Weather extremes have become a universal problem that everyone has to take part in order to prevent it from occurring. In further research to this serious issue, researchers at the Department of Energy’s Oak Ridge National Laboratory are trying to develop climate models to identify these weather extremes and their relationships with other climate extremes. […]

Mathematicians have confirmed that algorithms can be used to locate hidden oil traps which are sited many kilometers beneath the ground. We know that there is a surging demand for oil worldwide and therefore companies have to drill deeper and look for another new oil trap because oil is an exhaustible resource. This is considered […]

Collagen is a protein material with superior mechanical properties. It consists of collagen fibrils composed of a staggered array of ultra-long tropocollagen (TC) molecules. Theoretical and molecular modeling suggests that this natural design of collagen fibrils maximizes the strength and provides large energy dissipation during deformation, thus creating a tough and robust material.
The molecular model […]

Let q be a power of 2. We show by representation theory that there exists a q
x q unitary matrix of multiplicative order q+1 whose powers generate q+1
pairwise mutually unbiased base in C^q. When q is a power of an odd prime,
there is a q x q unitary matrix of multiplicative order q+1 whose first (q+1)/2
powers generate (q+1)/2 pairwise mutually unbiased bases. We also show how the
existence of these matrices implies the existence of a special type of
orthogonal decomposition with respect to the Killing form of the special linear
and symplectic Lie algebras.

We introduce and study symmetric and exterior algebras in braided monoidal
categories such as the category O for quantum groups. We relate our braided
symmetric algebras and braided exterior algebas with their classical
counterparts.

We study quivers with relations given by non-commutative analogs of Jacobian
ideals in the complete path algebra. This framework allows us to give a
representation-theoretic interpretation of quiver mutations at arbitrary
vertices. This gives a far-reaching generalization of
Bernstein-Gelfand-Ponomarev reflection functors. The motivations for this work
come from several sources: superpotentials in physics, Calabi-Yau algebras,
cluster algebras.

We give the defining structure of two-parameter quantum group of type G_2
defined over a field {\Bbb Q}(r,s) (with r\ne s), and prove the Drinfel’d
double structure as its upper and lower triangular parts, extending an earlier
result of [BW1] in type A and a recent result of [BGH1] in types B, C, D. We
further discuss the Lusztig’s Q-isomorphisms from U_{r,s}(G_2) to its
associated object U_{s^{-1},r^{-1}}(G_2), which give rise to the usual
Lusztig’s symmetries defined not only on U_q(G_2) but also on the centralized
quantum group U_q^c(G_2) only when r=s^{-1}=q. (This also reflects the
distinguishing difference between our newly defined two-parameter object and
the standard Drinfel’d-Jimbo quantum groups). Some interesting (r,s)-identities
holding in U_{r,s}(G_2) are derived from this discussion.

This paper is devoted to the analysis of a second order method for recovering
the \emph{a priori} unknown shape of an inclusion $\omega$ inside a body
$\Omega$ from boundary measurement. This inverse problem - known as electrical
impedance tomography - has many important practical applications and hence has
focussed much attention during the last years. However, to our best knowledge,
no work has yet considered a second order approach for this problem. This paper
aims to fill that void: we investigate the existence of second order derivative
of the state $u$ with respect to perturbations of the shape of the interface
$\partial\omega$, then we choose a cost function in order to recover the
geometry of $\partial \omega$ and derive the expression of the derivatives
needed to implement the corresponding Newton method. We then investigate the
stability of the process and explain why this inverse problem is severely
ill-posed by proving the compactness of the Hessian at the global minimizer.

Let $\Goo$ be a semisimple real Lie group with unitary dual $\Ghat$. The goal
of this note is to produce new upper bounds for the multiplicities with which
representations $\pi \in \Ghat$ of cohomological type appear in certain spaces
of cusp forms on $\Goo$.

Associated to a simple root of a finite-dimensional complex semisimple Lie
algebra, there are several endofunctors (defined by Arkhipov, Enright, Frenkel,
Irving, Jantzen, Joseph, Mathieu, Vogan and Zuckerman) on the BGG category
$\mathcal{O}_0$. We study their relations, compute cohomologies of their
derived functors and describe the monoid generated by Arkhipov’s and Joseph’s
functors and the monoid generated by Irving’s functors. It turns out that the
endomorphism rings of all elements in these monoids are isomorphic. We prove
that the functors give rise to an action of the singular braid monoid on the
bounded derived category of $\mathcal{O}_0$. We also use Arkhipov’s, Joseph’s
and Irving’s functors to produce new generalized tilting modules.

We study more general variational problems on time scales. Previous results
are generalized by proving necessary optimality conditions for (i) variational
problems involving delta derivatives of more than the first order, and (ii)
problems of the calculus of variations with delta-differential side conditions
(Lagrange problem of the calculus of variations on time scales).

Let $X$ be an equivariant embedding of a connected reductive group $G$ over
an algebraically closed field $k$ of positive characteristic. Let $B$ denote a
Borel subgroup of $G$. A $G$-Schubert variety in $X$ is a subvariety of the
form $\diag(G) \cdot V$, where $V$ is a $B \times B$-orbit closure in $X$. In
the case where $X$ is the wonderful compactification of a group of adjoint
type, the $G$-Schubert varieties are the closures of Lusztig’s $G$-stable
pieces. We prove that $X$ admits a Frobenius splitting that compatibly splits
all the $G$-Schubert varieties. Moreover, when $X$ is projective we prove that
$X$ admits a stable Frobenius splitting along an ample effective Cartier
divisor which compatibly splits all the $G$-Schubert varieties. Although this
indicates that $G$-Schubert varieties have nice singularities we give an
example, in the wonderful compactification of a group of adjoint type, which is
not normal. Finally we also extend the Frobenius splitting results to the more
general class of $R$-Schubert varieties.