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.

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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.

Lowest-weight hermitian irreducible ray representations (lowest-weight hermirreps) of the Lie algebra of SO(2,1) are used in quantum physics. A hermirrep of the Heisenberg-Weyl Lie algebra HW(1) is used in the paradigm description of a quantum mechanical one-dimensional oscillator.

Lowest-weight hermitian irreducible ray representations (lowest-weight hermirreps) of the Lie algebra of SO(2,2) are used in quantum physics.

The Lie algebra of SO(2,2) is a subalgebra in the chain of Lie algebras of SO(2,4) ⊃ SO(2,3) ⊃ SO(2,2) ⊃ SO(2,1) ⊃ SO(2). Irving Segal proposes to use lowest-weight (or highest-weight) unitary irreducible ray representations (lowest-weight unirreps) of the universal covering group of SO(2,4) to classify elementary particles. The universal covering group of SO(2,4) is the group of causality preserving transformations of Einstein spacetime and the group of transformations under which Maxwell's equations of electromagnetism are invariant. The only other spacetimes which meet conditions set by Segal are Minkowski (flat) spacetime, the group of causality preserving transformations of which is the dilatation extended Poincare subgroup of SO(2,4), and a spacetime of which the group of causality preserving transformations is the SO(2,3) subgroup of SO(2,4).

More to come ...

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