Thesis (Ph. D.)--University of Chicago, 1911. / Life. "Reprinted from American Journal of mathematics, vol. XXXVI, no. 2, April, 1914." Includes bibliographical references.

Thesis--Rijksuniversiteit te Groningen. / Half title: Verwantschappen naar aanleiding van de quadratische transformatie.

Proefschrift--Gronigen.

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Let A be an n x n hermitian matrix, let E₂(a₁, …, a_k) be the second elementary symmetric function of the letters a₁, …, a_k and let C₂(A) be the second compound matrix of A. In this thesis the maximum and minimum of det {(Ax_█, x_j)} and E₂ [(Ax₁, x₁), …, (Ax_█(k@), x_k)] the minimum of [formula omitted] (C₂(A)x_i ₁⋀x_i₂ , x_i₁ ⋀ax_i₂) are calculated. The maxima and minima are taken over all sets of k orthonormal vectors in unitary n-space and x_█(i@)₁ ⋀ x_i ₂ designates the Grassman exterior product. These results depend on the inequality E₂(a₁, …, a_k ) ≤ (k/2 ) [formula omitted] which is here established for arbitrary real numbers, and on the minimum of E₂ (x₁, …, x_(k)) where the minimum is taken over all values of x₁, …, x_█(k@) such that ∑_(i=1)^k▒xi = ∑_(i=1)^k▒〖∝i〗 and ∑_(i=1)^q▒xsi ≤ ∑_(i=1)^q▒〖∝i〗 for all sets of q distinct integers s₁, …, s_q taken from 1, …, k. Here α₁ ≥ … ≥ ∝_k. / Science, Faculty of / Mathematics, Department of / Graduate

This paper shall be mostly concerned with the development and the properties of three quadratic polynomials. The primary interest will by with n-ary quadratic polynomials, called forms.

Large integer factoring problem is a difficult computing problem. The security of many public-key cryptograohy system depend on the large interger factoring problem. Dr. Guan implement ¡uThe Multiple Polynomail Quadratic Sieve Algorithm¡v and name the program ¡uGQS¡v. The program successfully factor RSA-130 interger in 2004. It can reduce the time of sieving that the MPQS algorithm retain the smooth number with one or two prime. But finally the size of factor basis is large. We use some of the prime retained by the MPQS algorithm to match with the smooth number and reduce the size of factor basis. And then we can reduce the time of factoring. In this paper, we implement our idea in a AIX server and the result of this paper can be a suggestion of the improvement of MPQS.

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