On properties of different notions of centers for convex cones

The points on the revolution axis of a circular cone are somewhat special: they are the "most interior'' elements of the cone. This paper addresses the issue of formalizing the concept of center for a convex cone that is not circular. Four distinct proposals are studied in detail: the incenter, the circumcenter, the inner center, and the outer center. The discussion takes place in the context of a reflexive Banach space

Uniform boundedness of norms of convex and nonconvex processes

The lower limit of a sequence of closed convex processes is again a closed convex process. In this note we prove the following uniform boundedness principle: if the lower limit is nonempty-valued everywhere, then, starting from a certain index, the given sequence is uniformly norm-bounded. As shown with an example, the uniform boundedness principle is not true if one drops convexity. By way of illustration, we consider an application to the controllability analysis of differential inclusions

Inradius and circumradius of various convex cones arising in applications

This note addresses the issue of computing the inradius and the circumradius of a convex cone in a Euclidean space. It deals also with the related problem of finding the incenter and the circumcenter of the cone. We work out various examples of convex cones arising in applications

Condition number and eccentricity of a closed convex cone

We discuss some extremality issues concerning the circumradius, the inradius, and the condition number of a closed convex cone in $mathbbR^n$. The condition number refers to the ratio between the circumradius and the inradius. We also study the eccentricity of a closed convex cone, which is a coefficient that measures to which extent the circumcenter differs from the incenter

Distance to uncontrollability for convex processes

The classical study of controllability of linear systems assumes unconstrained control inputs. The 'distance to uncontrollability' measures the size of the smallest perturbation to the matrix description of the system rendering it uncontrollable, and is a key measure of system robustness. We extend the standard theory of this measure of controllability to the case where the control input must satisfy given linear inequalities. Specifically, we consider the control of differential inclusions, concentrating on the particular case where the control input takes values in a given convex cone

Marginality of bulk-edge correspondence for single-valley Hamiltonians

We study the correspondence between the non-trivial topological properties associated with the individual valleys of gapped bilayer graphene (BLG), as a prototypical multi-valley system, and the gapless modes at its edges and other interfaces. We find that the exact connection between the valley-specific Hall conductivity and the number of gapless edge modes does not hold in general, but is dependent on the boundary conditions, even in the absence of intervalley coupling. This non-universality is attributed to the absence of a well-defined topological invariant within a given valley of BLG; yet, a more general topological invariant may be defined in certain cases, which explains the distinction between the BLG-vacuum and BLG-BLG interfaces.Comment: 7 pages, 4 figure

Glucocorticoids Antagonize Ap-1 by Inhibiting the Activation/Phosphorylation of Jnk without Affecting Its Subcellular Distribution

The immunosuppressive and antiinflammatory actions of glucocorticoid hormones are mediated by their transrepression of activating protein-1 (AP-1) and nuclear factor-kappa B (NFκB) transcription factors. Inhibition of the c-Jun NH2-terminal kinase (JNK) signaling pathway, the main mediator of AP-1 activation, has been described in extracts of hormone-treated cells. Here, we show by confocal laser microscopy, enzymatic assays, and immunoblotting that the synthetic glucocorticoid dexamethasone inhibited tumor necrosis factor α (TNF-α)–induced phosphorylation and activation of JNK in the cytoplasm and nucleus of intact HeLa cells. As a result, c-Jun NH2-terminal domain phosphorylation and induction were impaired. Dexamethasone did not block the TNF-α–induced JNK nuclear translocation, but rather induced, per se, nuclear accumulation of the enzyme. Consistently with previous findings, a glucocorticoid receptor mutant (GRdim), which is deficient in dimerization, DNA binding, and transactivation, but retains AP-1 transrepressing activity, was as efficient as wild-type GR in mediating the same effects of dexamethasone on JNK in transfected Cos-7 cells. Our results show that glucocorticoids antagonize the TNF-α–induced activation of AP-1 by causing the accumulation of inactive JNK without affecting its subcellular distribution

Tyrosine hydroxylase deficiency: a treatable disorder of brain catecholamine biosynthesis

Tyrosine hydroxylase deficiency is an autosomal recessive disorder resulting from cerebral catecholamine deficiency. Tyrosine hydroxylase deficiency has been reported in fewer than 40 patients worldwide. To recapitulate all available evidence on clinical phenotypes and rational diagnostic and therapeutic approaches for this devastating, but treatable, neurometabolic disorder, we studied 36 patients with tyrosine hydroxylase deficiency and reviewed the literature. Based on the presenting neurological features, tyrosine hydroxylase deficiency can be divided in two phenotypes: an infantile onset, progressive, hypokinetic-rigid syndrome with dystonia (type A), and a complex encephalopathy with neonatal onset (type B). Decreased cerebrospinal fluid concentrations of homovanillic acid and 3-methoxy-4-hydroxyphenylethylene glycol, with normal 5-hydroxyindoleacetic acid cerebrospinal fluid concentrations, are the biochemical hallmark of tyrosine hydroxylase deficiency. The homovanillic acid concentrations and homovanillic acid/5-hydroxyindoleacetic acid ratio in cerebrospinal fluid correlate with the severity of the phenotype. Tyrosine hydroxylase deficiency is almost exclusively caused by missense mutations in the TH gene and its promoter region, suggesting that mutations with more deleterious effects on the protein are incompatible with life. Genotype-phenotype correlations do not exist for the common c.698G>A and c.707T>C mutations. Carriership of at least one promotor mutation, however, apparently predicts type A tyrosine hydroxylase deficiency. Most patients with tyrosine hydroxylase deficiency can be successfully treated with l-dop