Search Results (11)

This study presents, for the first time, a new class of interval-valued superquadratic stochastic processes and examines their core properties through the lens of the center-radius total order relation on intervals. These processes serve as a powerful tool for modeling uncertainty in stochastic systems involving interval-valued data. By utilizing their intrinsic structure, we derive sharpened versions of Jensen-type and Hermite–Hadamard-type inequalities, along with their fractional extensions, within the framework of mean-square stochastic Riemann–Liouville fractional integrals. The theoretical findings are validated through extensive graphical representations and numerical simulations. Moreover, the applicability of the proposed processes is demonstrated in the domain of information theory by constructing novel stochastic divergence measures and Shannon’s entropy grounded in interval calculus. The outcomes of this work lay a solid foundation for further exploration in stochastic analysis, particularly in advancing generalized integral inequalities and formulating new stochastic models under uncertainty. Full article

In this paper, we use the generalized version of convex functions, known as strongly convex functions, to derive improvements to the Jensen–Mercer inequality. We achieve these improvements through the newly discovered characterizations of strongly convex functions, along with some previously known results about strongly convex functions. We are also focused on important applications of the derived results in information theory, deducing estimates for $\chi$-divergence, Kullback–Leibler divergence, Hellinger distance, Bhattacharya distance, Jeffreys distance, and Jensen–Shannon divergence. Additionally, we prove some applications to Mercer-type power means at the end. Full article

In this paper, we study a three-layer wiretap network including the source node in the top layer, N nodes in the middle layer and L sink nodes in the bottom layer. Each sink node recovers the message generated from the source node correctly via the middle layer nodes that it has access to. Furthermore, it is required that an eavesdropper eavesdropping a subset of the channels between the top layer and the middle layer learns absolutely nothing about the message. For each pair of decoding and eavesdropping patterns, we are interested in finding the capacity region consisting of $(N+1)$-tuples, with the first element being the size of the message successfully transmitted and the remaining elements being the capacity of the N channels from the source node to the middle layer nodes. This problem can be seen as a generalization of the secret sharing problem. We show that when the number of middle layer nodes is no larger than four, the capacity region is fully characterized as a polyhedral cone. When such a number is 5, we find the capacity regions for 74,222 decoding and eavesdropping patterns. For the remaining 274 cases, linear capacity regions are found. The proving steps are: (1) Characterizing the Shannon region, an outer bound of the capacity region; (2) Characterizing the common information region, an outer bound of the linear capacity region; (3) Finding linear schemes that achieve the Shannon region or the common information region. Full article

In many areas of computer science, it is of primary importance to assess the randomness of a certain variable X. Many different criteria can be used to evaluate randomness, possibly after observing some disclosed data. A “sufficiently random” X is often described as “entropic”. Indeed, Shannon’s entropy is known to provide a resistance criterion against modeling attacks. More generally one may consider the Rényi $\alpha$-entropy where Shannon’s entropy, collision entropy and min-entropy are recovered as particular cases $\alpha =1$, 2 and $+\infty$, respectively. Guess work or guessing entropy is also of great interest in relation to $\alpha$-entropy. On the other hand, many applications rely instead on the “statistical distance”, also known as “total variation" distance, to the uniform distribution. This criterion is particularly important because a very small distance ensures that no statistical test can effectively distinguish between the actual distribution and the uniform distribution. In this paper, we establish optimal lower and upper bounds between $\alpha$-entropy, guessing entropy on one hand, and error probability and total variation distance to the uniform on the other hand. In this context, it turns out that the best known “Pinsker inequality” and recent “reverse Pinsker inequalities” are not necessarily optimal. We recover or improve previous Fano-type and Pinsker-type inequalities used for several applications. Full article

By using the symmetry of the Dunkl Laplacian operator, we prove a sharp Shannon-type inequality and a logarithmic Sobolev inequality for the Dunkl transform. Combining these inequalities, we obtain a new, short proof for Heisenberg-type uncertainty principles in the Dunkl setting. Moreover, by combining Nash’s inequality, Carlson’s inequality and Sobolev’s embedding theorems for the Dunkl transform, we prove new uncertainty inequalities involving the $L^{\infty }$-norm. Finally, we obtain a logarithmic Sobolev inequality in $L^p$-spaces, from which we derive an $L^p$-Heisenberg-type uncertainty inequality and an $L^p$-Nash-type inequality for the Dunkl transform. Full article

The spreading of the stationary states of the multidimensional single-particle systems with a central potential is quantified by means of Heisenberg-like measures (radial and logarithmic expectation values) and entropy-like quantities (Fisher, Shannon, Rényi) of position and momentum probability densities. Since the potential is assumed to be analytically unknown, these dispersion and information-theoretical measures are given by means of inequality-type relations which are explicitly shown to depend on dimensionality and state’s angular hyperquantum numbers. The spherical-symmetry and spin effects on these spreading properties are obtained by use of various integral inequalities (Daubechies–Thakkar, Lieb–Thirring, Redheffer–Weyl, ...) and a variational approach based on the extremization of entropy-like measures. Emphasis is placed on the uncertainty relations, upon which the essential reason of the probabilistic theory of quantum systems relies. Full article

Studying Bregman distance iterative methods for solving optimization problems has become an important and very interesting topic because of the numerous applications of the Bregman distance techniques. These applications are based on the type of convex functions associated with the Bregman distance. In this paper, two different extragraident methods were proposed for studying pseudomonotone variational inequality problems using Bregman distance in real Hilbert spaces. The first algorithm uses a fixed stepsize which depends on a prior estimate of the Lipschitz constant of the cost operator. The second algorithm uses a self-adaptive stepsize which does not require prior estimate of the Lipschitz constant of the cost operator. Some convergence results were proved for approximating the solutions of pseudomonotone variational inequality problem under standard assumptions. Moreso, some numerical experiments were also given to illustrate the performance of the proposed algorithms using different convex functions such as the Shannon entropy and the Burg entropy. In addition, an application of the result to a signal processing problem is also presented. Full article

Tree size diversity is an indicator of biodiversity values of a forest. Microsite conditions of a forest determine the survival and growth of trees. However, the contribution of variable habitats to tree size hierarchy and segregation is poorly understood. Tree size variation in a population is caused by different competition mechanisms. Therefore, the size distribution and spatial pattern of trees can identify the process governing resource utilisation in the forest. The objective of the study was to investigate the tree stem structural diversity in the Elephant Camp natural forest in the Omo Forest Reserve. Three and four 0.09 ha sample plots were established in Riparian (RF) and Old-growth forests (OF) in the Elephant Camp natural forest, respectively. The tree stems (Dbh ≥ 5cm) were identified to the species level and enumerated within each plot, and the stem density was computed. The diameter at breast height (Dbh) was measured with diameter tape. Species diversity was assessed using Shannon–Weiner (H’) and Simpson indices (1-D’), while size inequality was assessed using the Gini coefficient (GC), coefficient of variation (CV), H’ and I-D’. The performance of single two- and three-parameter Weibull models was evaluated using Kolmogorov–Smirnov (K-S) chi-square (χ²), root-mean-square error (RMSE), bias and the coefficient of determination (R²). Data were analysed using descriptive statistics. A total of 27 and 24 tree species were identified in RF and OF, respectively. The stem density of RF was significantly higher than that of OF. The values of species diversity (H’, 1-D’) and evenness (E’) were higher in OF than in RF, while richness (Margalef and number of species) was higher in RF than in OF. The Dbh was 38.30 ± 21.4 and 42.87 ± 19.2 cm in Riparian and Old-growth forests, respectively. Size-density distributions of both forests were positively skewed and expressed exponential pattern. The forest types of the Elephant Camp natural forest comprise the same size-density frequency shape but a different proportion of tree sizes and structural diversities. Full article

Fano’s inequality is one of the most elementary, ubiquitous, and important tools in information theory. Using majorization theory, Fano’s inequality is generalized to a broad class of information measures, which contains those of Shannon and Rényi. When specialized to these measures, it recovers and generalizes the classical inequalities. Key to the derivation is the construction of an appropriate conditional distribution inducing a desired marginal distribution on a countably infinite alphabet. The construction is based on the infinite-dimensional version of Birkhoff’s theorem proven by Révész [Acta Math. Hungar. 1962, 3, 188–198], and the constraint of maintaining a desired marginal distribution is similar to coupling in probability theory. Using our Fano-type inequalities for Shannon’s and Rényi’s information measures, we also investigate the asymptotic behavior of the sequence of Shannon’s and Rényi’s equivocations when the error probabilities vanish. This asymptotic behavior provides a novel characterization of the asymptotic equipartition property (AEP) via Fano’s inequality. Full article

This study presents a new metric for quantifying structural complexity using the diversity of tree damage types in forests that have experienced wind disturbance. Structural complexity studies of forests have to date not incorporated any protocol to address the variety of structural damage types experienced by trees in wind disturbances. This study describes and demonstrates such a protocol. Damage diversity, defined as the richness and evenness of types of tree damage, is calculated analogously to species diversity using two common indices, and termed a ‘Shannon Damage Heterogeneity Index’ (Sh-DHI) and an inverse Simpson Damage Heterogeneity Index (iSi-DHI). The two versions of the DHI are presented for >400 plots across 18 distinct wind disturbed forests of eastern North America. Relationships between DHI and pre-disturbance forest species diversity and size variability, as well as wind disturbance severity, calculated as the fraction of basal area downed in a wind disturbance event, are examined. DHIs are only weakly related to pre-disturbance tree species diversity, but are significantly positively related to pre-disturbance tree size inequality (size diversity). Damage diversity exhibits a robust curvilinear relationship to severity; both versions of the DHI show peaks at intermediate levels of wind disturbance severity, suggesting that in turn structural complexity may also peak at intermediate levels of severity. Full article