A note on traces of differential forms

A note on traces of set families

A family of sets F ⊆ 2[n] is defined to be l-trace k-Sperner if for any l-subset L of [n] the family of traces F|L = {F ∩L : F ∈ F} does not contain any chain of length k+ 1. In this paper we prove that for any positive integers l′, k with l′ < k if F is (n−l′)-trace k-Sperner, then |F| ≤ (k − l′ + o(1)) ( n bn/2c ) and this bound is asymptotically tight. AMS subject classification: 05D05

A note on quadratic forms

We extend an interesting theorem of Yuan [12] for two quadratic forms to three matrices. Let C1, C2, C3 be three symmetric matrices in <n×n , if max{xT C1x, xT C2x, xT C3x} ≥ 0 for all x ∈ <n , it is proved that there exist ti ≥ 0 (i = 1, 2, 3) such that ∑3 i=1 ti = 1 and ∑3 i=1 ti Ci has at most one negative eigenvalue.

A note on bilinear forms

A Note on Hermitian Forms

In this note we effect a reduction of the theory of hermitian forms of two particular types (coefficients in a quadratic field or in a quaternion algebra with the usual anti-automorphism) to that of quadratic forms. The main theorem (§2) enables us to apply directly the known results on quadratic forms. This is illustrated in the discussion in §3 of a number of special cases. Let <£ be an arbit...

A Note on Faithful Traces on a Von Neumann Algebra

It is known that a semifinite von Neumann algebra always has a faithful semifinite trace. This trace can be used, for instance, to build up a non-commutative integration and, consequently, to define non commutative L-spaces. In this note we give some techniques for constructing, starting from a family F of semifinite traces, a faithful one which is closely related to the family F. Let F = {ηα; ...