The rank aggregation problem consists in finding a consensus ranking on a set of alternatives, based on the preferences of individual voters. The alternatives are expressed by permutations, whose pairwise distance can be measured in many ways. In this work we study a collection of distances, including the Kendall tau, Spearman footrule, Minkowski, Cayley, Hamming, Ulam, and related edit distanc...

A new model of an evolution ranks employees due to staff turnover in organization is designed. If the rank determined only by performance, shift incumbents proportional initial rank: status high reduced slightly. This effect that has been observed literature. However, if also depends on some other variable, positions may be greatly deteriorated. In case, nonlinearly time $\tau$ between subseque...

This paper is the last of a series of papers devoted to the bispectral problem [3]–[6]. Here we examine the connection between the bispectral operators constructed in [6] and the Lie algebra W1+∞ (and its subalgebras). To give a more detailed idea of the contents of the present paper we briefly recall the results of [4]–[6] which we need. In [4] we built large families of representations of W1+...

For each affine Kac-Moody algebra $X_n^{(r)}$ of rank $\ell$, $r=1,2$, or $3$, and for every choice a vertex $c_m$, $m=0,\dots,\ell$, the corresponding Dynkin diagram, by using matrix-resolvent method we define gauge-invariant tau-structure associated Drinfeld-Sokolov hierarchy give explicit formulas generating series logarithmic derivatives tau-function in terms matrix resolvents, extending re...

We define the algebraic variety of almost intertwining matrices to be the set of triples ~X,Y,Z! of n3n matrices for which XZ5YX1T for a rank one matrix T. A surprisingly simple formula is given for tau functions of the KP hierarchy in terms of such triples. The tau functions produced in this way include the soliton and vanishing rational solutions. The induced dynamics of the eigenvalues of th...

Introduction: Alzheimer’s disease (AD) affects 35 million people worldwide (Dartigues, 2009). AD research using MRI has focused on early changes in medial temporal and cortical gray matter. Although caudate atrophy is more typical of motor disorders such as Parkinson’s disease (Nakamura et al. 2001), beta-amyloid and tau pathology accumulates in the caudate in AD (Braak and Braak, 1990). Here w...

‎Let $R$ be a $*$-prime ring with center‎ ‎$Z(R)$‎, ‎$d$ a non-zero $(sigma,tau)$-derivation of $R$ with associated‎ ‎automorphisms $sigma$ and $tau$ of $R$‎, ‎such that $sigma$‎, ‎$tau$‎ ‎and $d$ commute with $'*'$‎. ‎Suppose that $U$ is an ideal of $R$ such that $U^*=U$‎, ‎and $C_{sigma,tau}={cin‎ ‎R~|~csigma(x)=tau(x)c~mbox{for~all}~xin R}.$ In the present paper‎, ‎it is shown that if charac...

Sato theory provides a correspondence between solutions to the KP hierarchy and points in an infinite dimensional Grassmannian. In this correspondence, flows generated infinitesimally by powers of the “shift” operator give time dependence to the first coordinate of an arbitrarily selected point, making it a tau-function. These tau-functions satisfy a number of integrable equations, including th...

Comparing and ranking information is an important topic in social and information sciences, and in particular on the web. Its objective is to measure the difference of the preferences of voters on a set of candidates and to compute a consensus ranking. Commonly, each voter provides a total order of all candidates. In this work we consider the generalization of total orders and bucket orders to ...