Polar and singular value decomposition of 3×3 magic squares

In the literature there have been reported a number of studies of probabilities involved in Monopoly, the worldwide popular game designed by Charles B. Darrow in the 1930s. Simple assumptions often lead to inaccurate results. In this paper, having distinguished between ‘In Jail’ and ‘Just Visiting’, both theoretical computations and computer simulations are conducted to determine the probability of landing on ‘JAIL’. Theoretical and simulation results are consistent. The sum of the probabilities of landing on ‘In Jail’ and ‘Just Visiting’ is found to be 6.40%, both in theory and in simulation, compared with Ian Stewart's 5.89%. The long-term probabilities of landing on the other 39 spaces are computed and it is concluded that the best properties to buy are the orange ones.     

The Adomian decomposition method with Green’s function for solving nonlinear singular boundary value problems

In this paper, we present an efficient numerical algorithm for solving a general class of nonlinear singular boundary value problems. This present algorithm is based on the Adomian decomposition method (ADM) and Green’s function. The method depends on constructing Green’s function before establishing the recursive scheme. In contrast to the existing recursive schemes based on ADM, the proposed numerical algorithm avoids solving a sequence of transcendental equations for the undetermined coefficients. The approximate series solution is calculated in the form of series with easily computable components. Moreover, the convergence analysis and error estimation of the proposed method is given. Furthermore, the numerical examples are included to demonstrate the accuracy, applicability, and generality of the proposed scheme. The numerical results reveal that the proposed method is very effective.     

Numerical solution of singular boundary value problems using Green’s function and improved decomposition method

In this work, an effective technique for solving a class of singular two point boundary value problems is proposed. This technique is based on the Adomian decomposition method (ADM) and Green’s function. The technique relies on constructing Green’s function before establishing the recursive scheme for the solution components. In contrast to the existing recursive schemes based on ADM, the proposed recursive scheme avoids solving a sequence of nonlinear algebraic or transcendental equations for the undetermined coefficients. The approximate solution is obtained in the form of series with easily calculable components. For the completeness, the convergence and error analysis of the proposed scheme is supplemented. Moreover, the numerical examples are included to demonstrate the accuracy, applicability, and generality of the proposed scheme. The results reveal that the method is very effective, straightforward, and simple.     

The second generalized Frankl’ problem for the Chaplygin equation with a singular coefficient

In this paper we consider a generalized Frankl’ problem for the Chaplygin equation with a singular coefficient. By using the method of integral equations we prove the unique solvability of the mentioned problem.     

The distribution |f|λ, oscillating integrals and principal value integrals

We study the coefficients of asymptotic expansions of oscillating integrals. We also consider the connection with the coefficients of Laurent expansions at candidate poles of the distribution |f|^λ and show that some of these coefficients vanish. Next, we express some of the most important of these coefficients as the so-called principal value integrals, first introduced by Langlands. Together with our results on principal value integrals, this leads to new results on the vanishing of these coefficients.     

The generalized Krawtchouk polynomials and the fifth Painlevé equation

We study the recurrence coefficients of the orthogonal polynomials with respect to a semi-classical extension of the Krawtchouk weight. We derive a coupled discrete system for these coefficients and show that they satisfy the fifth Painlevé equation when viewed as functions of one of the parameters in the weight.     

Whose deletion does not affect your payoff? The difference between the Shapley value, the egalitarian value, the solidarity value, and the Banzhaf value

This study provides a unified axiomatic characterization method of one-point solutions for cooperative games with transferable utilities. Any one-point solution that satisfies efficiency, the balanced cycle contributions property (BCC), and the axioms related to invariance under a player deletion is characterized as a corollary of our general result. BCC is a weaker requirement than the well-known balanced contributions property. Any one-point solution that is both symmetric and linear satisfies BCC. The invariance axioms necessitate that the deletion of a specific player from games does not affect the other players’ payoffs, and this deletion is different with respect to solutions. As corollaries of the above characterization result, we are able to characterize the well-known one-point solutions, the Shapley, egalitarian, and solidarity values, in a unified manner. We also studied characterizations of an inefficient one-point solution, the Banzhaf value that is a well-known alternative to the Shapley value.     

The recursion operator method for generalized symmetries and Bäcklund transformations of the Burgers’ equations

In this paper, the generalized symmetries of the second-order Burgers’ equation are obtained through the symmetry transformation method. The Bäcklund transformations (BTs) of the two equations are constructed by the recursion operator method. Then, the infinite number of exact solutions to these equations are investigated in terms of the generalized symmetries and Bäcklund transformations. Furthermore, the Bäcklund transformations and conservation law of the general Burgers’ equations are discussed.     

The generalized Laguerre inequalities and functions in the Laguerre-Pólya class

The principal goal of this paper is to show that the various sufficient conditions for a real entire function, φ(x), to belong to the Laguerre-Pólya class (Definition 1.1), expressed in terms of Laguerre-type inequalities, do not require the a priori assumptions about the order and type of φ(x). The proof of the main theorem (Theorem 2.3) involving the generalized real Laguerre inequalities, is based on a beautiful geometric result, the Borel-Carathédodory Inequality (Theorem 2.1), and on a deep theorem of Lindelöf (Theorem 2.2). In case of the complex Laguerre inequalities (Theorem 3.2), the proof is sketched for it requires a slightly more delicate analysis. Section 3 concludes with some other cognate results, an open problem and a conjecture which is based on Cardon’s recent, ingenious extension of the Laguerre-type inequalities.     

The optimal exchange algorithm and comparisons with the generalized remes algorithm † ‡

This paper is concerned with the general problem of the determination of the best uniform approximation of a given function. A special case is the calculation of the minimax solution of an overdetermined linear system. Single point exchange algorithms produce successive approximate solutions for such problems. An example is furnished by the generalized Remes algorithm, which includes both the original Remes algorithm and the Stiefel algorithm as special cases. The optimal exchange algorithm is similar, but it has the important feature that every exchange is optimal in a certain desirable sense. It is proved that eventually the optimal exchange and generalized Remes algorithms coincide. However, early exchanges in the latter algorithm can be far from optimal and quite inefficient. A combination of the optimal exchange and generalized Remes algorithms is suggested as a reasonable strategy.