Stability of Tsallis antropy and instabilities of Renyi and normalized Tsallis entropies: A basis for q-exponential distributions

The q-exponential distributions, which are generalizations of the Zipf-Mandelbrot power-law distribution, are frequently encountered in complex systems at their stationary states. From the viewpoint of the principle of maximum entropy, they can apparently be derived from three different generalized entropies: the Renyi entropy, the Tsallis entropy, and the normalized Tsallis entropy. Accordingly, mere fittings of observed data by the q-exponential distributions do not lead to identification of the correct physical entropy. Here, stabilities of these entropies, i.e., their behaviors under arbitrary small deformation of a distribution, are examined. It is shown that, among the three, the Tsallis entropy is stable and can provide an entropic basis for the q-exponential distributions, whereas the others are unstable and cannot represent any experimentally observable quantities.Comment: 20 pages, no figures, the disappeared "primes" on the distributions are added. Also, Eq. (65) is correcte

A nonextensive approach to the dynamics of financial observables

We present results about financial market observables, specifically returns and traded volumes. They are obtained within the current nonextensive statistical mechanical framework based on the entropy $S_{q}=k\frac{1-\sum\limits_{i=1}^{W} p_{i} ^{q}}{1-q} (q\in \Re)$ ($S_{1} \equiv S_{BG}=-k\sum\limits_{i=1}^{W}p_{i} \ln p_{i}$). More precisely, we present stochastic dynamical mechanisms which mimic probability density functions empirically observed. These mechanisms provide possible interpretations for the emergence of the entropic indices $q$ in the time evolution of the corresponding observables. In addition to this, through multi-fractal analysis of return time series, we verify that the dual relation $q_{stat}+q_{sens}=2$ is numerically satisfied, $q_{stat}$ and $q_{sens}$ being associated to the probability density function and to the sensitivity to initial conditions respectively. This type of simple relation, whose understanding remains ellusive, has been empirically verified in various other systems.Comment: Invited paper to appear in special issue of Eur. Phys. J. B dedicated to econophysics, edited by T. Di Matteo and T. Aste. 7 page

Continuity and Stability of Partial Entropic Sums

Extensions of Fannes' inequality with partial sums of the Tsallis entropy are obtained for both the classical and quantum cases. The definition of kth partial sum under the prescribed order of terms is given. Basic properties of introduced entropic measures and some applications are discussed. The derived estimates provide a complete characterization of the continuity and stability properties in the refined scale. The results are also reformulated in terms of Uhlmann's partial fidelities.Comment: 9 pages, no figures. Some explanatory and technical improvements are made. The bibliography is extended. Detected errors and typos are correcte

Validity of the second law in nonextensive quantum thermodynamics

The second law of thermodynamics in nonextensive statistical mechanics is discussed in the quantum regime. Making use of the convexity property of the generalized relative entropy associated with the Tsallis entropy indexed by q, Clausius' inequality is shown to hold in the range of q between zero and two. This restriction on the range of the entropic index, q, is purely quantum mechanical and there exists no upper bound of q for validity of the second law in classical theory.Comment: 12 pages, no figure

Generalized entropy optimized by an arbitrary distribution

We construct the generalized entropy optimized by a given arbitrary statistical distribution with a finite linear expectation value of a random quantity of interest. This offers, via the maximum entropy principle, a unified basis for a great variety of distributions observed in nature, which can hardly be described by the conventional methods. As a simple example, we explicitly derive the entropy associated with the stretched exponential distribution. To include the distributions with the divergent moments (e.g., the Levy stable distributions), it is necessary to modify the definition of the expectation value.Comment: 10 pages, no figure

The prognostic significance of immune microenvironment in breast ductal carcinoma in situ

BackgroundThe role of different subtypes of tumour infiltrating lymphocytes (TILs) in breast ductal carcinoma in situ (DCIS) is still poorly defined. This study aimed to assess the prognostic significance of B and T lymphocytes and immune checkpoint proteins expression in DCIS.MethodsA well characterised DCIS cohort (n = 700) with long-term follow-up comprising pure DCIS (n = 508) and DCIS mixed with invasive carcinoma (IBC; n = 192) were stained immunohistochemically for CD20, CD3, CD4, CD8, FOXP3, PD1 and PDL1. Copy number variation and TP53 mutation status were assessed in a subset of cases (n = 58).ResultsCD3+ lymphocytes were the predominant cell subtype in the pure DCIS cohort, while FOXP3 showed the lowest levels. PDL1 expression was mainly seen in the stromal TILs. Higher abundance of TILs subtypes was associated with higher tumour grade, hormone receptor negativity and HER2 positivity. Mutant TP53 variants were associated with higher levels of stromal CD3+, CD4+ and FOXP3+ cells. DCIS coexisting with invasive carcinoma harboured denser stromal infiltrates of all immune cells and checkpoint proteins apart from CD4+ cells. Stromal PD1 was the most differentially expressed protein between DCIS and invasive carcinoma (Z = 5.8, p

On observability of Renyi's entropy

Despite recent claims we argue that Renyi's entropy is an observable quantity. It is shown that, contrary to popular belief, the reported domain of instability for Renyi entropies has zero measure (Bhattacharyya measure). In addition, we show the instabilities can be easily emended by introducing a coarse graining into an actual measurement. We also clear up doubts regarding the observability of Renyi's entropy in (multi--)fractal systems and in systems with absolutely continuous PDF's.Comment: 18 pages, 1 EPS figure, REVTeX, minor changes, accepted to Phys. Rev.

Relations for certain symmetric norms and anti-norms before and after partial trace

Changes of some unitarily invariant norms and anti-norms under the operation of partial trace are examined. The norms considered form a two-parametric family, including both the Ky Fan and Schatten norms as particular cases. The obtained results concern operators acting on the tensor product of two finite-dimensional Hilbert spaces. For any such operator, we obtain upper bounds on norms of its partial trace in terms of the corresponding dimensionality and norms of this operator. Similar inequalities, but in the opposite direction, are obtained for certain anti-norms of positive matrices. Through the Stinespring representation, the results are put in the context of trace-preserving completely positive maps. We also derive inequalities between the unified entropies of a composite quantum system and one of its subsystems, where traced-out dimensionality is involved as well.Comment: 11 pages, no figures. A typo error in Eq. (5.15) is corrected. Minor improvements. J. Stat. Phys. (in press

A Transplantable Phosphorylation Probe for Direct Assessment of G Protein-Coupled Receptor Activation

The newly developed multireceptor somatostatin analogs pasireotide (SOM230), octreotide and somatoprim (DG3173) have primarily been characterized according to their binding profiles. However, their ability to activate individual somatostatin receptor subtypes (sst) has not been directly assessed so far. Here, we transplanted the carboxyl-terminal phosphorylation motif of the sst2 receptor to other somatostatin receptors and assessed receptor activation using a set of three phosphosite-specific antibodies. Our comparative analysis revealed unexpected efficacy profiles for pasireotide, octreotide and somatoprim. Pasireotide was able to activate sst3 and sst5 receptors but was only a partial agonist at the sst2 receptor. Octreotide exhibited potent agonistic properties at the sst2 receptor but produced very little sst5 receptor activation. Like octreotide, somatoprim was a full agonist at the sst2 receptor. Unlike octreotide, somatoprim was also a potent agonist at the sst5 receptor. Together, we propose the application of a phosphorylation probe for direct assessment of G protein-coupled receptor activation and demonstrate its utility in the pharmacological characterization of novel somatostatin analogs

Phosphatidylinositol 3-Kinase Mediates Bronchioalveolar Stem Cell Expansion in Mouse Models of Oncogenic K-ras-Induced Lung Cancer

Background: Non-small cell lung cancer (NSCLC) is the most common cause of cancer-related death in Western countries. Developing more effective NSCLC therapeutics will require the elucidation of the genetic and biochemical bases for this disease. Bronchioalveolar stem cells (BASCs) are a putative cancer stem cell population in mouse models of oncogenic K-ras-induced lung adenocarcinoma, an histologic subtype of NSCLC. The signals activated by oncogenic K-ras that mediate BASC expansion have not been fully defined. Methodology/Principal Findings: We used genetic and pharmacologic approaches to modulate the activity of phosphatidylinositol 3-kinase (PI3K), a key mediator of oncogenic K-ras, in two genetic mouse models of lung adenocarcinoma. Oncogenic K-ras-induced BASC accumulation and tumor growth were blocked by treatment with a small molecule PI3K inhibitor and enhanced by inactivation of phosphatase and tensin homologue deleted from chromosome 10, a negative regulator of PI3K. Conclusions/Significance: We conclude that PI3K is a critical regulator of BASC expansion, supporting treatment strategies to target PI3K in NSCLC patients