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Displaying 51 - 60 of 85

The article supplies a short, elementary proof that for integers \(a_1 < a_2 < \cdots < a_n \), the expression \( \prod_{n \geq i > j \geq 1} \frac{a_i - a_j}{i-j} \) is an integer...

Consider a tournament with \(n\) players where each player plays every other player once, and ties are not allowed.  An \( n \times n\) tournament matrix \(A\) is constructed where...

The fact that the limit of the \(n\)-th power of a \(2\times 2\) matrix \(A\) tends to \(0\) if  \( \det A < 1\) and \( \mid 1 + \det(A) \mid > \mid\) tr\( (A) \mid \) is used...

The author shows that a finite group \(G\) can be represented as a group of invertible \( 2 \times 2\) integer matrices if and only if \(G\) is isomorphic to a subgroup of the dihedral groups...

The author shows that the volume of an ellipsoid can be determined by three parallel slices.

Given symmetric unit vectors \(u_i\), conditions on real numbers \(x_i\) are considered to be able to conclude that \(\sum x_i u_i \Rightarrow x_i = 0\) for all \(i\). Different kinds of...

The author discusses the relationship of two well-known, apparently unrelated supermultiplicative functions of nonnegative matrices, and shows they are special cases of a more general...

The author derives formulas for the coefficients of the characteristic polynomial of a square matrix in terms of the partial derivatives of its determinant, which is viewed as a function of...

Suppose we have several alternatives that we wish to rank.  For example, we may wish to rank five teachers according to their teaching excellence.  The author constructs a positive...

Consider \(n\) vectors issuing from the origin in \(n\)-dimensional space.  The author shows that the statement “any set of \(n\) vectors in \(n\)-space, no two of which meet at greater...

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