Saunders Mac Lane was born in 1909 in San Francisco, California. He was actually the co-founder of the category theory with Samuel Eilenberg. Saunders earned a BA from Yale University in 1930 and a MA from the University of Chicago, while attending the University he published his first paper in the category of physics with […]

We give an explicit handy (and cocycle-free) description of the groupoid of
weak maps between two crossed-modules using what we call a {\em papillon};
Theorem 8.3. We define composition of papillons and this way find a bicategory
that is naturally biequivalent to the 2-category of pointed homotopy 2-types.
This has applications in the the study of 2-group actions (say, on stacks).

This paper contains some basic results on 2-groupoids, with special emphasis
on computing derived mapping 2-groupoids between 2-groupoids and proving their
invariance under strictification. Some of the results proven here are
presumably folklore (but do not appear in the literature to the author’s
knowledge) and some of the results seem to be new. The main technical tool used
throughout the paper is the Quillen model structure on the category of
2-groupoids introduced by Moerdijk and Svensson.

Tom Lehrer wrote a satirical song named New Math which centered around the process of subtracting 173 from 342 in decimal and octal.

The song is in the style of a lecture about the general concept of subtraction in arbitrary number systems, illustrated by two simple calculations, and highlights the emphasis on insight and abstract concepts […]

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 […]

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 […]

Okay, here is some useful sites offering scholarships.
Scholarships in Math due April 24, 2007
Rich Scholarship
The Dr. Barnett and Jean Hollander Rich Mathematics Scholarships, currently in the amount of $5,000 each, are awarded annually to talented and needy graduates and undergraduates who have demonstrated superior ability in mathematics and who are preparing for careers in mathematics […]

We give a new procedure in Maple for finding the k-th power of a martix. The
algorithm is based on the article [1].

In this paper we give a generalization of Chebyshev polynomials and using
this we describe the M\”obius function of the generalized subword order from a
poset {a1,…as,c |ai<c}, which contains an affirmative answer for the
conjecture by Bj\”orner, Sagan, Vatter.[5,10]

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 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 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 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.

We establish an unusual second-order almost sure limit theorem for the
minimal position in a one-dimensional super-critical branching random walk, and
also prove a martingale convergence theorem which answers a question of Biggins
and Kyprianou [7]. Our method applies furthermore to the study of directed
polymers on a disordered tree. In particular, we give a rigorous proof of a
phase transition phenomenon for the partition function (from the point of view
of convergence in probability), already described by Derrida and Spohn [14].
Surprisingly, this phase transition phenomenon disappears in the sense of upper
almost sure limits.

We apply results of Malliavin-Thalmaier-Watanabe for strong and weak Taylor
expansions of solutions of perturbed stochastic differential equations (SDEs).
In particular, we work out weight expressions for the Taylor coefficients of
the expansion. The results are applied to LIBOR market models in order to deal
with the typical stochastic drift and with stochastic volatility. In contrast
to other accurate methods like numerical schemes for the full SDE, we obtain
easily tractable expressions for accurate pricing. In particular, we present an
easily tractable alternative to “freezing the drift” in LIBOR market models,
which has an accuracy similar to the full numerical scheme. Numerical examples
underline the results.