Group Classification Invariant Solutions of Burgers' Equation

The aims of the present paper is to solve the problem of the group classification of the general Burgers’ equation u_t=f(x,u) u_x^2+g(x,u)u_xx, where f and g are arbitrary smooth functions of the variables x and u, by using Lie method. The paper is one of the few applications of an algebraic approach to the problem of group classification: We followed the analysis mathematical method using the method of preliminary group classification. A number of new interesting nonlinear invariant models which have nontrivial invariance algebras are obtained. The result of the work is a wide class of equations summarized in table form.

“Choose your opponent”: A new knockout design for hybrid tournaments †

We present a new, simple knockout format for sports tournaments, that we call “Choose Your Opponent”, where the teams that have performed best during a preliminary group stage can choose their opponents during the subsequent knockout stage. The main benefit of this format is that it essentially solves a recently identified incentive compatibility problem when more than one teams from a group advance to the knockout stage, by effectively canceling the risk of tanking. This new design also makes the group stage more exciting, by giving teams a strong incentive to perform at their best level, and more fair, by limiting the risk of collusion and making sure that the best group winners are fairly rewarded in the knockout round. The choosing procedure would add a new, exciting strategic component to the competition. Advancing teams would choose their opponent during new, much anticipated TV shows which would attract a lot of media attention. We illustrate how this new format would work for the round of 16 of the UEFA Champions League, the most popular soccer club competition in the world.

Atomic absorption determination of Hg (II), Cd (II), and As (III) in natural and waste water after preliminary group preconcentration

The sorption of Hg (II), Cd (II), and As (III) by natural aluminosilicate is studied. It is shown that the mineral absorbs those toxicants in a rather wide pH range, quantitative extraction of analytes being achieved in a neutral or close to neutral medium (pH values range within 7.0 - 8.0; 6.3 - 7.5; 7.4 - 8.5 for Hg (II), As (III), and Cd (II), respectively). The effect of the time of phase contact on the degree of extraction of elements is shown. The sorption capacity of the mineral in optimal conditions of the medium acidity (0.06 mmol/g for mercury, 0.31 mmol/g for cadmium, and 0.52 mmol/g for arsenic) is determined. The distribution coefficients attain values of aboutnX 103-nX 104. A new combined method for determination of Hg (II), Cd (II), and As (III) in natural and waste water is developed and tested. The method consists in a preliminary group sorption concentration of the analytes by aluminosilicate, desorption of the analytes from the surface of the mineral and their subsequent atomic absorption determination. The correctness of the method is verified in analysis of spiked samples. The method is easy to use and exhibits high sensitivity, reproducibility and accuracy of analyte determination. The relative standard deviation does not exceed 0.13. Economic availability and possibility of using domestic sorption materials are the important advantages of the proposed procedure which can be used in the practice of laboratories monitoring the quality and safety of environmental objects.

GROUP PROPERTIES OF A ONE-DIMENSIONAL NONLINEAR POROELASTICITY EQUATION

This paper studies a class of partial differential equations of second order , with arbitrary functions and , with the help of the group classification. The main Lie algebra of infinitely infinitesimal symmetries is three-dimensional. We use the method of preliminary group classification for obtaining the classifications of these equations for a one-dimensional extension of the main Lie algebra.