Necessary and sufficient conditions are obtained for all solutions of a class of linear scalar neutral-integrodifferential equations to have at least one zero. An application to an "equilibrium level-crossing" of a logistic integrodifferential equation with infinite continuously distributed delay is briefly discussed.
It is shown that a certain algebraic identity involving a summation over partitions can be utilized to obtain a class of asymptotic expansions for large parameters A number of special formulas related to some well-known number sequences and classical polynomials are presented as illustrative examples.
In this article projective toric varieties are studied from the viewpoint of Gröbner basis theory and combinatorics. We characterize the radicals of all initial ideals of a toric variety X{\mathscr A} as the Stanley-Reisner ideals of regular triangulations of its set of weights {\mathscr A}. This implies that the secondary polytope Σ({\mathscr A}) is a Minkowski summand of the state polytope of X{\mathscr A}. Here the lexicographic (resp. reverse lexicographic) initial ideals of X{\mathscr A} arise from triangulations by placing (resp. pulling) vertices. We also prove that the state polytope of the Segre embedding of Pr-1×Ps-1 equals the secondary polytope Σ(Δr-1×Δs-1) of a product of simplices.