The classical approach of solvability using group theory is well known and one original motivation is to solve polynomials by radicals. Radicals are square, cube, square root, cube root etc of the original coefficients for the polynomial. A polynomial is solvable by radicals if the permutation group is solvable. This is exact solvability via group theory. With modern computers, we might need to...

In previous work of the author and others, max-plus methods have been explored for solution of first-order, nonlinear Hamilton-Jacobi-Bellman partial differential equations (HJB PDEs) and corresponding nonlinear control problems. These methods exploit the max-plus linearity of the associated semigroups. In particular, although the problems are nonlinear, the semigroups are linear in the max-plu...

In previous work of the first author and others, max-plus methods have been explored for solution of first-order, nonlinear Hamilton–Jacobi–Bellman partial differential equations (HJB PDEs) and corresponding nonlinear control problems. Although max-plus basis expansion and max-plus finite-element methods can provide substantial computational-speed advantages, they still generally suffer from th...

Let C ⊂ Z be an affine semigroup, R = K[C] its semigroup ring, and *modC R the category of finitely generated “C-graded” R-modules (i.e., Z -graded modules M with M = ⊕ c∈C Mc). When R is Cohen-Macaulay and simplicial, we show that information on M ∈ *modC R such as depth, CohenMacaulayness, and (Sn) condition, can be read off from numerical invariants of the minimal irreducible resolution (i.e...

1 Abstract. Let S ={si}i∈IN ⊆ IN be a numerical semigroup. For each i ∈ IN, let ν(si) denote the number of pairs (si−sj , sj) ∈ S: it is well-known that there exists an integer m such that the sequence {ν(si)}i∈IN is non-decreasing for i > m. The problem of finding m is solved only in special cases. By way of a suitable parameter t, we improve the known bounds for m and in several cases we dete...

Osaki and Yagi (2001) give a proof of global existence for the classical chemotaxis model in one space dimension with use of energy estimates. Here we present an alternative proof which uses the regularity properties of the heat-equation semigroup. With this method we can identify a large selection of admissible spaces, such that the chemotaxis model de nes a global semigroup on these spaces. W...

We construct a surface hopping semigroup, which asymptotically describes nuclear propagation through crossings of electron energy levels. The underlying time-dependent Schrödinger equation has a matrix-valued potential, whose eigenvalue surfaces have a generic intersection of codimension two, three or five in Hagedorn’s classification. Using microlocal normal forms reminiscent of the Landau-Zen...

In previous work of the author and others, max-plus methods have been explored for solution of first-order, nonlinear Hamilton-Jacobi-Bellman partial differential equations (HJB PDEs) and corresponding nonlinear control problems. Although max-plus basis expansion and max-plus finite-element methods provide computational-speed advantages, they still generally suffer from the curse-of-dimensional...

Let S(m,e) be a class of numerical semigroups with multiplicity m and embedding dimension e. We call graph GS an S(m,e)-graph if there exists semigroup S∈S(m,e) V(GS)={x:x∈g(S)} E(GS)={xy⇔x+y∈S}, where g(S) denotes the gap set S. The aim this article is to discuss planarity S(m,e)-graphs for some cases S irreducible semigroup.