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Articles 391 - 396 of 396

Full-Text Articles in Mathematics

On Polynomial Solutions Of A Class Of Linear Differential Equations Of The Second Order, W. C. Brenke Jan 1930

On Polynomial Solutions Of A Class Of Linear Differential Equations Of The Second Order, W. C. Brenke

Department of Mathematics: Faculty Publications

Certain well known polynomials have a number of common properties. They arise as coefficients of tn in the expansion of a generating function ; they may be obtained by means of orthogonalization of a set of functions xⁿg(x) when the function ρ(x) = g2(x) and the interval are properly chosen ; they may be regarded as polynomials which become orthogonal when multiplied by a proper factor g(x) ; they satisfy a certain type of difference equation ; they satisfy a certain type of differential equation. The results …

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Matrices Whose Characteristic Equations Are Cyclic, T. A. Pierce Jan 1930

Matrices Whose Characteristic Equations Are Cyclic, T. A. Pierce

Department of Mathematics: Faculty Publications

One of Sylvester's theorems f on matrices states that if the characteristic equation
(1) | M - λI| = f(λ) = 0
of a square matrix M has the roots λ1, λ2, … , λn, then the characteristic equation
(2) | φM - ρI| = = g(ρ) = 0
of any integral function of M, namely, φM, has the roots ρi = φ (λi), i = 1, 2, … , n. In this note an isomorphism is shown to exist between …

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Symmetric Functions Of N-Ic Residues (Mod P), T. A. Pierce Jan 1929

Symmetric Functions Of N-Ic Residues (Mod P), T. A. Pierce

Department of Mathematics: Faculty Publications

If p be an odd prime, q is said to be an n- ic residue of p if the congruence xn = q (mod p) has solutions; otherwise q is an n-ic non-residue of p. A necessary and sufficient condition that q be an n-ic residue of p is that
(1) q (p- 1)/δ ≡ 1, (mod p),
where δ = g.c.d. (p - 1, n). The numberf of n-ic residues of a given prime p is (p-1)/δ.
It is with the symmetric functions of these …

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An Approximation To The Least Root Of A Cubic Equation With Application To The Determination Of Units In Pure Cubic Fields, T. A. Pierce Jan 1926

An Approximation To The Least Root Of A Cubic Equation With Application To The Determination Of Units In Pure Cubic Fields, T. A. Pierce

Department of Mathematics: Faculty Publications

Approximation to the Least Root of a Cubic Bernoulli's method of approximating the largest root of an equation
(1) x3 = ax2 + bx + c,
with real coefficients, is to use (1) as a scale of relation for the recursion formula An = aAn-1 + bAn-2 + cAn-3. Successive A’s are calculated starting from any initial values. Then An+i/An for increasing values of n approximates that root of (1) which has the greatest absolute value if that root is real. The method here given for …

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A Set Of Completely Independent Postulates For The Linear Order Η*, M. G. Gaba Jan 1920

A Set Of Completely Independent Postulates For The Linear Order Η*, M. G. Gaba

Department of Mathematics: Faculty Publications

Professor E. V. Huntington has published three sets of completely independent postulates for serial order. His set A involves four postulates, which is as high a number of postulates as had been proved completely independent. In the present paper are given seven postulates which form a categorical and completely independent set for the linear order.
Our basis is a class of elements [p] and an undefined dyadic relation (called 'less than') among the elements. If we are given two elements p1p2 and if the relation p1 less than p2 holds, we will symbolize …

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A Theorem On Semi-Continuous Functions, Henry Blumberg Jan 1918

A Theorem On Semi-Continuous Functions, Henry Blumberg

Department of Mathematics: Faculty Publications

Recently G. C. Young and A. Denjoy have communicated theorems—those in Denjoy's memoir are of an especially comprehensive character—dealing, in particular, with point sets where the four derivatives of a given continuous function are identical. It is the purpose of this note to treat an analogous problem that arises when "derivative" is replaced by "saltus." However, instead of confining ourselves to "saltus," we prove a more general theorem that applies essentially to all semi-continuous functions. We preface the proof of this theorem with the following

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