Stephen R. Wassell

Semiclassical approximation for Schrödinger operators on a two‐sphere at high energy

Let H=−ΔS+V be a Schrodinger operator acting in L2(S), with S the two‐dimensional unit sphere, ΔS the spherical Laplacian, and V a smooth potential. Approximate eigenfunctions and eigenvalues for H are obtained involving expansions in inverse powers of the classical angular momentum variables, provided that these variables are in a region of phase space where the corresponding classical Hamiltonian is nearly integrable. The analysis is carried out in a Bargmann representation, where ΔS becomes a quadratic expression in the sum of two quantum harmonic oscillator Hamiltonians, and V becomes a pseudodifferential operator.

Stability of Hamiltonian systems at high energy

Let H=T(P)+U(P,φ) be the Hamiltonian for a classical mechanical system of l degrees of freedom, where T is an integrable Hamiltonian generating quasiperiodic motion in the first l−1 degrees of freedom which has the form

Semiclassical Approximation for Schrödinger Operators at High Energy

Let H ћ = −ћ 2 ∂ φ 2 /2+U(φ) be a Schrodinger operator acting in L 2(T) with T an l-dimensional torus and U an analytic periodic function on T. Approximate semiclassical expansions for the eigenfunctions and eigenvalues of H ћ are developed which are asymptotic in inverse powers of the classical action variables for the corresponding classical Hamiltonian. The leading term in the eigenvalue expansion is the energy associated with a KAM torus; the fact that KAM tori are abundant at high energy is exploited to show that the rank of the approximate eigenfunctions with energy ≤ E approaches the rank of the true eigenvalues of H ћ , for E large.

Art and Mathematics Before the Quattrocento: A Context for Understanding Renaissance Architecture

This present paper briefly discusses Neolithic speculative geometry; the beginnings of history in the Middle East; the Greeks, first true mathematicians; the Romans, masters of engineering; and the Middle Ages. In the years immediately preceding the Renaissance, the qualitative view of neo-Platonic metaphysics slowly gave way to a more quantitative view of reality that would eventually allow science to progress at a steady pace. This transition was slow, and Renaissance scholars were still heavily influenced by long-held ideas on number symbolism and sacred geometry. As did their predecessors, Renaissance artists sought to incorporate meaning into their design by using a rational approach towards aesthetics. Theorists such as Barbaro, Pacioli, and Durer focused their efforts largely on concerns of geometry and proportion, often doing so within the context of the ideas presented here.

Leon Battista Alberti, De componendis cifris

Those who are charged with the highest affairs know by experience how important it is to have a very trustworthy person with whom to reveal projects and decisions of the most secret nature, without ever having reasons to regret it.

Leon Battista Alberti, De lunularum quadratura

The way of measuring a two-cusped2 figure composed of two curved lines as shown in the figure3 Contrary to the opinion of the many who say that figures composed of lines that are curved and circular cannot be squared perfectly, most of all of those that are portions of circles, they say this in my opinion by the authority of Aristotle, who says that quadratura circuli est scibilis, sed non scita quia est impotentia naturœ,4 squaring the circle is knowable though not found but it is in nature’ s power; and not being able to give the squaring of the circle perfectly, they argue that it is impossible to give the perfect squaring of figures made of curved lines first and foremost circular ones; since I have found the perfect squaring of the figure shown here, that is, a figure with two cusps in the shape of the moon marked AB, I say that if we had had careful investigators, then if squaring the circle is in nature’ s power, it is likewise in men’ s power. Thus to demonstrate the squaring of said figure AB, after first noting two propositions of Euclid pertaining to the declaration, I will tell the way it is done.

Leon Battista Alberti, Ex ludis rerum mathematicarum

Se volete col veder[e] sendo in capo d’una piaza misurare quanto sia alta quella torre quale sia a pie d[e]lla piaza fate in questo modo fichate uno1 dardo in terra et fichatelo chegli stia a piombo fermo et poi scostatevi da questo dardo quanto pare a voi osei oocto piedi et indi mirata alla cima d[e]lla torre dirizando il vostro vedere a mira p[er] il diritto d[e]l dardo et li dove il vedere vostro batte nel dardo fatevi porre un poco di cera p[er] segno et chiamasi questa cera

Leon Battista Alberti, Elementi di pittura

Have you ever seen a blind man teach the way to someone who sees? Here with these briefest of notes, which we call Elements, you will see that someone who perhaps does not himself know how to draw can show the true and sure reasoning and the way to become a perfect draughtsman, as long as you don’t shy away from learning that which you judge to be impossible. First try to see if you can do it, and then judge our erudition and your acumen as you will. Trying, you’ll believe me, and believing me you’ll take delight in knowing them all. Do like this, and love me.1

Superexponentiation and- Fixed Points of Exponential and Logarithmic Functions

where exponentiation occurs n times. Superexponentiation simply continues the pattern of addition, multiplication, and exponentiation. It seems to have first appeared in the literature in [4], for the purpose of exhibiting extremely large (albeit finite) numbers (its implicit use in [7], an earlier work than [4], is shown in [5]). Superexponentiation is used in [5] and [6] to examine the logical foundation of mathematical induction. In [2] superexponentiation and its inverse operation, iteration of logarithms, are used to analyze the running time of certain algorithms. A general discussion of superexponentiation is given in [1]. For our purposes, superexponentiation naturally arises from analyzing fixed points of exponential functions (and hence of the corresponding logarithms). We approach this using orbit analysis as in [3], i.e., by iterating the exponential function F(x) = b, starting with input x = b. For b > 1 the resulting orbit clearly will be strictly increasing, and for bases such as 2 or 10 (those used in the applications cited above), the orbit will diverge rapidly to infinity. Will this divergence occur for all b > 1? What kind of behavior is observed if 0 < b < 1? Will the orbit converge to a single number for suitable values of b? In Section 2 we discuss the context under which the author arrived at the problem. In Section 3 we answer the question of convergence, finding the set of b for which the orbit does indeed converge; this result is stated as a theorem at the end of the section. In Section 4 we offer several suggestions for further exploration.