69 research outputs found
On the number of distinct block sizes in partitions of a set
AbstractThe average number of distinct block sizes in a partition of a set of n elements is asymptotic to e log n as n → ∞. In addition, almost all partitions have approximately e log n distinct block sizes. This is in striking contrast to the fact that the average total number of blocks in a partition is ∼n(log n)−1 as n → ∞
Zeta Function Zeros, Powers of Primes, and Quantum Chaos
We present a numerical study of Riemann's formula for the oscillating part of
the density of the primes and their powers. The formula is comprised of an
infinite series of oscillatory terms, one for each zero of the zeta function on
the critical line and was derived by Riemann in his paper on primes assuming
the Riemann hypothesis. We show that high resolution spectral lines can be
generated by the truncated series at all powers of primes and demonstrate
explicitly that the relative line intensities are correct. We then derive a
Gaussian sum rule for Riemann's formula. This is used to analyze the numerical
convergence of the truncated series. The connections to quantum chaos and
semiclassical physics are discussed
On some problems involving Hardy's function
Some problems involving the classical Hardy function are discussed. In particular we discuss the odd moments of
, the distribution of its positive and negative values and the primitive
of . Some analogous problems for the mean square of are
also discussed.Comment: 15 page
Certified Exact Transcendental Real Number Computation in Coq
Reasoning about real number expressions in a proof assistant is challenging.
Several problems in theorem proving can be solved by using exact real number
computation. I have implemented a library for reasoning and computing with
complete metric spaces in the Coq proof assistant and used this library to
build a constructive real number implementation including elementary real
number functions and proofs of correctness. Using this library, I have created
a tactic that automatically proves strict inequalities over closed elementary
real number expressions by computation.Comment: This paper is to be part of the proceedings of the 21st International
Conference on Theorem Proving in Higher Order Logics (TPHOLs 2008
Large N expansion of the 2-matrix model
We present a method, based on loop equations, to compute recursively all the
terms in the large topological expansion of the free energy for the
2-hermitian matrix model. We illustrate the method by computing the first
subleading term, i.e. the free energy of a statistical physics model on a
discretized torus.Comment: 41 pages, 9 figures eps
Hard Instances of the Constrained Discrete Logarithm Problem
The discrete logarithm problem (DLP) generalizes to the constrained DLP,
where the secret exponent belongs to a set known to the attacker. The
complexity of generic algorithms for solving the constrained DLP depends on the
choice of the set. Motivated by cryptographic applications, we study sets with
succinct representation for which the constrained DLP is hard. We draw on
earlier results due to Erd\"os et al. and Schnorr, develop geometric tools such
as generalized Menelaus' theorem for proving lower bounds on the complexity of
the constrained DLP, and construct sets with succinct representation with
provable non-trivial lower bounds
Accuracy and Stability of Computing High-Order Derivatives of Analytic Functions by Cauchy Integrals
High-order derivatives of analytic functions are expressible as Cauchy
integrals over circular contours, which can very effectively be approximated,
e.g., by trapezoidal sums. Whereas analytically each radius r up to the radius
of convergence is equal, numerical stability strongly depends on r. We give a
comprehensive study of this effect; in particular we show that there is a
unique radius that minimizes the loss of accuracy caused by round-off errors.
For large classes of functions, though not for all, this radius actually gives
about full accuracy; a remarkable fact that we explain by the theory of Hardy
spaces, by the Wiman-Valiron and Levin-Pfluger theory of entire functions, and
by the saddle-point method of asymptotic analysis. Many examples and
non-trivial applications are discussed in detail.Comment: Version 4 has some references and a discussion of other quadrature
rules added; 57 pages, 7 figures, 6 tables; to appear in Found. Comput. Mat
The strong thirteen spheres problem
The thirteen spheres problem is asking if 13 equal size nonoverlapping
spheres in three dimensions can touch another sphere of the same size. This
problem was the subject of the famous discussion between Isaac Newton and David
Gregory in 1694. The problem was solved by Schutte and van der Waerden only in
1953.
A natural extension of this problem is the strong thirteen spheres problem
(or the Tammes problem for 13 points) which asks to find an arrangement and the
maximum radius of 13 equal size nonoverlapping spheres touching the unit
sphere. In the paper we give a solution of this long-standing open problem in
geometry. Our computer-assisted proof is based on a enumeration of the
so-called irreducible graphs.Comment: Modified lemma 2, 16 pages, 12 figures. Uploaded program packag
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