Hardy-Carleman Type Inequalities for Dirac Operators

General Hardy-Carleman type inequalities for Dirac operators are proved. New inequalities are derived involving particular traditionally used weight functions. In particular, a version of the Agmon inequality and Treve type inequalities are established. The case of a Dirac particle in a (potential) magnetic field is also considered. The methods used are direct and based on quadratic form techniques

Existence and equilibration of global weak solutions to Hookean-type bead-spring chain models for dilute polymers

We show the existence of global-in-time weak solutions to a general class of coupled Hookean-type bead-spring chain models that arise from the kinetic theory of dilute solutions of polymeric liquids with noninteracting polymer chains. The class of models involves the unsteady incompressible Navier-Stokes equations in a bounded domain in two or three space dimensions for the velocity and the pressure of the fluid, with an elastic extra-stress tensor appearing on the right-hand side in the momentum equation. The extra-stress tensor stems from the random movement of the polymer chains and is defined by the Kramers expression through the associated probability density function that satisfies a Fokker-Planck-type parabolic equation, a crucial feature of which is the presence of a center-of-mass diffusion term. We require no structural assumptions on the drag term in the Fokker-Planck equation; in particular, the drag term need not be corotational. With a square-integrable and divergence-free initial velocity datum for the Navier-Stokes equation and a nonnegative initial probability density function for the Fokker-Planck equation, which has finite relative entropy with respect to the Maxwellian of the model, we prove the existence of a global-in-time weak solution to the coupled Navier-Stokes-Fokker-Planck system. It is also shown that in the absence of a body force, the weak solution decays exponentially in time to the equilibrium solution, at a rate that is independent of the choice of the initial datum and of the centre-of-mass diffusion coefficient.Comment: 86 page

White matter hyperintensities are an independent predictor of cognitive decline 3 years following first-ever stroke-results from the PROSCIS-B study

Background: White matter hyperintensities (WMH) are the result of cerebral small vessel disease and may increase the risk of cognitive impairment (CI), recurrent stroke, and depression. We aimed to explore the association between selected cerebrovascular risk factors (CVRF) and WMH load as well as the effect of increased WMH burden on recurrent vascular events, CI, and depression in first-ever ischemic stroke patients.Methods: 431 from the PROSpective Cohort with Incident Stroke (PROSCIS) were included; Age-Related White Matter Changes (ARWMC) score was used to assess WMH burden on FLAIR. The presence of CVRF (defined via blood pressure, body-mass-index, and serological markers of kidney dysfunction, diabetes mellitus, and hyperlipoproteinemia) was categorized into normal, borderline, and pathological profiles based on commonly used clinical definitions. The primary outcomes included recurrent vascular events (combined endpoint of recurrent stroke, myocardial infarction and/or death), CI 3 years post-stroke, and depression 1-year post-stroke.Results: There was no clear association between CVRF profiles and WMH burden. High WMH lesion load (ARWMC score ≥ 10) was found to be associated with CI (adjusted OR 1.05 [95% CI 1.00-1.11]; p Conclusion: Higher WMH burden was associated with a significant decline in cognition 3 years post-stroke in this cohort of first-ever stroke patients

Sharp constants in weighted trace inequalities on Riemannian manifolds

We establish some sharp weighted trace inequalities W^{1,2}(\rho^{1-2\sigma}, M)\hookrightarrow L^{\frac{2n}{n-2\sigma}}(\pa M) on $n+1$ dimensional compact smooth manifolds with smooth boundaries, where $\rho$ is a defining function of $M$ and $\sigma\in (0,1)$. This is stimulated by some recent work on fractional (conformal) Laplacians and related problems in conformal geometry, and also motivated by a conjecture of Aubin.Comment: 34 page

Toeplitz operators with radial symbols on weighted holomorphic Orlicz space

Peer reviewe

Hardy's inequality for functions vanishing on a part of the boundary

We develop a geometric framework for Hardy's inequality on a bounded domain when the functions do vanish only on a closed portion of the boundary.Comment: 26 pages, 2 figures, includes several improvements in Sections 6-8 allowing to relax the assumptions in the main results. Final version published at http://link.springer.com/article/10.1007%2Fs11118-015-9463-

The stability for the Cauchy problem for elliptic equations

We discuss the ill-posed Cauchy problem for elliptic equations, which is pervasive in inverse boundary value problems modeled by elliptic equations. We provide essentially optimal stability results, in wide generality and under substantially minimal assumptions. As a general scheme in our arguments, we show that all such stability results can be derived by the use of a single building brick, the three-spheres inequality.Comment: 57 pages, review articl

Sinteza i antihipoksično djelovanje alifatskih i arilalifatskih amida kofein-8-tioglikolne kiseline

The synthesis of some aliphatic and arylaliphatic amides of caffeine-8-thioglycolic acid was studied. The structures of synthesized compounds were proved by microanalyses, IR- and 1H NMR data. Values of acute p.o. and i.p. toxicity in mice show lower toxicity compared to caffeine. Declines in spontaneous locomotor activity support the idea of depressive CNS activity of the compounds. Two compounds exhibited brain antihypoxic activity (5a and 5b against haemic and circulatory hypoxia, respectively).U radu je opisana sinteza alifatskih i arilalifatskih amida kofein-8-tioglikolne kiseline i njihova karakterizacija elementarnom analizom, IR- i 1H NMR spektroskopijom. Testiranja na miševima pokazuju da su sintetizirani spojevi primijenjeni p.o. i i.p. manje toksični od kofeina. Smanjenje lokomotoričke aktivnosti podupire ideju o njihovom depresivnom djelovanju na SŽS. Spojevi 5a i 5b djeluju antihipoksički u uvjetima krvne i cirkulacijske hipoksije u mozgu

Differentiation of cerebral neoplasms with vessel size imaging (VSI)

PURPOSE: Cerebral neoplasms of various histological origins may show comparable appearances on conventional Magnetic Resonance Imaging (MRI). Vessel size imaging (VSI) is an MRI technique that enables noninvasive assessment of microvasculature by providing quantitative estimates of microvessel size and density. In this study, we evaluated the potential of VSI to differentiate between brain tumor types based on their microvascular morphology. METHODS: Using a clinical 3T MRI scanner, VSI was performed on 25 patients with cerebral neoplasms, 10 with glioblastoma multiforme (GBM), 8 with primary CNS lymphoma (PCNSL) and 7 with cerebral lung cancer metastasis (MLC). Following the postprocessing of VSI maps, mean vessel diameter (vessel size index, vsi) and microvessel density (Q) were compared across tumors, peritumoral areas, and healthy tissues. RESULTS: The MLC tumors have larger and less dense microvasculature compared to PCNSLs in terms of vsi and Q (p = 0.0004 and p < 0.0001, respectively). GBM tumors have higher yet non-significantly different vsi values than PCNSLs (p = 0.065) and non-significant differences in Q. No statistically significant differences in vsi or Q were present between GBMs and MLCs. GBM tumor volume was positively correlated with vsi (r = 0.502, p = 0.0017) and negatively correlated with Q (r = -0.531, p = 0.0007). CONCLUSION: Conventional MRI parameters are helpful in differentiating between PCNSLs, GBMs, and MLCs. Additionally incorporating VSI parameters into the diagnostic protocol could help in further differentiating between PCNSLs and metastases and potentially between PCNSLs and GBMs. Future studies in larger patient cohorts are required to establish diagnostic cut-off values for VSI