Classical and free Fourth Moment Theorems: universality and thresholds

Let $X$ be a centered random variable with unit variance, zero third moment, and such that $E[X^4] \ge 3$. Let $\{F_n : n\geq 1\}$ denote a normalized sequence of homogeneous sums of fixed degree $d\geq 2$, built from independent copies of $X$. Under these minimal conditions, we prove that $F_n$ converges in distribution to a standard Gaussian random variable if and only if the corresponding sequence of fourth moments converges to $3$. The statement is then extended (mutatis mutandis) to the free probability setting. We shall also discuss the optimality of our conditions in terms of explicit thresholds, as well as establish several connections with the so-called universality phenomenon of probability theory. Both in the classical and free probability frameworks, our results extend and unify previous Fourth Moment Theorems for Gaussian and semicircular approximations. Our techniques are based on a fine combinatorial analysis of higher moments for homogeneous sums.Comment: 26 page

Multidimensional limit theorems for homogeneous sums: a general transfer principle

The aim of the present paper is to establish the multidimensional counterpart of the \textit{fourth moment criterion} for homogeneous sums in independent leptokurtic and mesokurtic random variables (that is, having positive and zero fourth cumulant, respectively), recently established in \cite{NPPS} in both the classical and in the free setting. As a consequence, the transfer principle for the Central limit Theorem between Wiener and Wigner chaos can be extended to a multidimensional transfer principle between vectors of homogeneous sums in independent commutative random variables with zero third moment and with non-negative fourth cumulant, and homogeneous sums in freely independent non-commutative random variables with non-negative fourth cumulant

On two-dimensional nonlocal Venttsel' problems in piecewise smooth domains

We establish the regularity results for solutions of nonlocal Venttsel' problems in polygonal and piecewise smooth two-dimensional domains

mTOR regulation of metabolism in hematologic malignancies

Neoplastic cells rewire their metabolism, acquiring a selective advantage over normal cells and a protection from therapeutic agents. The mammalian Target of Rapamycin (mTOR) is a serine/threonine kinase involved in a variety of cellular activities, including the control of metabolic processes. mTOR is hyperactivated in a large number of tumor types, and among them, in many hematologic malignancies. In this article, we summarized the evidence from the literature that describes a central role for mTOR in the acquisition of new metabolic phenotypes for different hematologic malignancies, in concert with other metabolic modulators (AMPK, HIF1α) and microenvironmental stimuli, and shows how these features can be targeted for therapeutic purposes

Remediation of Lead-Contaminated Water by Virgin Coniferous Wood Biochar Adsorbent. Batch and Column Application

In this paper, RE-CHAR® biochar, produced by a wood biomass pyrolysis process, which is usually applied as a soil fertilizer, was investigated for a novel use, that was as adsorbent for remediating a lead-contaminated solution. Firstly, a deep physical and chemical characterization of RE-CHAR® biochar was carried out. Then, the adsorption capacity of lead from 50 to 100 mg/L solutions was determined under batch and continuous flow conditions. Kinetics of the batch adsorption process were very rapid and complete removal was achieved within 4-h contact time at both Pb concentrations, using a biochar dosage of 5 g/L. These data were best fitted by the pseudo-second-order model, with the rate constant and the equilibrium capacity equal to ks = 0.0091 g/min and qe = 9.9957 mg/g at 50 mg/L Pb and ks = 0.0128 g/min and qe = 20.1462 mg/g at 100 mg/L Pb, respectively. The Langmuir isotherm model best fitted the equilibrium data at both Pb concentrations, with the Langmuir constant and maximum adsorption capacity equal to b = 11.5804 L/mg and qmax = 4.6116 mg/g at 50 mg/L Pb and b = 2.8933 L/mg and qmax = 9.5895 mg/g at 100 mg/L Pb. Continuous flow column tests showed that adding biochar to the soil of the adsorbent bed significantly extended the breakthrough and exhaustion times, with respect to the column filled with soil only. The Thomas model best fitted the experimental data of the breakthrough curves, with the constant kTH = 5.28 × 10−5 mL/min/mg and the maximum adsorption capacity q0 = 334.57 mg/g which was comparable to the values reported for commercial adsorbents. Based on these results, it can be assessed that RE-CHAR® biochar can be used as an effective adsorbent for lead removal from water solutions even at high concentrations

Could the super-pulsed CO2 laser be used for oral excisional biopsies?

Background. The main purpose of a biopsy is microscopic examination and diagnosis. Keeping the margins of specimens safe and readable is always fundamental to detecting marginal infiltrations or malignant transformation. Numerous options and tools have been introduced for biopsy procedures. Lasers are one of these options that provide many enhancements to clinical and surgical biopsy procedures in comparison to scalpels. Objectives. The aim of the present study is to quantify the thermal artefacts in histological specimens obtained using a CO2 laser from different oral mucosal lesions and to evaluate if the resulting thermal effect hinders the histological examination. This aim is accomplished through quantitatively and qualitatively assessing the thermal effect in both the epithelium and connective tissue. Material and methods. A super-pulsed CO2 laser (10,600 nm) was used to obtain 10 excision biopsy samples. The parameters were a power of 4.2 W in focused mode and a frequency of 80 Hz in super-pulse mode. The histological analysis was performed with an optical microscope. Computerized imaging software was utilized to quantitatively evaluate the thermal effect in both the epithelium and connective tissue expressed in microns. Results. The thermal effect of the CO2 laser was limited to the surgical resection margins in all the specimens and did not hinder the histological analysis. Thermal artefacts were observed in 3 specimens. The range of thermal effects in the epithelial tissue was between 184 μm and 2,292 μm, while in the connective tissue it was between 133 μm and 2,958 μm. Conclusions. The resulting thermal effects of using a CO2 laser did not hamper the histological evaluation. Utilizing a laser in biopsy procedures should be tailored. Not only should laser parameters and safety margins be taken into consideration but also the working time, clinical accessibility, and the nature and water content of the tissue