The proof by Andrew Wiles of Fermat’s Last Theorem in 1995 resolved the existence question for non-trivial solutions in integers x,y,zto the equation xn+yn=znfor n>2. There are none. Surprisingly, there are infini...The proof by Andrew Wiles of Fermat’s Last Theorem in 1995 resolved the existence question for non-trivial solutions in integers x,y,zto the equation xn+yn=znfor n>2. There are none. Surprisingly, there are infinitely many solutions if the problem is recast in terms of modular arithmetic. Over a hundred years ago Issai Schur was able to show that for any n there is always a sufficiently large prime p0such that for all primes p≥p0the congruence xn+yn≡zn(modp)has a non-trivial solution. Schur’s argument wasnon-constructive, and there is no systematic method available at present to construct specific examples for small primes. We offer a simple method for constructing all possible solutions to a large class of congruences of this type.展开更多
There exist many iterative methods for computing the maximum likelihood estimator but most of them suffer from one or several drawbacks such as the need to inverse a Hessian matrix and the need to find good initial ap...There exist many iterative methods for computing the maximum likelihood estimator but most of them suffer from one or several drawbacks such as the need to inverse a Hessian matrix and the need to find good initial approximations of the parameters that are unknown in practice. In this paper, we present an estimation method without matrix inversion based on a linear approximation of the likelihood equations in a neighborhood of the constrained maximum likelihood estimator. We obtain closed-form approximations of solutions and standard errors. Then, we propose an iterative algorithm which cycles through the components of the vector parameter and updates one component at a time. The initial solution, which is necessary to start the iterative procedure, is automated. The proposed algorithm is compared to some of the best iterative optimization algorithms available on R and MATLAB software through a simulation study and applied to the statistical analysis of a road safety measure.展开更多
文摘The proof by Andrew Wiles of Fermat’s Last Theorem in 1995 resolved the existence question for non-trivial solutions in integers x,y,zto the equation xn+yn=znfor n>2. There are none. Surprisingly, there are infinitely many solutions if the problem is recast in terms of modular arithmetic. Over a hundred years ago Issai Schur was able to show that for any n there is always a sufficiently large prime p0such that for all primes p≥p0the congruence xn+yn≡zn(modp)has a non-trivial solution. Schur’s argument wasnon-constructive, and there is no systematic method available at present to construct specific examples for small primes. We offer a simple method for constructing all possible solutions to a large class of congruences of this type.
文摘There exist many iterative methods for computing the maximum likelihood estimator but most of them suffer from one or several drawbacks such as the need to inverse a Hessian matrix and the need to find good initial approximations of the parameters that are unknown in practice. In this paper, we present an estimation method without matrix inversion based on a linear approximation of the likelihood equations in a neighborhood of the constrained maximum likelihood estimator. We obtain closed-form approximations of solutions and standard errors. Then, we propose an iterative algorithm which cycles through the components of the vector parameter and updates one component at a time. The initial solution, which is necessary to start the iterative procedure, is automated. The proposed algorithm is compared to some of the best iterative optimization algorithms available on R and MATLAB software through a simulation study and applied to the statistical analysis of a road safety measure.
