Equidistribution Research Articles

A note on mass equidistribution of holomorphic Siegel modular forms

Let Ff∈Sk(Sp2n(Z)) be the Ikeda lifting of a Hecke eigenform f∈S2k−n(SL2(Z)) with the normalization 〈Ff,Ff〉=1. Let E(Z;s) denote the Klingen Eisenstein series. In this paper we verify thatlimk→∞⁡∫Sp2n(Z)\\HnE(Z;n2+it)|Ff(Z)|2(det⁡Y)kdμ=0 which is predicted by the mass equidistribution conjecture of Cogdell and Luo [CL].

Scaling of the Thue-Morse Diffraction Measure (Acta Physica Polonica A 125, 431 (2014)), ERRATUM

More importantly, our argument for Case B on the basis of uniform distribution modulo 1 of the sequence ( 2x ) n∈N for almost all x ∈ R is incomplete, because the function f defined by f(x) = log ( 1 − cos(2πx) ) is only Riemann integrable on [0, 1] in the generalised sense (meaning that it is an improper integral), which is insufficient here. However, f is properly integrable on [0, 1] in the Lebesgue sense and has an obvious 1-periodic extension to R. Consequently, rather than employing uniform distribution, one can argue with the dynamical system defined by the map T of the unit interval [0, 1] into itself, given by x 7→ 2x mod 1. It is well-known that T leaves Lebesgue measure invariant and is ergodic relative to it, so that the Birkhoff sums satisfy

Distribution modulo 1 and universality of the Hurwitz zeta-function

In the paper, a discrete universality theorem for the Hurwitz zeta-function ζ(s,α) on the approximation of analytic functions by shifts ζ(s+iτ,α) when τ takes values from the set {kβh:k=0,1,…} with fixed 0<β<1 and h>0 is obtained. For its proof, the uniform distribution modulo 1 is applied.

English

The paper contains a brief description of Yamasaki's remarkable investigation (1980) of the relationship between Moore-Yamasaki-Kharazishvili type measures and infinite powers of Borel diffused probability measures on ${\bf R}$. More precisely, we give Yamasaki's proof that no infinite power of the Borel probability measure with a strictly positive density function on $R$ has an equivalent Moore-Yamasaki-Kharazishvili type measure. A certain modification of Yamasaki's example is used for the construction of such a Moore-Yamasaki-Kharazishvili type measure that is equivalent to the product of a certain infinite family of Borel probability measures with a strictly positive density function on $R$. By virtue of the properties of equidistributed sequences on the real axis, it is demonstrated that an arbitrary family of infinite powers of Borel diffused probability measures with strictly positive density functions on $R$ is strongly separated and, accordingly, has an infinite-sample well-founded estimator of the unknown distribution function. This extends the main result established in [ Zerakidze Z., Pantsulaia G., Saatashvili G. On the separation problem for a family of Borel and Baire $G$-powers of shift-measures on $\mathbb{R}$ // Ukrainian Math. J. -2013.-65 (4).- P. 470--485 ].

Uniform distribution modulo 1 and the universality of zeta-functions of certain cusp forms

An universality theorem on the approximation of analytic functions by shifts ?(s+i?,F) of zeta-functions of normalized Hecke-eigen forms F, where ? takes values from the set {k?h:k=0,1,2,...} with fixed 0 &lt; ? &lt; 1 and h &gt; 0, is obtained.

Discrepancy estimates for some linear generalized monomials

We consider sequences modulo one that are generated using a generalized polynomial over the real numbers. Such polynomials may also contain the integer part operation [·] additionally to the addition and the multiplication. A well-studied example is the (nα) sequence defined by the monomial αx. Their most basic sister — ([nα]β)n≥0 — is less investigated. So far only the uniform distribution modulo one of these sequences is resolved. Completely new, however, are the discrepancy results proved in this paper. We show in particular that if the pair of real numbers (α, β) is in a certain sense badly approximable, then the discrepancy satisfies a bound of order Oα,β,e(N).

Common hypercyclic vectors for certain families of differential operators

Let (k(n)) n=1,2,... be a strictly increasing sequence of positive integers . We consider a specific sequence of differential operators Tk(n),{\lambda} , n=1,2,... on the space of entire functions , that depend on the sequence (k(n)) n=1,2,... and the non-zero complex number {\lambda} . We establish the existence of an entire function f , such that for every positive number {\lambda} the set {Tk(n),{\lambda} ,n=1,2,... } is dense in the space of entire functions endowed with the topology of uniform convergence on compact subsets of the complex plane . This provides the best possible strenghthened version of a corresponding result due to Costakis and Sambarino [9] . From this and using a non-trivial result of Weyl which concerns the uniform distribution modulo 1 of certain sequences and Cavalieri principle we can extend our result for a subset of the set of complex numbers with full 2-dimentional Lebesque measure .

Explorations and Expectations of Equidistribution Adaptations for Nonlinear Quenching Problems

Finite difference computations that involve spatial adaptation commonly employ an equidistribution principle. In these cases, a new mesh is constructed such that a given monitor function is equidistributed in some sense. Typical choices of the monitor function involve the solution or one of its many derivatives. This straightforward concept has proven to be extremely effective and practical. However, selections of core monitoring functions are often challenging and crucial to the computational success. This paper concerns six different designs of the monitoring function that targets a highly nonlinear partial differential equation that exhibits both quenching-type and degeneracy singularities. While the first four monitoring strategies are within the so-called primitive regime, the rest belong to a later category of the modified type, which requires the priori knowledge of certain important quenching solution characteristics. Simulated examples are given to illustrate our study and conclusions.

Analytically defined uniformly dense sequences

We introduce a notion of uniform density. This notion is an analogue of uniform distribution modulo 1, but is of interest mostly for non-compact spaces. Our main results show that various specific (families of) sequences or real numbers, defined by means of analytic formulas, are uniformly dense.

Genericity and UD-random reals

Avigad introduced the notion of UD-randomness based in Weyl's 1916 definition of uniform distribution modulo one. We prove that there exists a weakly 1-random real that is neither UD-random nor weakly 1-generic. We also show that no 2-generic real can Turing compute a UD-random real.

Managing the Redundancy of Actuation of a Mobile Robot

We present in this work the performances of a mobile omnidirectional robot through evaluating its management of the redundancy of actuation. The distribution of the wringer on the robot actions, through the inverse pseudo of Moore-Penrose, corresponds to a « geometric ›› distribution of efforts. We will show that the load on vehicle wheels would not be equi-distributed in terms of wheels configuration and of robot movement. Thus, the threshold of sliding is not the same for the three wheels of the vehicle. We suggest exploiting the redundancy of actuation to reduce the risk of wheels sliding and to ameliorate, thereby, its accuracy of displacement. This kind of approach was the subject of study for the legged robots.

Robust Numerical Scheme for Singularly Perturbed Parabolic Initial-Boundary-Value Problems on Equidistributed Mesh

Robust Numerical Scheme for Singularly Perturbed Parabolic Initial-Boundary-Value Problems on Equidistributed Mesh

Finite-state Markov Chains Obey Benford’s Law

nnite-state Markov chain with transition probability matrix P and limiting matrix P' is Benford if every com­ ponent of both sequences of matrices (pn - P') and (pn+1 - pn) is Benford or eventually zero. Using recent tools that established Benford behavior for finite-dimensional linear maps, via the classical theories of uniform distribution modulo 1 and Perron-Frobenius, this paper derives a simple sufficient condition (nonresonance) guaranteeing that P, or the Markov chain associated with it, is Benford. This result in turn is used to show that almost all Markov chains are Benford, in the sense that if the transition probability matrix is chosen in an absolutely continuous manner, then the resulting Markov chain is Benford with probability one. Concrete examples illustrate the various cases that arise, and the theory is complemented with simulations and potential applications.

The sum-of-digits function of polynomial sequences

Let q≥2 be an integer and sq(n) denote the sum of the digits in base q of the positive integer n. The goal of this work is to study a problem of Gelfond concerning the re-partition of the sequence (sq(P(n)))n∈ in arithmetic progressions when P∈[XS is such that P()⊂. We answer Gelfond's question and we show the uniform distribution modulo 1 of the sequence ( sq(P(n)))n∈ for ∈ , provided that q is a large enough prime number co-prime with the leading coefficient of P.

Sampling multidimensional signals by a new class of quasi-random sequences

A new method of equidistributed (EQD) sampling of nD signals is proposed and investigated. Equidistributed or quasi-random (QR) sequences are deterministic sequences, which cover nD space more evenly than uniformly distributed random sequences. We propose to generate quasi-random nD sample sites using space-filling curves, which propagate their measure and neighborhood preserving properties to sample points. It is proved that these points are not only equidistributed but also well distributed, which results in a good volume approximation from samples of 2D and 3D images. A general way of reconstructing output of a linear system from QR samples of its input nD signals is also proposed and proved to be convergent. As an example of applications of the theory, a double radial basis neural network is proposed, which allows for the reconstruction of 2D cross-sections from samples of 3D image or from a video sequence of 2D images.

Finite-State Markov Chains Obey Benford's Law

A sequence of real numbers (xn) is Benford if the significands, i.e. the fraction parts in the floating-point representation of (xn), are distributed logarithmically. Similarly, a discrete-time irreducible and aperiodic finite-state Markov chain with probability transition matrix P and limiting matrix P* is Benford if every component of both sequences of matrices (Pn − P*) and (Pn 1 − Pn) is Benford or eventually zero. Using recent tools that established Benford behavior both for Newton's method and for finite-dimensional linear maps, via the classical theories of uniform distribution modulo 1 and Perron-Frobenius, this paper derives a simple sufficient condition ('nonresonance') guaranteeing that P, or the Markov chain associated with it, is Benford. This result in turn is used to show that almost all Markov chains are Benford, in the sense that if the transition probabilities are chosen independently and continuously, then the resulting Markov chain is Benford with probability one. Concrete examples illustrate the various cases that arise, and the theory is complemented with several simulations and potential applications.

On Adaptive Choice of Shifts in Rational Krylov Subspace Reduction of Evolutionary Problems

We compute $u(t)=\exp(-tA)\varphi$ using rational Krylov subspace reduction for $0\leq t<\infty$, where $u(t),\varphi\in\mathbf{R}^N$ and $0<A=A^*\in\mathbf{R}^{N\times N}$. A priori optimization of the rational Krylov subspaces for this problem was considered in [V. Druskin, L. Knizhnerman, and M. Zaslavsky, SIAM J. Sci. Comput., 31 (2009), pp. 3760–3780]. There was suggested an algorithm generating sequences of equidistributed shifts, which are asymptotically optimal for the cases with uniform spectral distributions. Here we develop a recursive greedy algorithm for choice of shifts taking into account nonuniformity of the spectrum. The algorithm is based on an explicit formula for the residual in the frequency domain allowing adaptive shift optimization at negligible cost. The effectiveness of the developed approach is demonstrated in an example of the three-dimensional diffusion problem for Maxwell's equation arising in geophysical exploration. We compare our approach with the one using the above-mentioned equidistributed sequences of shifts. Numerical examples show that our algorithm is able to adapt to the spectral density of operator A. For examples with near-uniform spectral distributions, both algorithms show the same convergence rates, but the new algorithm produces superior convergence for cases with nonuniform spectra.

Hyperbolic Conservation Laws

Hyperbolic Conservation Laws

A property of the uniform distribution modulo 1

An interesting relationship between Farey fractions and the uniform distribution modulo 1 is discovered.

The sum of digits of primes in $${\mathbb{Z}}$$ [i

We study the distribution of the complex sum-of-digits function sq with basis q = –a±i, \({a \in \mathbb{Z}^+}\) for Gaussian primes p. Inspired by a recent result of Mauduit and Rivat (http://iml.univ-mrs.fr/~rivat/publications.html) for the real sum-of-digits function, we here get uniform distribution modulo 1 of the sequence (αsq(p)) provided \({\alpha \in \mathbb{R} \setminus \mathbb{Q}}\) and q is prime with a ≥ 28. We also determine the order of magnitude of the number of Gaussian primes whose sum-of-digits evaluation lies in some fixed residue class mod m.