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This paper presents an analogue of the rearrangement inequality, namely the circular
rearrangement inequality. It holds for any finite sequence of real numbers. A volume-invariant
packing problem and a combinatorial isoperimetric problem are addressed, as the geometric
interpretation of the inequality.
Contexts in source publication
Context 1
... a 's maxCP must contain (2, 1, 3). As (2, 1, 3) exists in a 's maxCP, we now focus on the maxCP of b = ([a 2 , a 1 , a 3 ], a 4 , ···, a k+1 ), where [.] imposes a 2 , a 1 , and a 3 in order (2, 1, 3) or (3, 1, 2). ...
Context 2
... the other hand, for any permutation x of ( [2,3], 4, ···, k + 1) and y = ([2, 1, 3], x(4, ···, k + 1)), it holds b(y) ≡ c(x) + (a 2 a 1 + a 1 a 3 ). That is, they attain maximum simultaneously . ...
Context 3
... the other hand, for any permutation z of ( [2, k], 3, ··· , k − 1) and f = ( [2, k + 1, 1, k], z(3, ··· , k − 1)), it holds b( f ) ≡ ≡c(z) + (a 2 a k+1 + a k+1 a 1 + a 1 a k ). That is, they attain maximum simultaneously . ...
Context 4
... an n -vector a with a 1 a 2 ··· a n , we denote its maxCP and minCP by r U = (1, 3, 5, ···, n, ··· , 6, 4, 2) (9) ...
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Citations
... The case r = 2 is not difficult to prove, and appears as early as [1]. Such facts are termed circular rearrangement inequalities in [14] (see also [2]). We present a simple proof for the case r = 2 and also give a (non-trivial) proof in the case r = 3. Observe that ...
... In other words we believe that the greedy permutation maximises P r for every r and every a m with decreasing entries. In the language of [14] this says that the circular symmetrical order maximises P r . In [14] it was shown that the so-called circular alternating order minimises P 2 . ...
... In the language of [14] this says that the circular symmetrical order maximises P r . In [14] it was shown that the so-called circular alternating order minimises P 2 . As noted earlier, Conjectures 5 and 8 have been verified in the special case p 0 = 1 in [3]. ...
A birth-death chain is a discrete-time Markov chain on the integers whose transition probabilities $p_{i,j}$ are non-zero if and only if $|i-j|=1$. We consider birth-death chains whose birth probabilities $p_{i,i+1}$ form a periodic sequence, so that $p_{i,i+1}=p_{i \mod m}$ for some $m$ and $p_0,\ldots,p_{m-1}$. The trajectory $(X_n)_{n=0,1,\ldots}$ of such a chain satisfies a strong law of large numbers and a central limit theorem. We study the effect of reordering the probabilities $p_0,\ldots,p_{m-1}$ on the velocity $v=\lim_{n\to\infty} X_n/n$. The sign of $v$ is not affected by reordering, but its magnitude in general is. We show that for Lebesgue almost every choice of $(p_0,\ldots,p_{m-1})$, exactly $(m-1)!/2$ distinct speeds can be obtained by reordering. We make an explicit conjecture of the ordering that minimises the speed, and prove it for all $m\leq 7$. This conjecture is implied by a purely combinatorial conjecture that we think is of independent interest.
... This is a variant of rearrangement inequalities considered in Ref. [19]. Let σ m1 denote the permutation p1, n´1, 3, n´3, 5,¨¨¨, n´6, 6, n´4, 4, n´2, 2, nq and σ m2 denote the permutationp1, 3, 5,¨¨¨, n,¨¨¨, 6, 4, 2q. ...
The rearrangement inequality states that the sum of products of permutations of 2 sequences of real numbers are maximized when the terms are similarly ordered and minimized when the terms are ordered in opposite order. We show that similar inequalities exists for multi-valued logic with the multiplication and addition operation replaced with various T-norms and T-conorms respectively.
... The case r = 2 is not difficult to prove, and appears as early as [1]. Such facts are termed circular rearrangement inequalities in [10] (see also [2]). We present a simple proof for the case r = 2 and also give a (non-trivial) proof in the case r = 3. Observe that ...
... In other words we believe that the greedy permutation maximises P r for every r and every a m with decreasing entries. In the language of [10] this says that the circular symmetrical order maximises P r . In [10] it was shown that the so-called circular alternating order minimises P 2 . ...
... In the language of [10] this says that the circular symmetrical order maximises P r . In [10] it was shown that the so-called circular alternating order minimises P 2 . The following two examples (which can be verified by simply evaluating the cyclic products for all possible permutations) show that the minimal ordering is neither constant over r for fixed a m , nor constant over a m for fixed r. ...
A birth-death chain is a discrete-time Markov chain on the integers whose transition probabilities $p_{i,j}$ are non-zero if and only if $|i-j|=1$. We consider birth-death chains whose birth probabilities $p_{i,i+1}$ form a periodic sequence, so that $p_{i,i+1}=p_{i \mod m}$ for some $m$ and $p_0,\ldots,p_{m-1}$. The trajectory $(X_n)_{n=0,1,\ldots}$ of such a chain satisfies a strong law of large numbers and a central limit theorem. We study the effect of reordering the probabilities $p_0,\ldots,p_{m-1}$ on the velocity $v=\lim_{n\to\infty} X_n/n$. The sign of $v$ is not affected by reordering, but its magnitude in general is. We show that for Lebesgue almost every choice of $(p_0,\ldots,p_{m-1})$, exactly $(m-1)!/2$ distinct speeds can be obtained by reordering. We make an explicit conjecture of the ordering that minimises the speed, and prove it for all $m\leq 7$. This conjecture is implied by a purely combinatorial conjecture that we think is of independent interest.
... The minimum of (1.3) for the special case F (x, y) = xy was dealt with recently by H. Yu in [8]. There the author introduced the circular alternating order arrangement of a set (x) and proved that ∑ n i=1 x i x i+1 , x n+1 = x 1 gets its minimum for this specific arrangement. ...
... When the ordered set (x) = (x 1 , ..., x n ) of n terms satisfies (1.4), then the ordered set (x 2 , ..., x n ) of n − 1 terms satisfies (1.5). DEFINITION 2. (See [5]) A circular rearrangement of an ordered set (x) is a cyclic rearrangement of (x) or a cyclic rearrangement followed by inversion; For example, the circular rearrangements of the ordered set (1,2,3,4) are the sets [8] and the illustrations in Figure 3 and Figure 4). An ordered set (x) = (x 1 , ..., x n ) of n real numbers is arranged in alternating order if (1.6) or if ...
... DEFINITION 5. A set (x) is arranged in circular alternating order if one of its circular rearrangements is arranged in an alternating order. DEFINITION 6. (See [8]). Given the set ...
The rearrangement inequality states that the sum of products of permutations of 2 sequences of real numbers are maximized when the terms are similarly ordered and minimized when the terms are ordered in opposite order. We show that similar inequalities exist in algebras of multi-valued logic when the multiplication and addition operations are replaced with various T-norms and T-conorms respectively. For instance, we show that the rearrangement inequality holds when the T-norms and T-conorms are derived from Archimedean copulas.
We consider Sturm-Liouville problems with a boundary condition linearly dependent on the eigenparameter. We concentrate the study on the cases where non-real or non-simple (multiple) eigenvalues are possible. We prove that the system of root (i.e. eigen and associated) functions of the corresponding operator, with an arbitrary function removed, form a minimal system in L2(0, 1), except some cases where this system is neither complete nor minimal. The method used is based on the determination of the explicit form of the biorthogonal system. These minimality results can be extended to basis properties in L2(0, 1).
In this paper we extend the well known Heinz inequality which says that 2a1a2≤H(t)≤a1+a2, a1, a2 > 0, 0 ≤ t ≤ 1, where H(t)=a1ta21−t+a11−ta2t. We discuss the bounds of H(t) in the intervals t ∈ [1, 2] and t ∈ [2, ∞) using the subquadracity and the superquadracity of φ(x) = xt, x ≥ 0 respectively. Further, we extend H(t) to get results related to ∑i=1nHi(t)=∑i=1naitai+11−t+ai1−tai+1t, an+1 = a1, ai > 0, i = 1, …, n, where H1(t) = H(t). These results, obtained by using rearrangement techniques, show that the minimum and the maximum of the sum ∑i=1nHi(t) for a given t, depend only on the specific arrangements called circular alternating order rearrangement and circular symmetrical order rearrangement of a given set a=a1,a2,…,an, ai > 0, i = 1, 2, …, n. These lead to extended Heinz type inequalities of ∑i=1nHi(t) for different intervals of t. The results may also be considered as special cases of Jensen type inequalities for concave, convex, subquadratic and superquadratic functions, which are also discussed in this paper.
