On the football pool problem for 6 matches: A new upper bound

Consider the set (F₃)⁶ of all 6 tuples x=(x₁,…, x₆) with x_i? {0, 1, 2}. We show that there exists a subset $\text{T}$ of (F₃)⁶ with 79 elements such that for any x of (F₃)⁶ there is an element if $\text{T}$ which differs from x in at most one coordinate.     

The neumann problem for equations of monge-ampère type

We prove the existence of a classical solution to a Neumann type problem for nonlinear elliptic equations of Monge-Ampère type. The methods depend upon the establishment of a priori derivative estimates up to order two and yield a sharp result for the equation of prescribed Gauss curvature. We also present applications to asymptotic problems which approach the non-unique case and maximum principles for linear equations of Aleksandrov-Bakelman type.     

An adaptive ejection pool with toggle-rule diversification approach for the capacitated team orienteering problem

In the capacitated team orienteering problem (CTOP), we are given a set of homogeneous vehicles and a set of customers each with a service demand value and a profit value. A vehicle can get the profit of a customer by satisfying its demand, but the total demand of all customers in its route cannot exceed the vehicle capacity and the length of the route must be within a specified maximum. The problem is to design a set of routes that maximizes the total profit collected by the vehicles. In this article, we propose a new heuristic algorithm for the CTOP using the ejection pool framework with an adaptive strategy and a diversification mechanism based on toggling between two priority rules. Experimental results show that our algorithm can match or improve all the best known results on the standard CTOP benchmark instances proposed by Archetti et al. (2008).     

New central configurations for the planar 7-body problem

In this paper we study numerically the existence of families of planar central configurations for the 7-body problem with the following properties: five bodies are on the vertices of a regular pentagon and the other two bodies are symmetrically positioned in the interior of the pentagon.     

Regularity for the fractional Gelfand problem up to dimension 7

We study the problem (−Δ)^su=λe^u${(-\mathrm{\Delta })}^{s}u=\lambda e^u$ in a bounded domain Ω⊂Rⁿ$\Omega \subset {\mathbb{R}}^n$, where λ   is a positive parameter. More precisely, we study the regularity of the extremal solution to this problem. Our main result yields the boundedness of the extremal solution in dimensions n≤7$n\le 7$ for all s∈(0,1)$s\in (0,1)$ whenever Ω   is, for every i=1,...,n$i=1,...,n$, convex in the _xi$x_i$-direction and symmetric with respect to {_xi=0}$\{x_i=0\}$. The same holds if n=8$n=8$ and s?0.28206...$s?0.28206...$, or if n=9$n=9$ and s?0.63237...$s?0.63237...$. These results are new even in the unit ball Ω=_B1$\Omega =B_1$.     

Ramanujan-type congruences modulo 8 for 7- and 23-core partition functions

Recently, Radu and Sellers proved numerous congruences modulo powers of 2 for \( (2k+1)\)-core partition functions by employing the theory of modular forms. In this paper, employing Ramanujan’s theta function identities, we prove many infinite families of congruences modulo 8 for 7-core partition function. Our results generalize the congruences modulo 8 for 7-core partition function discovered by Radu and Sellers. Furthermore, we present new proofs of congruences modulo 8 for 23-core partition function. These congruences were first proved by Radu and Sellers.     

The professional foul in football: Tactics and deterrents

We demonstrate the use of a model calibrated using data from every 1999–2000 Premiership match to determine the circumstances under which a player ‘should’ commit a professional foul in order to benefit his team. The results are illustrated using two hypothetical matches, one between evenly balanced teams and one where one of the teams is much stronger than the other. These circumstances turn out to be wide-ranging and, in some cases, somewhat counterintuitive. The many drawbacks of the current system for punishing such fouls are discussed, and a simple remedy is proposed that would not only be much fairer to all involved but would also, at a stroke, render the professional foul virtually obsolete.     

The mesa problem for Neumann boundary value problem

In this paper, we study the singular limit of the Porous Medium equation u_t=Δu^m+g(x,u), as m→∞, in a bounded domain with Neumann boundary condition.     

A metaheuristic based on a pool of routes for the vehicle routing problem with multiple trips and time windows

This paper studies the vehicle routing problem with multiple trips and time windows, in which vehicles are allowed to perform multiple trips during a scheduling period and each customer must be served within a given time interval. The problem is of particular importance for planning fleets of hired vehicles in common practices, such as e-grocery distributions, but this problem has received little attention in the literature. As a result of the multi-layered structure characteristic of the problem solution, we propose a pool-based metaheuristic in which various routes are first constructed to fill a pool, following which some of the routes are selected and combined to form vehicle working schedules. Finally, we conduct a series of experiments over a set of benchmark instances to evaluate and demonstrate the effectiveness of the proposed metaheuristic.     

A simulation model for American football plays

A Monte Carlo model is presented which simulates the time history of the players' positions and velocities. Passing plays are excluded. The play is broken into constant-interval epochs at which the players select their respective strategies. These in turn are used in conjuctions with kinematic equations to update the players' state variables. Various objectives are formulated via nonlinear programming problems for the three classes of players (offensive ball-carrier, defensive tacklers and offensive blockers). Interactions between players (blocking and tackling) are postulated from considerations of particle dynamics. Sample results are presented and methods of validating the model are discussed.     

The obstacle problem for beams and plates

The numerical solution of the obstacle problem for beams and plates by means of variational inequalities and finite elements is examined. Algorithms for the solution of the discrete problem are discussed and particular attention is paid to different methods of approximating the constraint. The results of some numerical experiments for beams and plates are included.     

The Minkowski problem for polytopes

The traditional solution to the Minkowski problem for polytopes involves two steps. First, the existence of a polytope satisfying given boundary data is demonstrated. In the second step, the uniqueness of that polytope (up to translation) is then shown to follow from the equality conditions of Minkowski's inequality, a generalized isoperimetric inequality for mixed volumes that is typically proved in a separate context. In this article we adapt the classical argument to prove both the existence theorem of Minkowski and his mixed volume inequality simultaneously, thereby providing a new proof of Minkowski's inequality that demonstrates the equiprimordial relationship between these two fundamental theorems of convex geometry.     

The smallness problem for -algebras

Akemann showed that any von Neumann algebra with a weak^* separable dual space has a faithful normal representation on a separable Hilbert space. He posed the question: If a C^*-algebra has a weak^* separable state space, must it have a faithful representation on a separable Hilbert space? Wright solved this question negatively and showed that a unital C^*-algebra has the weak^* separable state space if and only if it has a unital completely positive map, into a type I factor on a separable Hilbert space, whose restriction to the self-adjoint part induces an order isomorphism. He called such a C^*-algebra almost separably representable. We say that a unital C^*-algebra is small if it has a unital complete isometry into a type I factor on a separable Hilbert space. In this paper we show that a unital C^*-algebra is small if and only if the state spaces of all n by n matrix algebras over the C^*-algebra are weak^*-separable. It is natural to ask whether almost separably representable algebras are small or not. We settle this question positively for simple C^*-algebras but the general question remains open.     

Resolvable BIBDs with block size 8 and index 7

The necessary conditions for the existence of a resolvable BIBD RB(k,λ; v) are λ(v ? 1) = 0(mod k ? 1) and v = 0(mod k). In this article, it is proved that these conditions are also sufficient for k = 8 and λ = 7, with at most 36 possible exceptions. © 1994 John Wiley & Sons, Inc.     

Local structure of 7- and 8-connected graphs

We show that if graph on n vertices is minimally and contraction critically k-connected, then it has at least n/2 vertices of degree k for k = 7,8. Bibliography: 17 titles.     

The weighted Fermat-Torricelli problem for tetrahedra and an “inverse” problem

We study the weighted Fermat-Torricelli problem for tetrahedra in R³ and solve an “inverse” problem by introducing a method of differentiation. The solution of the inverse problem is the main result which states that: Given the Fermat-Torricelli point A₀ with the vertices lie on four prescribed rays, find the ratios between every pair of non-negative weights of two corresponding rays such that the sum of the four non-negative weights is a constant number. An application of the inverse weighted Fermat-Torricelli problem is the strong invariance principle of the weighted Fermat-Torricelli point which gives some classes of tetrahedra that could be named “evolutionary tetrahedra”.     

Global Error Estimators for Order 7, 8 Runge–Kutta Pairs

Dormand, Prince and their colleagues [3–5] showed in a sequence of papers that the approximation of an initial value differential system propagated by a Runge–Kutta pair, together with a continuous approximation obtained using additional derivative values could be utilized to obtain estimates of the global error. They illustrated the results using pairs of orders p–1 and p for several values of p. The current authors [13] have developed a more direct representation of the order conditions, characterized families of global error estimators for Runge–Kutta pairs of arbitrary values of p, and showed that efficient global error estimating Runge–Kutta methods are based on the nodes of a Lobatto quadrature formula. Here, formulas for a good 7, 8 pair, interpolants of each of orders 7 and 8, and global error estimators of orders 10 and 12 illustrate how to obtain global error estimates of orders 9, 10, or 11, for arbitrary initial value systems. One set of graphs indicates that the stated order of the global error estimators is achieved numerically, and a second set illustrates the relative efficiency for several global error estimators when the approximation is propagated with a variable stepsize.