We know that Mathematics is useful in sciences and other subjects, but the recent development has shown that mathematics is a useful concept underlying arts. A major subject in Mathematics, algebra, is now used to retrace a picture so that the clear picture without dirt or flaws can be obtained. This new concept is that […]

The notion of topological free entropy dimension of $n-$tuples of elements in
a C$^*$ algebra was introduced by Voiculescu. In the paper, we compute
topological free entropy dimension of one self-adjoint element and topological
orbit dimension of one self-adjoint element in a C$^*$ algebra.

We prove that the Pauli representation of the quantum permutation algebra
$A(S_4)$ is faithful. This provides the second known model for a free quantum
algebra. We use this model for performing some computations, with the result
that at level of laws of diagonal coordinates, the Lebesgue measure appears
between the Dirac mass and the free Poisson law.

Combining a lattice path integral formulation for thermodynamics with the
solution of the quantum inverse scattering problem for local spin operators, we
derive a multiple integral representation for the time-dependent longitudinal
correlation function of the spin-1/2 Heisenberg XXZ chain at finite temperature
and in an external magnetic field. Our formula reproduces the previous results
in the following three limits: the static, the zero-temperature and the XY
limits.

Let $n \geq 2$ be an integer. An \emph{$n$-potent} is an element $e$ of a
ring $R$ such that $e^n = e$. In this paper, we study $n$-potents in matrices
over $R$ and use them to construct an abelian group $K_0^n(R)$. If $A$ is a
complex algebra, there is a group isomorphism $K_0^n(A) \cong
\bigl(K_0(A)\bigr)^{n-1}$ for all $n \geq 2$. However, for algebras over
cyclotomic fields, this is not true in general. We consider $K_0^n$ as a
covariant functor, and show that it is also functorial for a generalization of
homomorphism called an \emph{$n$-homomorphism}.

Using the computational algebra system GAP (this http URL) and
the GAP package LAGUNA (this http URL), we
checked that all 2-groups of order not greater than 32 are determined by
normalized unit groups of their modular group algebras over the field of two
elements.

We develop a systematic approach to the study of independence in topological
dynamics with an emphasis on combinatorial methods. One of our principal aims
is to combinatorialize the local analysis of topological entropy and related
mixing properties. We also reframe our theory of dynamical independence in
terms of tensor products and thereby expand its scope to C*-dynamics.

We develop some applications of certain algebraic and combinatorial
conditions on the elements of Coxeter groups, such as elementary proofs of the
positivity of certain structure constants for the associated Kazhdan–Lusztig
basis. We also explore some consequences of the existence of a Jones-type trace
on the Hecke algebra of a Coxeter group, such as simple procedures for
computing leading terms of certain Kazhdan–Lusztig polynomials.

The sequel to this paper is math.QA/0509363.

Onsager conjectured that weak solutions of the Euler equations for
incompressible fluids in 3D conserve energy only if they have a certain minimal
smoothness, (of order of 1/3 fractional derivatives) and that they dissipate
energy if they are rougher. In this paper we prove that energy is conserved for
velocities in the function space $B^{1/3}_{3,c(\NN)}$. We show that this space
is sharp in a natural sense. We phrase the energy spectrum in terms of the
Littlewood-Paley decomposition and show that the energy flux is controlled by
local interactions. This locality is shown to hold also for the helicity flux;
moreover, every weak solution of the Euler equations that belongs to
$B^{2/3}_{3,c(\NN)}$ conserves helicity. In contrast, in two dimensions, the
strong locality of the enstrophy holds only in the ultraviolet range.

A $\mathbf{Q}$-Cartier divisor $D$ on a projective variety $M$ is {\it almost
nup}, if $(D, C) > 0$ for every very general curve $C$ on $M$. An algebraic
variety $X$ is of {\it almost general type}, if there exists a projective
variety $M$ with only terminal singularities such that the canonical divisor
$K_M$ is almost nup and that $M$ is birationally equivalent to $X$. We prove
that, a complex algebraic variety is of almost general type if and only if it
is neither uniruled nor covered by any family of varieties being birationally
equivalent to minimal varieties with numerically trivial canonical divisors,
under the minimal model conjecture. Furthermore we prove that, for a projective
variety $X$ with only terminal singularities, $X$ is of almost general type if
and only if the canonical divisor $K_X$ is almost nup, under the minimal model
conjecture.

We give an introduction to Tropical Geometry and prove some results in
Tropical Intersection Theory. The first part of this paper is an introduction
to tropical geometry aimed at researchers in Algebraic Geometry from the point
of view of degenerations of varieties using projective not-necessarily-normal
toric varieties. The second part is a foundational account of tropical
intersection theory with proofs of some new theorems relating it to classical
intersection theory.

Revised version includes corrections and new proofs relating on lattice
indices and the associated cycle.

In this note, we give a description of the graded Lie algebra of double
derivations of a path algebra as a graded version of the necklace Lie algebra
equipped with the Kontsevich bracket. Furthermore, we formally introduce the
notion of double Poisson-Lichnerowicz cohomology for double Poisson algebras,
and give some elementary properties. We introduce the notion of a linear double
Poisson tensor on a quiver and show that it induces the structure of a finite
dimensional algebra on the vector spaces V_v generated by the loops in the
vertex v. We show that the Hochschild cohomology of the associative algebra can
be recovered from the double Poisson cohomology. Then, we use the description
of the graded necklace Lie algebra to determine the low-dimensional double
Poisson-Lichnerowicz cohomology groups for three types of (linear and
non-linear) double Poisson brackets on the free algebra in two variables. This
allows us to develop some useful techniques for the computation of the double
Poisson-Lichnerowicz cohomology.

Under some positivity assumptions, extension properties of rationally
connected fibrations from a submanifold to its ambient variety are studied.
Given a family of rational curves on a complex projective manifold X inducing a
covering family on a submanifold Y with ample normal bundle in X, the main
results relate, under suitable conditions, the associated rational connected
fiber structures on X and on Y. Applications of these results include an
extension theorem for Mori contractions of fiber type and a classification
theorem in the case Y has a structure of projective bundle or quadric
fibration.