scholarly journals New refinements of the discrete Jensen’s inequality generated by finite or infinite permutations

2019 ◽  
Vol 94 (6) ◽  
pp. 1109-1121
Author(s):  
László Horváth
Keyword(s):  
Banach Spaces ◽  
Jensen’S Inequality ◽  
Vector Spaces ◽  
Real Vector ◽  
Finite Sets ◽  
Jensen's Inequality

AbstractIn this paper some new refinements of the discrete Jensen’s inequality are obtained in real vector spaces. The idea comes from some former refinements determined by cyclic permutations. We essentially generalize and extend these results by using permutations of finite sets and bijections of the set of positive numbers. We get refinements of the discrete Jensen’s inequality for infinite convex combinations in Banach spaces. Similar results are rare. Finally, some applications are given on different topics.

scholarly journals Functional Equations Related to Inner Product Spaces

2009 ◽  
Vol 2009 ◽  
pp. 1-11 ◽  
Author(s):  
Choonkil Park ◽  
Won-Gil Park ◽  
Abbas Najati
Keyword(s):  
Functional Equation ◽  
Banach Spaces ◽  
Functional Equations ◽  
Vector Spaces ◽  
Inner Product ◽  
Ulam Stability ◽  
Product Spaces ◽  
Real Vector ◽  
Inner Product Spaces

LetV,Wbe real vector spaces. It is shown that an odd mappingf:V→Wsatisfies∑i−12nf(xi−1/2n∑j=12nxj)=∑i=12nf(xi)−2nf(1/2n∑i=12nxi)for allx1,…,x2n∈Vif and only if the odd mappingf:V→Wis Cauchy additive. Furthermore, we prove the generalized Hyers-Ulam stability of the above functional equation in real Banach spaces.

On a global upper bound for Jensen's inequality

2009 ◽  
Vol 50 ◽  
Author(s):  
Julije Jaksetic ◽  
Bogdan Gavrea ◽  
Josip Pecaric
Keyword(s):  
Upper Bound ◽  
Jensen’S Inequality ◽  
Jensen's Inequality

scholarly journals On the Number of Rich Lines in High Dimensional Real Vector Spaces

2016 ◽  
Vol 55 (4) ◽  
pp. 955-962
Author(s):  
Márton Hablicsek ◽  
Zachary Scherr
Keyword(s):  
Vector Spaces ◽  
High Dimensional ◽  
Real Vector

WEIGHTED GRIDS IN COMPLEX JORDAN* TRIPLES

2010 ◽  
Vol 03 (01) ◽  
pp. 155-184
Author(s):  
L. L. STACHÓ
Keyword(s):  
Independent Sets ◽  
Vector Spaces ◽  
Complete List ◽  
Real Vector ◽  
Linearly Independent

Weighted grids are linearly independent sets {gw : w ∈ W} of signed tripotents in Jordan* triples indexed by figures W in real vector spaces such that {gugvgw} ∈ ℂgu-v+w (= 0 if u - v + w ∉ W). They arise naturally as systems of weight vectors of certain abelian families of Jordan* derivations. Based on Neher's grid theory, a classification of association free non-nil weighted grids is given. As a first step beyond the setting of classical grids, the complete list of complex weighted grids of pairwise associated signed tripotents indexed by ℤ2 is established.

Topological equivalence and rigidity of flows on certain solvmanifolds

1999 ◽  
Vol 19 (3) ◽  
pp. 559-569
Author(s):  
D. BENARDETE ◽  
S. G. DANI
Keyword(s):  
Lie Group ◽  
Vector Spaces ◽  
Complex Numbers ◽  
Real Vector ◽  
Orbit Equivalent ◽  
Finite Dimensional ◽  
Generic Class ◽  
Induced Flow ◽  
Simply Connected ◽  
Solvable Lie Group

Given a Lie group $G$ and a lattice $\Gamma$ in $G$, a one-parameter subgroup $\phi$ of $G$ is said to be rigid if for any other one-parameter subgroup $\psi$, the flows induced by $\phi$ and $\psi$ on $\Gamma\backslash G$ (by right translations) are topologically orbit-equivalent only if they are affinely orbit-equivalent. It was previously known that if $G$ is a simply connected solvable Lie group such that all the eigenvalues of $\mathrm{Ad} (g) $, $g\in G$, are real, then all one-parameter subgroups of $G$ are rigid for any lattice in $G$. Here we consider a complementary case, in which the eigenvalues of $\mathrm{Ad} (g)$, $g\in G$, form the unit circle of complex numbers.Let $G$ be the semidirect product $N \rtimes M$, where $M$ and $N$ are finite-dimensional real vector spaces and where the action of $M$ on the normal subgroup $N$ is such that the center of $G$ is a lattice in $M$. We prove that there is a generic class of abelian lattices $\Gamma$ in $G$ such that any semisimple one-parameter subgroup $\phi$ (namely $\phi$ such that $\mathrm{Ad} (\phi_t)$ is diagonalizable over the complex numbers for all $t$) is rigid for $\Gamma$ (see Theorem 1.4). We also show that, on the other hand, there are fairly high-dimensional spaces of abelian lattices for which some semisimple $\phi$ are not rigid (see Corollary 4.3); further, there are non-rigid semisimple $\phi$ for which the induced flow is ergodic.

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