Global solutions of the Boltzmann equation on a lattice

The nonlinear Boltzmann equation with a discretized spatial variable is studied in a Banach space of absolutely integrable functions of the velocity variables. Conservation laws and positivity are utilized to extend weak local solutions to a global solution. This is shown to be a strong solution by analytic semigroup techniques.Supported by National Science Foundation Grant ENG-7515882.     

Non-linear Semigroup of a Class of Abstract Semilinear Functional Differential Equations with a Non-Dense Domain

In this work, we are concerned with a general class of abstract semilinear autonomous functional differential equations with a non-dense domain on a Banach space. Our objective is to study, using the Crandall-Liggett approach, the solutions as a semigroup of non-linear operators.     

Iterative Algebras without Projections

We deal with iterative algebras of functions of -valued logic lacking projections, which we call algebras without projections. It is shown that a partially ordered set of algebras of functions of -valued logic, for , without projections contains an interval isomorphic to the lattice of all iterative algebras of functions of -valued logic. It is found out that every algebra without projections is contained in some maximal algebra without projections, which is the stabilizer of a semigroup of non-surjective transformations of the basic set. It is proved that the stabilizer of a semigroup of all monotone non-surjective transformations of a linearly ordered 3-element set is not a maximal algebra without projections, but the stabilizer of a semigroup of all transformations preserving an arbitrary non one-element subset of the basic set is.     

Hardy spaces of spaces of homogeneous type

Let be a space of homogeneous type, and be the generator of a semigroup with Gaussian kernel bounds on . We define the Hardy spaces of for a range of , by means of area integral function associated with the Poisson semigroup of , which is proved to coincide with the usual atomic Hardy spaces on spaces of homogeneous type.

     

Some Conditions for a C0-Semigroup to Be Asymptotically Finite-Dimensional

We study the class of bounded C ₀-semigroups T=(T _t ) _t0 on a Banach space X satisfying the asymptotic finite dimensionality condition: codim X ₀(T)<, where X ₀(T):={x X:lim_t T _t x=0}. We prove a theorem which provides some necessary and sufficient conditions for asymptotic finite dimensionality.     

Divisorial Linear Algebra of Normal Semigroup Rings

We investigate the minimal number of generators and the depth of divisorial ideals over normal semigroup rings. Such ideals are defined by the inhomogeneous systems of linear inequalities associated with the support hyperplanes of the semigroup. The main result is that for every bound C there exist, up to isomorphism, only finitely many divisorial ideals I such that (I)C. It follows that there exist only finitely many Cohen–Macaulay divisor classes. Moreover, we determine the minimal depth of all divisorial ideals and the behaviour of and depth in arithmetic progressions in the divisor class group.The results are generalized to more general systems of linear inequalities whose homogeneous versions define the semigroup in a not necessarily irredundant way. The ideals arising this way can also be considered as defined by the nonnegative solutions of an inhomogeneous system of linear diophantine equations.We also give a more ring-theoretic approach to the theorem on minimal number of generators of divisorial ideals: it turns out to be a special instance of a theorem on the growth of multigraded Hilbert functions.     

Additive Kernels and Integral Representation of Potentials

Let P=(P _t ) _t>0 be a submarkovian semigroup of kernels on a measurable space (X,). An additive kernel of P is a kernel K from X into ]0,[ such that P _t K(x,A)=K(x,A+t) for every t>0,xX and every Borel subset A of ]0,[. It is proved in this paper that for every potential f of P, there exits an additive kernel K of P, unique (up to equivalence) such that f=K1=₀ K(,dt). This result is already well known for regular potentials of right processes. If U=(U _p ) _p>0 is a sub-Markovian resolvent of kernels on (X,), we give a notion of additive kernel of U and we prove a similar integral representation of potentials of U.     

Bi-Invariant Domains in Semisimple Groups: Nonprincipal Orbits and Boundaries

Let G be a complex semisimple group with real form G . For the action of G × G on G by left and right translation, we define and study an orbit-type stratification of G. In particular, we compute the dimension of an arbitrary stratum; we identify certain strata of low codimension in G, the union of whose closures contains every nonprincipal stratum; and we describe the boundaries and adjacency relations of the maximal connected domains in G that contain only principal orbits.     

Poincaré Semigroup Symmetry as an Emergent Property of Unstable Systems

The notion that elementary systems correspond to irreducible representations of the Poincaré group is the starting point for this paper, which then goes on to discuss how a semigroup for the time evolution of unstable states and resonances could emerge from the underlying Poincaré symmetry. Important tools in this analysis are the Clebsch-Gordan coefficients for the Poincaré group.