An Example of a Positive Semidefinite Double Sequence Which is Not a Moment Sequence

The first explicit example of a positive semidefinite double sequence which is not a moment sequence was given by Friedrich. We present an example with a simpler definition and more moderate growth as (m, n) .     

Stieltjes moment sequences and positive definite matrix sequences

For a certain constant (a little less than ), every function satisfying , , is a Stieltjes indeterminate Stieltjes moment sequence. For every indeterminate moment sequence there is a positive definite matrix sequence which is not of positive type and which satisfies , . For a certain constant (a little greater than ), for every function satisfying , , there is a convolution semigroup of measures on , with moments of all orders, such that , , and for every such convolution semigroup the measure is Stieltjes indeterminate for all .

     

Mode field diameter preserving fiber tapers

An approach for preserving the mode field diameter (MFD) in fiber tapers is demonstrated. The approach utilizes concentric dual-core fibers, which couple light from an inner core to an outer core through a taper. Fibers with a 6 μm MFD feedthrough and a 15 μm polarization maintaining feedthrough are demonstrated experimentally. Simulations of the MFD in the tapered dual-core fibers are also presented.     

Factoring of positive definite functions: A counterexample based on an extension theorem

No abstract. December 24, 1999     

On the continuity of convolution semigroups

Communicated by Karl H. Hofmann     

Sharp bounds on the ranks of negativity of certain sums

If M is a complex vector space and 〈·, ·〉 a Hermitian sesquilinear form on M with a finite rank of negativity k (i.e., k is the maximal dimension of any linear subspace E of M satisfying 〈x, x〉 < 0 for each nonzero x in E), if n is a positive integer, and if a ₁, …, a _n are endomorphisms of M, then it is easy to see that the Hermitian sesquilinear form $ (x,y) \mapsto \sum\limits_{v = 1}^n {\left\langle {a_v x,a_v y} \right\rangle } $ on M has rank of negativity at most nk. It is also fairly easy to see that the bound nk cannot be improved in general. Less trivial is the fact that it cannot be improved by making the following assumption (a) the space M is the *-algebra A:= (C[[w ₁, w ₂]] of polynomials in two self-adjoint non-commuting indeterminates; there is a (necessarily Hermitian) linear form φ on A such that 〈x, y〉 = φ(y* x) (x, y ∈ A); and a _v is just left multiplication by some element of A (which we may denote by ‘a _v ’ at no great risk of confusion). Now suppose that, with M, 〈·, ·〉, k, n, and a ₁ , …, a _n as initially, the following two conditions are satisfied:

1. each a _v has a formal adjoint a* _v , being an endomorphism of M such that $ \left\langle {a_v x,y} \right\rangle = \left\langle {x,a_v^* y} \right\rangle (x,y \in M); $
2. the mappings a ₁, …, a _n , a*₁, …, a* _n commute pairwise.

Then the bound nk can be replaced by k (regardless of how large n may be). This result cannot be improved in general since it may happen that each a _v is a scalar multiple of the identical mapping of M into itself (not all a _v equal to 0), in which case the form (1) is a positive multiple of 〈·, ·〉 itself. There are ties with the subjects of ‘positive semidefinite submodules’ (‘positive semidefinite left ideals’) and ‘definitisation’.     

Empowering the configuration-IP: new PTAS results for scheduling with setup times

Mathematical Programming - Integer linear programs of configurations, or configuration IPs, are a classical tool in the design of algorithms for scheduling and packing problems where a set of items...     

Stieltjes Perfect Semigroups are Perfect

An abelian *-semigroup S is perfect (resp. Stieltjes perfect) if every positive definite (resp. completely so) function on S admits a unique disintegration as an integral of hermitian multiplicative functions (resp. nonnegative such). We prove that every Stieltjes perfect semigroup is perfect. The converse has been known for semigroups with neutral element, but is here shown to be not true in general. We prove that an abelian *-semigroup S is perfect if for each s ∈ S there exist t ∈ S and m, n ∈ ℕ₀ such that m + n ≥ 2 and s + s* = s* + mt + nt*. This was known only with s = mt + nt* instead. The equality cannot be replaced by s + s* + s = s + s* + mt + nt* in general, but for semigroups with neutral element it can be replaced by s + p(s + s*) = p(s + s*) + mt + nt* for arbitrary p ∈ ℕ (allowed to depend on s).