Exponential Sums Research Articles

Few-time-points time-integrated activity coefficients calculation using non-linear mixed-effects modeling: Proof of concept for [111In]In-DOTA-TATE in kidneys

PurposeThe purpose of this study is to investigate the accuracy of few-time-points (FTP) time-integrated activity coefficients (TIACs) in peptide-receptor radionuclide therapy (PRRT) using non-linear mixed-effects (NLME) modeling. MethodsBiokinetic data of [111In]In-DOTA-TATE in kidneys at T-1 = (2.9 ± 0.6) h, T-2 = (4.6 ± 0.4) h, T-3 = (22.8 ± 1.6) h, T-4 = (46.7 ± 1.7) h, and T-5 = (70.9 ± 1.0) h after injection were obtained from eight patients using planar imaging. The Sum-Of-Exponentials (SOE) function with four parameters was used, which was selected as the best model for the renal biokinetic data of [111In]In-DOTA-TATE. The parameters of the SOE function were fitted to the all-time-point data in the NLME framework to derive reference (rTIACs). FTP fits, which consist of all combinations of time points, are done to calculate the estimated TIACs (eTIACs). The accuracy of the FTP-NLME TIACs calculations was assessed by calculating the relative deviations (RDs) and relative root-mean-square errors (RMSEs) between the eTIACs and rTIACs. ResultsThe lowest (mean ± SD) of RDs for the single-, two-, three-, four-time point FTPs were (0 ± 8) % (T-4), (1 ± 6) % (T-3 and T-4), (3 ± 5) % (T-2, T-3 and T-4), and (0 ± 2) % (T-2, T-3, T-4, and T-5), respectively. The lowest RMSEs for the one-, two-, three-, and four-time point FTPs were 8 % (T-4), 6 % (T-3 and T-4), 5 % (T-2, T-3 and T-4), and 2 % (T-2, T-3, T-4, and T-5), respectively. ConclusionOur results showed that FTP-NLME in an example of [111In]In-DOTA-TATE could lead to a high accuracy of eTIAC across various time points, when incorporating time point T-4 = (46.7 ± 1.7) h.

Estimates for Smooth Weyl Sums on Major Arcs

Abstract We present estimates for smooth Weyl sums of use on sets of major arcs in applications of the Hardy–Littlewood method. In particular, we derive mean value estimates on major arcs for smooth Weyl sums of degree $k$ delivering essentially optimal bounds for moments of order $u$ whenever $u>2\lfloor k/2\rfloor +4$.

Estimates for a three-dimensional exponential sum with monomials

Estimates for a three-dimensional exponential sum with monomials

Wave propagation in poroviscoelastic vertical transversely isotropic media: A sum-of-exponentials approximation for the simulation of fractional Biot-squirt equations

The presence of porous, viscoelastic, and anisotropic structures profoundly influences the propagation of seismic waves. Poroelasticity, based on the Biot-squirt (BISQ) theory, can characterize the impact of fluid flow on seismic waves. However, an improved mechanistic understanding and mathematical representation of seismic wave propagation in porous formations require the accounting of velocity and attenuation anisotropy. We derive the wave equations based on an efficient sum-of-exponentials (SOE) approximation by combining the porous BISQ equations with Kjartansson’s constant- Q dissipative model to describe seismic wave propagation in poroviscoelastic vertical transversely isotropic media. The approximation represents the fractional differential equations as an SOE, enabling a reduction in computational costs and storage compared with the traditional L2 scheme. Numerical examples prove that the extended BISQ model can describe the wave propagation characteristics in poroviscoelastic anisotropic media and effectively captures potentially strong, direction-dependent attenuation phenomena.

A novel fast tempered algorithm with high-accuracy scheme for 2D tempered fractional reaction-advection-subdiffusion equation

In this article, we propose a novel and computationally efficient difference scheme for evaluating the Caputo tempered fractional derivative (TFD). Our approach introduces a new fast tempered FλL2−1σ difference method, achieving a higher convergence rate of order O(Δt3−α). Specifically, we apply this method to a class of two-dimensional tempered-fractional reaction-advection-subdiffusion equations (2D-TF-RASDE) characterized by the tempered parameter λ and a tempered fractional derivative of order α, (where 0<α<1). The novel fast tempered scheme is based on the sum of exponents (SOE) technique. Additionally, we develop a novel high order compact finite difference (CFD) scheme using an alternating direction implicit (ADI) formulation for the 2D-TF-RASDE. We investigate the stability and convergence of this FLλ2−1σ-ADI-CD scheme. Numerical simulations reveal a convergence order of O(Δt1+α+hϰ4+hy4+ϵ) under robust regularity assumptions underlining the scheme's superior computational efficiency. Furthermore, our computational results align well with theoretical analysis, demonstrating both accuracy and reduced computational complexity and storage requirements as compared to the standard tempered Lλ2−1σ-ADI-CD scheme. Notably, the fast tempered FLλ2−1σ-ADI-CD scheme exhibits competitive performance and reduced CPU time relative to the standard Lλ2−1σ-ADI-CD scheme, thereby offering a distinct advantage for efficiently solving complex fractional differential equations.

On some Airy sheaves of Laurent type

We study certain one-parameter families of exponential sums of Airy–Laurent type. Their general theory was developed in Katz and Tiep (Airy sheaves of Laurent type: an introduction, https://web.math.princeton.edu/~nmk/kt31_11sept.pdf). In the present paper, we make use of that general theory to compute monodromy groups in some particularly simple families (in the sense of “simple to remember"), realizing Weyl groups of type E6\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$E_6$$\\end{document} and E8\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$E_8$$\\end{document}.

Regular Polygonal Vortex Filament Evolution and Exponential Sums

When taking a regular planar polygon of M\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$M$\\end{document} sides and length 2π\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$2\\pi $\\end{document} as the initial datum of the vortex filament equation, Xt=Xs∧Xss\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\mathbf{X}_{t}= \\mathbf{X}_{s}\\wedge \\mathbf{X}_{ss}$\\end{document}, the solution becomes polygonal at times of the form tpq=(p/q)(2π/M2)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$t_{pq} = (p/q)(2\\pi /M^{2})$\\end{document}, with gcd(p,q)=1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\gcd (p,q)=1$\\end{document}, and the corresponding polygon has Mq\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$Mq$\\end{document} sides, if q\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$q$\\end{document} is odd, and Mq/2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$Mq/2$\\end{document} sides, if q\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$q$\\end{document} is even. Moreover, that polygon is skew (except when q=1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$q = 1$\\end{document} or q=2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$q = 2$\\end{document}, where the initial shape is recovered), and the angle ρ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\rho $\\end{document} between two adjacent sides is a constant. In this paper, we give a rigorous proof of the conjecture that states that, at a time tpq\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$t_{pq}$\\end{document}, cosq(ρ/2)=cos(π/M)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\cos ^{q}(\\rho /2) = \\cos (\\pi /M)$\\end{document}, if q\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$q$\\end{document} is odd, and cosq(ρ/2)=cos2(π/M)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\cos ^{q}(\\rho /2) = \\cos ^{2}(\\pi /M)$\\end{document}, if q\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$q$\\end{document} is even. Since the transition of one side of the polygon to the next one is given by a rotation in R3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\mathbb{R}^{3}$\\end{document} determined by a generalized Gauss sum, the idea of the proof consists in showing that a certain product of those rotations is a rotation of angle 2π/M\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$2\\pi /M$\\end{document}, which is equivalent to proving that some exponential sums with arithmetic content are purely imaginary.

Bounds for Exponential Sums With Random Multiplicative Coefficient

Abstract For $f$ a Rademacher or Steinhaus random multiplicative function, we prove that $ \max _{\theta \in [0,1]} \frac{1}{\sqrt{N}} \bigl | \sum _{n \leq N} f(n) \mathrm{e} (n \theta ) \bigr | \gg \sqrt{\log N}$, asymptotically almost surely as $N \rightarrow \infty $. Furthermore, for $f$ a Steinhaus random multiplicative function, and any $\varepsilon&amp;gt; 0$, we prove the partial upper bound result $ \max _{\theta \in [0,1]} \frac{1}{\sqrt{N}} \bigl | \sum _{\substack{n \leq N \\ P(n) \geq N^{0.8}}} f(n) \mathrm{e} (n \theta ) \bigr | \ll{(\log N)}^{7/4 + \varepsilon }$, asymptotically almost surely as $N \rightarrow \infty $, where $P(n)$ denotes the largest prime factor of $n$.

A novel explicit fast numerical scheme for the Cauchy problem for integro-differential equations with a difference kernel and its application

The present study focuses on designing a second-order novel explicit fast numerical scheme for the Cauchy problem incorporating memory associated with an evolutionary equation, where the integral term's kernel is a discrete difference operator. The Cauchy problem under consideration is related to a real finite-dimensional Hilbert space and includes a self-adjoint operator that is both positive and definite. We introduce a transformative technique for converting the Cauchy problem incorporating memory, into a local evolutionary system of equations by approximating the difference kernel using the sum of exponentials (SoE) approach. A second-order explicit scheme is then constructed to solve the local system. We thoroughly investigate the stability of this explicit scheme, and present the necessary conditions for the stability of the scheme. Moreover, we extended our investigation to encompass time-fractional diffusion-wave equations (TFDWEs) involving a fractional Caputo derivative with an order ranging between (1,2). Initially, we transform the main TFDWE model into a new model that incorporates the fractional Riemann-Liouville integral. Subsequently, we expand the applicability of our idea to develop an explicit fast numerical algorithm for approximating the model. The stability properties of this fast scheme for solving TFDWEs are assessed. Numerical simulations including a two-dimensional Cauchy problem as well as one-dimensional and two-dimensional TFDWE models are provided to validate the accuracy and experimental order of convergence of the schemes.

Development of adaptive exponential min sum decoding algorithm

This paper presents an optimized min sum (MS) decoding algorithm with low complexity and high decoding performance for LDPC short codes. The MS algorithm has low computational complexity and is simple to deploy. The MS decoding algorithm, while demonstrating a performance gap compared to the belief propagation (BP) and likelihood ratio BP (LLR-BP) decoding algorithms, shows significant potential for optimization. To improve the decoding performance of traditional MS algorithm, secondary external information is introduced into the control node (CNs) update operations of MS algorithm and optimized as adaptive exponential correction factor (AECF). The optimized MS algorithm is named as adaptive exponential exponential MS decoding algorithm (AEMS). The decoding efficiency of the AEMS algorithm for regular, irregular and LDPC codes of the Consultative Committee on Space Data Systems (CCSDS) was extensively tested, then the complexity of the AEMS algorithm was analyzed and compared with other decoding algorithms. The results show that the AEMS algorithm outperforms the offset MS (OMS) and normalized MS (NMS) algorithms in decoding performance, and outperforms the BP algorithm as the signal-to-noise ratio (SNR) gradually increases. В статье представлен оптимизированный алгоритм декодирования минимальной суммы (MS) с низкой сложностью и высокой производительностью декодирования для коротких кодов LDPC. Алгоритм MS имеет низкую вычислительную сложность и прост в развертывании. По сравнению с алгоритмом декодирования распространения убеждения (BP) и отношения правдоподобия BP (LLR-BP) он показывает разрыв в производительности декодирования, но алгоритм декодирования MS имеет высокий потенциал оптимизации. Для улучшения производительности декодирования традиционного алгоритма MS в операции обновления контрольных узлов (CN) алгоритма MS вводится вторичная внешняя информация и оптимизируется как адаптивный экспоненциальный поправочный коэффициент (AECF). Оптимизированный алгоритм MS назван адаптивным экспоненциальным алгоритмом декодирования MS (AEMS). Эффективность декодирования алгоритма AEMS для обычных, нерегулярных и LDPC-кодов консультативного комитета по системам космических данных (CCSDS) была всесторонне протестирована, затем был проведен анализ и сравнение сложности алгоритма AEMS с другими алгоритмами декодирования. Результаты показывают, что алгоритм AEMS превосходит алгоритмы смещенного MS (OMS) и нормализованного MS (NMS) по производительности декодирования, а также превосходит алгоритм BP по мере постепенного увеличения отношения сигнал/шум (SNR).

On a family of sparse exponential sums

AbstractWe investigate exponential sums modulo primes whose phase function is a sparse polynomial, with exponents growing with the prime. In particular, such sums model those which appear in the study of the quantum cat map. While they are not amenable to treatment by algebro‐geometric methods such as Weil's bounds, Bourgain gave a nontrivial estimate for these and more general sums. In this work, we obtain explicit bounds with reasonable savings over various types of averaging. We also initiate the study of the value distribution of these sums.

The Riemann zeta function and exact exponential sum identities of divisor functions

We prove an explicit integral formula for computing the product of two shifted Riemann zeta functions everywhere in the complex plane. We show that this formula implies the existence of infinite families of exact exponential sum identities involving the divisor functions, and we provide examples of these identities. We conjecturally propose a method to compute divisor functions by matrix inversion, without employing arithmetic techniques.

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.

Symmetric SAGE and SONC forms, exactness and quantitative gaps

The classes of sums of arithmetic-geometric exponentials (SAGE) and of sums of nonnegative circuit polynomials (SONC) provide nonnegativity certificates which are based on the inequality of the arithmetic and geometric means. We study the cones of symmetric SAGE and SONC forms and their relations to the underlying symmetric nonnegative cone.As main results, we provide several symmetric cases where the SAGE or SONC property coincides with nonnegativity and we present quantitative results on the differences in various situations. The results rely on characterizations of the zeroes and the minimizers for symmetric SAGE and SONC forms, which we develop. Finally, we also study symmetric monomial mean inequalities and apply SONC certificates to establish a generalized version of Muirhead's inequality.

Infinitude of Palindromic Almost-Prime Numbers

Abstract It is proven that, in any given base, there are infinitely many palindromic numbers having at most six prime divisors, each relatively large. The work involves equidistribution estimates for the palindromes in residue classes to large moduli, offering upper bounds for moments and averages of certain products closely related to exponential sums over palindromes.

Compressing the memory variables in constant-Q viscoelastic wave propagation via an improved sum-of-exponentials approximation

Earth introduces strong attenuation and dispersion to propagating waves. The time-fractional wave equation with very small fractional exponent, based on Kjartansson's constant-Q theory, is widely recognized in the field of geophysics as a reliable model for frequency-independent Q anelastic behavior. Nonetheless, the numerical resolution of this equation poses considerable challenges due to the requirement of storing a complete time history of wavefields. To address this computational challenge, we present a novel approach: a nearly optimal sum-of-exponentials (SOE) approximation to the Caputo fractional derivative with very small fractional exponent, utilizing the machinery of generalized Gaussian quadrature. This method minimizes the number of memory variables needed to approximate the power attenuation law within a specified error tolerance. We establish a mathematical equivalence between this SOE approximation and the continuous fractional stress-strain relationship, relating it to the generalized Maxwell body model. Furthermore, we prove an improved SOE approximation error bound to thoroughly assess the ability of rheological models to replicate the power attenuation law. Numerical simulations on constant-Q viscoacoustic equation in 3D homogeneous media and variable-order P- and S- viscoelastic wave equations in 3D inhomogeneous media are performed. These simulations demonstrate that our proposed technique accurately captures changes in amplitude and phase resulting from material anelasticity. This advancement provides a significant step towards the practical usage of the time-fractional wave equation in seismic inversion.

Legendre-signed partition numbers

Let f:N→{0,±1}, for n∈N let Π[n] be the set of partitions of n, and for all partitions π=(a1,a2,…,ak)∈Π[n] letf(π):=f(a1)f(a2)⋯f(ak). With this we define the f-signed partition numbersp(n,f)=∑π∈Π[n]f(π).In this paper, for odd primes p we derive asymptotic formulae for p(n,χp) as n→∞, where χp(n) is the Legendre symbol (np) associated to p. A similar asymptotic formula for p(n,χ2) is also established, where χ2(n) is the Kronecker symbol (n2). Special attention is paid to the sequence (p(n,χ5))N, and a formula for p(n,χ5) supporting the recent discovery that p(10j+2,χ5)=0 for all j≥0 is discussed. As a corollary, our main results imply that the periodic vanishing displayed by (p(n,χ5))N does not occur in any sequence (p(n,χp))N for p≠5 such that p≢1(mod8). In addition, work of Montgomery and Vaughan on exponential sums with multiplicative coefficients is applied to establish a uniform upper bound on certain doubly infinite series involving multiplicative functions f with |f|≤1.

On Artin’s Primitive Root Conjecture for Function Fields over 𝔽q

ABSTRACT In 1927, E. Artin proposed a conjecture for the natural density of primes p for which g generates $(\mathbb{Z}/p\mathbb{Z})^\times$. By carefully observing numerical deviations from Artin’s originally predicted asymptotic, Derrick and Emma Lehmer (1957) identified the need for an additional correction factor, leading to a modified conjecture which was eventually proved to be correct by Hooley (1967) under the assumption of the generalized Riemann hypothesis. An appropriate analogue of Artin’s primitive root conjecture may moreover be formulated for an algebraic function field K of r variables over $\mathbb{F}_{q}$. Relying on a soon to be established theorem of Weil (1948), Bilharz (1937) provided a proof in the particular case that K is a global function field (that is, r = 1), which is correct under the assumption that $g \in K$ is a geometric element. Under the same assumptions, Pappalardi and Shparlinski (1995) established a quantitative version of Bilharz’s result. In this paper we build upon these works by both generalizing to function fields in r variables over $\mathbb{F}_{q}$ and removing the assumption that $g \in K$ is geometric, thereby completing a proof of Artin’s primitive root conjecture for function fields over $\mathbb{F}_{q}$. In doing so, we moreover identify an interesting correction factor which emerges when g is not geometric. A crucial feature of our work is an exponential sum estimate over varieties that we derive from Weil’s Theorem.

On the accuracy of Prony's method for recovery of exponential sums with closely spaced exponents

In this paper we establish accuracy bounds of Prony's method (PM) for recovery of sparse measures from incomplete and noisy frequency measurements, or the so-called problem of super-resolution, when the minimal separation between the points in the support of the measure may be much smaller than the Rayleigh limit. In particular, we show that PM is optimal with respect to the previously established min-max bound for the problem, in the setting when the measurement bandwidth is constant, with the minimal separation going to zero. Our main technical contribution is an accurate analysis of the inter-relations between the different errors in each step of PM, resulting in previously unnoticed cancellations. We also prove that PM is numerically stable in finite-precision arithmetic. We believe our analysis will pave the way to providing accurate analysis of known algorithms for the super-resolution problem in full generality.

First-order compartment model solutions – Exponential sums and beyond

First-order compartment models are common tools for modelling many biological processes, including pharmacokinetics. Given the compartments and the transfer rates, solutions for the time-dependent quantity (or concentration) curves can normally be described by a sum of exponentials. This paper investigates cases that go beyond simple sums of exponentials. With specific relations between the transfer rate constants, two exponential rate constants can be equal, in which case the normal solution cannot be used. The conditions for this to occur are discussed, and advice is provided on how to circumvent these cases. An example of an analytic solution is given for the rare case where an exact equality is the expected result. Furthermore, for models with at least three compartments, cases exist where the solution to a real-valued model involves complex-valued exponential rate constants. This leads to solutions with an oscillatory element in the solution for the tracer concentration, i.e., there are cases where the solution is not a simple sum of (real-valued) exponentials but also includes sine and cosine functions. Detailed solutions for three-compartment cases are given. As a tentative conclusion of the analysis, oscillatory solutions appear to be tied to cases with a cyclic element in the model itself.