Truncated K-moment problems in several variables

Let $\beta\equiv\beta^{(2n)}$ be an N-dimensional real multi-sequence of degree 2n, with associated moment matrix $\mathcal{M}(n)\equiv \mathcal{M}(n)(\beta)$, and let $r:=rank \mathcal{M}(n)$. We prove that if $\mathcal{M}(n)$ is positive semidefinite and admits a rank-preserving moment matrix extension $\mathcal{M}(n+1)$, then $\mathcal{M}(n+1)$ has a unique representing measure \mu, which is r-atomic, with supp \mu$equal to$\mathcal{V}(\mathcal{M}(n+1))$, the algebraic variety of$\mathcal{M}(n+1)$. Further, \beta has an r-atomic (minimal) representing measure supported in a semi-algebraic set$K_{\mathcal{Q}}$subordinate to a family$\mathcal{Q}% \equiv\{q_{i}\}_{i=1}^{m}\subseteq\mathbb{R}[t_{1},...,t_{N}]$if and only if$\mathcal{M}(n)$is positive semidefinite and admits a rank-preserving extension$\mathcal{M}(n+1)$for which the associated localizing matrices$\mathcal{M}_{q_{i}}(n+[\frac{1+\deg q_{i}}{2}])$are positive semidefinite$(1\leq i\leq m)$; in this case, \mu (as above) satisfies supp \mu\subseteq K_{\mathcal{Q}}$, and \mu has precisely rank \mathcal{M}(n)-rank \mathcal{M}_{q_{i}}(n+[\frac{1+\deg q_{i}}{2}])$atoms in$\mathcal{Z}(q_{i})\equiv {t\in\mathbb{R}^{N}:q_{i}(t)=0}$,$1\leq i\leq m$.Comment: 33 pages; to appear in J. Operator Theor

Flat extensions of positive moment matrices: recursively generated relations

We develop new computational tests for existence and uniqueness of representing measures in the Truncated Complex Moment Problem: γij = Z zizj d (0 i + j 2n):(TCMP) We characterize the existence of nitely atomic representing measures in terms of positivity and extension properties of the moment matrix M(n)(γ) associated with γ γ(2n): γ00; : : : ; γ0;2n; : : : ; γ2n;0, γ00> 0 (Theorem 1.5). We study conditions for flat (i.e., rank-preserving) extensions M(n + 1) of M(n) 0; each such extension corresponds to a distinct rank M(n)-atomic representing measure, and each such measure is minimal among representing measures in terms of the cardinality of its support. For a natural class of moment matrices satisfying the tests of recursive generation, recursive consis-tency, and normal consistency, we reduce the existence problem for minimal representing measures to the solubility of small systems of multivariable alge

Constitutive cytoplasmic localization of p21Waf1/Cip1 affects the apoptotic process in monocytic leukaemia

In the present study, we analysed the expression and localization of p21Waf1/Cip1 in normal and malignant haematopoietic cells. We demonstrate that in normal monocytic cells, protein kinase C (PKC)-induced p21 gene activation, which is nuclear factor-κB (NF-κB) independent, results in predominantly cytoplasmic localized p21 protein. In acute monocytic leukaemia (M4, M5), monocytic blasts (N=12) show constitutive cytoplasmic p21 expression in 75% of the cases, while in myeloid leukaemic blasts (N=10), low nuclear and cytoplasmic localization of p21 could be detected, which is also PKC dependent. Constitutive p21 expression in monocytic leukaemia might have important antiapoptotic functions. This is supported by the finding that in U937 cells overexpressing p21, VP16-induced apoptosis is significantly reduced (20.0±0.9 vs 55.8±3.8%, P<0.01, N=5), reflected by a reduced phosphorylation of p38 and JNK. Similarly, AML blasts with high cytoplasmic p21 were less sensitive to VP16-induced apoptosis as compared to AML cases with low or undetectable p21 expression (42.25 vs 12.3%, P<0.01). Moreover, complex formation between p21 and ASK1 could be demonstrated in AML cells, by means of coimmunoprecipitation. In summary, these results indicate that p21 has an antiapoptotic role in monocytic leukaemia, and that p21 expression is regulated in a PKC-dependent and NF-κB independent manner.

Downregulation of Mcl-1 has anti-inflammatory pro-resolution effects and enhances bacterial clearance from the lung

Phagocytes not only coordinate acute inflammation and host defense at mucosal sites, but also contribute to tissue damage. Respiratory infection causes a globally significant disease burden and frequently progresses to acute respiratory distress syndrome, a devastating inflammatory condition characterized by neutrophil recruitment and accumulation of protein-rich edema fluid causing impaired lung function. We hypothesized that targeting the intracellular protein myeloid cell leukemia 1 (Mcl-1) by a cyclin-dependent kinase inhibitor (AT7519) or a flavone (wogonin) would accelerate neutrophil apoptosis and resolution of established inflammation, but without detriment to bacterial clearance. Mcl-1 loss induced human neutrophil apoptosis, but did not induce macrophage apoptosis nor impair phagocytosis of apoptotic neutrophils. Neutrophil-dominant inflammation was modelled in mice by either endotoxin or bacteria (Escherichia coli). Downregulating inflammatory cell Mcl-1 had anti-inflammatory, pro-resolution effects, shortening the resolution interval (R(i)) from 19 to 7 h and improved organ dysfunction with enhanced alveolar–capillary barrier integrity. Conversely, attenuating drug-induced Mcl-1 downregulation inhibited neutrophil apoptosis and delayed resolution of endotoxin-mediated lung inflammation. Importantly, manipulating lung inflammatory cell Mcl-1 also accelerated resolution of bacterial infection (R(i); 50 to 16 h) concurrent with enhanced bacterial clearance. Therefore, manipulating inflammatory cell Mcl-1 accelerates inflammation resolution without detriment to host defense against bacteria, and represents a target for treating infection-associated inflammation