Minkowski inequality for nearly spherical domains

In this paper, we prove a sharp anisotropic Lp Minkowski inequality involving the total Lp anisotropic mean curvature and the anisotropic p-capacity for any bounded domains with smooth boundary in ℝn. As consequences, we obtain an anisotropic Willmore inequality, a sharp anisotropic Minkowski inequality for outward F-minimising sets and a sharp volumetric anisotropic Minkowski inequality. For the proof, we utilize a nonlinear potential theoretic approach which has been recently developed by Agostiniani et al. (2019).

In this paper, we review some recent works on sharp fractional inequalities on nilpotent Lie groups, studying their sharp constants, extremizers, stability and other improvements. We also prove a new sharp subcritical inequality in Section sub . The conformal symmetry, compactness, spectral estimates and flow methods are discussed. We remark that a positivity-type restriction on the singular exponent is required in cases for groups with high dimensional centres, which brings extra difficulty.

A nonlinear Korn inequality on a surface is any estimate of the distance, up to a proper isometry of [Formula: see text], between two surfaces measured by some appropriate norms (the “left-hand side” of the inequality) in terms of the distances between their three fundamental forms measured by some appropriate norms (the “right-hand side” of the inequality). The first objective of this paper is to provide several extensions of a nonlinear Korn inequality on a surface obtained in 2006 by the first and third authors and Gratie, then measured by means of [Formula: see text]-norms on the left-hand side and [Formula: see text]-norms on the right-hand side. First, we extend this inequality to [Formula: see text]-norms on the left-hand side and [Formula: see text]-norms on the right-hand side for any [Formula: see text] and [Formula: see text] that satisfy [Formula: see text]; second, we show how the third fundamental forms can be disposed in the right-hand side; and third, we show that there is no need to introduce proper isometries of [Formula: see text] in the left-hand side if the surfaces satisfy appropriate boundary conditions. The second objective is to provide nonlinear Korn inequalities on a surface where the left-hand sides are now measured by means of [Formula: see text]-norms while the right-hand sides are measured by means of [Formula: see text]-norms, for any [Formula: see text]. These nonlinear Korn inequalities on a surface themselves rely on various nonlinear Korn inequalities in a domain in [Formula: see text], recently obtained by the first and third authors in 2015 and by the first author and Sorin Mardare in 2016.

We use norm inequalities for operators, such as Cauchy–Schwarz inequality and Minkowski inequality, to establish sharp norm inequalities for the absolute value of operators. Among other inequalities, it is shown that if A, B and X are operators on a Hilbert space, such that X is self-adjoint, and r ≥ 1, then This inequality generalizes some earlier related results. Other related inequalities are also discussed.

To clarify the interhemispheric asymmetrical change in gray matter volume (GMV) in unilateral hippocampal sclerosis (HS), we compared changes in GMV relative to normal subjects between the HS and contralateral or non-HS sides. Forty-five patients with unilateral HS and 30 healthy subjects were enrolled. We quantified changes in GMV in the patients with HS as compared to GMV in the normal subjects by introducing the Z-score (Z-GMV) in each region or region of interest in unilateral HS. Then, we assessed the asymmetrically decreased regions, that is, regions with significantly higher Z-GMV on the HS side than the contralateral or non-HS side. Z-GMV was calculated according to the two templates of 58 regions per hemisphere covering the whole brain by anatomical automatic labeling (AAL) and 78 regions per cerebral hemisphere using the Anatomy Toolbox. Seven and four regions in AAL and 17 and 11 regions in Anatomy Toolbox were asymmetrically decreased in the Left Hand Side (LHS) and Right Hand Side (RHS), respectively. Hippocampus and Caudate in AAL, five subregions of the hippocampus (CA1-3, Dentate Gyrus and hippocampus-amygdala-transition-area and 4 extrahippocampal regions including two subregions in amygdala (CM: Centromedial, SF: Superficial), basal forebrain (BF) (Ch4), and thalamus (temporal) in anatomy toolbox were common among LHS and RHS concerning asymmetrically decreased regions. By introducing Z-GMV, we demonstrated the regions with asymmetrically decreased GMV in LHS and RHS, and found that the hippocampus and extrahippocampal regions, including the BF, were the common asymmetrically decreased regions among LHS and RHS.

The security of the large database that contains certain crucial information, it will become a serious issue when sharing data to the network against unauthorized access. Privacy preserving data mining is a new research trend in privacy data for data mining and statistical database. Association analysis is a powerful tool for discovering relationships which are hidden in large database. Association rules hiding algorithms get strong and efficient performance for protecting confidential and crucial data. Data modification and rule hiding is one of the most important approaches for secure data. The objective of the proposed Association rulehiding algorithm for privacy preserving data mining is to hide certain information so that they cannot be discovered through association rule mining algorithm. The main approached of association rule hiding algorithms to hide some generated association rules, by increase or decrease the support or the confidence of the rules. The association rule items whether in Left Hand Side (LHS) or Right Hand Side (RHS) of the generated rule, that cannot be deduced through association rule mining algorithms. The concept of Increase Support of Left Hand Side (ISL) algorithm is decrease the confidence of rule by increase the support value of LHS. It doesnEt work for both side of rule; it works only for modification of LHS. In Decrease Support of Right Hand Side (DSR) algorithm, confidence of the rule decrease by decrease the support value of RHS. It works for the modification of RHS. We proposed a new algorithm solves the problem of them. That can increase and decrease the support of the LHS and RHS item of the rule correspondingly so that more rule hide less number of modification. The efficiency of the proposed algorithm is compared with ISL algorithms and DSR algorithms using real databases, on the basis of number of rules hide, CPU time and the number of modifies entries and got better results.

Concepts of convexity and related techniques are widely used in various branches of mathematics and in applications such as functional analysis, calculus of variations, and mathematical physics, geometry, number theory, theory of games, optimization and other computational methods and convex programming, integral geometry (tomography), and many others. In this chapter we will focus mainly on the basic elements of the theory of convex functions, which can be taught to readers with limited experience. The remaining problems are directed toward specific applications such as convex and linear programming, hierarchy of power means, geometric inequalities, the Hölder and Minkowski inequalities, Young’s inequality and the Legendre transform, and functions on metric and normed vector spaces. The necessary introductory material is provided in these sections. A specially added section discusses the widely known Cauchy-Schwarz-Bunyakovskii (CSB) inequality and Lagrange-type identities in connection with the hierarchy of the power means.KeywordsConvex FunctionConvex HullConvex DomainVector SubspaceConvex PolyhedronThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Dynamic programming and penalty functions

- Preface to 'Means and their Inequalities'. Preface to the Handbook. Basic References. - Notations. 1. Referencing. 2. Bibliographic References. 3. Symbols for some Important Inequalities. 4. Numbers, Sets and Set Functions. 5. Intervals. 6. n-tuples. 7. Matrices. 8. Functions. 9. Various. A List of Symbols. An Introductory Survey. - I: Introduction. 1. Properties of Polynomials. 2. Elementary Inequalities. 3. Properties of Sequences. 4. Convex Functions. - II: The Arithmetic, Geometric and Harmonic Means. 1. Definitions and Simple Properties. 2. The Geometric Mean-Arithmetic Mean Inequality. 3. Refinements of the Geometric Mean-Arithmetic Mean Inequality. 4. Converse Inequalities. 5. Some Miscellaneous Results. - III: The Power Means. 1. Definitions and Simple Properties. 2. Sums of Powers. 3. Inequalities between the Power Means. 4. Converse Inequalities. 5. Other Means Defined Using Powers. 6. Some Other Results. - IV: Quasi-Arithmetic Means. 1. Definitions and Basic Properties. 2. Comparable Means and Functions. 3. Results of Rado Popoviciu Type. 4. Further Inequalities. 5. Generalizations of the Hoelder and Minkowski Inequalities. 6. Converse Inequalities. 7. Generalizations of the Quasi-arithmetic Means. - V: Symmetric Polynomial Means. 1. Elementary Symmetric Polynomials and Their Means. 2. The Fundamental Inequalities. 3.Extensions of S(r s) of Rado Popoviciu Type. 4. The Inequalities of Marcus and Lopes. 5. Complete Symmetric Polynomial Means: Whiteley Means. 6. The Muirhead Means. 7. Further Generalizations. - VI: Other Topics. 1. Integral Means and Their Inequalities. 2. Two Variable Means. 3. Compounding of Means. 4. Some General Approaches to Means. 5. Mean Inequalities for Matrices. 6. Axiomatization of Means. Bibliography. Name Index. Index.

Two Morrey-Sobolev inequalities (with support-bound and $L^1-$bound, respectively) are investigated on complete Riemannian manifolds with their sharp constants in $\mathbb R^n$. We prove the following results in both cases: $\bullet$ If $(M,g)$ is a {\it Cartan-Hadamard manifold} which verifies the $n-$dimensional Cartan-Hadamard conjecture, sharp Morrey-Sobolev inequalities hold on $(M,g)$. Moreover, extremals exist if and only if $(M,g)$ is isometric to the standard Euclidean space $(\mathbb R^n,e)$. $\bullet$ If $(M,g)$ has {\it non-negative Ricci curvature}, $(M,g)$ supports the sharp Morrey-Sobolev inequalities if and only if $(M,g)$ is isometric to $(\mathbb R^n,e)$.

We first establish a family of sharp Caffarelli–Kohn–Nirenberg type inequalities (shortly, sharp CKN inequalities) on the Euclidean spaces and then extend them to the setting of Cartan–Hadamard manifolds with the same best constant. The quantitative version of these inequalities also is proved by adding a non-negative remainder term in terms of the sectional curvature of manifolds. We next prove several rigidity results for complete Riemannian manifolds supporting the Caffarelli–Kohn–Nirenberg type inequalities with the same sharp constant as in the Euclidean space of the same dimension. Our results illustrate the influence of curvature to the sharp CKN inequalities on the Riemannian manifolds. They extend recent results of Kristály (J. Math. Pures Appl. 119 (2018), 326–346) to a larger class of the sharp CKN inequalities.

This paper consists of two parts. One is that for a kind of self-shrinker in a manifold with warped product metric, we prove that under some conditions on ambient space, the mean convex self-shrinker must have parallel second fundamental form. The other one is a generalization of Brendle’s Minkowski inequality for weighted mean curvature.

We prove generalized Hardy's type inequalities with sharp constants and Caffarelli-Kohn-Nirenberg inequalities with sharp constants on Riemannian manifolds $M$. When the manifold is Euclidean space we recapture the sharp Caffarelli-Kohn-Nirenberg inequality. By using a double limiting argument, we obtain an inequality that implies a sharp Hardy's inequality, for functions with compact support on the manifold $M $ (that is, not necessarily on a punctured manifold $ M \backslash \{ x_0 \} $ where $x_0$ is a fixed point in $M$). Some topological and geometric applications are discussed.

Improved Caffarelli–Kohn–Nirenberg inequalities in unit ball and sharp constants in dimension three

We prove a sharp inequality for toroidal hypersurfaces in three- and four-dimensional Horowitz–Myers geon. This extend previous results on Minkowski inequality in the static spacetime to toroidal surfaces in asymptotically hyperbolic manifold with flat toroidal conformal infinity.