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One of the fundamental inequalities which is used in many inequities is Jensen inequality. In fact, it is a base of some inequality such as the arithmetic mean, harmonic mean inequality also in inequality with respect to entropies and information theory. The purpose of this research paper is to give a new interesting refinement of Jensen inequality for two particular finite sequences by using uniformly convex function. Also, we give some applications of this in information theory. [ABSTRACT FROM AUTHOR]

In recent years, many researches have been done on the tgs-convex functions and their applications. In this article, we present some properties of the tgs-convex functions by interesting examples. Then we investigate the non-positive property of the tgs-convex functions. Also, we derive types of the Jensen’s inequality for the tgs-convex functions and obtain several inequalities with respect to the Jensen’s inequality. Finally, we give some applications of these inequalities.

In recent years, many researches have been done on the tgsconvex functions and their applications. In this article, we present some properties of the tgs-convex functions by interesting examples. Then we investigate the non-positive property of the tgs-convex functions. Also, we derive types of the Jensen's inequality for the tgs-convex functions and obtain several inequalities with respect to the Jensen's inequality. Finally, we give some applications of these inequalities. [ABSTRACT FROM AUTHOR]

The paper presents a rigorous perturbation analysis of the QR decomposition A = Q R of an n × m matrix A using the method of splitting operators. New asymptotic componentwise perturbation bounds are derived for the elements of Q and R and the subspaces spanned by the first p ≤ m columns of A. The new bounds are less conservative than the known bounds and are significantly better than the normwise bounds. An iterative scheme is proposed to determine global componentwise bounds in the case of perturbations for which such bounds are valid. Several numerical results are given that illustrate the analysis and the quality of the bounds obtained. [ABSTRACT FROM AUTHOR]

This paper shows the global existence and boundedness of solutions of a reaction diffusion system modeling liver infections. The existence proof is presented step by step and the focus lies on the interpretation of intermediate results in the context of liver infections which is modeled. Non-local effects in the dynamics between the virus and the immune system cells coming from the immune response in the lymphs lead to an integro-partial differential equation. While existence theorems for parabolic partial differential equations are textbook examples in the field, the additional integral term requires new approaches to proving the global existence of a solution. This allows to set up an existence proof with a focus on interpretation leading to more insight in the system and in the modeling perspective at the same time.We show the boundedness of the solution in the L1(Ω)- and the L2(Ω)-norms, and use these results to prove the global existence and boundedness of the solution. A core element of the proof is the handling of oppositely acting mechanisms in the reaction term, which occur in all population dynamics models and which results in reaction terms with opposite monotonicity behavior. In the context of modeling liver infections, the boundedness in the L∞(Ω)-norm has practical relevance: Large immune responses lead to strong inflammations of the liver tissue. Strong inflammations negatively impact the health of an infected person and lead to grave secondary diseases. The gained rough estimates are compared with numerical tests.

The aim of this paper is to present the global estimate for gradient of renormalized solutions to the following quasilinear elliptic problem: − div (A (x , ∇ u)) = μ in Ω , u = 0 on ∂ Ω , in Lorentz–Morrey spaces, where Ω ⊂ ℝ n (n ≥ 2), μ is a finite Radon measure, A is a monotone Carathéodory vector-valued function defined on W 0 1 , p (Ω) and the p -capacity uniform thickness condition is imposed on the complement of our domain Ω. It is remarkable that the local gradient estimates have been proved first by Mingione in [Gradient estimates below the duality exponent, Math. Ann.346 (2010) 571–627] at least for the case 2 ≤ p ≤ n , where the idea for extending such result to global ones was also proposed in the same paper. Later, the global Lorentz–Morrey and Morrey regularities were obtained by Phuc in [Morrey global bounds and quasilinear Riccati type equations below the natural exponent, J. Math. Pures Appl.102 (2014) 99–123] for regular case p > 2 − 1 n . Here in this study, we particularly restrict ourselves to the singular case 3 n − 2 2 n − 1 < p ≤ 2 − 1 n . The results are central to generalize our technique of good- λ type bounds in the previous work [M.-P. Tran, Good- λ type bounds of quasilinear elliptic equations for the singular case, Nonlinear Anal.178 (2019) 266–281], where the local gradient estimates of solution to this type of equation were obtained in the Lorentz spaces. Moreover, the proofs of most results in this paper are formulated globally up to the boundary results. [ABSTRACT FROM AUTHOR]

In this paper, inspired by the work by A. Iserles and G. Söoderlind [Global bounds on numerical error for ordinary differential equations, J. Com- plexity, 9 (1993), pp. 97-112 ], we present comprehensive discussion on Dirichlet series for dynamical systems of first-order ordinary differential equations (ODEs). We first derive the scheme of Dirichlet approximation for scalar dynamical systems and present the bounds on the terms of Dirichlet series. The global error and the right choice of a term in Dirichlet series are analysed and two numerical experiments are carried out to demonstrate the efficiency of Dirichlet approximation. Then we consider applying Dirichlet series to multivariate dynamical systems and present a new scheme of Dirichlet approximation for such systems. Some discussion and a numerical experiment are accordingly carried out for the new Dirichlet approximation. Compared with routine time-stepping algorithms, Dirichlet series does not need time stepping and yields a continuous solution that is equally valid along an interval, which is significant for obtaining long-time numerical solution. As a result of the special nature of Dirichlet series, the Dirichlet approximation delivers considerable information on dynamical systems of first-order ODEs and provides a novel and effective approach to numerical solutions of these dynamical systems. [ABSTRACT FROM AUTHOR]

We propose an improved model explaining the occurrence of high stresses due to the difference in specific volumes during phase transitions between water and ice. The unknowns of the resulting evolution problem are the absolute temperature, the volume increment, and the liquid fraction. The main novelty here consists in including the dependence of the specific heat and of the speed of sound upon the phase. These additional nonlinearities bring new mathematical difficulties which require new estimation techniques based on Moser iteration. We establish the existence of a global solution to the corresponding initial-boundary value problem, as well as lower and upper bounds for the absolute temperature. Assuming constant heat conductivity, we also prove uniqueness and continuous data dependence of the solution. [ABSTRACT FROM AUTHOR]

We establish global Gaussian estimates for the Green's matrix of divergence form, second order parabolic systems in a cylindrical domain under the assumption that weak solutions of the system vanishing on a portion of the boundary satisfy a certain local boundedness estimate and a local Hölder estimate. From these estimates, we also derive global estimates for the Green's matrix for elliptic systems with bounded measurable coefficients in two dimensional domains. We present a unified approach valid for both the scalar and vectorial cases and discuss several applications of our result. [ABSTRACT FROM AUTHOR]

Abstract: We establish global pointwise bounds for the Green''s matrix for divergence form, second order elliptic systems in a domain under the assumption that weak solutions of the system vanishing on a portion of the boundary satisfy a certain local boundedness estimate. Moreover, we prove that such a local boundedness estimate for weak solutions of the system is equivalent to the usual global pointwise bound for the Green''s matrix. In the scalar case, such an estimate is a consequence of De Giorgi–Moser–Nash theory and holds for equations with bounded measurable coefficients in arbitrary domains. In the vectorial case, one need to impose certain assumptions on the coefficients of the system as well as on domains to obtain such an estimate. We present a unified approach valid for both the scalar and vectorial cases and discuss several applications of our result. [Copyright &y& Elsevier]

Abstract: In this article we found the form of best possible global upper bound for Jensen’s inequality. Thereby, previous results on this topic are essentially improved. We also give some applications in Analysis and Information Theory. [Copyright &y& Elsevier]

Abstract: We establish new lower and upper bounds for Jensen’s discrete inequality. Applying those results in information theory, we obtain new and more precise bounds for Shannon’s entropy. [Copyright &y& Elsevier]

We give another global upper bound for Jensen's discrete inequality which is better than already existing ones. For instance, we determine a new converses for generalized A-G and G-H inequalities. [ABSTRACT FROM AUTHOR]

The blind source separation problem arises when one attempts to recover source signals from their linear mixtures without detailed knowledge of the mixing process. Solutions are nonunique and have degrees of freedom in scaling and permutation. One may impose equality (hard) constraints to fix these scaling parameters; however, small divisor problems may appear. In this paper, an iterative method is introduced based on information maximization and auxiliary equations for the scaling parameters. The method dynamically selects scaling parameters and avoids divisions, and it is called the soft-constrained method. Global boundedness of the algorithm is proved. The convergence of solutions in the large time and small step size regimes is analyzed. An upscaled, dynamically averaged equation for the separating matrix is derived. The stable and accurate separation performance is illustrated by examples of instantaneous random mixtures of two and eight sound signals. [ABSTRACT FROM AUTHOR]

Abstract: In this paper we shall give a global upper bound for Jensen''s inequality without restrictions on the target convex function f. We also introduce a characteristic i.e. an absolute constant depending only on f, by which the global bound is improved. [Copyright &y& Elsevier]

In this article we give sharp global bounds for the generalized Jensen functional J n (g , h ; p , x) . In particular, exact bounds are determined for the generalized power mean in terms from the class of Stolarsky means. As a consequence, we obtain the best possible global converses of quotients and differences of the generalized arithmetic, geometric and harmonic means. [ABSTRACT FROM AUTHOR]