This paper surveys {\it tropical modifications}, which have already become a
folklore in tropical geometry. Tropical modifications are used in tropical
intersection theory, tropical Hodge theory, and in the study of singularities.
They admi...
Asymptotic Differential Algebra and Model Theory of Transseries
September 8, 2015
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Mathematics
Logic
Commutative Algebra
Classical Analysis and ODEs
Logic
Commutative Algebra
Classical Analysis and ODEs
We develop here the algebra of the differential field of transseries and of
related valued differential fields. This book contains in particular our
recently obtained decisive positive results on the model theory of these
structures....
The complex structure on the six dimensional sphere
September 8, 2015
Mathematics
Differential Geometry
Algebraic Geometry
Proof of existence of a complex structure on the six-sphere, followed by an
explicit computation of its underlying integrable almost complex tensor by the
aid of inner automorphisms of the octonions, is exhibited. Both are elementary
and se...
The six operations in equivariant motivic homotopy theory
September 7, 2015
Mathematics
Algebraic Geometry
Algebraic Topology
K-Theory and Homology
We introduce and study the homotopy theory of motivic spaces and spectra
parametrized by quotient stacks [X/G], where G is a linearly reductive linear
algebraic group. We extend to this equivariant setting the main foundational
results of m...
The subalgebras of A2
September 2, 2015
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Mathematics
Rings and Algebras
A classification of the semisimple subalgebras of the Lie algebra of
traceless 3×3 matrices with complex entries, denoted A2, is
well-known. We classify its nonsemisimple subalgebras, thus completing the
classification of the sub...
The rectilinear local crossing number of Kn
August 31, 2015
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Mathematics
Combinatorics
We determine lcrˉ(Kn), the rectilinear local crossing number
of the complete graph Kn for every n. More precisely, for every n∈/{8,14}, \[ {\bar{\rm{lcr}}}(K_n)=\left\lceil \frac{1}{2} \left(
n-3-\left\lceil...
Limit Models in Strictly Stable Abstract Elementary Classes
August 19, 2015
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Mathematics
Logic
In this paper, we examine the locality condition for non-splitting and
determine the level of uniqueness of limit models that can be recovered in some
stable, but not superstable, abstract elementary classes. In particular we
prove (note th...
On the minimal modules for exceptional Lie algebras: Jordan blocks and
stabilisers
August 12, 2015
Mathematics
Representation Theory
Group Theory
Rings and Algebras
Let G be a simple simple-connected exceptional algebraic group of type G_2,
F_4, E_6 or E_7 over an algebraically closed field k of characteristic p>0 with
\g=Lie(G). For each nilpotent orbit G.e of \g, we list the Jordan blocks of the
acti...
What is the force on a magnetic dipole?
August 6, 2015
Physics
Classical Physics
We show that attempts to modify the force on a magnetic dipole by introducing
either hidden momentum or internal forces are not correct. The standard
textbook result {\bf F=\nabla(\bmu\cdot B)} is correct even in the presence
of time depe...
The Divergence Index: A Decomposable Measure of Segregation and
Inequality
August 5, 2015
Physics
Mathematics
Computer Science
Statistics
Physics and Society
Information Theory
Information Theory
Methodology
Decomposition analysis is a critical tool for understanding the social and
spatial dimensions of segregation and diversity. In this paper, I highlight the
conceptual, mathematical, and empirical distinctions between segregation and
diversit...