George Polya

Mathematical discovery : on understanding, learning, and teaching problem solving

PATTERNS. The Pattern of Two Loci. The Cartesian Pattern. Recursion. Superposition. TOWARD A GENERAL METHOD. Problems. Widening the Scope. Solutions. Appendix. Bibliography. Index.

CONVESITY OF FUNCTIONALS BY TRANSPLANTATION

Abstract : The dependence of various functionals on their domain of definition is discussed. The functionals are defined by certain extremum problems. The methods of transplanting extremum functions and of variation are applied to the problem of utilizing the knowledge of the functional for a few special domains to obtain knowledge about the same functional in the general case. Functionals such as torsional rigidity, virtual mass, outer conformal radius, and electrostatic capacity are treated. A discussion is given of a theorem of Poincare which permits an easy simultaneous estimation of the N first eigenvalues of a general type of eigenvalue problem. The convexity of various combinations of eigenvalues is studied for the case in which the domain of definition is deformed by stretching or by conformal transformation. The usefulness of the fact that the initial domain D(1) has symmetry properties is indicated. The invariance of the class of harmonic functions in D under conformal mapping can be used to derive convexity statements for some functionals connected with the Greens function for Laplaces equation. A numerical application is given for the torsional rigidity of isosceles triangles and rectangles.

On Picture-Writing*

To write “sun”, “moon” and “tree” in picture-writing, one draws simply a circle, a crescent and some simplified, conventionalized picture of a tree, respectively. Picture-writing was used by some tribes of red Indians and it may well be that more advanced systems of writing evolved everywhere from this primitive system. And so picture-writing may be the ultimate source of the Greek, Latin and Gothic alphabets, the letters of which we currently use as mathematical symbols. I wish to observe that also the primitive picture-writing may be of some use in mathematics.

Graeffe's method for eigenvalues

Let an entire functionF(z) of finite genus have infinitely many zeros which are all positive, and take real values for realz. Then it is shown how to give two-sided bounds for all the zeros ofF in terms of the coefficients of the power series ofF, in fact in terms of the coefficients obtained byGraeffes algorithm applied toF. A simple numerical illustration is given for a Bessel function.

An Elementary Analogue to the Gauss-Bonnet Theorem

(1954). An Elementary Analogue to the Gauss-Bonnet Theorem. The American Mathematical Monthly: Vol. 61, No. 9, pp. 601-603.

Combinations and Permutations

January 5. In his first lecture, Polya discussed in general terms what combinatorics is about: The study of counting various combinations or configurations. He stated with a problem based on the mystical sign known, appropriately, as an “abracadabra”.

Determinants and Quadratic Forms

Interchanging two numbers in the numbering of the vertices is equivalent to the simultaneous interchange of two rows and two columns. If we give opposite vertices of the octahedron numbers that differ by 3, then the determinant

How to Solve It. A New Aspect of Mathematical Method.

\left| {\begin{array}{*{20}c} 0 \hfill & 1 \hfill & 1 \hfill & 0 \hfill & 1 \hfill & 1 \hfill \\ 1 \hfill & 0 \hfill & 1 \hfill & 1 \hfill & 0 \hfill & 1 \hfill \\ 1 \hfill & 1 \hfill & 0 \hfill & 1 \hfill & 1 \hfill & 0 \hfill \\ 0 \hfill & 1 \hfill & 1 \hfill & 0 \hfill & 1 \hfill & 1 \hfill \\ 1 \hfill & 0 \hfill & 1 \hfill & 1 \hfill & 0 \hfill & 1 \hfill \\ 1 \hfill & 1 \hfill & 0 \hfill & 1 \hfill & 1 \hfill & 0 \hfill \\ \end{array} } \right| = 0.