Multiplicative Coefficients Research Articles

АППРОКСИМАЦИОННЫЙ ПОДХОД В НЕКОТОРЫХ ЗАДАЧАХ ТЕОРИИ РЯДОВ ДИРИХЛЕ С МУЛЬТИПЛИКАТИВНЫМИ КОЭФФИЦИЕНТАМИ

In this paper we consider a class of Dirichlet series with multiplicative coefficients which define functions holomorphic in the right half of the complex plane, and for which there are sequences of Dirichlet polynomials that converge uniformly to these functions in any rectangle within the critical strip. We call such polynomials approximating Dirichlet polynomials. We study the properties of the approximating polynomials, in particular, for those Dirichlet series, whose coefficients are determined by nonprincipal generalized characters, i.e. finite-valued numerical characters which do not vanish on almost all prime numbers and whose summatory function is bounded. These developments are interesting in connection with the problem of the analytical continuation of such Dirichlet series to the entire complex plane, which in turn is tied with the solution of a well-known Chudakov hypothesis about every generalized character being a Dirichlet character.

Shifted character sums with multiplicative coefficients, II

Let f(n) be a multiplicative function with |f(n)|≤1,q be a prime number and a be an integer with (a,q)=1, χ be a non-principal Dirichlet character modulo q. Let ε be a sufficiently small positive constant, A be a large constant, q12+ε≪N≪qA. In this paper, we shall prove that∑n≤Nf(n)χ(n+a)≪Nlog⁡log⁡qlog⁡q and that∑n≤Nf(n)χ(n+a1)⋯χ(n+at)≪Nlog⁡log⁡qlog⁡q, where t≥2,a1,…,at are distinct integers modulo q.

Asymptotic expansion of certain power series with multiplicative coefficients near the unit circle

An asymptotic theorem important for the study of many power series with multiplicative coefficients is proved. Examples of concrete series to which the theorem can be applied are given. It is shown that power series of many classical arithmetic sequences can be expanded in asymptotic series as the variable tends to the roots of unity along the radii of the unit circle.

О ГРАНИЧНОМ ПОВЕДЕНИИ ОДНОГО КЛАССА РЯДОВ ДИРИХЛЕ С МУЛЬТИПЛИКАТИВНЫМИ КОЭФФИЦИЕНТАМИ

In this paper we consider the behavior of funcions defined by Dirichlet series with multiplicative coefficients and with bounded summatory function when approaching the imaginary axis. We show that the points of the imaginary axis are also the points of continuity in a broad sense of functions defined by Dirichlet series with multiplicative coefficients which are determined by nonprincipal generalized characters. This result is particularly interesting in its connection with a solution of Chudakov hyphotesis, which states that any finite-valued numerical character, which does not vanish on all prime numbers and has bounded summatory function, is a Dirichlet character. The proof of the main result in this paper is based on the method of reduction to power series, basic principles of which were developed by prof. Kuznetsov in the early 1980s. Ths method establishes a connection between analytical properties of Dirichlet series and boundary properties of the corresponding power series (i.e. a power series with the same coefficients as the Dirichlet series). This allows to obtain new results both for the Dirichlet series and for the power series. In our case this method allowed us to prove the main result using the properties of the power series with multiplicative coefficients determined by the nonprincipal generalized characters, which also were obtained in this work.

A robust image registration method based on total variation regularization under complex illumination changes

Background and objectiveImage registration is one of the fundamental and essential tasks for medical imaging and remote sensing applications. One of the most common challenges in this area is the presence of complex spatially varying intensity distortion in the images. The widely used similarity metrics, such as MI (Mutual Information), CC (Correlation Coefficient), SSD (Sum of Square Difference), SAD (Sum of Absolute Difference) and CR (Correlation Ratio), are not robust against this kind of distortion because stationarity assumption and the pixel-wise independence cannot be obeyed and captured by these metrics. MethodsIn this paper, we propose a new intensity-based method for simultaneous image registration and intensity correction. We assume that the registered moving image can be reconstructed by the reference image through a linear function that consists of multiplicative and additive coefficients. We also assume that the illumination changes in the images are spatially smooth in each region, so we use weighted Total Variation as a regularization term to estimate the aforesaid multiplicative and additive coefficients. Using weighted Total Variation leads to reduce the smoothness-effect on the coefficients across the edges and causes low level segmentation on the coefficients. For minimizing the reconstruction error, as a dissimilarity term, we use l1norm which is more robust against illumination change and non-Gaussian noises than the l2 norm. Primal-Dual method is used for solving the optimization problem. ResultsThe proposed method is applied to simulated as well as real-world data consisting of clinically 4-D Computed Tomography, retina, Digital Subtraction Angiography (DSA), and iris image pairs. Then, the comparisons are made to MI, CC, SSD, SAD and RC qualitatively and sometimes quantitatively. ConclusionsThe experiment results are demonstrating that the proposed method produces more accurate registration results than conventional methods.

Racial isolation and exposure to airborne particulate matter and ozone in understudied US populations: Environmental justice applications of downscaled numerical model output

BackgroundResearchers and policymakers are increasingly focused on combined exposures to social and environmental stressors, especially given how often these stressors tend to co-locate. Such exposures are equally relevant in urban and rural areas and may accrue disproportionately to particular communities or specific subpopulations. ObjectivesTo estimate relationships between racial isolation (RI), a measure of the extent to which minority racial/ethnic group members are exposed to only one another, and long-term particulate matter with an aerodynamic diameter of <2.5μ (PM2.5) and ozone (O3) levels in urban and nonurban areas of the eastern two-thirds of the US. MethodsLong-term (5year average) census tract-level PM2.5 and O3 concentrations were calculated using output from a downscaler model (2002–2006). The downscaler uses a linear regression with additive and multiplicative bias coefficients to relate ambient monitoring data with gridded output from the Community Multi-scale Air Quality (CMAQ) model. A local, spatial measure of RI was calculated at the tract level, and tracts were classified by urbanicity, RI, and geographic region. We examined differences in estimated pollutant exposures by RI, urbanicity, and demographic subgroup (e.g., race/ethnicity, education, socioeconomic status, age), and used linear models to estimate associations between RI and air pollution levels in urban, suburban, and rural tracts. ResultsHigh RI tracts (≥80th percentile) had higher average PM2.5 levels in each category of urbanicity compared to low RI tracts (<20th percentile), with the exception of the rural West. Patterns in O3 levels by urbanicity and RI differed by region. Linear models indicated that PM2.5 concentrations were significantly and positively associated with RI. The largest association between PM2.5 and RI was observed in the rural Midwest, where a one quintile increase in RI was associated with a 0.90μg/m3 (95% confidence interval: 0.83, 0.99μg/m3) increase in PM2.5 concentration. Associations between O3 and RI in the Northeast, Midwest and West were positive and highest in suburban and rural tracts, even after controlling for potential confounders such as percentage in poverty. ConclusionRI is associated with higher 5year estimated PM2.5 concentrations in urban, suburban, and rural census tracts, adding to evidence that segregation is broadly associated with disparate air pollution exposures. Disproportionate burdens to adverse exposures such as air pollution may be a pathway to racial/ethnic disparities in health.

Optimal Information Criteria Minimizing Their Asymptotic Mean Square Errors

Methods of minimizing the asymptotic mean square errors of information criteria using the Kullback-Leibler distance are shown. First, optimal multiplicative coefficients of bias adjustment for the Akaike and Takeuchi information criteria are given as solutions of quadratic equations. Then, similar optimal coefficients are obtained to yield shrinkage estimators for the Kullback-Leibler distance. Three examples under possible model misspecification or correct model specification are shown. A simulation and a real example for model selection in linear regression are also given, which show that the proportion of selecting a true model can be increased by some optimal information criteria. The problem of the dependence of the optimal coefficients on unknown parameters is discussed.

Asymptotic Expansion of Certain Power Series with Multiplicative Coefficients near the Unit Circle

В данной статье доказывается асимптотическая теорема, важная для исследования многих степенных рядов с мультипликативными коэффициентами. Приводятся примеры конкретных рядов, к которым применима теорема. Степенные ряды многих классических арифметических последовательностей раскладываются в асимптотические ряды при стремлении переменной к корням из единицы по радиусам единичной окружности. Библиография: 10 названий.

ON REPRESENTATION OF MODULAR FORMS AS HOMOGENEOUS POLYNOMIALS

In the article we study the spaces of modular forms such that each element of them is a homogeneous polynomial of modular forms of low weights of the same level. It is a classical fact that it is true for the level 1. N. Koblitz point out that it is true for cusp forms of level 4. In this article we show that the analogous situation takes place for the levels corresponding to the eta-products with multiplicative coefficients. In all cases under consideration the base functions are eta-products. In each case the base functions are written explicitly. Dimensions of spaces are calculated by the Cohen - Oesterle formula, the orders in cusps are calculated by the Biagioli formula.

Kloosterman sums with multiplicative coefficients

Let f(n) be a multiplicative function satisfying |f(n)| ≤ 1, q (≤ N2) be a positive integer and a be an integer with (a, q) = 1. In this paper, we shall prove that $$\sum\limits_{n \leqslant N(n,q) = 1} {f(n)e\left( {\frac{{a\bar n}} {q}} \right) \ll \sqrt {\frac{{\tau (q)}} {q}} N\log \log (6N) + q\tfrac{1} {4} + \tfrac{\varepsilon } {2}N\tfrac{1} {2}(\log (6N))\tfrac{1} {2} + \frac{N} {{\log \log (6N)}},} $$ where \(\overline n \) is the multiplicative inverse of {itn} such that \(\overline n n\) ≡ 1 (mod {itq}), {ite}({itx}) = exp(2πi{itx}), and τ(·) is the divisor function.

Fonctions multiplicatives, sommes d’exponentielles, et loi des grands nombres

We provide essentially optimal, effective conditions to ensure that, when available, the Halberstam–Richert upper bound for the mean value of a non-negative multiplicative function actually furnishes the true order of magnitude. This is applied, in particular, to short sums of multiplicative functions over arithmetic progressions, to exponential sums with multiplicative coefficients, and to strong law of large numbers with multiplicative weights.

Convergence abscissas for Dirichlet series with multiplicative coefficients

This note deals with the relationship between the abscissas of simple, uniform and absolute convergence for the Dirichlet series f(s)=∑n=1∞ann−s, when the coefficients an are either multiplicative or completely multiplicative.

Exponential Growth of Bifurcating Processes with Ancestral Dependence

Branching processes are classical growth models in cell kinetics. In their construction, it is usually assumed that cell lifetimes are independent random variables, which has been proved false in experiments. Models of dependent lifetimes are considered here, in particular bifurcating Markov chains. Under the hypotheses of stationarity and multiplicative ergodicity, the corresponding branching process is proved to have the same type of asymptotics as its classic counterpart in the independent and identically distributed supercritical case: the cell population grows exponentially, the growth rate being related to the exponent of multiplicative ergodicity, in a similar way as to the Laplace transform of lifetimes in the i.i.d. case. An identifiable model for which the multiplicative ergodicity coefficients and the growth rate can be explicitly computed is proposed.

Exponential Growth of Bifurcating Processes with Ancestral Dependence

Branching processes are classical growth models in cell kinetics. In their construction, it is usually assumed that cell lifetimes are independent random variables, which has been proved false in experiments. Models of dependent lifetimes are considered here, in particular bifurcating Markov chains. Under the hypotheses of stationarity and multiplicative ergodicity, the corresponding branching process is proved to have the same type of asymptotics as its classic counterpart in the independent and identically distributed supercritical case: the cell population grows exponentially, the growth rate being related to the exponent of multiplicative ergodicity, in a similar way as to the Laplace transform of lifetimes in the i.i.d. case. An identifiable model for which the multiplicative ergodicity coefficients and the growth rate can be explicitly computed is proposed.

Summation of Rational Series Twisted by Strongly $B$-multiplicative Coefficients

We evaluate in closed form series of the type $\sum u(n) R(n)$, with $(u(n))_n$ a strongly $B$-multiplicative sequence and $R(n)$ a (well-chosen) rational function. A typical example is:$$\sum_{n \geq 1} (-1)^{s_2(n)} \frac{4n+1}{2n(2n+1)(2n+2)} = -\frac{1}{4}$$where $s_2(n)$ is the sum of the binary digits of the integer $n$. Furthermore closed formulas for series involving automatic sequences that are not strongly $B$-multiplicative, such as the regular paperfolding and Golay-Shapiro-Rudin sequences, are obtained; for example, for integer $d \geq 0$:$$\sum_{n \geq 0} \frac{v(n)}{(n+1)^{2d+1}} = \frac{\pi^{2d+1} |E_{2d}|}{(2^{2d+2}-2)(2d)!}$$where $(v(n))_n$ is the $\pm 1$ regular paperfolding sequence and $E_{2d}$ is an Euler number.

Shifted character sums with multiplicative coefficients

Let f(n) be a multiplicative function satisfying |f(n)|≤1, q(≤N2) be a prime number and a be an integer with (a,q)=1, χ be a non-principal Dirichlet character modulo q. In this paper, we shall prove that∑n≤Nf(n)χ(n+a)≪Nq14log⁡log⁡(6N)+q14N12log⁡(6N)+Nlog⁡log⁡(6N) and that∑n≤Nf(n)χ(n+a1)⋯χ(n+at)≪Nq14log⁡log⁡(6N)+q14N12log⁡(6N)+Nlog⁡log⁡(6N), where t≥2, a1,…,at are pairwise distinct integers modulo q.

Estimate of an arithmetic sum with multiplicative coefficients

The class \(\mathcal{F}\) consisting of all multiplicative functions f satisfying the inequality |f(p)| ≤ A for some constant A ≥ 1 and all prime p and Σn=1N|f(n)|2≤A2N is considered. It is proved that for any real irrational algebraic α and for all natural numbers k and N the following estimate holds uniformly over all multiplicative functions f from \(\mathcal{F}\): $$S(\alpha ) = \sum\limits_{n = 1}^N {f(n)\rho (n\alpha ) \ll _A \frac{N} {{\ln N}},}$$ where ρ(t) = 0.5 − {t}.

Multi-GPU Accelerated Parallel Algorithm of Wallis Transformation for Image Enhancement

With the development of satellite remote sensing technology, satellite remote sensing data obtained by the amount will increase rapidly. Consequently, the process of Wallis transformation is faced with such challenges as large data size, high intensity, high computational complexity and large computational quantity, and so on. A fast algorithm and efficient implementation of Wallis filtering based on Compute Unified Device Architecture (CUDA) is proposed. The parallel hardware architecture and software development process of multiple graphic processing unit (multi-GPU) is introduced firstly. On the basis of the parallel architecture and hardware characteristic of GPU, some algorithms are focused on computation and optimization of multiplicative coefficients, additive coefficients and updated gray values of the image pixels to improve the computing speed, and the shared memory is used to improve the data access efficiency. The experimental results clearly show that, in the same image quality, the average acceleration rate is 107 times, and the algorithm on multi-GPU can achieve better performance.

A ROM‐less reverse RNS converter for moduli set {2 q ± 1, 2 q ± 3}

Numerous contributions on reverse conversion methods based on the Chinese Remainder Theorem (CRT), for residue number systems (RNS), have been regularly appearing in relevant literature. Reverse conversion is known as a slow RNS operation that becomes more complicated and slower for larger moduli sets. In this study, the authors examine four previous reverse converters for moduli set ƒ = {2q − 1, 2q + 3, 2q + 1, 2q − 3}. Three of these converters heavily utilise Read Only Memories (ROM), and the other one uses a Montgomery multiplier. In order to cut the costs and improve performance, the authors propose an adder-only two-stage New CRT conversion scheme that uses conjugate grouping of the moduli as {{2q ± 1, 2q ± 3}}. Also, manipulation of multiplicative inverse coefficients that are expressed as a series of power-of-two terms takes place via multi-operand addition instead of using ROMs and/or multipliers. This leads to roughly 22, 19 and 8% improvement in delay, area consumption and power dissipation, respectively, in comparison to the only previous ROM-less design for ƒ. Moreover, use of no ROMs allows for pipelining, if desired. They also address four state-of-the-art converters and compare their performance with the authors (i.e. that of ƒ), where none is faster than the proposed converter. They support their claims with analytical gate-level comparisons and via synthesis results.

On Zeros of Periodic Zeta Functions

We consider zeta functions ζ(s; $$ \mathfrak{a} $$ ) given by Dirichlet series with multiplicative periodic coefficients and prove that, for some classes of functions F , the functions F(ζ(s; $$ \mathfrak{a} $$ )) have infinitely many zeros in the critical strip. For example, this is true for sin(ζ(s; $$ \mathfrak{a} $$ )).