{GilLli1}. is a seminal paper in the field that has been
cited by many authors. It contains, however, a minor technical mistake in
Theorem 14 that is important to fix. In this short note, we will provide a
correction to that result.Comment: 8page
Holomorphic vector fields and minimal Lagrangian submanifolds
The purpose of this note is to establish the following theorem: Let N be a
Kahler manifold, L be a compact oriented immersed minimal Lagrangian
submanifold in N and V be a holomorphic vector field in a neighbourhood of L in
N. Let div(V) be the (complex) divergence of V. Then the integral of div(V)
over L is 0. Vice versa let N^2n be Kahler-Einstein with non-zero scalar
curvature and L^n be a totally real oriented embedded n-dimensional
real-analytic submanifold of N s.t. the divergence of any holomorphic vector
field defined in a neighbourhood of L in N integrates to 0 on L. Then L is a
minimal Lagrangian submanifold of N.Comment: 7 page
Minimal fields of canonical dimensionality are free
It is shown that in a scale-invariant relativistic field theory, any field
belonging to the or representations of the Lorentz
group and with dimensionality is a free field. For other field types
there is no value of the dimensionality that guarantees that the field is free.
Conformal invariance is not used in the proof of these results, but it gives
them a special interest; as already known and as shown here in an appendix, the
only fields in a conformal field theory that can describe massless particles
belong to the or representations of the Lorentz group and have
dimensionality . Hence in conformal field theories massless particles
are free.Comment: 14 page
Measuring definable sets in o-minimal fields
We introduce a non real-valued measure on the definable sets contained in the
finite part of a cartesian power of an o-minimal field . The measure takes
values in an ordered semiring, the Dedekind completion of a quotient of . We
show that every measurable subset of with non-empty interior has positive
measure, and that the measure is preserved by definable -diffeomorphisms
with Jacobian determinant equal to
On VC-minimal fields and dp-smallness
In this paper, we show that VC-minimal ordered fields are real closed. We
introduce a notion, strictly between convexly orderable and dp-minimal, that we
call dp-small, and show that this is enough to characterize many algebraic
theories. For example, dp-small ordered groups are abelian divisible and
dp-small ordered fields are real closed.Comment: 17 page
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