Mixed-Up-Ness or Entropy?

Simple SummaryThe second law of thermodynamics has a mystical appeal in disciplines with tenuous connections to its origins. We hypothesize that many of these appeals instead should be to another principle heretofore unrecognized: the law of mixed-up-ness (LOM). Instead of using a number such as entropy to characterize randomness, non-thermodynamic systems can be arranged in simple diagrams according to their degree of mixed-up-ness. Curiously, the evolution of such systems from degrees of low to high mixed-up-ness is both consistent with and richer than the principle of increasing entropy.AbstractMixed-up-ness can be traced to unpublished notes by Josiah Gibbs. Subsequently, the concept was developed independently, and under somewhat different names, by other investigators. The central idea of mixed-up-ness is that systems states can be organized in a hierarchy by their degree of mixed-up-ness. In its purest form, the organizing principle is independent of thermodynamic and statistical mechanics principles, nor does it imply irreversibility. Yet, Gibbs and subsequent investigators kept entropy as the essential concept in determining system evolution, thus retaining the notion that systems evolve from states of perfect “order” to states of total “disorder”. Nevertheless, increasing mixed-up-ness is consistent with increasing entropy; however, there is no unique one-to-one connection between the two. We illustrate the notion of mixed-up-ness with an application to the permutation function of integer partitions and then formalize the notion of mixed-up-ness as a fundamental hierarchal principle, the law of mixed-up-ness (LOM), for non-thermodynamic systems.     

Schur-Ostrowski type theorems revisited

In this paper we extend Schur-Ostrowski theorem on Schur-convex functions from majorized vectors to separable ones. For this, we introduce a generalized Schur-Ostrowski?s condition. We apply the obtained result for cone orderings and group-induced cone orderings. Finally, we give some interpretations for absolutely weak majorization and for group majorization on the space of complex matrices.     

n维单形的纳斯必特—彼得洛维奇不等式

利用优超理论将平面上关于三角形的纳斯必特彼得洛维奇不等式推广到 n维欧几里得空间中的 n维单形上 ,得到N 2n( N -1 ) d+nN ≤∑Nk=1sd+ak∑Ni=1,i≠ kak≤ N -nn +nn-1 ( d+1 ) ,式中 ai i=1 ,… ,N ;N =n( n+1 )2 为 n维单形 ∑A的棱长 ,d为任一非负实数 ,s=1n∑Ni=1ai     

Adamovic''''s不等式的推广与加细

In this article, by means of the theory of majorization, Adamovic's inequality is extended to the cases of the general elementary symmetric functions and its duals, and the refined and reversed forms are also given. As applications, some new inequalities for simplex are established.     

Some New Results on Stochastic Comparisons of Spacings from Heterogeneous Exponential Distributions

Some new results are obtained on stochastic orderings between random vectors of spacings from heterogeneous exponential distributions and homogeneous ones. LetD₁, …, D_nbe the normalized spacings associated with independent exponential random variablesX₁, …,X_n, whereX_ihas hazard rateλ_i,i=1, 2, …, n. LetD*₁, …, D*_nbe the normalized spacings of a random sampleY₁, …, Y_nof sizenfrom an exponential distribution with hazard rateλ=∑ⁿ_i=1 λ_i/n. It is shown that for anyn2, the random vector (D₁, …, D_n) is greater than the random vector (D*₁, …, D*_n) in the sense of multivariate likelihood ratio ordering. It also follows from the results proved in this paper that for anyjbetween 2 andn, the survival function ofX_j:n−X_1:nis Schur convex.     

n维单形的伍德几何不等式

利用优超理论将平面上关于三角形的伍德(Wood)不等式推广到n维欧几里得空间中的n维单形上,得到2NN-1≤∑Ni=1ai2∑Ni=1ai∑Ni相似文献   

Feedback invariants of pairs of matrices and restrictions

If (A,B) εF ^n×n ×F ^×m is a given pair and S is an (A,B)-invariant subspace we investigate the relationship between the feedback invariants of (A, B) and those of its restrictions

to S.     

单项式的控制序及逐次差分代换算法的终止性

We introduce a concept for the majorization order on monomials. With the help of this order, we derive a necessary condition on the positive termination of a general successive difference substitution algorithm(KSDS) for an input form f.     

Stochastic orders of scalar products with applications

In this paper, we study stochastic orders of scalar products of random vectors. Based on the study of Ma [Ma, C., 2000. Convex orders for linear combinations of random variables. J. Statist. Plann. Inference 84, 11-25], we first obtain more general conditions under which linear combinations of random variables can be ordered in the increasing convex order. As an application of this result, we consider the scalar product of two random vectors which separates the severity effect and the frequency effect in the study of the optimal allocation of policy limits and deductibles. Finally, we obtain the ordering of the optimal allocation of policy limits and deductibles when the dependence structure of the losses is unknown. This application is a further study of Cheung [Cheung, K.C., 2007. Optimal allocation of policy limits and deductibles. Insurance: Math. Econom. 41, 382-391].     

DISPERSION COMPARISONS OF TWO PROBABILITY VECTORS UNDER MULTINOMIAL SAMPLING

We consider testing hypotheses concerning comparing dispersions between two parameter vectors of multinomial distributions in both one-sample and two-sample cases. The comparison criterion is the concept of Schur majorization. A new dispersion index is proposed for testing the hypotheses. The corresponding test for the one-sample problem is an exact test. For the two-sample problem, the bootstrap is used to approximate the null distribution of the test statistic and the p-value. We prove that the bootstrap test is asymptotically correct and consistent. Simulation studies for the bootstrap test are reported and a real life example is presented.