On Unbounded Composition Operators in L2L^2-Spaces

Fundamental properties of unbounded composition operators in $L^2$-spaces are studied. Characterizations of normal and quasinormal composition operators are provided. Formally normal composition operators are shown to be normal. Composition operators generating Stieltjes moment sequences are completely characterized. The unbounded counterparts of the celebrated Lambert's characterizations of subnormality of bounded composition operators are shown to be false. Various illustrative examples are supplied

Relativity of quantum states and observables

Under the principle that quantum mechanical observables are invariant under relevant symmetry transformations, we explore how the usual, non-invariant quantities may capture measurement statistics. Using a relativisation mapping, viewed as the incorporation of a quantum reference frame, we show that the usual quantum description approximates the relative one precisely when the reference system admits an appropriate localisable quantity and a localised state. From this follows a new perspective on the nature and reality of quantum superpositions and optical coherence

Orthogonalities and functional equations

In this survey we show how various notions of orthogonality appear in the theory of functional equations. After introducing some orthogonality relations, we give examples of functional equations postulated for orthogonal vectors only. We show their solutions as well as some applications. Then we discuss the problem of stability of some of them considering various aspects of the problem. In the sequel, we mention the orthogonality equation and the problem of preserving orthogonality. Last, but not least, in addition to presenting results, we state some open problems concerning these topics. Taking into account the big amount of results concerning functional equations postulated for orthogonal vectors which have appeared in the literature during the last decades, we restrict ourselves to the most classical equations

Bioelectrical Impedance Phase Angle as a Prognostic Indicator of Survival in Head-and-Neck Cancer

Background: Phase angle could be an alternative to subjective global assessment for the assessment of nutrition status in patients with head-and-neck cancer. Methods: We prospectively evaluated a cohort of 75 stage IIIB and IV head-and-neck patients treated at the Otolaryngology Department, Head and Neck Surgery, Medical University of Lublin, Poland. Bioelectrical impedance analysis was performed in all patients using an analyzer that operated at 50 kHz. The phase angle was calculated as reactance divided by resistance (Xc/R) and expressed in degrees. The Kaplan–Meier method was used to calculate survival. Results: Median overall survival in the cohort was 32.0 months. At the time of analysis, 47 deaths had been recorded in the cohort (62.7%). The risk of shortened overall survival was significantly higher in patients whose phase angle was less than 4.733 degrees than in the remaining patients (19.6 months vs. 45 months, p = 0.0489; chi-square: 3.88; hazard ratio: 1.8856; 95% confidence interval: 1.0031 to 3.5446). Conclusions: Phase angle might be prognostic of survival in patients with advanced head-and-neck cancer. Further investigation in a larger population is required to confirm our results