4,014 research outputs found
Isotropy of unitary involutions
We prove the so-called Unitary Isotropy Theorem, a result on isotropy of a
unitary involution. The analogous previously known results on isotropy of
orthogonal and symplectic involutions as well as on hyperbolicity of
orthogonal, symplectic, and unitary involutions are formal consequences of this
theorem. A component of the proof is a detailed study of the quasi-split
unitary grassmannians.Comment: final version, to appear in Acta Mat
Ellipsoidal flows in relativistic hydrodynamics of finite systems
A new class of 3D anisotropic analytic solutions of relativistic
hydrodynamics with constant pressure is found. We analyse, in particular,
solutions corresponding to ellipsoidally symmetric expansion of finite systems
into vacuum. They can be utilized for relativistic description of the system
evolution in thermodynamic region near the softest point and at the final stage
of the hydrodynamic expansion in A+A collisions. The solutions can be used also
for testing of numerical hydrodynamic codes solving relativistic hydrodynamic
equations for anisotropic expansion of finite systemsComment: 7 pages, talk given at RHIC school 2004 (Budapest, december 2004
Unitary grassmannians
We study projective homogeneous varieties under an action of a projective
unitary group (of outer type). We are especially interested in the case of
(unitary) grassmannians of totally isotropic subspaces of a hermitian form over
a field, the main result saying that these grassmannians are 2-incompressible
if the hermitian form is generic. Applications to orthogonal grassmannians are
provided.Comment: 25 page
Holes in I^n
Let F be an arbitrary field of characteristic not 2. We write W(F) for the
Witt ring of F, consisting of the isomorphism classes of all anisotropic
quadratic forms over F. For any element x of W(F), dimension dim x is defined
as the dimension of a quadratic form representing x. The elements of all even
dimensions form an ideal denoted I(F). The filtration of the ring W(F) by the
powers I(F)^n of this ideal plays a fundamental role in the algebraic theory of
quadratic forms. The Milnor conjectures, recently proved by Voevodsky and
Orlov-Vishik-Voevodsky, describe the successive quotients I(F)^n/I(F)^{n+1} of
this filtration, identifying them with Galois cohomology groups and with the
Milnor K-groups modulo 2 of the field F. In the present article we give a
complete answer to a different old-standing question concerning I(F)^n, asking
about the possible values of dim x for x in I(F)^n. More precisely, for any
positive integer n, we prove that the set dim I^n of all dim x for all x in
I(F)^n and all F consisists of 2^{n+1}-2^i, i=1,2,...,n+1 together with all
even integers greater or equal to 2^{n+1}. Previously available partial
informations on dim I^n include the classical Arason-Pfister theorem, saying
that no integer between 0 and 2^n lies in dim I^n, as well as a recent Vishik's
theorem, saying the same on the integers between 2^n and 2^n+2^{n-1} (the case
n=3 is due to Pfister, n=4 to Hoffmann). Our proof is based on computations in
Chow groups of powers of projective quadrics (involving the Steenrod
operations); the method developed can be also applied to other types of
algebraic varieties.Comment: 29 page
Incompressibility of orthogonal grassmannians
We prove the following conjecture due to Bryant Mathews (2008). Let Q be the
orthogonal grassmannian of totally isotropic i-planes of a non-degenerate
quadratic form q over an arbitrary field (where i is an integer in the interval
[1, (\dim q)/2]). If the degree of each closed point on Q is divisible by 2^i
and the Witt index of q over the function field of Q is equal to i, then the
variety Q is 2-incompressible.Comment: 5 page
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