Abstract:
Wegener granulomatosis (WG) is a type of vasculitis characterized by the presence of anti-neutrophil cy-toplasmic antibodies (ANCA) and inflammation of small and medium sized vessels with granulomas formation. Most commonly affected organs include upper and lower respiratory tract, kidneys, eyes, nervous system and skin. Kidneys’ involvement has a central position in the classification, diagnosis, treatment and prognosis of patients with WG, and is characterized by the presence of necrotic glomerulonephritis and clinical manifestations that vary from microscopic hematuria to acute renal insufficiency. We describe a case report of a ten year old boy presenting with microscopic hematuria of glomerular origin and a medical history of orbital pseudotumor two years before his hospitalization due to renal symptoms. Renal biopsy revealed lesions of pauci-immune glome-rulonephritis and findings of granulomatous inflam-mation and necrotizing vasculitis. Serum was positive for p-ANCA antibodies (perinuclear staining pattern ANCA antibodies). These findings led to the diagnosis of WG of generalized form (according to EULAR/ PRINTO/PRES criteria). The patient has been treated with aggressive immunotherapy with the use of ster-oids, cyclophosphamide and mycophenolate mofetil. Disease remission has been established and retained one year after initial diagnosis. Orbital pseudotumor, which is a diagnosis of exclusion, has been the initial disease’s clinical manifestation, even though at that time neither the ocular biopsy nor the immunologic workup had been indicative in terms of WG. Although WG is very rare in children, this disease should always been included in the differential diagnosis in patients with similar clinical manifestations and clinicians should emphasize on the recognition of granulomatous vasculitis in biopsies as well as on repeated tests for ANCA antibodies’ detection in serum. High morbidity and mortality rates [1] of this clinical entity necessitates the early recognition of atypical disease’s forms and the close follow up in cases of uncertain initial diagnosis.

Abstract:
A class of examples concerning the relationship of linear regression and maximal correlation is provided. More precisely, these examples show that if two random variables have (strictly) linear regression on each other, then their maximal correlation is not necessarily equal to their (absolute) correlation.

Abstract:
Consider an i.i.d. sample X^*_1,X^*_2,...,X^*_n from a location-scale family, and assume that the only available observations consist of the partial maxima (or minima)sequence, X^*_{1:1},X^*_{2:2},...,X^*_{n:n}, where X^*_{j:j}=max{X^*_1,...,X^*_j}. This kind of truncation appears in several circumstances, including best performances in athletics events. In the case of partial maxima, the form of the BLUEs (best linear unbiased estimators) is quite similar to the form of the well-known Lloyd's (1952, Least-squares estimation of location and scale parameters using order statistics, Biometrika, vol. 39, pp. 88-95) BLUEs, based on (the sufficient sample of) order statistics, but, in contrast to the classical case, their consistency is no longer obvious. The present paper is mainly concerned with the scale parameter, showing that the variance of the partial maxima BLUE is at most of order O(1/log n), for a wide class of distributions.

Abstract:
In this article we derive the best possible upper bound for $E[\max{X_i}-\min_i{X_i}]$ under given means and variances on $n$ random variables $X_i$. The random vector $(X_1,...,X_n)$ is allowed to have any dependence structure, provided $E X_i=\mu_i$ and $Var X_i=\sigma_i^2$, $0<\sigma_i<\infty$. We provide an explicit characterization of the $n$-variate distributions that attain the equality (extremal random vectors), and the tight bound is compared to other existing results. Key words and phrases: Range; Dependent Observations; Tight Expectation Bounds; Extremal Random Vectors; Probability Matrices; Characterizations.

Abstract:
A continuum theory is used to predict scaling laws for the morphological relaxation of crystal surfaces in two independent space dimensions. The goal is to unify previously disconnected experimental observations of decaying surface profiles. The continuum description is derived from the motion of interacting atomic steps. For isotropic diffusion of adatoms across each terrace, induced adatom fluxes transverse and parallel to step edges obey different laws, yielding a tensor mobility for the continuum surface flux. The partial differential equation (PDE) for the height profile expresses an interplay of step energetics and kinetics, and aspect ratio of surface topography that plausibly unifies observations of decaying bidirectional surface corrugations. The PDE reduces to known evolution equations for axisymmetric mounds and one-dimensional periodic corrugations.

Abstract:
In Bose-Einstein condensation, a macroscopically large number of particles occupy the same single-particle quantum state. Our goal is to study time-dependent aspects of particle excitations to states other than the single-particle macroscopic state in trapped dilute atomic gases. We adopt the view that atoms are excited in pairs so that their scattering from the single-particle state to vector positions x and y at time t is described by the pair-excitation function, K0(x,y,t). We solve a nonlocal equation for K0 under a slowly varying external potential by assuming that the wave function of the macroscopic state satisfies a time-independent nonlinear Schroedinger equation. For zero initial excitation (K0=0 at t=0) and sufficiently large t, we evaluate asymptotically K0 in terms of the one-variable Lommel function for any distance |x-y|.

Abstract:
At finite temperatures below the phase transition point, the Bose-Einstein condensation, the macroscopic occupation of a single quantum state by particles of integer spin, is not complete. In the language of superfluid helium, this means that the superfluid coexists with the normal fluid. Our goal is to describe this coexistence in trapped, dilute atomic gases with repulsive interactions via mean field laws that account for a {\em spatially varying} particle interaction strength. By starting with the $N$-body Hamiltonian, $N\gg 1$, we formally derive a system of coupled, nonlinear evolution equations in $3+1$ dimensions for the following quantities: (i) the wave function of the macroscopically occupied state; and (ii) the single-particle wave functions of thermally excited states. For stationary (bound) states and a scattering length with {\em periodic microstructure} of subscale $\epsilon$, we heuristically extract effective equations of motion via periodic homogenization up to second order in $\epsilon$.

Abstract:
We identify the complex plane C with the open unit disc D={z:|z|<1} by the homeomorphism z --> z/(1+|z|). This leads to a compactification $\bar{C}$ of C, homeomorphic to the closed unit disc. The Euclidean metric on the closed unit disc induces a metric d on $\bar{C}$. We identify all uniform limits of polynomials on $\bar{D}$ with respect to the metric d. The class of the above limits is an extension of the disc algebra and it is denoted by $\bar{A}(D)$. We study properties of the elements of $\bar{A}(D)$ and topological properties of the class $\bar{A}(D)$ endowed with its natural topology. The class $\bar{A}(D)$ is different and, from the geometric point of view, richer than the class $\tilde{A}(D)$ introduced in Nestoridis (2010), Arxiv:1009.5364, on the basis of the chordal metric.

Abstract:
In this note we introduce the notion of factorial moment distance for non-negative integer-valued random variables and we compare it with the total variation distance. Furthermore, we study the rate of convergence in the classical matching problem and in a generalized matching distribution.

Abstract:
A random variable Z will be called self-inverse if it has the same distribution as its reciprocal 1/Z. It is shown that if Z is defined as a ratio, X/Y, of two rv's X and Y (with Pr[X=0]=Pr[Y=0]=0), then Z is self-inverse if and only if X and Y are (or can be chosen to be) exchangeable. In general, however, there may not exist iid X and Y in the ratio representation of Z.