The Approximate Functional Formula for the Theta Function and Diophantine Gauss Sums

By introducing the discrete curvature of the polygonal line, and by exploiting the similarity of segments of the line, for small w, to Cornu spirals (C-spirals), we prove the precise renormalization formula. This formula, which sharpens Hardy and Littlewood\u27s approximate functional formula for the theta function, generalizes to irrationals, as a Diophantine inequality, the well-known sum formula of Gauss. The geometrical meaning of the relation between the two limits is that the first sum is taken to a point of inflection of the corresponding C-spirals. The second sum replaces whole C-spirals of the first by unit vectors times scale and phase factors. The block renormalization procedure implied by this replacement is governed by the circle map whose orbits are analyzed by expressing w as an even continued fraction

Asymptotic theory of second order differential equations with two simple turning points

Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/46172/1/205_2004_Article_BF00277924.pd

Remarks on bang-bang control in Hilbert space

In this note, a natural definition of bang-bang control in Hilbert space is given, and some of the theory of the authors' paper (Ref. 1) is rebuilt upon it. An elliptic boundary-value problem illustrating the theory is given. In the last part of this note, the results of Ref. 1 are extended to nonlinear perturbations of linear operators and to homogeneous nonlinear operators.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/45189/1/10957_2004_Article_BF00934808.pd

Sufficient conditions for bang-bang control in Hilbert space

Sufficient conditions for bang-bang and singular optimal control are established in the case of linear operator equations with cost functionals which are the sum of linear and quadratic terms, that is, Ax = u , J ( u )=( r,x )+β( x,x ), β>0. For example, if A is a bounded operator with a bounded inverse from a Hilbert space H into itself and the control set U is the unit ball in H , then an optimal control is bang-bang (has norm l) if 0⩽β1/2∥ A −1 * r ∥·∥ A ∥ 2 .Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/45225/1/10957_2004_Article_BF00928120.pd

Pathologies of the large-N limit for RP^{N-1}, CP^{N-1}, QP^{N-1} and mixed isovector/isotensor sigma-models

We compute the phase diagram in the N\to\infty limit for lattice RP^{N-1}, CP^{N-1} and QP^{N-1} sigma-models with the quartic action, and more generally for mixed isovector/isotensor models. We show that the N=\infty limit exhibits phase transitions that are forbidden for any finite N. We clarify the origin of these pathologies by examining the exact solution of the one-dimensional model: we find that there are complex zeros of the partition function that tend to the real axis as N\to\infty. We conjecture the correct phase diagram for finite N as a function of the spatial dimension d. Along the way, we prove some new correlation inequalities for a class of N-component sigma-models, and we obtain some new results concerning the complex zeros of confluent hypergeometric functions.Comment: LaTeX, 88 pages, 33 figure

Efficient Homomorphic Comparison Methods with Optimal Complexity

Comparison of two numbers is one of the most frequently used operations, but it has been a challenging task to efficiently compute the comparison function in homomorphic encryption (HE) which basically support addition and multiplication. Recently, Cheon et al. (Asiacrypt 2019) introduced a new approximate representation of the comparison function with a rational function, and showed that this rational function can be evaluated by an iterative algorithm. Due to this iterative feature, their method achieves a logarithmic computational complexity compared to previous polynomial approximation methods; however, the computational complexity is still not optimal, and the algorithm is quite slow for large-bit inputs in HE implementation. In this work, we propose new comparison methods with optimal asymptotic complexity based on composite polynomial approximation. The main idea is to systematically design a constant-degree polynomial $f$ by identifying the \emph{core properties} to make a composite polynomial $f\circ f \circ \cdots \circ f$ get close to the sign function (equivalent to the comparison function) as the number of compositions increases. We additionally introduce an acceleration method applying a mixed polynomial composition $f\circ \cdots \circ f\circ g \circ \cdots \circ g$ for some other polynomial $g$ with different properties instead of $f\circ f \circ \cdots \circ f$. Utilizing the devised polynomials $f$ and $g$, our new comparison algorithms only require $\Theta(\log(1/\epsilon)) + \Theta(\log\alpha)$ computational complexity to obtain an approximate comparison result of $a,b\in[0,1]$ satisfying $|a-b|\ge \epsilon$ within $2^{-\alpha}$ error. The asymptotic optimality results in substantial performance enhancement: our comparison algorithm on encrypted $20$-bit integers for $\alpha = 20$ takes $1.43$ milliseconds in amortized running time, which is $30$ times faster than the previous work

On the validity of the geometrical theory of diffraction by convex cylinders

In this paper we consider the scattering of a wave from an infinite line source by an infinitely long cylinder C. The line source is parallel to the axis of C , and the cross section C of this cylinder is smooth, closed and convex. C is formed by joining a pair of smooth convex arcs to a circle C 0 , one on the illuminated side, and one on the dark side, so that C is circular near the points of diffraction. By a rigorous argument we establish the asymptotic behavior of the field at high frequencies, in a certain portion of the shadow S that is determined by the geometry of C in S. The leading term of our asymptotic expansion is the field predicted by the geometrical theory of diffraction.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/46181/1/205_2004_Article_BF00248157.pd