Integer Relation Research Articles

Asymptotically stable optimal multi-rate rigid body attitude estimation based on lagrange-d'alembert principle

<abstract><p>The rigid body attitude estimation problem is treated using the discrete-time Lagrange-d'Alembert principle. Three different possibilities are considered for the multi-rate relation between angular velocity measurements and direction vector measurements for attitude: 1) integer relation between sampling rates, 2) time-varying sampling rates, 3) non-integer relation between sampling rates. In all cases, it is assumed that angular velocity measurements are sampled at a higher rate compared to the inertial vectors. The attitude determination problem from two or more vector measurements in the body-fixed frame is formulated as Wahba's problem. At instants when direction vector measurements are absent, a discrete-time model for attitude kinematics is used to propagate past measurements. A discrete-time Lagrangian is constructed as the difference between a kinetic energy-like term that is quadratic in the angular velocity estimation error and an artificial potential energy-like term obtained from Wahba's cost function. An additional dissipation term is introduced and the discrete-time Lagrange-d'Alembert principle is applied to the Lagrangian with this dissipation to obtain an optimal filtering scheme. A discrete-time Lyapunov analysis is carried out to show that the optimal filtering scheme is asymptotically stable in the absence of measurement noise and the domain of convergence is almost global. For a realistic evaluation of the scheme, numerical experiments are conducted with inputs corrupted by bounded measurement noise. These numerical simulations exhibit convergence of the estimated states to a bounded neighborhood of the actual states.</p></abstract>

Inference in High-Dimensional Linear Regression via Lattice Basis Reduction and Integer Relation Detection

We consider the high-dimensional linear regression problem, where the algorithmic goal is to efficiently infer an unknown feature vector <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\beta ^{*}\in \mathbb {R}^{p}$ </tex-math></inline-formula> from its linear measurements, using a small number <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$n$ </tex-math></inline-formula> of samples. Unlike most of the literature, we make no sparsity assumption on <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\beta ^{*}$ </tex-math></inline-formula> , but instead adopt a different regularization: In the noiseless setting, we assume <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\beta ^{*}$ </tex-math></inline-formula> consists of entries, which are either rational numbers with a common denominator <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$Q\in \mathbb {Z}^{+}$ </tex-math></inline-formula> (referred to as <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$Q-$ </tex-math></inline-formula> rationality); or irrational numbers taking values in a rationally independent set of bounded cardinality, known to learner; collectively called as the mixed-range assumption. Using a novel combination of the Partial Sum of Least Squares (PSLQ) integer relation detection, and the Lenstra-Lenstra-Lovász (LLL) lattice basis reduction algorithms, we propose a polynomial-time algorithm which provably recovers a <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\beta ^{*}\in \mathbb {R}^{p}$ </tex-math></inline-formula> enjoying the mixed-range assumption, from its linear measurements <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$Y=X\beta ^{*}\in \mathbb {R}^{n}$ </tex-math></inline-formula> for a large class of distributions for the random entries of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$X$ </tex-math></inline-formula> , even with one measurement ( <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$n=1$ </tex-math></inline-formula> ). In the noisy setting, we propose a polynomial-time, lattice-based algorithm, which recovers a <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\beta ^{*}\in \mathbb {R}^{p}$ </tex-math></inline-formula> enjoying the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$Q-$ </tex-math></inline-formula> rationality property, from its noisy measurements <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$Y=X\beta ^{*}+W\in \mathbb {R}^{n}$ </tex-math></inline-formula> , even from a single sample ( <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$n=1$ </tex-math></inline-formula> ). We further establish that for large <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$Q$ </tex-math></inline-formula> , and normal noise, this algorithm tolerates information-theoretically optimal level of noise. We then apply these ideas to develop a polynomial-time, single-sample algorithm for the phase retrieval problem. Our methods address the single-sample ( <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$n=1$ </tex-math></inline-formula> ) regime, where the sparsity-based methods such as the Least Absolute Shrinkage and Selection Operator (LASSO) and the Basis Pursuit are known to fail. Furthermore, our results also reveal algorithmic connections between the high-dimensional linear regression problem, and the integer relation detection, randomized subset-sum, and shortest vector problems.

A sequence of algebraic integer relation numbers which converges to 4

Let α∈R and letA=[1101]andBα=[10α1]. The subgroup Gα of SL2(R) is a group generated by the matrices A and Bα. In this paper, we investigate the property of the group Gα. We construct a generalization of the Farey graph for the subgroup Gα. This graph determines whether the group Gα is a free group of rank 2. More precisely, the group Gα is a free group of rank 2 if and only if the graph is tree. In particular, we show that if 1/2 is a vertex of the graph, then Gα is not a free group of rank 2. Using this, we construct a sequence of real numbers so that the sequence converges to 4 and each number has the corresponding group that is not a free group of rank 2. It turns out that the real numbers are algebraic integers.

Optimal Sampling of a Bandlimited Noisy Multiple Output Channel

This paper deals with an extension of Papoulis' generalized sampling expansion (GSE) of bandlimited signals to a case where a white noise, filtered to the signal's band, is added before sampling. We look for the best sampling scheme that maximizes the capacity or minimizes the mean-squared error (MSE) of the sampled channel between the input signal and the M sampled outputs signals, e.g. channels. Due to the noise it is beneficial to sample at a total rate higher than Nyquist, yet there is no gain in sampling each signal above Nyquist; thus the total rate considered, normalized by Nyquist rate, is between 1 to M. To avoid preference to any channel, the paper assumes that the channels are composed of all-pass linear time-invariant (LTI) systems. For the case where the total normalized rate is between M-1 and M, we show that the best scheme samples M-1 outputs at Nyquist rate and the last output at the remaining rate. Surprisingly, equal sampling is suboptimal in general, but when there is an integer relation between M and the total normalized rate, equal sampling achieves the optimal performance. We also make a conjecture on other rates that the best scheme either samples some channels at Nyquist and one another channel at the remaining rate, or sample the channels equally. This conjecture is supported by simulation for M ≤ 5. Finally, we discuss the relation between maximizing the capacity and minimizing the MSE.

A Study on Computer Consciousness on Intuitive Geometry Based on Mathematics Experiments and Statistical Analysis

In this paper, we present our research on building computing machines consciousness about intuitive geometry based on mathematics experiments and statistical inference. The investigation consists of the following five steps. At first, we select a set of geometric configurations and for each configuration we construct a large amount of geometric data as observation data using dynamic geometry programs together with the pseudo-random number generator. Secondly, we refer to the geometric predicates in the algebraic method of machine proof of geometric theorems to construct statistics suitable for measuring the approximate geometric relationships in the observation data. In the third step, we propose a geometric relationship detection method based on the similarity of data distribution, where the search space has been reduced into small batches of data by pre-searching for efficiency, and the hypothetical test of the possible geometric relationships in the search results has be performed. In the fourth step, we explore the integer relation of the line segment lengths in the geometric configuration in addition. At the final step, we do numerical experiments for the pre-selected geometric configurations to verify the effectiveness of our method. The results show that computer equipped with the above procedures can find out the hidden geometric relations from the randomly generated data of related geometric configurations, and in this sense, computing machines can actually attain certain consciousness of intuitive geometry as early civilized humans in ancient Mesopotamia.

Conjectures about the ground-state energy of the Lieb-Liniger model at weak repulsion

We develop an alternative description to solve the problem of the ground-state energy of the Lieb-Liniger model that describes one-dimensional bosons with contact repulsion. For this integrable model we express the Lieb integral equation in the representation of Chebyshev polynomials. The latter form is convenient to efficiently obtain very precise numerical results in the singular limit of weak interaction. Such highly precise data enable us to use the integer relation algorithm to discover the analytical form of the coefficients in the expansion of the ground-state energy for small values of the interaction parameter. We obtained the first nine terms of the expansion using quite moderate numerical efforts. The detailed knowledge of behavior of the ground-state energy on the interaction immediately leads to exact perturbative results for the excitation spectrum.

The PSLQ algorithm for empirical data

The celebrated integer relation finding algorithm PSLQ has been successfully used in many applications. PSLQ was only analyzed theoretically for exact input data, however, when the input data are irrational numbers, they must be approximate ones due to the finite precision of the computer. When the algorithm takes empirical data (inexact data with error bounded) instead of exact real numbers as its input, how do we theoretically ensure the output of the algorithm to be an exact integer relation? In this paper, we investigate the PSLQ algorithm for empirical data as its input. Firstly, we give a termination condition for this case. Secondly, we analyze a perturbation on the hyperplane matrix constructed from the input data and hence disclose a relationship between the accuracy of the input data and the output quality (an upper bound on the absolute value of the inner product of the exact data and the computed integer relation), which naturally leads to an error control strategy for PSLQ. Further, we analyze the complexity bound of the PSLQ algorithm for empirical data. Examples on transcendental numbers and algebraic numbers show the meaningfulness of our error control strategy.

Constructing exact symmetric informationally complete measurements from numerical solutions

Recently, several intriguing conjectures have been proposed connecting symmetric informationally complete quantum measurements (SIC POVMs, or SICs) and algebraic number theory. These conjectures relate the SICs to their minimal defining algebraic number field. Testing or sharpening these conjectures requires that the SICs are expressed exactly, rather than as numerical approximations. While many exact solutions of SICs have been constructed previously using Gröbner bases, this method has probably been taken as far as is possible with current computer technology (except in special cases where there are additional symmetries). Here, we describe a method for converting high-precision numerical solutions into exact ones using an integer relation algorithm in conjunction with the Galois symmetries of an SIC. Using this method, we have calculated 69 new exact solutions, including nine new dimensions, where previously only numerical solutions were known—which more than triples the number of known exact solutions. In some cases, the solutions require number fields with degrees as high as 12 288. We use these solutions to confirm that they obey the number-theoretic conjectures, and address two questions suggested by the previous work.

Experimental Math for Math Monthly Problems

Experimental mathematics is a newly developed approach to discovering mathematical truths through the use of computers. In this paper, we look at how these techniques can be applied to help solve six problems that have appeared in the Problems section of the MONTHLY. The paper has examples of constant recognition, sequence recognition, and integer relation detection.

Extrapolation methods and Bethe ansatz for the asymmetric exclusion process

The one-dimensional asymmetric simple exclusion process (ASEP), where N hard-core particles hop forward with rate 1 and backward with rate , is considered on a periodic lattice of L site. Using KPZ universality and previous results for the totally asymmetric model , precise conjectures are formulated for asymptotics at finite density of ASEP eigenstates close to the stationary state. The conjectures are checked with high precision using extrapolation methods on finite size Bethe ansatz numerics. For weak asymmetry , double extrapolation combined with an integer relation algorithm gives an exact expression for the spectral gap up to 10th order in the asymmetry.

Computer Discovery and Analysis of Large Poisson Polynomials

ABSTRACTIn two earlier studies of lattice sums arising from the Poisson equation of mathematical physics, it was established that the lattice sum 1/π · ∑m, n oddcos (mπx)cos (nπy)/(m2 + n2) = log A, where A is an algebraic number, and explicit minimal polynomials associated with A were computed for a few specific rational arguments x and y. Based on these results, one of us (Kimberley) conjectured a number-theoretic formula for the degree of A in the case x = y = 1/s for some integer s. These earlier studies were hampered by the enormous cost and complexity of the requisite computations. In this study, we address the Poisson polynomial problem with significantly more capable computational tools. As a result of this improved capability, we have confirmed that Kimberley’s formula holds for all integers s up to 52 (except for s = 41, 43, 47, 49, 51, which are still too costly to test), and also for s = 60 and s = 64. As far as we are aware, these computations, which employed up to 64,000-digit precision, producing polynomials with degrees up to 512 and integer coefficients up to 10229, constitute the largest successful integer relation computations performed to date. By examining the computed results, we found connections to a sequence of polynomials defined in a 2010 paper by Savin and Quarfoot. These investigations subsequently led to a proof, given in the Appendix, of Kimberley’s formula and the fact that when s is even, the polynomial is palindromic (i.e., coefficient ak = am − k, where m is the degree).

Experimental mathematics meets gravitational self-force

It is now possible to compute linear in mass-ratio terms in the post-Newtonian (PN) expansion for compact binaries to very high orders using black hole perturbation theory applied to various invariants. For instance, a computation of the redshift invariant of a point particle in a circular orbit about a black hole in linear perturbation theory gives the linear-in-mass-ratio portion of the binding energy of a circular binary with arbitrary mass ratio. This binding energy, in turn, encodes the system's conservative dynamics. We give a method for extracting the analytic forms of these PN coefficients from high-accuracy numerical data using experimental mathematics techniques, notably an integer relation algorithm. Such methods should be particularly important when the calculations progress to the considerably more difficult case of perturbations of the Kerr metric. As an example, we apply this method to the redshift invariant in Schwarzschild. Here we obtain analytic coefficients to 12.5PN, and higher-order terms in mixed analytic-numerical form to 21.5PN, including analytic forms for the complete 13.5PN coefficient, and all the logarithmic terms at 13PN. At these high orders, an individual coefficient can have over 30 terms, including a wide variety of transcendental numbers, when written out in full. We are still able to obtain analytic forms for such coefficients from the numerical data through a careful study of the structure of the expansion. The structure we find also allows us to predict certain "leading logarithm"-type contributions to all orders. The additional terms in the expansion we obtain improve the accuracy of the PN series for the redshift observable, even in the very strong-field regime inside the innermost stable circular orbit, particularly when combined with exponential resummation.

Computing Maximal Copies of Polyhedra Contained in a Polyhedron

Kepler (1619) and Croft (1980) considered the problem of finding the largest homothetic copies of one regular polyhedron contained in another regular polyhedron. For arbitrary pairs of polyhedra, we propose to model this as a quadratically constrained optimization problem. These problems can then be solved numerically; in case the optimal solutions are algebraic, exact optima can be recovered by solving systems of equations to very high precision and then using integer relation algorithms. Croft solved the special cases concerning maximal inclusions of Platonic solids for 14 out of 20 pairs. For the six remaining cases, we give numerical solutions and conjecture exact algebraic solutions.

Automated simplification of large symbolic expressions

We present a set of algorithms for automated simplification of symbolic constants of the form ∑iαixi with αi rational and xi complex. The included algorithms, called SimplifySum2 and implemented in Mathematica, remove redundant terms, attempt to make terms and the full expression real, and remove terms using repeated application of the multipair PSLQ integer relation detection algorithm. Also included are facilities for making substitutions according to user-specified identities. We illustrate this toolset by giving some real-world examples of its usage, including one, for instance, where the tool reduced a symbolic expression of approximately 100 000 characters in size enough to enable manual manipulation to one with just four simple terms.2Available from https://github.com/alexkaiser/SimplifySum.

A complete algorithm to find exact minimal polynomial by approximations

Based on an improved parameterized integer relation construction method, a complete algorithm is proposed for finding an exact minimal polynomial from its approximate root. It relies on a study of the error controlling for its approximation. We provide a sufficient condition on the precision of the approximation, depending only on the degree and the height of its minimal polynomial. Our result is superior to the existent error controlling on obtaining an exact rational or algebraic number from its approximation. Moreover, some applications are presented and compared with the subsistent methods.

On the Euler scale and the μEuclidean integer relation algorithm

The equal-tempered 10-tone scale e n/10 (n=0,±1,±2, …), using the Euler number e=2.71828… as a pseudo-octave is shown to approximate well the prime number harmonics 2, 3, 5, and 11. Equal-tempered scales simultaneously approximating certain frequency ratios, shall be called tonal scales.1 Some of the properties of the Euler scale and its relation to other tonal scales are explored. The general mathematical problem of identifying tonal scales can be solved by investigating integer relations, using the μEuclidean algorithm, a modification of the PSLQ algorithm. If restricted to two numbers, the μEuclidean algorithm goes over identically into the ancient Euclidean algorithm, contrary to the PSLQ algorithm. The μEuclidean is able to solve a certain class of higher dimensional integer relations where the PSLQ (not the PPSLQ) algorithm breaks down. In general, the μEuclidean algorithm finds smaller integer relations than the (P)PSLQ algorithm. In an appendix, a simple alternative procedure is presented for determining tonal scales based on continued fractions.

Continued fractions and the hydrogen molecular ion H2+

The Jaffé continued fraction solution for the hydrogen molecular ion has been the basis for most H2+ energy calculations at physically relevant nuclear separation R. I show that its slow convergence at small R can be overcome so it can also be used to obtain the power series expansion of the energy about the united atom limit R = 0. I restrict the discussion to the H2+ ground state and show, by a generalization of Wynn's accelerated convergence technique and an analytical calculation, that the continued fraction yields the O(R5) series term which is the first term in which log(R) makes an appearance. For higher order terms I apply the convergence method numerically to obtain very accurate energies at small R and, by fitting, numerical coefficients in the R series. The analytic representations of the coefficients are identified using the integer relation algorithm PSLQ. All terms to O(R14) are obtained exactly and to O(R20) in a mixed analytic/numeric form. Comparisons with previous work based on wavefunction perturbation theory are made.

CLUSTER EFFECTS IN NUCLEAR BINDING ENERGIES

Stable character of differences of nuclear binding energies of nuclei differing by 6 He or 4 He clusters is used to check the tuning effect in particle masses which consists in integer relation between electromagnetic mass differences of leptons and pion as well as rational relations between other particle masses.

Hand-to-hand combat with thousand-digit integrals

In this paper we describe numerical investigations of definite integrals that arise by considering the moments of multi-step uniform random walks in the plane, together with a closely related class of integrals involving the elliptic functions K, K′, E and E′. We find that in many cases such integrals can be “experimentally” evaluated in closed form or that intriguing linear relations exist within a class of similar integrals. Discovering these identities and relations often requires the evaluation of integrals to extreme precision, combined with large-scale runs of the “PSLQ” integer relation algorithm. This paper presents details of the techniques used in these calculations and mentions some of the many difficulties that can arise.

High-precision numerical integration: Progress and challenges

One of the most fruitful advances in the field of experimental mathematics has been the development of practical methods for very high-precision numerical integration, a quest initiated by Keith Geddes and other researchers in the 1980s and 1990s. These techniques, when coupled with equally powerful integer relation detection methods, have resulted in the analytic evaluation of many integrals that previously were beyond the realm of symbolic techniques. This paper presents a survey of the current state-of-the-art in this area (including results by the present authors and others), mentions some new results, and then sketches what challenges lie ahead.