Stability for Borell-Brascamp-Lieb inequalities

We study stability issues for the so-called Borell-Brascamp-Lieb inequalities, proving that when near equality is realized, the involved functions must be $L^1$-close to be $p$-concave and to coincide up to homotheties of their graphs.Comment: to appear in GAFA Seminar Note

Closure properties of solutions to heat inequalities

We prove that if $u_1,u_2 : (0,\infty) \times \R^d \to (0,\infty)$ are sufficiently well-behaved solutions to certain heat inequalities on $\R^d$ then the function $u: (0,\infty) \times \R^d \to (0,\infty)$ given by $u^{1/p}=u_1^{1/p_1} * u_2^{1/p_2}$ also satisfies a heat inequality of a similar type provided $\tfrac{1}{p_1} + \tfrac{1}{p_2} = 1 + \tfrac{1}{p}$. On iterating, this result leads to an analogous statement concerning $n$-fold convolutions. As a corollary, we give a direct heat-flow proof of the sharp $n$-fold Young convolution inequality and its reverse form.Comment: 12 page

Projective re-normalization for improving the behavior of a homogeneous conic linear system

In this paper we study the homogeneous conic system F : Ax = 0, x ∈ C \ {0}. We choose a point ¯s ∈ intC∗ that serves as a normalizer and consider computational properties of the normalized system F¯s : Ax = 0, ¯sT x = 1, x ∈ C. We show that the computational complexity of solving F via an interior-point method depends only on the complexity value ϑ of the barrier for C and on the symmetry of the origin in the image set H¯s := {Ax : ¯sT x = 1, x ∈ C}, where the symmetry of 0 in H¯s is sym(0,H¯s) := max{α : y ∈ H¯s -->−αy ∈ H¯s} .We show that a solution of F can be computed in O(sqrtϑ ln(ϑ/sym(0,H¯s)) interior-point iterations. In order to improve the theoretical and practical computation of a solution of F, we next present a general theory for projective re-normalization of the feasible region F¯s and the image set H¯s and prove the existence of a normalizer ¯s such that sym(0,H¯s) ≥ 1/m provided that F has an interior solution. We develop a methodology for constructing a normalizer ¯s such that sym(0,H¯s) ≥ 1/m with high probability, based on sampling on a geometric random walk with associated probabilistic complexity analysis. While such a normalizer is not itself computable in strongly-polynomialtime, the normalizer will yield a conic system that is solvable in O(sqrtϑ ln(mϑ)) iterations, which is strongly-polynomialtime. Finally, we implement this methodology on randomly generated homogeneous linear programming feasibility problems, constructed to be poorly behaved. Our computational results indicate that the projective re-normalization methodology holds the promise to markedly reduce the overall computation time for conic feasibility problems; for instance we observe a 46% decrease in average IPM iterations for 100 randomly generated poorly-behaved problem instances of dimension 1000 × 5000.Singapore-MIT Allianc

Influence of acute pancreatitis on the in vitro responsiveness of rat mesenteric and pulmonary arteries

<p>Abstract</p> <p>Background</p> <p>Acute pancreatitis is an inflammatory disease characterized by local tissue injury and systemic inflammatory response leading to massive nitric oxide (NO) production and haemodynamic disturbances. Therefore, the aim of this work was to evaluate the vascular reactivity of pulmonary and mesenteric artery rings from rats submitted to experimental pancreatitis.</p> <p>Male Wistar rats were divided into three groups: saline (SAL); tauracholate (TAU) and phospholipase A₂(PLA₂). Pancreatitis was induced by administration of TAU or PLA₂from <it>Naja mocambique mocambique </it>into the common bile duct of rats, and after 4 h of duct injection the animals were sacrificed. Concentration-response curves to acetylcholine (ACh), sodium nitroprusside (SNP) and phenylephrine (PHE) in isolated mesenteric and pulmonary arteries were obtained. Potency (pEC₅₀) and maximal responses (E_MAX) were determined. Blood samples were collected for biochemical analysis.</p> <p>Results</p> <p>In mesenteric rings, the potency for ACh was significantly decreased from animals treated with TAU (about 4.2-fold) or PLA₂(about 6.9-fold) compared to saline group without changes in the maximal responses. Neither pEC₅₀nor E_MAXvalues for Ach were altered in pulmonary rings in any group. Similarly, the pEC₅₀and the E_MAXvalues for SNP were not changed in both preparations in any group. The potency for PHE was significantly decreased in rat mesenteric and pulmonary rings from TAU group compared to SAL group (about 2.2- and 2.69-fold, for mesenteric and pulmonary rings, respectively). No changes were seen in the E_MAXfor PHE. The nitrite/nitrate (NO_x^-) levels were markedly increased in animals submitted to acute pancreatitis as compared to SAL group, approximately 76 and 68% in TAU and PLA₂protocol, respectively.</p> <p>Conclusion</p> <p>Acute pancreatitis provoked deleterious effects in endothelium-dependent relaxing response for ACh in mesenteric rings that were strongly associated with high plasma NO_x^-levels as consequence of intense inflammatory responses. Furthermore, the subsensitivity of contractile response to PHE in both mesenteric and pulmonary rings might be due to the complications of this pathological condition in the early stage of pancreatitis.</p