E8 is first of all the largest exceptional root system, which is a set of vectors in an 8-dimensional real vector space satisfying certain properties. Root systems were classified by Wilhelm Killing in the 1890s. He found 4 infinite classes of Lie algebras, labelled An, Bn, Cn, and Dn, where n=1,2,3…. He […]

Given a remarkable representation of the generalized Pauli operators of
two-qubits in terms of the points of the generalized quadrangle of order two,
W(2), it is shown that specific subsets of these operators can also be
associated with the points and lines of the four-dimensional projective space
over the Galois field with two elements - the so-called Veldkamp space of W(2).
An intriguing novelty is the recognition of (uni- and tri-centric) triads and
specific pentads of the Pauli operators in addition to the “classical” subsets
answering to geometric hyperplanes of W(2).

The spectral action on noncommutative torus is obtained, using a
Chamseddine–Connes formula via computations of zeta functions. The importance
of a Diophantine condition is outlined. Several results on holomorphic
continuation of series of holomorphic functions are obtained in this context.

The Faddeev-Volkov solution of the star-triangle relation is connected with
the modular double of the quantum group U_q(sl_2). It defines an Ising-type
lattice model with positive Boltzmann weights where the spin variables take
continuous values on the real line. The free energy of the model is exactly
calculated in the thermodynamic limit. The model describes quantum fluctuations
of circle patterns and the associated discrete conformal transformations
connected with the Thurston’s discrete analogue of the Riemann mappings
theorem. In particular, in the quasi-classical limit the model precisely
describe the geometry of integrable circle patterns with prescribed
intersection angles.

In this paper we are presenting the theory of balance equations of the
Continuum Thermodynamics (balance systems) in a geometrical form using
Poincare-Cartan formalism of the Multisymplectic Field Theory. A constitutive
relation $\mathcal{C}$ of a balance system $B_{C}$ is realized as a mapping
between a (partial) 1-jet bundle of the configurational bundle $\pi:Y\to X$ and
the dual bundle similar to the Legendre mapping of the Lagrangian Field Theory.
Invariant (variational) form of the balance system $B_{C}$ is presented in
three different forms and the space of admissible variations is defined and
studied. Action of automorphisms of the bundle $\pi$ on the constitutive
mappings $C$ is studied and it is shown that the symmetry group $Sym(C)$ of the
constitutive relation $C$ acts on the space of solutions of the balance system
$B_{C}$. Suitable version of Noether Theorem for an action of a symmetry group
is presented with the usage of conventional multimomentum mapping. Finally, the
geometrical (bundle) picture of the Rational Extended Thermodynamics in terms
of Lagrange-Liu fields is developed and the entropy principle is shown to be
equivalent to the holonomicy of the current component of the constitutive
section.

We discuss the asymptotic behaviour of models of lattice polygons, mainly on
the square lattice. In particular, we focus on limiting area laws in the
uniform perimeter ensemble where, for fixed perimeter, each polygon of a given
area occurs with the same probability. We relate limit distributions to the
scaling behaviour of the associated perimeter and area generating functions,
thereby providing a geometric interpretation of scaling functions. To a major
extent, this article is a pedagogic review of known results.

We give an extended review of recent work on the extended weak coupling
limit. Background material on completely positive semigroups and their unitary
dilations is given, as well as a particularly easy construction of `quadratic
noises’.

We study a heavy piston of mass $M$ that moves in one dimension. The piston
separates two gas chambers, each of which contains finitely many ideal, unit
mass gas particles moving in $d$ dimensions, where $ d\geq 1$. Using averaging
techniques, we prove that the actual motions of the piston converge in
probability to the predicted averaged behavior on the time scale $M^ {1/2} $
when $M$ tends to infinity while the total energy of the system is bounded and
the number of gas particles is fixed. Neishtadt and Sinai previously pointed
out that an averaging theorem due to Anosov should extend to this situation.

When $ d=1$, the gas particles move in just one dimension, and we prove that
the rate of convergence of the actual motions of the piston to its averaged
behavior is $\mathcal{O} (M^ {-1/2}) $ on the time scale $M^ {1/2} $. The
convergence is uniform over all initial conditions in a compact set. We also
investigate the piston system when the particle interactions have been
smoothed. The convergence to the averaged behavior again takes place uniformly,
both over initial conditions and over the amount of smoothing.

In addition, we prove generalizations of our results to $N$ pistons
separating $N+1$ gas chambers. We also provide a general discussion of
averaging theory and the proofs of a number of previously known averaging
results. In particular, we include a new proof of Anosov’s averaging theorem
for smooth systems that is primarily due to Dolgopyat.

We present a compared analysis of some properties of 3-Sasakian and
3-cosymplectic manifolds. We construct a canonical connection on an almost
3-contact metric manifold which generalises the Tanaka-Webster connection of a
contact metric manifold and we use this connection to show that a 3-Sasakian
manifold does not admit any Darboux-like coordinate system. Moreover, we prove
that any 3-cosymplectic manifold is Ricci-flat and admits a Darboux coordinate
system if and only it is flat.

We consider the defocusing, $\dot{H}^1$-critical Hartree equation for the
radial data in all dimensions $(n\geq 5)$. We show the global well-posedness
and scattering results in the energy space. The new ingredient in this paper is
that we first take advantage of the estimate of the term $\displaystyle -
\int_{I}\int_{|x|\leq A|I|^{1/2}}|u|^{2}\Delta \Big(\frac{1}{|x|}\Big)dxdt$ in
the localized Morawetz inequality to rule out the possibility of energy
concentration, instead of the usual Morawetz estimate dependent of the
nonlinearity.

We argue that the critical behaviour near the point of “gradient
catastrophe” of the solution to the Cauchy problem for the focusing nonlinear
Schr\”odinger equation $ i\epsilon \psi_t +\frac{\epsilon^2}2\psi_{xx}+
|\psi|^2 \psi =0$ with analytic initial data of the form $\psi(x,0;\epsilon)
=A(x) e^{\frac{i}{\epsilon} S(x)}$ is approximately described by a particular
solution to the Painlev\’e-I equation.

This paper is devoted to describe the finite-dimensionality of a
two-dimensional micropolar fluid flow with periodic boundary conditions. We
define the notions of determining modes and nodes and estimate the number of
them, we also estimate the dimension of the global attractor. Finally we
compare our results with analogous results for Navier-Stokes equation.