Sum Of Functions Research Articles

Within-chain parallelization-Giving Stan Jet Fuel for population modeling in pharmacometrics.

Stan is a powerful probabilistic programming language designed mainly for Bayesian data analysis. Torsten is a collection of Stan functions that handles the events (e.g., dosing events) and solves the ODE systems that are frequently present in pharmacometric models. To perform a Bayesian data analysis, most models in pharmacometrics require Markov Chain Monte Carlo (MCMC) methods to sample from the posterior distribution. However, MCMC is computationally expensive and can be time-consuming, enough so that people will often forgo Bayesian methods for a more traditional approach. This paper shows how to speed up the sampling process in Stan by within-chain parallelization through both multi-threading using Stan's reduce_sum() function and multi-processing using Torsten's group ODE solver. Both methods show substantial reductions in the time necessary to sufficiently sample from the posterior distribution compared with a basic approach with no within-chain parallelization.

Solvability of $$\psi $$-Hilfer Fractional Differential Equations in the Space of Summable Functions

Solvability of $$\psi $$-Hilfer Fractional Differential Equations in the Space of Summable Functions

Fractional integration of summable functions: Maz'ya's~$\Phi$-inequalities

We study the inequalities of the type $|\int_{\mathbb{R}^d} \Phi(K*f)| \lesssim \|f\|_{L_1(\mathbb{R}^d)}^p$, where the kernel $K$ is homogeneous of order $\alpha - d$ and possibly vector-valued, the function $\Phi$ is positively $p$-homogeneous, and $p = d/(d-\alpha)$. Under mild regularity assumptions on $K$ and $\Phi$, we find necessary and sufficient conditions on these functions under which the inequality holds true with a uniform constant for all sufficiently regular functions $f$.

Slalom at the Carnival: Privacy-preserving Inference with Masks from Public Knowledge

Machine learning applications gain more and more access to highly sensitive information while simultaneously requiring more and more computation resources. Hence, the need for outsourcing these computational expensive tasks while still ensuring security and confidentiality of the data is imminent. In their seminal work, Tramer and Boneh presented the Slalom protocol for privacy-preserving inference by splitting the computation into a data-independent preprocessing phase and a very efficient online phase. In this work, we present a new method to significantly speed up the preprocessing phase by introducing the Carnival protocol. Carnival leverages the pseudo-randomness of the Subset sum problem to also enable efficient outsourcing during the preprocessing phase. In addition to a security proof we also include an empirical study analyzing the landscape of the uniformity of the output of the Subset sum function for smaller parameters. Our findings show that Carnival is a great candidate for real-world implementations.

On some geometric properties of sequence spaces of generalized arithmetic divisor sum function

Recently, some new sequence spaces ℓp(Aα)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\ell _{p}(\\mathfrak{A}^{\\alpha })$\\end{document}(0<p<∞)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$(0< p<\\infty )$\\end{document}, c0(Aα)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$c_{0}(\\mathfrak{A}^{\\alpha })$\\end{document}, c(Aα)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$c(\\mathfrak{A}^{\\alpha })$\\end{document}, and ℓ∞(Aα)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\ell _{\\infty }(\\mathfrak{A}^{\\alpha })$\\end{document} have been studied by Yaying et al. (Forum Math., 2024, https://doi.org/10.1515/forum-2023-0138) as matrix domains of Aα=(an,vα)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\mathfrak{A}^{\\alpha }=(a_{n,v}^{\\alpha })$\\end{document}, where am,vα={vαρ(α)(m),v∣m,0,v∤m,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ a_{\\mathfrak{m},v}^{\\alpha }=\\left \\{ \ extstyle\\begin{array}{c@{\\quad}c@{\\quad}c} \\dfrac{v^{\\alpha }}{\\rho ^{(\\alpha )}(\\mathfrak{m})} & , & v\\mid \\mathfrak{m}, \\\\ 0 & , & v\ mid \\mathfrak{m},\\end{array}\\displaystyle \\right . $$\\end{document}and ρ(α)(m):=\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\rho ^{(\\alpha )}(\\mathfrak{m}):=$\\end{document} sum of the αth\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\alpha ^{\ ext{th}}$\\end{document} power of the positive divisors of m∈N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\mathfrak{m}\\in \\mathbb{N}$\\end{document}. They obtained their duals, matrix transformations and associated compact matrix operators for these matrix classes.This article deals with some geometric properties of these sequence spaces.

Family-related risk indicators and dental attendance in association with dental caries in preschool children

BackgroundDetermining risk indicators behind dental caries is important for identifying children in need of enhanced dental care. The aim of this register-based study was to investigate the association of family-related risk indicators and dental attendance in the development of dental caries in preschool children.MethodsThe data for this study were collected from the medical records of 206 randomly chosen preschool children who lived in the city of Oulu, Finland, during 2014–2020. Data on challenges in family life, missing appointments and utilization of oral health care were collected. Sum functions were formed of risk indicators. Analyses were carried out for three age groups (1- to 2-, 3- to 4- and 5- to 6-year-olds) born between 2014 and 2018.ResultsThere was a significant association between the number of family-related risk indicators and the prevalence of manifested caries lesions in the age group of 5- to 6-year-olds. All family-related risk indicators and information about health care utilization were associated with dental caries risk. Challenges in a child’s family life were present among 20.3% of the 5- to 6-year-olds. In all age groups, the most common challenges in family life were parental exhaustion and problems in the parents’ relationship or divorce.ConclusionFamily-related risk indicators and dental attendance should be considered when determining caries risk. The caries risk indicators investigated here are associated with each other.

Irrational Outputs from Rational Inputs in Theta-3 Functions Using Dirichlet’s Theorem

Theta Functions are not an easy subject of mathematics because new researches about this wide topic are still being published regularly. The findings about Theta Functions lead directly to many applications in different fields of applied mathematics and help also to develop old interesting mathematical topics such as Poisson summation and Riemann Zeta Function. This work proves by using a theorem of Dirichlet and a previously demonstrated useful approximation that the rational inputs of Theta-3 Functions give only irrationals as outputs of these functions. This paper applies only simple notions of sequences and represents an opportunity for the students of mathematics to easily understand the steps of the demonstrations.

Melatonin Regulates Neuronal Synaptic Plasticity in the Supramammillary Nucleus and Attenuates Methamphetamine-Induced Conditioned Place Preference and Sensitization in Mice.

Methamphetamine (METH) is an addictive drug that threatens human health. The supramammillary nucleus (SuM) and its neural circuits play key roles in the regulation of spatial memory retrieval, and hippocampal contextual or social memory. Melatonin (MLT), a pineal hormone, can regulate hypothalamic-neurohypophysial activity. Our previous study showed that MLT attenuates METH-induced locomotor sensitization. However, whether MLT regulates SuM function and participates in METH-induced contextual memory retrieval remains unclear. Using a mouse model of METH-conditioned place preference (CPP) and sensitization, we found that METH activated c-Fos expression and elevated calcium (Ca²⁺) levels in SuM neurons. Chemogenetic inhibition of SuM attenuates CPP and sensitization. Pretreatment with MLT decreased c-Fos expression and Ca2+ levels in the SuM and reversed METH-induced addictive behavior, effects that were blocked with the selective MT2 receptors antagonist 4P-PDOT and the MT1 receptors antagonist S26131. Furthermore, MLT reduced SuM synaptic plasticity, glutamate (Glu) release, and neuronal oscillations caused by METH, which were blocked by 4P-PDOT. In conclusion, our data revealed that MLT regulates neuronal synaptic plasticity in the SuM, likely through the MLT receptors (MTs), and plays a role in modulating METH-addictive behavior.

Diversity of short-term DC outcomes in bulk-fill RBCs subjected to a 3 s high-irradiance protocol

ObjectivesTo determine the short-term (5 min) initial effects of a high-irradiance light-curing (LC) protocol on light transmission (LT%), radiant exposure (RE) and degree of conversion (DC%) of different bulk-fill resin-based composites (RBCs). Materials and methodsSix bulk-fill composites with different viscosities were investigated: OBF (One Bulk Fill, 3 M), EB (Estelite bulkfill,Tokuyama), PFill, PFlow, ECeram and EFlow (PowerFill, Poweflow, Tetric EvoCeram bulkfill, Tetric Evoflow bulkfill, Ivoclar), subjected to different LC protocols: one ultra-high-intensity (3 W/cm2 −3 s via PowerCure LCU) and two conventional (1.2 W/cm2 −10 s and 20 s via PowerCure and Elipar S10 LCUs). Specimens (n = 5) were polymerized within their molds (ϕ5 mm × 4 mm depth) to determine LT% and RE at 4 mm using a MARC-LC spectrometer. For real-time DC% measurements by FTIR, similar molds were utilized. Data were analyzed by one-way ANOVA and Tukey post-hoc tests at 5 % significance. ResultsRegardless of the applied LC protocols, OBF and low-viscosity RBCs (EB, PFlow and EFlow) had the lowest and highest LT%, RE, DC% and RPmax, respectively. RE results of all RBCs were in the same sequence: Elipar-20 s > PCure-10 s > PCure-3 s. DC% of PFill and PFlow displayed no significant difference between the applied LC protocols (p > 0.05). The polymerization kinetic in all materials was well described by an exponential sum function (r2 varied between 0.85 and 0.98), showing a faster polymerization with the PCure-3 s protocol. SignificanceThe measurement of LT% and DC% at 5 min gave an insight into the developing polymerization process. The initial response of these bulk-fill composite to a high-irradiation protocol varied depending on their composition and viscosity, being faster for low viscosity materials. Nevertheless, even though multiple resin composites are designed to be efficient during photopolymerization, care should be taken when selecting materials/curing protocol.

Interval Type-2 enhanced possibilistic fuzzy C-means noisy image segmentation algorithm amalgamating weighted local information

The fuzzy clustering algorithms based on interval type-2 are effective methods for data clustering and image segmentation with some potential advantages in dealing with higher-order uncertainty. However, the fuzzy clustering algorithms based on interval type-2 have limited accuracy in noisy image segmentation. Therefore, we propose a new noisy image segmentation algorithm based on weighted local information for interval type-2 enhanced possibilistic fuzzy C-means clustering. Firstly, a new possibilistic fuzzy weighted local information factor, which amalgamates the mutually guided image filtering, is devised to control the relationship between mutually guided image filtering and the original image utilizing the absolute difference image between the original image and mutually guided image filtering. Additionally, two weighted sum functions are also introduced into the new possibilistic fuzzy weighted local information factor to calculate the grayscale differences between the current pixel and its neighborhood pixels. Secondly, the distance of the objective function is corrected using mutually guided image filtering to obtain the local features and global information of the image. Finally, this paper introduces the new possibilistic fuzzy weighted local information factor into the interval type-2 possibilistic fuzzy C-means and proposes a new interval type-2 enhanced possibilistic fuzzy C-means clustering algorithm with local information and mutually guided image filtering constraints to improve noisy image segmentation results. Through extensive experiments on noisy synthetic images and noisy real images, the proposed algorithm can achieve higher performance and more accurate segmentation results compared with several existing algorithms.

Особенности оценки вредоносной активности в инфраструктуре Умного города на основе гранулирования информации и гранулярных моделей вычислений

The object of the research is a new methodological approach to information granulation and fuzzy granular calculations, as a mathematical and methodological tool for improving the reliability of assessing the level of information security of the Smart City infrastructure. The proposed approach is one of the options for the practical application of elements of the theory of fuzzy sets in the tasks of search, identification and current assessment of signs of time-bearing activity. A detailed analysis of the features of this approach has been carried out, determining the expediency and conditions of its application for assessing malicious activity in the infrastructure of a Smart City. The theoretical aspects of the application of information granulation and fuzzy granular computing to the assessment of malicious activity combining various signs for various categories of potential threats to the infrastructure and subjects of a Smart City - the categories “cyberattack”, “malicious virus threat” or “data leakage (loss)” are studied and described. The analysis of the features of the proposed approach is carried out, which allows taking into account the opinions of experts and eliminating the vagueness associated with noise, disorder and lack of formalization of surveillance data collected and pre-processed in the interests of assessing threats and consequences of negative manifestations of malicious activity. A sequence of calculations and analytical expressions for calculating the estimated values of signs of harmful activity for various categories of potential threats to the infrastructure and subjects of a Smart City has been developed and described in detail. The approach assumes the practical possibility of evaluating signs of malicious activity using information granules formed on the basis of a minimum numerical distance between the values of membership functions characterizing vaguely specified data on the presence or absence of observed signs (attributes) of malicious activity, as well as granular summation and determination of the trace function of the granular sum. At the same time, the proposed approach makes it possible to obtain estimates of signs of malicious activity that are adequate to the tasks of monitoring the Smart City security policy and, ultimately, provides increased reliability of proactive threat control and analysis of the possible consequences of a negative manifestation of suspicious activity.

Lattice properties of strength functions

AbstractThis paper investigates an important functional representation of the cone of bounded positive semidefinite operators. It is known that the representation by strength functions turns the Löwner order into the pointwise order. However, very little is known about the structure of strength functions. Our main result says that the representation behaves naturally with the infimum and supremum operations. More precisely, we show that the pointwise minimum of two strength functions $$f_A$$ f A and $$f_B$$ f B is a strength function if and only if the infimum of A and B exists. This complements a recent result of L. Molnár stating that the pointwise maximum of $$f_A$$ f A and $$f_B$$ f B exists if and only if A and B are comparable, as this latter statement is equivalent to the existence of the supremum. The cornerstone of each argument in this paper is a fact that was discovered recently, namely that the strength function of the parallel sum A : B (which is half of the harmonic mean) equals the parallel sum of the strength functions $$f_A$$ f A and $$f_B$$ f B . We provide a new proof for this statement, and as a byproduct, in some special cases, we describe the strength function of the so-called (generalized) short.

Symmetric operator extensions of composites of higher order difference operators

In this paper we have considered two higher order difference operators generated by two higher order difference functions on the Hilbert space of square summable functions. By allowing the leading coefficients to be unbounded and the other coefficients as constant functions, we have shown that the composites of two symmetric difference operators are symmetric if the leading coefficients are scalar multiple of each other and the common divisor of their orders is 1. Using examples, we have shown that these conditions of symmetry cannot be weakened. Furthermore, We have shown that the deficiency indices of the composites is equal to the sum of the deficiency indices of the individual operators and that the spectra of the self-adjoint operator extensions is the whole of the real line.

Generating Function of q- and Elliptic Multiple Polylogarithms of Hurwitz Type

Ohno and Zagier (Indag Math 12:483–487, 2001) found that a generating function of sums of multiple polylogarithms can be written in terms of the Gauss hypergeometric function 2F1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${}_2F_1$$\\end{document}. As a generalization of the Ohno and Zagier formula, Ihara et al. (Can J Math 76:1–17, 2022) showed that a generating function of sums of interpolated multiple polylogarithms of Hurwitz type can be expressed in terms of the generalized hypergeometric function r+1Fr\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${}_{r+1}F_r$$\\end{document}. In this paper, we establish q- and elliptic analogues of this result. We introduce elliptic q-multiple polylogarithms of Hurwitz type and study a generating function of sums of them. By taking the trigonometric and classical limits in the main theorem, we can obtain q- and elliptic generalizations of the Ihara, Kusunoki, Nakamura and Saeki formula.

The error term of the sum of digital sum functions in arbitrary bases

Let $k$ be a non-negative integer and $q &gt; 1$ be a positive integer. Let $s_q(k)$ be the sum of digits of $k$ written in base $q.$ In 1940, Bush proved that $A_q(x)=\sum_{k \leq x} s_q (k)$ is asymptotic to $\frac{q-1}{2}x \log_q x.$ In 1968, Trollope proved an explicit formula for the error term of $A_2(n-1),$ labeled by $-E_2(n),$ where $n$ is a positive integer. In 1975, Delange extended Trollope's result to an arbitrary base $q$ by another method and labeled the error term $nF_q(\log_q n).$ When $q=2,$ the two formulas of the error term are supposed to be equal, but they look quite different. We proved directly that those two formulas are equal. More interestingly, Cooper and Kennedy in 1999 applied Trollope's method to extend $-E_2(n)$ to $-E_q(n)$ with a general base $q,$ and we also proved directly that $nF_q(\log_q n)$ and $-E_q(n)$ are equal for any $q.$

Summing the sum of digits

We revisit and generalize inequalities for the summatory function of the sum of digits in a given integer base. We prove that several known results can be deduced from a theorem in a 2023 paper by Mohanty, Greenbury, Sarkany, Narayanan, Dingle, Ahnert, and Louis, whose primary scope is the maximum mutational robustness in genotype-phenotype maps.

An Improved Sobol Sensitivity Analysis Method

The Sobol method is a variance-based global sensitivity analysis method that evaluates single-input and multiple-input interaction effects by calculating the contribution of a single input to the output variance and the contribution of multiple inputs to the output variance. The Sobol method requires each input to obey a uniform distribution of U [0,1], but it is difficult to meet the requirements in practice. Taking the sum function as an example, this paper analyzes the inapplicability of the existing Sobol method when the input does not obey the uniform distribution U [0,1]. To solve the inapplicability of the Sobol method and broaden the application scope, an improved Sobol sensitivity analysis method is proposed. First, the effect of the joint probability density function not 1 on sensitivity calculation is studied; second, the input parameters are changed to uniform distribution U [0,1] through variable substitution; finally, a complete algorithm model is presented and logical sensitivity analysis results are obtained. Application verification shows that the improved Sobol method is more scientific, applicable and practical.

The Small-N Series in the Zero-Dimensional O(N) Model: Constructive Expansions and Transseries

AbstractWe consider the zero-dimensional quartic O(N) vector model and present a complete study of the partition function Z(g, N) and its logarithm, the free energy W(g, N), seen as functions of the coupling g on a Riemann surface. We are, in particular, interested in the study of the transseries expansions of these quantities. The point of this paper is to recover such results using constructive field theory techniques with the aim to use them in the future for a rigorous analysis of resurgence in genuine quantum field theoretical models in higher dimensions. Using constructive field theory techniques, we prove that both Z(g, N) and W(g, N) are Borel summable functions along all the rays in the cut complex plane $$\mathbb {C}_{\pi } =\mathbb {C}{\setminus } \mathbb {R}_-$$ C π = C \ R - . We recover the transseries expansion of Z(g, N) using the intermediate field representation. We furthermore study the small-N expansions of Z(g, N) and W(g, N). For any $$g=|g| e^{\imath \varphi }$$ g = | g | e ı φ on the sector of the Riemann surface with $$|\varphi |&lt;3\pi /2$$ | φ | &lt; 3 π / 2 , the small-N expansion of Z(g, N) has infinite radius of convergence in N, while the expansion of W(g, N) has a finite radius of convergence in N for g in a subdomain of the same sector. The Taylor coefficients of these expansions, $$Z_n(g)$$ Z n ( g ) and $$W_n(g)$$ W n ( g ) , exhibit analytic properties similar to Z(g, N) and W(g, N) and have transseries expansions. The transseries expansion of $$Z_n(g)$$ Z n ( g ) is readily accessible: much like Z(g, N), for any n, $$Z_n(g)$$ Z n ( g ) has a zero- and a one-instanton contribution. The transseries of $$W_n(g)$$ W n ( g ) is obtained using Möbius inversion, and summing these transseries yields the transseries expansion of W(g, N). The transseries of $$W_n(g)$$ W n ( g ) and W(g, N) are markedly different: while W(g, N) displays contributions from arbitrarily many multi-instantons, $$W_n(g)$$ W n ( g ) exhibits contributions of only up to n-instanton sectors.

Task offloading method based on CNN-LSTM-attention for cloud–edge–end collaboration system

Most of the existing task offloading methods in edge computing environments do not fully utilize the processing capabilities of cloud servers and have high computational complexity, making them unsuitable for real-time processing tasks. To address this problem, we propose a cloud–edge–end collaborative task offloading method based on deep learning methods, combining Convolutional Neural Networks (CNN), Long Short-Term Memory (LSTM), and attention mechanisms. This method models the total cost of the cloud–edge–end collaboration system as a weighted sum function of the time delay and energy consumption of task execution. Then, with the objective of minimizing the total cost of the system, the task offloading problem is formulated as a mixed-integer joint optimization problem with offloading decision and bandwidth allocation, and two subproblems are decomposed from the optimization problem: one focuses on offloading decision and the other on bandwidth allocation. A CNN-LSTM-Attention neural network-based method is proposed to solve the optimal offloading decision efficiently. Based on this, the differential evolution algorithm is used to generate the optimal bandwidth allocation, resulting in efficient task offloading. The simulation experiment results demonstrate that our method enhances system performance and reduces the overall cost of the system, which is significantly better than existing methods.

Laplace convolutions of weighted averages of arithmetical functions

Abstract Let G ⁢ ( g ; x ) := ∑ n ≤ x g ⁢ ( n ) {G(g;x):=\sum_{n\leq x}g(n)} be the summatory function of an arithmetical function g ⁢ ( n ) {g(n)} . In this paper, we prove that we can write weighted averages of an arbitrary fixed number N of arithmetical functions g j ⁢ ( n ) , j ∈ { 1 , … , N } {g_{j}(n),\,j\in\{1,\dots,N\}} as an integral involving the convolution (in the sense of Laplace) of G j ⁢ ( x ) {G_{j}(x)} , j ∈ { 1 , … , N } {j\in\{1,\dots,N\}} . Furthermore, we prove an identity that allows us to obtain known results about averages of arithmetical functions in a very simple and natural way, and overcome some technical limitations for some well-known problems.