Purchasing a property or a car requires a lot of money and sometimes we do not have it. Our attention is turned to borrowing loans of money, but sometimes it turns out that the lenders charged a very high interest and some of them are untrustworthy. However, now you can compare personal loans offers so […]

Bullying has always become a problem that happens among schoolmates. Although we have tried to prevent it, but it always occur no matter whether it is just in the form of jokes or serious allusions that despise a group of people. A brawl can break out if the bullied students fight back for equal treatments […]

Hundreds of people are suffering from dyslexia, but they do not get any help to recover from this problem. That is what happens to students who study in Texas. Dyslexia is a learning disability that has the symptoms of difficulty in reading, pronouncing words and therefore they cannot absorb the lessons taught by the teachers. […]

Competition for resources and power between countries can result in war, which will adversely affect people in those countries. However, we sometimes see that the outcome of war is unpredictable; the more powerful countries lose to the supposedly weaker countries. A scientist in the University of Georgia then comes up with a theory that […]

Republican presidential candidate Rudy Giuliani stated that he was keen to use statistics to solve national crime problems if he is chosen as the president. In recent debates, he repeatedly cited the word ’stat’ - short term for statistics - as an effective weapon that he will use to lower the increasing crime rate in […]

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.

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).

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 […]

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 […]

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 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.