Mark Loewe's mathematical interests

Mark Loewe, Libertarian, for Congress, United States Representative, District 35, Texas
This is a shelf of good to excellent physics, chemistry, and mathematics books, several of which may be obtained at low cost.

For real coefficients *a* ≠ 0, *b*, *c*, and *d*, this note expresses the roots of the cubic equation 0 = *ax*^{3} + *bx*^{2} + *cx* + *d* as algebraic, cosine, and arccosine functions of the coefficients.

This sum is used when Planck's law of radiation is integrated to obtain the expression *σ* = 2π^{5}*k*^{4}/(15*h*^{3}*c*^{2}) for the Stefan-Boltzmann constant *σ* in terms of the speed of light *c*, Boltzmann's constant *k*, and Planck's constant *h*.

Except for special cases, no general algebraic solution exists for the line with minimum weighted mean squared distance from a set of data points (*x*_{i},*y*_{i}) with horizontal weights *w*_{xi} and vertical weights *w*_{yi}. This note derives the solution for the case that each vertical weight is the same multiple *c* of the corresponding horizontal weight, *w*_{yi} = *c**w*_{xi}. Vertical least squares, horizontal least squares, and perpendicular least squares lines are obtained when *c* = 0, *c* = ∞, and *c* = 1.

This note expresses lengths of vectors, areas of parallelograms, and volumes of parallelepipeds as square roots of determinants of 1-by-1, 2-by-2, and 3-by-3 Gram matrices, whose matrix elements are scalar products (dot products) of edge vectors *a*, *b*, and *c*. The expressions extend to higher dimensions and complex numbers, have many practical applications in science, engineering, and other areas of applied mathematics, and are important connections between geometry and algebra.

Unitary irreducible ray representations (unirreps) of the group SO(2,1) are used in quantum physics. HW(1) is the Heisenberg-Weyl group used to describe a quantum mechanical one-dimensional oscillator.

More to come ...

Mark Loewe, Libertarian, for Congress, United States Representative, District 35, Texas