scholarly journals The Dual Orlicz–Aleksandrov–Fenchel Inequality

Mathematics ◽  
2020 ◽  
Vol 8 (11) ◽  
pp. 2005
Author(s):  
Chang-Jian Zhao
Keyword(s):  
Orlicz Space ◽  
Minkowski Inequality ◽  
Mixed Volume ◽  
Mixed Volumes ◽  
Geometric Quantity ◽  
First Order ◽  
Special Cases ◽  
Star Bodies

In this paper, the classical dual mixed volume of star bodies V˜(K1,⋯,Kn) and dual Aleksandrov–Fenchel inequality are extended to the Orlicz space. Under the framework of dual Orlicz-Brunn-Minkowski theory, we put forward a new affine geometric quantity by calculating first order Orlicz variation of the dual mixed volume, and call it Orlicz multiple dual mixed volume. We generalize the fundamental notions and conclusions of the dual mixed volume and dual Aleksandrov-Fenchel inequality to an Orlicz setting. The classical dual Aleksandrov-Fenchel inequality and dual Orlicz-Minkowski inequality are all special cases of the new dual Orlicz-Aleksandrov-Fenchel inequality. The related concepts of Lp-dual multiple mixed volumes and Lp-dual Aleksandrov-Fenchel inequality are first derived here. As an application, the dual Orlicz–Brunn–Minkowski inequality for the Orlicz harmonic addition is also established.

scholarly journals Orlicz-Aleksandrov-Fenchel Inequality for Orlicz Multiple Mixed Volumes

2018 ◽  
Vol 2018 ◽  
pp. 1-16 ◽  
Author(s):  
Chang-Jian Zhao
Keyword(s):  
Orlicz Space ◽  
Convex Bodies ◽  
Minkowski Inequality ◽  
Mixed Volume ◽  
Mixed Volumes ◽  
Geometric Quantity ◽  
First Order ◽  
Special Cases ◽  
Order Variation

Our main aim is to generalize the classical mixed volumeV(K1,…,Kn)and Aleksandrov-Fenchel inequality to the Orlicz space. In the framework of Orlicz-Brunn-Minkowski theory, we introduce a new affine geometric quantity by calculating the Orlicz first-order variation of the mixed volume and call itOrlicz multiple mixed volumeof convex bodiesK1,…,Kn, andLn, denoted byVφ(K1,…,Kn,Ln), which involves(n+1)convex bodies inRn. The fundamental notions and conclusions of the mixed volume and Aleksandrov-Fenchel inequality are extended to an Orlicz setting. The related concepts and inequalities ofLp-multiple mixed volumeVp(K1,…,Kn,Ln)are also derived. The Orlicz-Aleksandrov-Fenchel inequality in special cases yieldsLp-Aleksandrov-Fenchel inequality, Orlicz-Minkowski inequality, and Orlicz isoperimetric type inequalities. As application, a new Orlicz-Brunn-Minkowski inequality for Orlicz harmonic addition is established, which implies Orlicz-Brunn-Minkowski inequalities for the volumes and quermassintegrals.

Orlicz dual affine quermassintegrals

2018 ◽  
Vol 30 (4) ◽  
pp. 929-945 ◽  
Author(s):  
Chang-Jian Zhao
Keyword(s):  
Orlicz Space ◽  
Minkowski Inequality ◽  
Mixed Volumes ◽  
Geometric Quantity ◽  
First Order ◽  
Order Variation ◽  
Dual Affine ◽  
Dual Minkowski Inequality ◽  
Dual Mixed Volumes ◽  
Special Case

Abstract In the paper, our main aim is to generalize the dual affine quermassintegrals to the Orlicz space. Under the framework of Orlicz dual Brunn–Minkowski theory, we introduce a new affine geometric quantity by calculating the first-order variation of the dual affine quermassintegrals, and call it the Orlicz dual affine quermassintegral. The fundamental notions and conclusions of the dual affine quermassintegrals and the Minkoswki and Brunn–Minkowski inequalities for them are extended to an Orlicz setting, and the related concepts and inequalities of Orlicz dual mixed volumes are also included in our conclusions. The new Orlicz–Minkowski and Orlicz–Brunn–Minkowski inequalities in a special case yield the Orlicz dual Minkowski inequality and Orlicz dual Brunn–Minkowski inequality, which also imply the {L_{p}} -dual Minkowski inequality and Brunn–Minkowski inequality for the dual affine quermassintegrals.

scholarly journals Orlicz Mean Dual Affine Quermassintegrals

2018 ◽  
Vol 2018 ◽  
pp. 1-13 ◽  
Author(s):  
Chang-Jian Zhao ◽  
Wing-Sum Cheung
Keyword(s):  
Orlicz Space ◽  
Minkowski Inequality ◽  
Isoperimetric Inequalities ◽  
Mixed Volumes ◽  
Geometric Quantity ◽  
Dual Affine ◽  
The Mean ◽  
Dual Minkowski Inequality ◽  
Special Case

Our main aim is to generalize the mean dual affine quermassintegrals to the Orlicz space. Under the framework of dual Orlicz-Brunn-Minkowski theory, we introduce a new affine geometric quantity by calculating the first Orlicz variation of the mean dual affine quermassintegrals and call it the Orlicz mean dual affine quermassintegral. The fundamental notions and conclusions of the mean dual affine quermassintegrals and the Minkowski and Brunn-Minkowski inequalities for them are extended to an Orlicz setting. The related concepts and inequalities of dual Orlicz mixed volumes are also included in our conclusions. The new Orlicz isoperimetric inequalities in special case yield theLp-dual Minkowski inequality and Brunn-Minkowski inequality for the mean dual affine quermassintegrals, which also imply the dual Orlicz-Minkowski inequality and dual Orlicz-Brunn-Minkowski inequality.

scholarly journals Extensions in graph normal form

2020 ◽  
Author(s):  
Michał Walicki
Keyword(s):  
Normal Form ◽  
Fixed Points ◽  
Propositional Logic ◽  
Order Logic ◽  
First Order Logic ◽  
First Order ◽  
Special Cases

Abstract Graph normal form, introduced earlier for propositional logic, is shown to be a normal form also for first-order logic. It allows to view syntax of theories as digraphs, while their semantics as kernels of these digraphs. Graphs are particularly well suited for studying circularity, and we provide some general means for verifying that circular or apparently circular extensions are conservative. Traditional syntactic means of ensuring conservativity, like definitional extensions or positive occurrences guaranteeing exsitence of fixed points, emerge as special cases.

Geometric, Spatial Path Tracking Using Non-Redundant Manipulators via Speed-Ratio Control

2010 ◽  
Author(s):  
Satyajit Ambike ◽  
James P. Schmiedeler ◽  
Michael M. Stanisˇic´
Keyword(s):  
Path Tracking ◽  
Speed Ratio ◽  
Redundant Manipulators ◽  
Computationally Efficient ◽  
Third Order ◽  
Practical Advantage ◽  
First Order ◽  
Canonical Frame ◽  
Special Cases ◽  
Six Degree Of Freedom

Path tracking can be accomplished by separating the control of the desired trajectory geometry and the control of the path variable. Existing methods accomplish tracking of up to third-order geometric properties of planar paths and up to second-order properties of spatial paths using non-redundant manipulators, but only in special cases. This paper presents a novel methodology that enables the geometric tracking of a desired planar or spatial path to any order with any non-redundant regional manipulator. The governing first-order coordination equation for a spatial path-tracking problem is developed, the repeated differentiation of which generates the coordination equation of the desired order. In contrast to previous work, the equations are developed in a fixed global frame rather than a configuration-dependent canonical frame, providing a significant practical advantage. The equations are shown to be linear, and therefore, computationally efficient. As an example, the results are applied to a spatial 3-revolute mechanism that tracks a spatial path. Spatial, rigid-body guidance is achieved by applying the technique to three points on the end-effector of a six degree-of-freedom robot. A spatial 6-revolute robot is used as an illustration.

First-order Theory of Fields with Arbitrary Spin

2020 ◽  
pp. 135-146
Author(s):  
Daniel Canarutto
Keyword(s):  
Order Theory ◽  
Arbitrary Spin ◽  
General Description ◽  
First Order ◽  
First Order Theory ◽  
Special Cases ◽  
Spinor Geometry ◽  
Theory Of Fields ◽  
Angular Momentum Algebra ◽  
Dirac Spinors

By exploiting the previously exposed results in 2-spinor geometry, a general description of fields of arbitrary spin is exposed and shown to admit a first-order Lagrangian which extends the theory of Dirac spinors. The needed bundle is the fibered direct product of a symmetric ‘main sector’—carrying an irreducible representation of the angular-momentum algebra—and an induced sequence of ‘ghost sectors’. Several special cases are considered; in particular, one recovers the Bargmann-Wigner and Joos-Weinberg equations.

Orlicz Addition for Measures and an Optimization Problem for the -divergence

2019 ◽  
Vol 72 (2) ◽  
pp. 455-479
Author(s):  
Shaoxiong Hou ◽  
Deping Ye
Keyword(s):  
Optimization Problem ◽  
Jensen’S Inequality ◽  
Minkowski Inequality ◽  
Isoperimetric Inequalities ◽  
Jensen's Inequality ◽  
Dual Functional ◽  
Two Measures ◽  
Star Bodies

AbstractThis paper provides a functional analogue of the recently initiated dual Orlicz–Brunn–Minkowski theory for star bodies. We first propose the Orlicz addition of measures, and establish the dual functional Orlicz–Brunn–Minkowski inequality. Based on a family of linear Orlicz additions of two measures, we provide an interpretation for the famous $f$-divergence. Jensen’s inequality for integrals is also proved to be equivalent to the newly established dual functional Orlicz–Brunn–Minkowski inequality. An optimization problem for the $f$-divergence is proposed, and related functional affine isoperimetric inequalities are established.

scholarly journals Hyers-Ulam stability of first-order homogeneous linear dynamic equations on time scales

2018 ◽  
Vol 51 (1) ◽  
pp. 198-210 ◽  
Author(s):  
Douglas R. Anderson ◽  
Masakazu Onitsuka
Keyword(s):  
Time Scales ◽  
Dynamic Equations ◽  
Ulam Stability ◽  
Step Size ◽  
First Order ◽  
Linear Dynamic ◽  
Discrete Step ◽  
Special Cases ◽  
Parameter Values ◽  
Homogeneous Linear

Abstract We establish theHyers-Ulam stability (HUS) of certain first-order linear constant coefficient dynamic equations on time scales, which include the continuous (step size zero) and the discrete (step size constant and nonzero) dynamic equations as important special cases. In particular, for certain parameter values in relation to the graininess of the time scale, we find the minimum HUS constants. A few nontrivial examples are provided. Moreover, an application to a perturbed linear dynamic equation is also included.

scholarly journals Generalized second-order slip for unsteady convective flow of a nanofluid: a utilization of Buongiorno’s two-component nonhomogeneous equilibrium model

2020 ◽  
Vol 9 (1) ◽  
pp. 156-168
Author(s):  
Seyed Mahdi Mousavi ◽  
Saeed Dinarvand ◽  
Mohammad Eftekhari Yazdi
Keyword(s):  
Boundary Layer ◽  
Equilibrium Model ◽  
Second Order ◽  
Model Parameters ◽  
Slip Parameter ◽  
Two Phase ◽  
Convective Boundary ◽  
First Order ◽  
Special Cases ◽  
Two Component

AbstractThe unsteady convective boundary layer flow of a nanofluid along a permeable shrinking/stretching plate under suction and second-order slip effects has been developed. Buongiorno’s two-component nonhomogeneous equilibrium model is implemented to take the effects of Brownian motion and thermophoresis into consideration. It can be emphasized that, our two-phase nanofluid model along with slip concentration at the wall shows better physical aspects relative to taking the constant volume concentration at the wall. The similarity transformation method (STM), allows us to reducing nonlinear governing PDEs to nonlinear dimensionless ODEs, before being solved numerically by employing the Keller-box method (KBM). The graphical results portray the effects of model parameters on boundary layer behavior. Moreover, results validation has been demonstrated as the skin friction and the reduced Nusselt number. We understand shrinking plate case is a key factor affecting non-uniqueness of the solutions and the range of the shrinking parameter for which the solution exists, increases with the first order slip parameter, the absolute value of the second order slip parameter as well as the transpiration rate parameter. Besides, the second-order slip at the interface decreases the rate of heat transfer in a nanofluid. Finally, the analysis for no-slip and first-order slip boundary conditions can also be retrieved as special cases of the present model.

On the reduction of the decision problem. First paper. Ackermann prefix, a single binary predicate

1939 ◽  
Vol 4 (1) ◽  
pp. 1-9 ◽  
Author(s):  
László Kalmár
Keyword(s):  
Decision Problem ◽  
Reduction Process ◽  
The Other ◽  
First Order ◽  
Other Hand ◽  
Special Cases ◽  
Binary Predicate ◽  
Ternary Variables ◽  
Order Predicate Calculus ◽  
Predicate Variable

1. Although the decision problem of the first order predicate calculus has been proved by Church to be unsolvable by any (general) recursive process, perhaps it is not superfluous to investigate the possible reductions of the general problem to simple special cases of it. Indeed, the situation after Church's discovery seems to be analogous to that in algebra after the Ruffini-Abel theorem; and investigations on the reduction of the decision problem might prepare the way for a theory in logic, analogous to that of Galois.It has been proved by Ackermann that any first order formula is equivalent to another having a prefix of the form(1) (Ex1)(x2)(Ex3)(x4)…(xm).On the other hand, I have proved that any first order formula is equivalent to some first order formula containing a single, binary, predicate variable. In the present paper, I shall show that both results can be combined; more explicitly, I shall prove theTheorem. To any given first order formula it is possible to construct an equivalent one with a prefix of the form (1) and a matrix containing no other predicate variable than a single binary one.2. Of course, this theorem cannot be proved by a mere application of the Ackermann reduction method and mine, one after the other. Indeed, Ackermann's method requires the introduction of three auxiliary predicate variables, two of them being ternary variables; on the other hand, my reduction process leads to a more complicated prefix, viz.,(2) (Ex1)…(Exm)(xm+1)(xm+2)(Exm+3)(Exm+4).

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