On the Lieb-Thirring constants L_gamma,1 for gamma geq 1/2

Let $E_i(H)$ denote the negative eigenvalues of the one-dimensional Schr\"odinger operator $Hu:=-u^{\prime\prime}-Vu,\ V\geq 0,$ on $L_2({\Bbb R})$. We prove the inequality \sum_i|E_i(H)|^\gamma\leq L_{\gamma,1}\int_{\Bbb R} V^{\gamma+1/2}(x)dx, (1) for the "limit" case $\gamma=1/2.$ This will imply improved estimates for the best constants $L_{\gamma,1}$ in (1), as $1/2<\gamma<3/2.Comment: AMS-LATEX, 15 page

A simple proof of Hardy-Lieb-Thirring inequalities

We give a short and unified proof of Hardy-Lieb-Thirring inequalities for moments of eigenvalues of fractional Schroedinger operators. The proof covers the optimal parameter range. It is based on a recent inequality by Solovej, Soerensen, and Spitzer. Moreover, we prove that any non-magnetic Lieb-Thirring inequality implies a magnetic Lieb-Thirring inequality (with possibly a larger constant).Comment: 12 page

Spectroscopy and Electrochemistry of Cobalt(III) Schiff Base Complexes

The structural, spectroscopic, and electrochemical properties of cobalt(III) derivatives of acacen (H_2acacen = bis(acetylacetone) ethylenediimine) and related ligands have been investigated. Electronic structure calculations indicate that the absorption between 340 and 378 nm in Co^(III)(acacen) spectra is attributable to the lowest π−π* intraligand charge-transfer transition. Equatorial ligand substitutions affect reduction potentials less than axial ligand changes, consistent with an electronic structural model in which d_(z^2) is populated in forming cobalt(II). The crystal structure of [Co(3-Cl-acacen)(NH_3)_2]BPh_4 has been determined:  The compound crystallizes in the monoclinic space group (P2_1)/m (No. 11) with a = 9.720(2) Å, b = 18.142(4) Å, c = 10.046(2) Å, β = 100.11(3)°, D_c = 1.339 g cm^(-3), and Z = 2; the complex cation, [Co(3-Cl-acacen)(NH_3)_2]^+, exhibits a slightly distorted octahedral coordination geometry. The distances between the cobalt atom and the two axial nitrogen donor atoms differ only slightly (1.960(6) and 1.951(6) Å) and are similar to Co−N distances found in cobalt−ammine complexes as well as the axial Co−N distances in [Co(acacen)(4-MeIm)_2]Br·1.5H_2O; the latter compound crystallizes in the triclinic space group P1̄ (No. 2) with a = 18.466(9) Å, b = 14.936(7) Å, c = 10.111(5)Å, α = 96.27(5)°, β = 94.12(5)°, γ = 112.78(5)°, D_c = 1.447 g cm^(-3), and Z = 4

Towards a unified theory of Sobolev inequalities

We discuss our work on pointwise inequalities for the gradient which are connected with the isoperimetric profile associated to a given geometry. We show how they can be used to unify certain aspects of the theory of Sobolev inequalities. In particular, we discuss our recent papers on fractional order inequalities, Coulhon type inequalities, transference and dimensionless inequalities and our forthcoming work on sharp higher order Sobolev inequalities that can be obtained by iteration.Comment: 39 pages, made some changes to section 1

Hormonal regulation of plasminogen activator in rat hepatoma cells

Plasminogen activators are membrane-associated, arginine-specific serine proteases which convert the inactive plasma zymogen plasminogen to plasmin, an active, broad-spectrum serine protease. Plasmin, the major fibrinolytic enzyme in blood, also participates in a number of physiologic functions involving protein processing and tissue remodelling, and may play an important role in tumor invasion and metastasis. In HTC rat hepatoma cells in. tissue culture, glucocorticoids rapidly decrease plasminogen activator (PA) activity. We have shown that this decrease is mediated by induction of a soluble inhibitor of PA activity rather than modulation of the amount of PA. The hormonally-induced inhibitor is a cellular product which specifically inhibits PA but not plasmin. We have isolated variant lines of HTC cells which are selectively resistant to the glucocorticoid inhibition of PA but retain other glucocorticoid responses. These variants lack the hormonally-induced inhibitor; PA from these variants is fully sensitive to inhibition by inhibitor from steroid-treated wild-type cells. Cyclic nucleotides dramatically stimulate PA activity in HTC cells in a time- and concentration-dependent manner. Paradoxically, glucocorticoids further enhance this stimulation. Thus glucocorticoids exert two separate and opposite effects on PA activity. The availability of glucocorticoid-resistant variant cell lines, together with the unique regulatory interactions of steroids and cyclic nucleotides, make HTC cells a useful experimental system in which to study the multihormonal regulation of plasminogen activator.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/45346/1/11010_2004_Article_BF00225243.pd