On the α-q-Mutual Information and the α-q-Capacities

The measures of information transfer which correspond to non-additive entropies have intensively been studied in previous decades. The majority of the work includes the ones belonging to the Sharma–Mittal entropy class, such as the Rényi, the Tsallis, the Landsberg–Vedral and the Gaussian entropies. All of the considerations follow the same approach, mimicking some of the various and mutually equivalent definitions of Shannon information measures, and the information transfer is quantified by an appropriately defined measure of mutual information, while the maximal information transfer is considered as a generalized channel capacity. However, all of the previous approaches fail to satisfy at least one of the ineluctable properties which a measure of (maximal) information transfer should satisfy, leading to counterintuitive conclusions and predicting nonphysical behavior even in the case of very simple communication channels. This paper fills the gap by proposing two parameter measures named the α-q-mutual information and the α-q-capacity. In addition to standard Shannon approaches, special cases of these measures include the α-mutual information and the α-capacity, which are well established in the information theory literature as measures of additive Rényi information transfer, while the cases of the Tsallis, the Landsberg–Vedral and the Gaussian entropies can also be accessed by special choices of the parameters α and q. It is shown that, unlike the previous definition, the α-q-mutual information and the α-q-capacity satisfy the set of properties, which are stated as axioms, by which they reduce to zero in the case of totally destructive channels and to the (maximal) input Sharma–Mittal entropy in the case of perfect transmission, which is consistent with the maximum likelihood detection error. In addition, they are non-negative and less than or equal to the input and the output Sharma–Mittal entropies, in general. Thus, unlike the previous approaches, the proposed (maximal) information transfer measures do not manifest nonphysical behaviors such as sub-capacitance or super-capacitance, which could qualify them as appropriate measures of the Sharma–Mittal information transfer.     

A Novel Algebraic Structure of (α,β)-Complex Fuzzy Subgroups

A complex fuzzy set is a vigorous framework to characterize novel machine learning algorithms. This set is more suitable and flexible compared to fuzzy sets, intuitionistic fuzzy sets, and bipolar fuzzy sets. On the aspects of complex fuzzy sets, we initiate the abstraction of (α,β)-complex fuzzy sets and then define -complex fuzzy subgroups. Furthermore, we prove that every complex fuzzy subgroup is an (α,β)-complex fuzzy subgroup and define (α,β)-complex fuzzy normal subgroups of given group. We extend this ideology to define (α,β)-complex fuzzy cosets and analyze some of their algebraic characteristics. Furthermore, we prove that (α,β)-complex fuzzy normal subgroup is constant in the conjugate classes of group. We present an alternative conceptualization of (α,β)-complex fuzzy normal subgroup in the sense of the commutator of groups. We establish the (α,β)-complex fuzzy subgroup of the classical quotient group and show that the set of all (α,β)-complex fuzzy cosets of this specific complex fuzzy normal subgroup form a group. Additionally, we expound the index of -complex fuzzy subgroups and investigate the (α,β)-complex fuzzification of Lagrange’s theorem analog to Lagrange’ theorem of classical group theory.     

Error Exponents and α-Mutual Information

Over the last six decades, the representation of error exponent functions for data transmission through noisy channels at rates below capacity has seen three distinct approaches: (1) Through Gallager’s E0 functions (with and without cost constraints); (2) large deviations form, in terms of conditional relative entropy and mutual information; (3) through the α-mutual information and the Augustin–Csiszár mutual information of order α derived from the Rényi divergence. While a fairly complete picture has emerged in the absence of cost constraints, there have remained gaps in the interrelationships between the three approaches in the general case of cost-constrained encoding. Furthermore, no systematic approach has been proposed to solve the attendant optimization problems by exploiting the specific structure of the information functions. This paper closes those gaps and proposes a simple method to maximize Augustin–Csiszár mutual information of order α under cost constraints by means of the maximization of the α-mutual information subject to an exponential average constraint.     

Robust Aggregation for Federated Learning by Minimum γ-Divergence Estimation

Federated learning is a framework for multiple devices or institutions, called local clients, to collaboratively train a global model without sharing their data. For federated learning with a central server, an aggregation algorithm integrates model information sent from local clients to update the parameters for a global model. Sample mean is the simplest and most commonly used aggregation method. However, it is not robust for data with outliers or under the Byzantine problem, where Byzantine clients send malicious messages to interfere with the learning process. Some robust aggregation methods were introduced in literature including marginal median, geometric median and trimmed-mean. In this article, we propose an alternative robust aggregation method, named γ-mean, which is the minimum divergence estimation based on a robust density power divergence. This γ-mean aggregation mitigates the influence of Byzantine clients by assigning fewer weights. This weighting scheme is data-driven and controlled by the γ value. Robustness from the viewpoint of the influence function is discussed and some numerical results are presented.     

Adaptive Diagnosis for Fault Tolerant Data Fusion Based on α-Rényi Divergence Strategy for Vehicle Localization

When applying a diagnostic technique to complex systems, whose dynamics, constraints, and environment evolve over time, being able to re-evaluate the residuals that are capable of detecting defaults and proposing the most appropriate ones can quickly prove to make sense. For this purpose, the concept of adaptive diagnosis is introduced. In this work, the contributions of information theory are investigated in order to propose a Fault-Tolerant multi-sensor data fusion framework. This work is part of studies proposing an architecture combining a stochastic filter for state estimation with a diagnostic layer with the aim of proposing a safe and accurate state estimation from potentially inconsistent or erroneous sensors measurements. From the design of the residuals, using α-Rényi Divergence (α-RD), to the optimization of the decision threshold, through the establishment of a function that is dedicated to the choice of α at each moment, we detail each step of the proposed automated decision-support framework. We also dwell on: (1) the consequences of the degree of freedom provided by this α parameter and on (2) the application-dictated policy to design the α tuning function playing on the overall performance of the system (detection rate, false alarms, and missed detection rates). Finally, we present a real application case on which this framework has been tested. The problem of multi-sensor localization, integrating sensors whose operating range is variable according to the environment crossed, is a case study to illustrate the contributions of such an approach and show the performance.     

ϕ-Informational Measures: Some Results and Interrelations

In this paper, we focus on extended informational measures based on a convex function ϕ: entropies, extended Fisher information, and generalized moments. Both the generalization of the Fisher information and the moments rely on the definition of an escort distribution linked to the (entropic) functional ϕ. We revisit the usual maximum entropy principle—more precisely its inverse problem, starting from the distribution and constraints, which leads to the introduction of state-dependent ϕ-entropies. Then, we examine interrelations between the extended informational measures and generalize relationships such the Cramér–Rao inequality and the de Bruijn identity in this broader context. In this particular framework, the maximum entropy distributions play a central role. Of course, all the results derived in the paper include the usual ones as special cases.     

λ-Deformation: A Canonical Framework for Statistical Manifolds of Constant Curvature

This paper systematically presents the λ-deformation as the canonical framework of deformation to the dually flat (Hessian) geometry, which has been well established in information geometry. We show that, based on deforming the Legendre duality, all objects in the Hessian case have their correspondence in the λ-deformed case: λ-convexity, λ-conjugation, λ-biorthogonality, λ-logarithmic divergence, λ-exponential and λ-mixture families, etc. In particular, λ-deformation unifies Tsallis and Rényi deformations by relating them to two manifestations of an identical λ-exponential family, under subtractive or divisive probability normalization, respectively. Unlike the different Hessian geometries of the exponential and mixture families, the λ-exponential family, in turn, coincides with the λ-mixture family after a change of random variables. The resulting statistical manifolds, while still carrying a dualistic structure, replace the Hessian metric and a pair of dually flat conjugate affine connections with a conformal Hessian metric and a pair of projectively flat connections carrying constant (nonzero) curvature. Thus, λ-deformation is a canonical framework in generalizing the well-known dually flat Hessian structure of information geometry.     

α-Geodesical Skew Divergence

The asymmetric skew divergence smooths one of the distributions by mixing it, to a degree determined by the parameter λ, with the other distribution. Such divergence is an approximation of the KL divergence that does not require the target distribution to be absolutely continuous with respect to the source distribution. In this paper, an information geometric generalization of the skew divergence called the α-geodesical skew divergence is proposed, and its properties are studied.     

A Statistical Journey through the Topological Determinants of the β2 Adrenergic Receptor Dynamics

Activation of G-protein-coupled receptors (GPCRs) is mediated by molecular switches throughout the transmembrane region of the receptor. In this work, we continued along the path of a previous computational study wherein energy transport in the β2 Adrenergic Receptor (β2-AR) was examined and allosteric switches were identified in the molecular structure through the reorganization of energy transport networks during activation. In this work, we further investigated the allosteric properties of β2-AR, using Protein Contact Networks (PCNs). In this paper, we report an extensive statistical analysis of the topological and structural properties of β2-AR along its molecular dynamics trajectory to identify the activation pattern of this molecular system. The results show a distinct character to the activation that both helps to understand the allosteric switching previously identified and confirms the relevance of the network formalism to uncover relevant functional features of protein molecules.     

New Parameterized Inequalities for η-Quasiconvex Functions via (p,q)-Calculus

In this work, first, we consider novel parameterized identities for the left and right part of the (p,q)-analogue of Hermite–Hadamard inequality. Second, using these new parameterized identities, we give new parameterized (p,q)-trapezoid and parameterized (p,q)-midpoint type integral inequalities via η-quasiconvex function. By changing values of parameter μ∈[0,1], some new special cases from the main results are obtained and some known results are recaptured as well. Finally, at the end, an application to special means is given as well. This new research has the potential to establish new boundaries in comparative literature and some well-known implications. From an application perspective, the proposed research on the η-quasiconvex function has interesting results that illustrate the applicability and superiority of the results obtained.     

Photoluminescence in γ-irradiated α-quartz investigated by synchrotron radiation

We report an experimental investigation of the photoluminescence, under excitation by synchrotron radiation within the absorption band at 7.6 eV, induced in γ-irradiated α-quartz. Two emissions centered at 4.9 and 2.7 eV are observed at low temperature: the former decreases above 40 K, whereas the second band exhibits an initial slight increase and its quenching is effective above 100 K. Furthermore, the decay kinetics of both emissions occur in a time scale of nanoseconds: at T=17.5 K we measured a lifetime τ1.0 ns for the photoluminescence at 4.9 eV and τ3.6 ns for that at 2.7 eV. These results give new insight on the optical properties associated with defects peculiar of crystalline matrix, also on the basis of their comparison with previous studies on silica.     

A comparative study on the gas‐phase and liquid‐phase thermal isomerization reaction of α‐pinene

In this paper, a method of preparation of ocimene is investigated, which is obtained from isomerization reaction of α‐pinene. Two kinds of experimental apparatus are established for the investigation of the thermal isomerization reaction of α‐pinene. The behavior of thermal isomerization reaction of α‐pinene is respectively discussed in the gas phase and in the liquid phase. Under gas phase conditions, the conversion of α‐pinene is 80% and the selectivity of ocimene is 30%–33%. Under liquid phase conditions, the conversion of α‐pinene is 60% and the selectivity of ocimene is 50%–54%. According to the kinetic‐molecular theory of ideal gases, two kinds of reaction models are proposed to visualize the reaction process. In addition, the mechanism and kinetics of thermal isomerization reaction of α‐pinene are respectively discussed. The conclusion is that the gas phase reaction temperature is calculated to be 390–450 °C and the liquid phase reaction temperature is calculated to be 450–550 °C. From a bond dissociation energy point of view, results support the hypothesis that the reaction involves biradical intermediates. Copyright © 2012 John Wiley & Sons, Ltd.     

Retrospective Change-Points Detection for Multidimensional Time Series of Arbitrary Nature: Model-Free Technology Based on the ϵ-Complexity Theory

We consider a retrospective change-point detection problem for multidimensional time series of arbitrary nature (in particular, panel data). Change-points are the moments at which the changes in generating mechanism occur. Our method is based on the new theory of ϵ-complexity of individual continuous vector functions and is model-free. We present simulation results confirming the effectiveness of the method.     

Kernel Estimation of Cumulative Residual Tsallis Entropy and Its Dynamic Version under ρ-Mixing Dependent Data

Tsallis introduced a non-logarithmic generalization of Shannon entropy, namely Tsallis entropy, which is non-extensive. Sati and Gupta proposed cumulative residual information based on this non-extensive entropy measure, namely cumulative residual Tsallis entropy (CRTE), and its dynamic version, namely dynamic cumulative residual Tsallis entropy (DCRTE). In the present paper, we propose non-parametric kernel type estimators for CRTE and DCRTE where the considered observations exhibit an ρ-mixing dependence condition. Asymptotic properties of the estimators were established under suitable regularity conditions. A numerical evaluation of the proposed estimator is exhibited and a Monte Carlo simulation study was carried out.     

Relative stabilities and molecular structures of the isomeric enol ethers and carboxylic esters derived from α‐acetyl‐γ‐butyrolactone and α‐acetyl‐δ‐valerolactone

Recently recorded ¹⁷O NMR spectra of compounds studied in a previous work (Taskinen E. Acta Chem. Scand. 1985; B39 : 489–494) dealing with the thermodynamics of isomerization of the enol ethers of α‐acetyl‐γ‐butyrolactone reveal an error in compound identification, caused by an unexpected isomerization reaction during the synthetic procedure. Thus, acid‐catalyzed treatment of the lactone with HC(OR)₃ in the respective alcohol ROH is shown to lead initially to the desired enol ethers which, however, are gradually isomerized to a mixture of the enol ethers and an ester of 2‐methyl‐4,5‐dihydrofuran‐3‐carboxylic acid. As a result, only one of the two isomeric compounds detected in the previous equilibration study was the expected enol ether (the thermodynamically more stable E isomer) of α‐acetyl‐γ‐butyrolactone, while the other, dominating species was the respective carboxylic ester. In the present work, the evidence provided by the ¹⁷O NMR spectra is presented, and the relative stabilities of the isomeric compounds are discussed on the basis of computational enthalpy data. The treatment is also extended to the respective isomeric compounds derived from α‐acetyl‐δ‐valerolactone. Copyright © 2007 John Wiley & Sons, Ltd.     

Determination of microquantities of Be in samples using α-sensitive track etch detectors

A new technique of determining Be content in samples using 4.66 MeV γ-rays is proposed. This energy range is sufficient to knock out the neutron from the ⁹Be nucleaus and to excite the first 2.9 MeV level of the ⁸Be nuclei. 1.5 MeV alpha particles are registered by means of CR-39 or cellulose nitrate detectors. Alpha particles produced in the reaction of γ-rays with Li nuclei which are contained in samples may cause the background. The latter is excluded by heating of samples before irradiation for Li molecules to be evaporated. ¹¹B nuclei do not produce alpha particles at the energy range of γ-rays presented above.The results obtained in the experiments of determining Be content in natural beryl crystal samples are in good agreement with their known chemical composition. Microparticles containing Beryllium were found in soil samples from the Ust-Kamenogorsk are deposit region, which contains 10⁻³ − 10⁻⁴ g/g of Be. The sensitivity of the method is 10⁻⁶ g/g of Be in solids.     

β‐Elimination of an aziridine to an allylic amine: a mechanistic study

The base‐induced rearrangement of aziridines has been examined using a combination of calculations and experiment. The calculations show that the substituent on nitrogen is a critical feature that greatly affects the favorability of both α‐deprotonation, and β‐elimination to form an allylic amine. Experiments were carried out to determine whether E2‐like rearrangement to the allylic amine with lithium diisopropyl amide (LDA) is possible. N‐tosyl aziridines were found to deprotonate on the tosyl group, preventing further reaction. A variety of N‐benzenesulfonyl aziridines having both α‐ and β‐protons decomposed when treated with LDA in either tetrahydrofuran or hexamethylphosphoramide. However, when α‐protons were not present, allylic amine was formed, presumably via β‐elimination. Copyright © 2011 John Wiley & Sons, Ltd.     

Stability of Non-Linear Dirichlet Problems with ϕ-Laplacian

We study the stability and the solvability of a family of problems −(ϕ(x′))′=g(t,x,x′,u)+f* with Dirichlet boundary conditions, where ϕ, u, f* are allowed to vary as well. Applications for boundary value problems involving the p-Laplacian operator are highlighted.     

Computational study of the proton affinity and basicity of structurally diverse α1‐adrenoceptor ligands

The α₁‐adrenoceptor is a target for the treatment of several conditions from hypertension to benign prostatic hyperplasia. In this paper, we describe a new analysis approach to explore the conformational space of several ligands of the α₁‐adrenoceptor and we also present the calculation of their proton affinity and basicity. For each compound a conformational search followed by a semi‐empirical optimisation was performed and a selection of conformations for each ligand was subjected to further optimisation using density functional theory methods. Different positions were explored to determine the favoured site of protonation, and then, the proton affinity (in the gas phase) and basicity (using the polarisable continuum model for the aqueous solution) were calculated for each of them. In addition, an alternative method using one explicit water molecule in combination with the polarisable continuum model for aqueous solvent was explored. Moreover, the acid dissociation constant (pK_a) in water of these 26 compounds was calculated because this is an important parameter for a ligand when binding to its receptor. The experimental pK_a values of six of these ligands and those of two compounds with a very low and a very large pK_a were used to validate the theoretical methodology. Copyright © 2011 John Wiley & Sons, Ltd.     

Molecular structure,experimental and theoretical 1H and 13C NMR chemical shift assignment of cyclic and acyclic α,β‐unsaturated esters

The experimental ¹H and ¹³C NMR spectra of 13 phenyl cinnamates and four 4‐methylcoumarins were investigated and their chemical shifts assigned on the basis of the two‐dimensional spectra. For the unsubstituted cinnamic acid phenyl ester, optimized molecular structures were calculated at a B3LYP/6‐311++G(d,p) level of theory. ¹H and ¹³C NMR chemical shifts were also calculated with the GIAO method at the B3LYP/6‐311 + G(2d,p) level of theory. The comparison between experimental and calculated NMR chemical shift suggests that the experimental spectra are formed from the superposition spectra of the two lowest energy conformers of the compound in solution. The most stable s‐cis configuration found in our studies is also the conformation adopted for a related phenyl cinnamate in solid state. The experimental results were analyzed in terms of the substituent effects. Copyright © 2013 John Wiley & Sons, Ltd.