A new look at the optimal assignment problem

Let E=[e_ij] be a matrix with integral elements. We study matrices of the form X=[e^e_ij], where x is an indeterminate defined over the rational field Q. There is a fascinating interplay between the combinatorial structures of the matrices E and X. For example, in the case of square matrices the number of optimal assignments in the matrix E is precisely the coefficient of the term of highest degree in the polynomial per(X). We allow the multiplication of rows and columns of X by arbitrary integral powers of x, and we study topics such as the diagonal products of X. We apply certain familiar results to this “exponential” setting, and we also introduce some new concepts, such as the class of matrices S(X) generated by a matrix X and the notion of an invariant 1 within such a class.     

A problem on brachistochrone as invariant variational problem

Based on the modified theory of invariant variational problems developed by the author, we describe a theoretical group approach to the problem of brachistochrone.     

A new look at the doubling algorithm for a structured palindromic quadratic eigenvalue problem

Recently, Guo and Lin [SIAM J. Matrix Anal. Appl., 31 (2010), 2784–2801] proposed an efficient numerical method to solve the palindromic quadratic eigenvalue problem (PQEP) (λ²A^T+λQ + A)z = 0 arising from the vibration analysis of high speed trains, where have special structures: both Q and A are, among others, m × m block matrices with each block being k × k (thus, n = mk), and moreover, Q is block tridiagonal, and A has only one nonzero block in the (1,m)th block position. The key intermediate step of the method is the computation of the so‐called stabilizing solution to the n × n nonlinear matrix equation X + A^TX⁻¹A = Q via the doubling algorithm. The aim of this article is to propose an improvement to this key step through solving a new nonlinear matrix equation having the same form but of only k × k in size. This new and much smaller matrix equation can also be solved by the doubling algorithm. For the same accuracy, it takes the same number of doubling iterations to solve both the larger and the new smaller matrix equations, but each doubling iterative step on the larger equation takes about 4.8 as many flops than the step on the smaller equation. Replacing Guo's and Lin's key intermediate step by our modified one leads to an alternative method for the PQEP. This alternative method is faster, but the improvement in speed is not as dramatic as just for solving the respective nonlinear matrix equations and levels off as m increases. Numerical examples are presented to show the effectiveness of the new method. Copyright © 2014 John Wiley & Sons, Ltd.     

A new look at quarks confinement

Quarks confinement is an experimental fact. ‘tHooft and later on Gross, Wilczek and Politzer have contributed in various ways to our present considerable theoretical understanding of the problem. However, an exact water proof theoretical derivation of the problem is at best still in progress. The present note argues that to understand quarks confinement, a deeper understanding of the Planck scale physics is indispensable and shows using analytical topological arguments that absolute confinement is a result of a phase transition of quantum spacetime at the Planck scale.     

A new look at fractional programming

The parametric approach to fractional programming problems is examined and a new format is proposed for it. The latter reflects the fact that the approach as a whole capitalizes on a first-order necessary and sufficient optimality condition pertaining to differentiable pseudolinear functions.     

The brachistochrone problem for a disc

The motion of a vertical disc along a curve under the influence of gravity is investigated. On the assumption of regular rolling without slip and separation of contact points, the problem of plotting the curve of most rapid motion of the disc centre from the origin of coordinates to an arbitrary fixed point of the lower half-plane is solved. As usual, the velocity at the initial instant of time is zero, and at the final instant of time it is not fixed. In explicit parametric form, the classical brachistochrone for contact points of the disc is plotted and investigated. The response time, trajectory and kinematic and dynamic characteristics of motion are calculated analytically. Previously unknown qualitative properties of regular rolling are established. In particular, it is shown that the disc centre moves along a cycloid connecting specified points. The envelopes of the boundary points of the disc, produced as its centre moves along the cycloid, are brachistochrones. The feasibility of mechanical coupling of the disc and the curve by reaction forces at the contact point (the normal pressure and dry friction) is investigated.     

A new look at solving inequalities

Several ways for solving inequalities have been suggested in the literature, among them graphical, verbal and algebraic solutions. A new look at solving inequalities via use of the intermediate value theorem is presented here, and this method may prove to be straightforward for many students.     

A new look at the homogeneous risk model

The present paper aims to revisit the homogeneous risk model investigated by [De Vylder and Goovaerts, 1999] and [De Vylder and Goovaerts, 2000]. First, a claim arrival process is defined on a fixed time interval by assuming that the arrival times satisfy an order statistic property. Then, the variability and the covariance of an aggregate claim amount process is discussed. The distribution of the aggregate discounted claims is also examined. Finally, a closed-form expression for the non-ruin probability is derived in terms of a family of Appell polynomials. This formula holds for all claim distributions, even dependent. It generalizes several results obtained so far.     

A look at a problem in heat transfer

Consideration and careful investigation of a simple physical problem in heat transfer shows that even such a problem may not always have solutions and meaningful solutions exist only under restricted conditions. In other words, not all physical problems are well posed. The analysis given here shows that in many practical problems in fluid mechanics and heat transfer the question ‘how well posed is it?’ is non-trivial.     

A new look at the homogeneous risk model

The present paper aims to revisit the homogeneous risk model investigated by De Vylder and Goovaerts, 1999, De Vylder and Goovaerts, 2000. First, a claim arrival process is defined on a fixed time interval by assuming that the arrival times satisfy an order statistic property. Then, the variability and the covariance of an aggregate claim amount process is discussed. The distribution of the aggregate discounted claims is also examined. Finally, a closed-form expression for the non-ruin probability is derived in terms of a family of Appell polynomials. This formula holds for all claim distributions, even dependent. It generalizes several results obtained so far.     

A new look at the chemical master equation

This paper discusses the numerical simulation of stochastic chemical kinetics from a point of view that is opposite to the one prevalent in the literature. Specifically, we re-derive existing methods by first discretizing in time the chemical master equation, then sampling the resulting numerical solution (which is a probability density). This analysis reveals the hidden approximations made by the stochastic simulation algorithm and by the tau-leaping method, and opens the way for constructing new families of solvers.     

Nonclassical pseudospectral method for the solution of brachistochrone problem

In this paper, nonclassical pseudospectral method is proposed for solving the classic brachistochrone problem. The brachistochrone problem is first formulated as a nonlinear optimal control problem. Properties of nonclassical pseudospectral method are presented, these properties are then utilized to reduce the computation of brachistochrone problem to the solution of algebraic equations. Using this method, the solution to the brachistochrone problem is compared with those in the literature.     

A new look at localic interpolation theorems

This paper presents a new treatment of the localic Katětov-Tong interpolation theorem, based on an analysis of special properties of normal frames, which shows that it does not hold in full generality. Besides giving us the conditions under which the localic Katětov-Tong interpolation theorem holds, this approach leads to a especially transparent and succinct proof of it. It is also shown that this pointfree extension of Katětov-Tong theorem still covers the localic versions of Urysohn's Lemma and Tietze's Extension Theorem.     

A new look at totally positive matrices

A close relationship between the class of totally positive matrices and anti-Monge matrices is used for suggesting a new direction for investigating totally positive matrices. Some questions are posed and a partial answer in the case of Vandermonde-like matrices is given.     

A new look at Jarvis' distribution formula

Starting from the well-known transformation law for the Klein functions, we give a proof of a fairly general multiplicative distribution formula for the Siegel functions associated to isogenous complex lattices. This formula has as an immediate consequence the remarkable distribution formula proved by Jarvis in 2000 on the occasion of Rolshausen's thesis on the second K-group of an elliptic curve.     

A new look at interpretability and saturation

We investigate the interpretability ordering ${?}^{?}$ using generalized Ehrenfeucht–Mostowski models. This gives a new approach to proving inequalities and investigating the structure of types.     

A new look at probability-weighted moments estimators

In this article, we propose to study, in more generality, the probability-weighted moments method used by Hosking and Wallis (1987) in the case of generalized Pareto distributions which depend on two parameters γ and σ. The objective is to extend the domain of validity: γ<1/2 required in order to obtain the asymptotic properties of their estimators. By simulations, we show the efficiency of our technique. To cite this article: J. Diebolt et al., C. R. Acad. Sci. Paris, Ser. I 338 (2004).     

A new look at KK-theory

We describe the Kasparov group KK(A, B) as the set of homotopy classes of homomorphisms from an algebra qA associated with A into K B. The algebra qA consists of K-theory differential forms over A. Its construction is dual to that of M ₂(A). The analysis of qA and of its interplay with M ₂(A) gives the basic results of KK-theory.Partially supported by NSF.