‎the gutman index and degree distance of a connected graph $g$ are defined as‎ ‎begin{eqnarray*}‎ ‎textrm{gut}(g)=sum_{{u,v}subseteq v(g)}d(u)d(v)d_g(u,v)‎, ‎end{eqnarray*}‎ ‎and‎ ‎begin{eqnarray*}‎ ‎dd(g)=sum_{{u,v}subseteq v(g)}(d(u)+d(v))d_g(u,v)‎, ‎end{eqnarray*}‎ ‎respectively‎, ‎where‎ ‎$d(u)$ is the degree of vertex $u$ and $d_g(u,v)$ is the distance between vertices $u$ and $v$‎. ‎in th...

We point out that the matrix description of M-theory compactified on Calabi-Yau threefolds is in many respects simpler than the matrix description of a T 6 compactification. This is largely because of the differences between D6 branes wrapped on Calabi-Yau threefolds and D6 branes wrapped on six-tori. In particular, if we define the matrix theory following the prescription of Sen and Seiberg, w...

A link in production network is usually a creditordebtor relationship. If a firm goes into financial insolvency state or bankruptcy, then firms on its upstream can have secondary effect from the bankruptcy. By using the recent 10 years data of bankruptcy in japan, we show that these causes of “link effect” are by no means negligible. Indeed, nearly 20% of total debt (a few percent of GDP) is du...

2 Local theory 3 2.1 The endomorphism μ of V0(n) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.2 Double coset representatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.3 The matrix of μ at level p2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 2.4 Eigenvalues of μ at level p2 for Iwahori-spherical representations . . . . . . . . . 17 2.5 Pr...

We show that the adjoint A+ of a matrix A with respect to a given inner product is a rational function in A, if and only if A is normal with respect to the inner product. We consider such matrices and analyze the McMillan degrees of the rational functions r such that A+ = r(A). We introduce the McMillan degree of A as the smallest among these degrees, characterize this degree in terms of the nu...