Let A be a real continuous nxn matrix on an interval T3 and let the n-vector x be a solution of the differential equation x = Ax on r. If [oc,g]er, g is called a conjugate point of a if the equation has a nontrivial solution vector x = (x1,...,xn) such that x1(a) = ... = xk(a) = x k+1(P) = ... = x (0) = 0 for some ke[l,n-l]. It is shown that the absence on (t..,t ) of a point conjugate to t, wi...

The relationships between the values taken by ordinary characters of symmetric groups are exploited to prove two theorems in the modular representation theory of the symmetric group. 1. The decomposition matrices of symmetric groups in odd characteristic have distinct rows. In characteristic 2 the rows of a decomposition matrix labelled by the different partitions λ and μ are equal if and only ...

In this paper we study multivariate polynomial functions in complex variables and the corresponding associated symmetric tensor representations. The focus is on finding conditions under which such complex polynomials/tensors always take real values. We introduce the notion of symmetric conjugate forms and general conjugate forms, and present characteristic conditions for such complex polynomial...

The theory of conjugate points in the calculus of variations is reconsidered with a perspective emphasizing the connection to finite-dimensional optimization. The object of central importance is the spectrum of the second-variation operator, analogous to the eigenvalues of the Hessian matrix in finite dimensions. With a few basic properties of this spectrum, one can gain a new perspective on th...

Background: The dosimetric performance of Eclipse 6.5 three dimensional treatment planning system (3DTPS) is evaluated by comparing the calculated and measured dose in two dimensions following the guide lines of American Association for Physicists in Medicine Task Group 53. Materials and Methods: The calculations were performed by the 3DTPS for symmetric as well as asymmetric fields for standar...