In two previous papers we computed cohomology groups H(Γ0(N);C) for a range of levels N , where Γ0(N) is the congruence subgroup of SL4(Z) consisting of all matrices with bottom row congruent to (0, 0, 0, ∗) mod N . In this note we update this earlier work by carrying it out for prime levels up to N = 211. This requires new methods in sparse matrix reduction, which are the main focus of the pap...

In this paper some properties of the complement of  a new graph  associated with a commutative ring  are investigated ....

In this work, by employing the Krasnosel'skii fixed point theorem, we study the existence of positive solutions of a three-point boundary value problem for the following fourth-order differential equation begin{eqnarray*} left { begin{array}{ll} u^{(4)}(t) -f(t,u(t),u^{prime prime }(t))=0 hspace{1cm} 0 leq t leq 1, & u(0) = u(1)=0, hspace{1cm} alpha u^{prime prime }(0) - beta u^{prime prime pri...

Suppose $G$ is a split connected‎ ‎reductive orthogonal or symplectic group over an infinite field‎ ‎$F,$ $P=MN$ is a maximal parabolic subgroup of $G,$ $frak{n}$ is‎ ‎the Lie algebra of the unipotent radical $N.$ Under the adjoint‎ ‎action of its stabilizer in $M,$ every maximal prehomogeneous‎ ‎subspaces of $frak{n}$ is determined‎.

1. I n t r o d u c t i o n In 1640, Pierre de Fermat conjectured that all numbers F m = 2 2 ' " + 1 for m = 0, 1, 2 , . . . , (1) are prime, which was later found to be incorrect. The numbers Fm are called Fermat numbers after him. If Fm is prime, we say that it is a Fermat prime. Until the end of the 18th century, Fermat numbers were most likely a mathe­ matical curiosity. The interest in the ...