Theory Of Uniform Distribution Research Articles

Uniform-Density Micro-Textured Ball-End Milling Cutter Model Based on Fractal and Uniform Distribution Theory

At present, for micro-textured tools, the determination of the micro-texture placement area depends on the derivation of the cutting geometric model. The micro-texture distribution form applies geometric methods, and the research methods and accuracy are limited. Therefore, in this paper, the ball-end milling cutter is taken as the research object. Based on fractal theory, the morphology of the tool before and after wear is compared to determine the tool–chip contact area. The uniform-density micro-texture distribution model is established using the uniform distribution point theorem, and the synergistic mechanism of the edge and the micro-texture is revealed. The strength of the micro-textured tool with uniform density under the action of the edge is studied by simulation. Finally, the determination of the tool–chip contact area and the establishment of a uniform-density micro-texture model is realized. It is proved that the synergistic effect of the cutting edge and the micro-texture has a positive effect on the milling behavior of the tool. When comparing the non-edge and non-texture tools with the cutting-edge tools, the maximum strain and stress of the cutting-edge micro-textured tools increased by 12% and 30%, and 30% and 20%, respectively, without affecting the normal use of the tool. This research provides a new method for the design of micro-textured tools.

Optimum Geometric Distribution of Stellar Observations in Astronomical Positioning

Astronomical positioning can determine the astronomical longitude and latitude of a station based on observations of celestial bodies. The accuracy of astronomical positioning depends on the observation instruments, surveying environment, and geometric distribution of stellar observations. For astronomical theodolite, existing theories support that the best positioning accuracy can only be obtained with uniformly distributed stars on an equal-altitude circle. In fact, inhomogeneous distribution can also obtain the best positioning accuracy. To illustrate this problem, a quantitative index to determine the optimum geometric distribution of stellar observations by introducing the mean resultant length of vectors in directional statistics is proposed, providing a theoretical basis for studying high-precision astronomical positioning and automated star selection algorithms. Multiple optimum distributions of observations with constant numbers of stars during simulation experiments using the equal-altitude method have been demonstrated. Regarding the approximate equal-altitude method, the inhomogeneous distribution of stars guided by the conclusion described in this paper was also optimum. It could achieve the same accuracy as the uniform distribution. These results can aid in overcoming the limitations of the existing uniform distribution theory for astronomical positioning and provide a theoretical basis for achieving high-precision and high-efficiency astronomical positioning by inhomogeneous distribution stellar observations, especially when only a small number of stars can be observed.

Weyl’s Uniform Distribution Under Periodic Perturbation

Abstract We examine the uniform distribution theory of H. Weyl when there is a periodic perturbation present. As opposed to the classical setting, in this case the conditions for (mod 1) density and (mod 1) uniform distribution turn out to be different.

Improving Deep Learning-Based Recommendation Attack Detection Using Harris Hawks Optimization

Recommendation attack attempts to bias the recommendation results of collaborative recommender systems by injecting malicious ratings into the rating database. A lot of methods have been proposed for detecting such attacks. Among these works, the deep learning-based detection methods get rid of the dependence on hand-designed features of recommendation attack besides having excellent detection performance. However, most of them optimize the key hyperparameters by manual analysis which relies too much on domain experts and their experience. To address this issue, in this paper we propose an approach based on the Harris Hawks Optimization (HHO) algorithm to improve the deep learning-based detection methods. Being different from the original detection methods which optimize the key hyperparameters manually, the improved deep learning-based detection methods can optimize the key hyperparameters automatically. We first convert the key hyperparameters of discrete type to continuous type according to the uniform distribution theory to expand the application scope of HHO algorithm. Then, we use the detection stability as an early stop condition to reduce the optimization iterations to improve the HHO algorithm. After that, we use the improved HHO algorithm to automatically optimize the key hyperparameters for the deep learning-based detection methods. Finally, we use the optimized key hyperparameters to train the deep learning-based detection methods to generate classifiers for detecting the recommendation attack. The experiments conducted on two benchmark datasets illustrate that the improved deep learning-based detection methods have effective performance.

Uniform distribution and geometric incidence theory

A celebrated unit distance conjecture due to Erdős says that the unit distances cannot arise more than Cϵn1+ϵ times (for any ϵ>0) among n points in the Euclidean plane (see e.g. [8] and the references contained therein). In three dimensions, the conjectured bound is Cn43 (see e.g. [6] and [11]). In dimensions four and higher, this problem, in its general formulation, loses meaning because the Lens example shows that one can construct a set of n points in dimension 4 and higher where the unit distance arises ≈n2 times (see e.g. [1]). However, the Lens example is one-dimension in nature, which raises the possibility that the unit distance conjecture is still quite interesting in higher dimensions under additional structural assumptions on the point set. This point of view was explored in [4], [5], [7] and has led to some interesting connections between the unit distance problem and its continuous counterparts, especially the Falconer distance conjecture ([3]). In this paper, we study the unit distance problem and its variants under the assumption that the underlying family of point sets is uniformly distributed. We prove several incidence bounds in this setting and clarify some key properties of uniformly distributed sequences in the context of incidence problems in combinatorial geometry.

On a Class of Lacunary Almost Newman Polynomials Modulo P and Density Theorems

Abstract The reduction modulo p of a family of lacunary integer polynomials, associated with the dynamical zeta function ζβ (z)of the β-shift, for β> 1 close to one, is investigated. We briefly recall how this family is correlated to the problem of Lehmer. A variety of questions is raised about their numbers of zeroes in 𝔽 p and their factorizations, via Kronecker’s Average Value Theorem (viewed as an analog of classical Theorems of Uniform Distribution Theory). These questions are partially answered using results of Schinzel, revisited by Sawin, Shusterman and Stoll, and density theorems (Frobenius, Chebotarev, Serre, Rosen). These questions arise from the search for the existence of integer polynomials of Mahler measure > 1 less than the smallest Salem number 1.176280. Explicit connection with modular forms (or modular representations) of the numbers of zeroes of these polynomials in 𝔽 p is obtained in a few cases. In general it is expected since it must exist according to the Langlands program.

Second Moment Of Dirichlet L-Functions, Character Sums Over Subgroups And Upper Bounds On Relative Class Numbers

Abstract We prove an asymptotic formula for the mean-square average of L-functions associated with subgroups of characters of sufficiently large size. Our proof relies on the study of certain character sums ${\mathcal{A}}(p,d)$ recently introduced by E. Elma, where p ≥ 3 is prime and d ≥ 1 is any odd divisor of p − 1. We obtain an asymptotic formula for ${\mathcal{A}}(p,d),$ which holds true for any odd divisor d of p − 1, thus removing E. Elma’s restrictions on the size of d. This answers a question raised in Elma’s paper. Our proof relies on both estimates on the frequency of large character sums and techniques from the theory of uniform distribution. As an application, in the range $1\leq d\leq\frac{\log p}{3\log\log p}$, we obtain a significant improvement $h_{p,d}^- \leq 2(\frac{(1+o(1))p}{24})^{m/4}$ over the trivial bound $h_{p,d}^- \ll (\frac{dp}{24} )^{m/4}$ on the relative class numbers of the imaginary number fields of conductor $p\equiv 1\mod{2d}$ and degree $m=(p-1)/d$, where d ≥ 1 is odd.

Inhomogeneous Diophantine Approximation on M0-Sets with Restricted Denominators

Abstract Let $F \subseteq [0,1]$ be a set that supports a probability measure $\mu $ with the property that $ |\widehat{\mu }(t)| \ll (\log |t|)^{-A}$ for some constant $ A> 0 $. Let $\mathcal{A}= (q_n)_{n\in{\mathbb{N}}} $ be a sequence of natural numbers. If $\mathcal{A}$ is lacunary and $A>2$, we establish a quantitative inhomogeneous Khintchine-type theorem in which (1) the points of interest are restricted to $F$ and (2) the denominators of the “shifted” rationals are restricted to $\mathcal{A}$. The theorem can be viewed as a natural strengthening of the fact that the sequence $(q_nx \textrm{mod} 1)_{n\in{\mathbb{N}}} $ is uniformly distributed for $\mu $ almost all $x \in F$. Beyond lacunary, our main theorem implies the analogous quantitative result for sequences $\mathcal{A}$ for which the prime divisors are restricted to a finite set of $k$ primes and $A> 2k$. Loosely speaking, for such sequences, our result can be viewed as a quantitative refinement of the fundamental theorem of Davenport, Erdös, and LeVeque (1963) in the theory of uniform distribution.

Application of An Improved Artificial Bee Colony Algorithm

An improved artificial bee colony algorithm which overcome the shortcomings, including slow convergence speed, easily falling into local optimum value, Mechanism of other bionic intelligent optimization algorithms is put forward. The algorithm with number theory of uniform distribution of good point set, first established algorithm initialization model, in order to maintain the diversity of the population; Then, the location is updated iteratively to maintain the diversity of the population and prevent the algorithm from falling into the local optimum. Finally, the seven standard test functions are selected to carry out the experiment and simulation. The results show that compared with the ABC algorithm, GABC algorithm and so on. The algorithm (GCABC) the convergence speed and accuracy have been improved significantly, and to improve the overall searching ability, effectively avoid the population into a local optimum, to solve the problem of multi-modal optimization.

On Proinov’s Lower Bound for the Diaphony

Abstract In 1986, Proinov published an explicit lower bound for the diaphony of finite and infinite sequences of points contained in the d−dimensional unit cube [Proinov, P. D.:On irregularities of distribution, C. R. Acad. Bulgare Sci. 39 (1986), no. 9, 31–34]. However, his widely cited paper does not contain the proof of this result but simply states that this will appear elsewhere. To the best of our knowledge, this proof was so far only available in a monograph of Proinov written in Bulgarian [Proinov, P. D.: Quantitative Theory of Uniform Distribution and Integral Approximation, University of Plovdiv, Bulgaria (2000)]. The first contribution of our paper is to give a self contained version of Proinov’s proof in English. Along the way, we improve the explicit asymptotic constants implementing recent, and corrected results of [Hinrichs, A.—Markhasin, L.: On lower bounds for the ℒ2 -discrepancy, J. Complexity 27 (2011), 127–132.] and [Hinrichs, A.—Larcher, G.: An improved lower bound for the ℒ2 -discrepancy, J. Complexity 34 (2016), 68–77]. (The corrections are due to a note in [Hinrichs, A.—Larcher, G. An improved lower bound for the ℒ2 -discrepancy, J. Complexity 34 (2016), 68–77].) Finally, as a main result, we use the method of Proinov to derive an explicit lower bound for the dyadic diaphony of finite and infinite sequences in a similar fashion.

Quantitative behavior of non-integrable systems. I

The theory of Uniform Distribution started with the equidistribution of the irrational rotation of the circle, proved around 1905 independently by Bohl, Sierpinski and Weyl. The quantitative upgrading of this qualitative result was done by Hardy, Littlewood, Ostrowski, Weyl, and others around 1920. Their work pointed out the crucial role of continued fractions in uniform distribution (for a good treatment of this, see e.g. the book of Drmota–Tichy [11]). This quantitative theory immediately solves—via “discretization”—the problem of quantitative equidistribution of the torus line flow in a square. Due to a simple equivalence, the torus line also settles the case of all “integrable” flat dynamical systems. In the last forty years a group of (mostly) ergodic theorists worked out a qualitative theory of nonintegrable flat systems (see Section 2.1). Using a new nonergodic approach that we call “shortline method”, here we start to develop a quantitative theory of nonintegrable flat dynamical systems. In this long work—which is a series of several papers—we discuss the similarities and the fundamental quantitative differences between “integrable” and “nonintegrable” systems.

Resolution Analysis of Inverting the Generalized Radon Transform from Discrete Data in $\mathbb{R}^3$

A number of practically important imaging problems involve inverting the generalized Radon transform (GRT) ${\mathcal R}$ of a function $f$ in $\mathbb{R}^3$. On the other hand, not much is known about the spatial resolution of the reconstruction from discretized data. In this paper we study how accurately and with what resolution the singularities of $f$ are reconstructed. The GRT integrates over a fairly general family of surfaces ${\mathcal S}_y$ in $\mathbb{R}^3$. Here $y$ is the parameter in the data space, which runs over an open set ${\mathcal V}\subset\mathbb{R}^3$. Assume that the data $g(y)=({\mathcal R} f)(y)$ are known on a regular grid $y_j$ with step-sizes $O(\epsilon)$ along each axis, and suppose ${\mathcal S}=\text{singsupp}(f)$ is a piecewise smooth surface. Let $f_\epsilon$ denote the result of reconstruction from the discrete data. We obtain explicitly the leading singular behavior of $f_\epsilon$ in an $O(\epsilon)$-neighborhood of a generic point $x_0\in{\mathcal S}$, where $f$ has a jump discontinuity. We also prove that under some generic conditions on ${\mathcal S}$ (which include, e.g., a restriction on the order of tangency of ${\mathcal S}_y$ and ${\mathcal S}$), the singularities of $f$ do not lead to nonlocal artifacts. For both computations, a connection with the uniform distribution theory turns out to be important. Finally, we present a numerical experiment, which demonstrates a good match between the theoretically predicted behavior and actual reconstruction.

The Sixth International Conference on Uniform Distribution Theory (UDT 2018)

Abstract This volume contains papers originally presented or inspired by the Sixth International Conference on Uniform Distribution Theory, which was held at CIRM in Luminy, Marseilles, France, October 1–5, 2016.

A generalization of Kátai's orthogonality criterion with applications

We study properties of arithmetic sets coming from multiplicative number theory and obtain applications in the theory of uniform distribution and ergodic theory. Our main theorem is a generalization of K\'atai's orthogonality criterion. Here is a special case of this theorem: Let $a\colon\mathbb{N}\to\mathbb{C}$ be a bounded sequence satisfying $$ \sum_{n\leq x} a(pn)\overline{a(qn)} = {\rm o}(x),~\text{for all distinct primes $p$ and $q$.} $$ Then for any multiplicative function $f$ and any $z\in\mathbb{C}$ the indicator function of the level set $E=\{n\in\mathbb{N}:f(n)=z\}$ satisfies $$ \sum_{n\leq x} \mathbb{1}_E(n)a(n)={\rm o}(x). $$ With the help of this theorem one can show that if $E=\{n_1<n_2<\ldots\}$ is a level set of a multiplicative function having positive upper density, then for a large class of sufficiently smooth functions $h\colon(0,\infty)\to\mathbb{R}$ the sequence $(h(n_j))_{j\in\mathbb{N}}$ is uniformly distributed $\bmod~1$. This class of functions $h(t)$ includes: all polynomials $p(t)=a_kt^k+\ldots+a_1t+a_0$ such that at least one of the coefficients $a_1,a_2,\ldots,a_k$ is irrational, $t^c$ for any $c>0$ with $c\notin \mathbb{N}$, $\log^r(t)$ for any $r>2$, $\log(\Gamma(t))$, $t\log(t)$, and $\frac{t}{\log t}$. The uniform distribution results, in turn, allow us to obtain new examples of ergodic sequences, i.e. sequences along which the ergodic theorem holds.

Analysis of Reconstruction from Discrete Radon Transform Data in R^3 When the Function Has Jump Discontinuities

In this paper we study reconstruction of a function $f$ from its discrete Radon transform data in $\mathbb{R}^3$ when $f$ has jump discontinuities. Consider a conventional parametrization of the Radon data in terms of the affine and angular variables. The step size along the affine variable is $\epsilon$, and the density of measured directions on the unit sphere is $O(\epsilon^2)$. Let $f_\epsilon$ denote the result of reconstruction from the discrete data. Pick any generic point $x_0$ (i.e., satisfying some mild conditions), where $f$ has a jump. Our first result is an explicit leading term behavior of $f_\epsilon$ in an $O(\epsilon)$-neighborhood of $x_0$ as $\epsilon\to0$. A closely related question is why can we accurately reconstruct functions with discontinuities at all? This is a fundamental question, which has not been studied in the literature in dimensions three and higher. We prove that the discrete inversion formula “works,” i.e., if $x_0\not\in S:=\text{singsupp}(f)$ is generic, then $f_\epsilon(x_0)\to f(x_0)$ as $\epsilon\to0$. The proof of this result reveals a surprising connection with the theory of uniform distribution. This is a new phenomenon that has not been known previously. We also present some numerical experiments, which confirm the validity of the developed theory.

Crystallographic explorations into uniform distribution theory

Crystallographic explorations into uniform distribution theory

Bounded error uniformity of the linear flow on the torus

A linear flow on the torus $$\mathbb {R}^d / \mathbb {Z}^d$$ is uniformly distributed in the Weyl sense if the direction of the flow has linearly independent coordinates over $$\mathbb {Q}$$ . In this paper we combine Fourier analysis and the subspace theorem of Schmidt to prove bounded error uniformity of linear flows with respect to certain polytopes if, in addition, the coordinates of the direction are all algebraic. In particular, we show that there is no van Aardenne–Ehrenfest type theorem for the mod 1 discrepancy of continuous curves in any dimension, demonstrating a fundamental difference between continuous and discrete uniform distribution theory.

The Fifth International Conference on Uniform Distribution Theory (UDT 2016) Sopron, Hungary, July 5–8, 2016

Abstract This volume contains papers originally presented or inspired by the Fifth International Conference on Uniform Distribution Theory which was held in Sopron, Hungary, July 5-8, 2016.

A summation formula for divergent continued fractions

Paradoxical formulas which have been proposed by a number of authors for evaluating divergent continued fractions are discussed. The point of the paradoxes is that limit passages (which are to be rigorously defined) result in the convergence real sequences to complex values. These formulas are refined and substantiated on the basis of the theory of uniform distribution supplemented with certain statements of complex analysis.

An Extremal Problem in Uniform Distribution Theory

Abstract In this paper we consider an optimization problem for Cesàro means of bivariate functions. We apply methods from uniform distribution theory, calculus of variations and ideas from the theory of optimal transport.