6,828 research outputs found
The Polish topology of the isometry group of the infinite dimensional hyperbolic space
We consider the isometry group of the infinite dimensional separable
hyperbolic space with its Polish topology. This topology is given by the
pointwise convergence. For non-locally compact Polish groups, some striking
phenomena like automatic continuity or extreme amenability may happen. Our
leading idea is to compare this topological group with usual Lie groups on one
side and with non-Archimedean infinite dimensional groups like
, the group of all permutations of a countable set on the
other side. Our main results are
Automatic continuity (any homomorphism to a separable group is continuous),
minimality of the Polish topology, identification of its universal Furstenberg
boundary as the closed unit ball of a separable Hilbert space with its weak
topology, identification of its universal minimal flow as the completion of
some suspension of the action of the additive group of the reals on its
universal minimal flow.
All along the text, we lead a parallel study with the sibling group of
isometries of a separable Hilbert space.Comment: After a first version of this paper, Todor Tsankov asked if the
topology is minimal. A positive answer has been added to this second versio
Topological properties of Wazewski dendrite groups
Homeomorphism groups of generalized Wa\.zewski dendrites act on the infinite
countable set of branch points of the dendrite and thus have a nice Polish
topology. In this paper, we study them in the light of this Polish topology.
The group of the universal Wa\.zewski dendrite is more
characteristic than the others because it is the unique one with a dense
conjugacy class. For this group , we show some of its topological
properties like existence of a comeager conjugacy class, the Steinhaus
property, automatic continuity and the small index subgroup property. Moreover,
we identify the universal minimal flow of . This allows us to prove
that point-stabilizers in are amenable and to describe the universal
Furstenberg boundary of .Comment: Slight modifications about the expositio
Superrigidity In Infinite Dimension And Finite Rank Via Harmonic Maps
We prove geometric superrigidity for actions of cocompact lattices in
semisimple Lie groups of higher rank on infinite dimensional Riemannian
manifolds of nonpositive curvature and finite telescopic dimension
Walks of bubbles on a hot wire in a liquid bath
When a horizontal resistive wire is heated up to the boiling point in a
subcooled liquid bath, some vapor bubbles nucleate on its surface. Traditional
nucleate boiling theory predicts that bubbles generated from active nucleate
sites, grow up and depart from the heating surface due to buoyancy and inertia.
However, we observed here a different behavior: the bubbles slide along the
heated wire. In this situation, unexpected regimes are observed; from the
simple sliding motion to bubble clustering. We noticed that bubbles could
rapidly change their moving direction and may also interact. Finally, we
propose an interpretation for both the attraction between the bubbles and the
wire and for the motion of the bubbles on the wire in term of Marangoni
effects
Symmetry classes of alternating sign matrices in the nineteen-vertex model
The nineteen-vertex model on a periodic lattice with an anti-diagonal twist
is investigated. Its inhomogeneous transfer matrix is shown to have a simple
eigenvalue, with the corresponding eigenstate displaying intriguing
combinatorial features. Similar results were previously found for the same
model with a diagonal twist. The eigenstate for the anti-diagonal twist is
explicitly constructed using the quantum separation of variables technique. A
number of sum rules and special components are computed and expressed in terms
of Kuperberg's determinants for partition functions of the inhomogeneous
six-vertex model. The computations of some components of the special eigenstate
for the diagonal twist are also presented. In the homogeneous limit, the
special eigenstates become eigenvectors of the Hamiltonians of the integrable
spin-one XXZ chain with twisted boundary conditions. Their sum rules and
special components for both twists are expressed in terms of generating
functions arising in the weighted enumeration of various symmetry classes of
alternating sign matrices (ASMs). These include half-turn symmetric ASMs,
quarter-turn symmetric ASMs, vertically symmetric ASMs, vertically and
horizontally perverse ASMs and double U-turn ASMs. As side results, new
determinant and pfaffian formulas for the weighted enumeration of various
symmetry classes of alternating sign matrices are obtained.Comment: 61 pages, 13 figure
Fusion hierarchies, -systems and -systems for the dilute loop models
The fusion hierarchy, -system and -system of functional equations are
the key to integrability for 2d lattice models. We derive these equations for
the generic dilute loop models. The fused transfer matrices are
associated with nodes of the infinite dominant integral weight lattice of
. For generic values of the crossing parameter , the -
and -systems do not truncate. For the case
rational so that
is a root of unity, we find explicit closure
relations and derive closed finite - and -systems. The TBA diagrams of
the -systems and associated Thermodynamic Bethe Ansatz (TBA) integral
equations are not of simple Dynkin type. They involve nodes if is
even and nodes if is odd and are related to the TBA diagrams of
models at roots of unity by a folding which originates
from the addition of crossing symmetry. In an appropriate regime, the known
central charges are . Prototypical examples of the
loop models, at roots of unity, include critical dense polymers
with central charge , and loop
fugacity and critical site percolation on the triangular lattice
with , and . Solving
the TBA equations for the conformal data will determine whether these models
lie in the same universality classes as their counterparts. More
specifically, it will confirm the extent to which bond and site percolation lie
in the same universality class as logarithmic conformal field theories.Comment: 34 page
Constant Froude number in a circular hydraulic jump and its implication on the jump radius selection
The properties of a standard hydraulic jump depend critically on a Froude
number Fr defined as the ratio of the flow velocity to the gravity waves speed.
In the case of a horizontal circular jump, the question of the Froude number is
not well documented. Our experiments show that Fr measured just after the jump
is locked on a constant value that does not depend on flow rate Q, kinematic
viscosity {\nu} and surface tension {\gamma}. Combining this result to a
lubrication description of the outer flow yields, under appropriate conditions,
a new and simple law ruling the jump radius RJ : , in excellent agreement
with our experimental data. This unexpected RJ result asks an unsolved question
to all available models.Comment: 5 pages, 3 figure
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