Estimates For Sums Research Articles

Primes as Sums of Fibonacci Numbers

The purpose of this paper is to discuss the relationship between prime numbers and sums of Fibonacci numbers. One of our main results says that for every sufficiently large integer k k there exists a prime number that can be represented as the sum of k k different and non-consecutive Fibonacci numbers. This property is closely related to, and based on, a prime number theorem for certain morphic sequences. In our case, these morphic sequences are based on the Zeckendorf expansion of a positive integer n n — we write n n as the sum of non-consecutive Fibonacci numbers. More precisely, we are concerned with the Zeckendorf sum-of-digits function z \mathsf {z} , which returns the minimal number of Fibonacci numbers needed to write a positive integer as their sum. The proof of such a prime number theorem for z \mathsf {z} , combined with a corresponding local result, constitutes the central contribution of this paper, from which the result stated in the beginning follows. Problems of this type have been discussed intensively in the context of the base- q q expansion of integers. The driving forces of this development were the Gelfond problems (1968/1969), more specifically the behavior of the sum-of-digits function in base q q along the sequence of primes and along integer-valued polynomials, and the Sarnak conjecture. Mauduit and Rivat resolved the question on the sum of digits of prime numbers (2010) and the sum of digits of squares (2009). Later the second author (2017) proved Sarnak’s conjecture for the class of automatic sequences, which are based on the q q -ary expansion of integers, and which generalize the sum-of-digits function in base q q considerably. For the (partial) solution of the Gelfond problems (1967/1968), Mauduit and Rivat have developed a powerful method that is based on techniques for “cutting off digits”, on sophisticated estimates for Fourier terms, and on estimates for exponential sums. These techniques — together with a new decomposition of finite automata — were also the basis for the second author’s result on automatic sequences. In order to obtain corresponding results for Fibonacci numbers, we have to extend Mauduit and Rivat’s method considerably. In fact, we are departing significantly from this method, proving the statement that exp ⁡ ( 2 π i ϑ z ( n ) ) \exp (2\pi i \vartheta \hspace {0.5pt}\mathsf {z}(n)) has level of distribution 1 1 . This latter result forms an essential part of our treatment of the occurring sums of type I \mathrm {I} and I I \mathrm {II} and uses Gowers norms related to the Zeckendorf sum-of-digits function as a central technical tool. Gowers norms are a higher order generalization of the above-mentioned Fourier terms, and their appearance in our method is intimately tied to the iterated application of a new generalization of van der Corput’s inequality.

Local Limit Theorems for Hook Lengths in Partitions

AbstractPartition hook lengths have wide-ranging applications in combinatorics, number theory, physics, and representation theory. We study two infinite families of random variables associated with t-hooks. For fixed $$t\ge 1,$$ t ≥ 1 , if $$Y_{t;\,n}$$ Y t ; n counts the number of hooks of length t in a random integer partition of n, we prove a uniform local limit theorem for $$Y_{t;\,n}$$ Y t ; n on any bounded set of $${\mathbb {R}}.$$ R . To achieve this, we establish an asymptotic formula with a power-saving error term for the number of partitions of n with m many t-hooks. In contrast, we define $${\widehat{Y}}_{t;\,n}$$ Y ^ t ; n as the count of hooks divisible by t in a randomly chosen partition of n. While $${\widehat{Y}}_{t;\,n}$$ Y ^ t ; n converges in distribution, we show that it fails to satisfy the local limit theorem for any $$t \ge 2$$ t ≥ 2 . The proofs employ the multivariable saddle-point method, asymptotic formulas for the number of t-core partitions from Anderson and Lulov–Pittel, and estimates of certain exponential sums. Notably, for $$t=4,$$ t = 4 , the analysis involves the asymptotic behavior of class numbers of imaginary quadratic fields.

On the Power Series Method for Solitons and Peakons of Nonlinear PDE

Abstract In this paper we discuss the power series method to obtain traveling wave solutions of nonlinear PDE in one space variable. The method, including a recursive procedure for the coefficients and error estimate for the partial sums, is presented for the Camassa-Holm equation. Numerical examples of approximate single soliton and single peakon solutions are included.

Improved weighted sum estimation of total radiated power at W7-X

AbstractAs magnetic confinement devices move toward higher fusion powers, moderating the heat load to the plasma-facing components becomes increasingly challenging. Efficient power dissipation can be achieved through control of the plasma radiation. However, defining a reliable proxy for the total radiated power is particularly challenging for non-axisymmetric devices such as stellarators. To address this problem, the radiated power can be estimated through a sum of the individual line-integrated bolometer measurements with weights properly calculated to account for the three-dimensional magnetic geometry. The present contribution aims to apply this weighted sum approach to Wendelstein 7-X (W7-X) and quantitatively validate it. First, we generate synthetic radiated power phantoms with characteristic W7-X radiation features to derive a set of optimized line-of-sight weights. Then, we test the weights on mock-ups and EMC3-EIRENE radiation patterns, including acquisition and analysis errors such as random noise fluctuations, camera misalignments, and field errors. Compared to other methods, the optimized weighted sum technique exhibited the best performance in all the presented synthetic test cases. When applied to experimental bolometer data, the optimized weights provided a proxy that is both reliable and real-time capable. Further validation is foreseen for the next experimental campaign.

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 Sums of Eigenfunctions of Elliptic Pseudo-differential Operators on Compact Lie Groups

We extend the estimates proved by Donnelly and Fefferman, and by Lebeau, Robbiano and Zuazua, for sums of eigenfunctions of the Laplacian (on a compact manifold) to estimates for sums of eigenfunctions of any positive and elliptic pseudo-differential operator of positive order on a compact Lie group. Our criteria are imposed in terms of the positivity of the corresponding matrix-valued symbol of the operator. As an application of these inequalities in control theory, we obtain the null-controllability for diffusion models for elliptic pseudo-differential operators on compact Lie groups.

Rational points on a class of cubic hypersurfaces

Abstract Let r ⩾ 3 r\geqslant 3 be an integer and 𝑄 any positive definite quadratic form in 𝑟 variables. We establish asymptotic formulae with power-saving error terms for the number of rational points of bounded height on singular hypersurfaces S Q S_{Q} defined by x 3 = Q ⁢ ( y 1 , … , y r ) ⁢ z x^{3}=Q(y_{1},\dots,y_{r})z . This confirms Manin’s conjecture for any S Q S_{Q} . Our proof is based on analytic methods, and uses some estimates for character sums and moments of 𝐿-functions. In particular, one of the ingredients is Siegel’s mass formula in the argument for the case r = 3 r=3 .

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.

Better Sum Estimation via Weighted Sampling

Given a large set U where each item a ∈ U has weight w ( a ), we want to estimate the total weight W = ∑ a ∈ U w ( a ) to within factor of 1 ± ε with some constant probability > 1/2. Since n = | U | is large, we want to do this without looking at the entire set U . In the traditional setting in which we are allowed to sample elements from U uniformly, sampling Ω ( n ) items is necessary to provide any non-trivial guarantee on the estimate. Therefore, we investigate this problem in different settings: in the proportional setting we can sample items with probabilities proportional to their weights, and in the hybrid setting we can sample both proportionally and uniformly. These settings have applications, for example, in sublinear-time algorithms and distribution testing. Sum estimation in the proportional and hybrid setting has been considered before by Motwani, Panigrahy, and Xu [ICALP, 2007]. In their paper, they give both upper and lower bounds in terms of n . Their bounds are near-matching in terms of n , but not in terms of ε. In this paper, we improve both their upper and lower bounds. Our bounds are matching up to constant factors in both settings, in terms of both n and ε. No lower bounds with dependency on ε were known previously. In the proportional setting, we improve their \(\tilde{O}(\sqrt {n}/\varepsilon ^{7/2}) \) algorithm to \(O(\sqrt {n}/\varepsilon) \) . In the hybrid setting, we improve \(\tilde{O}(\sqrt [3]{n}/ \varepsilon ^{9/2}) \) to \(O(\sqrt [3]{n}/\varepsilon ^{4/3}) \) . Our algorithms are also significantly simpler and do not have large constant factors. We then investigate the previously unexplored scenario in which n is not known to the algorithm. In this case, we obtain a \(O(\sqrt {n}/\varepsilon + \log n / \varepsilon ^2) \) algorithm for the proportional setting, and a \(O(\sqrt {n}/\varepsilon) \) algorithm for the hybrid setting. This means that in the proportional setting, we may remove the need for advice without greatly increasing the complexity of the problem, while there is a major difference in the hybrid setting. We prove that this difference in the hybrid setting is necessary, by showing a matching lower bound. Our algorithms have applications in the area of sublinear-time graph algorithms. Consider a large graph G = ( V , E ) and the task of (1 ± ε)-approximating | E |. We consider the (standard) settings where we can sample uniformly from E or from both E and V . This relates to sum estimation as follows: we set U = V and the weights to be equal to the degrees. Uniform sampling then corresponds to sampling vertices uniformly. Proportional sampling can be simulated by taking a random edge and picking one of its endpoints at random. If we can only sample uniformly from E , then our results immediately give a \(O(\sqrt {|V|} / \varepsilon) \) algorithm. When we may sample both from E and V , our results imply an algorithm with complexity \(O(\sqrt [3]{|V|}/\varepsilon ^{4/3}) \) . Surprisingly, one of our subroutines provides an (1 ± ε)-approximation of | E | using \(\tilde{O}(d/\varepsilon ^2) \) expected samples, where d is the average degree, under the mild assumption that at least a constant fraction of vertices are non-isolated. This subroutine works in the setting where we can sample uniformly from both V and E . We find this remarkable since it is O (1/ε 2 ) for sparse graphs.

L2 to Lp bounds for spectral projectors on the Euclidean two-dimensional torus

AbstractWe consider spectral projectors associated to the Euclidean Laplacian on the two-dimensional torus, in the case where the spectral window is narrow. Bounds for their L2 to Lp operator norm are derived, extending the classical result of Sogge; a new question on the convolution kernel of the projector is introduced. The methods employed include $\ell^2$ decoupling, small cap decoupling and estimates of exponential sums.

A Fast and Simple Machine Vision Framework for Approximating the Volume of Axi-Symmetric Objects Using Shadow Ray Casting

The volume measurement using machine vision system is contactless techniques that play an important role in industries now a day. Basically, three-dimensional reconstruction is required to determine a depth using a special lighting system or multiple cameras. This increases the complexity of the measurement system. A fast and simple machine vision framework called RayVol for estimating the volume of axisymmetric objects in near real-time using a single camera and simple illumination is presented. The RayVol framework employs a shadow casting method to reconstruct the 3D shape of the object by tracing rays from the object’s shadow pixels to the light source location. The result of this technique shows a significant accuracy improvement from the area-projection method. A virtual slice representing the cross-section of an object is reconstructed using a cubic spline approximation from baseline points derived from the boundary pixels of the object image and a shadow casting method. The volume estimation was calculated by restricted integration using the Riemann sum estimation algorithm, and the closed area of the virtual slices was calculated using the shoestring algorithm. Mangoes were used as a case study of the RayVol framework. The volume estimation provides the correlation coefficient of 0.9849 between the developed system and the water replacement method.

Large sums of high‐order characters

AbstractLet be a primitive character modulo a prime , and let . It has previously been observed that if has large order then for some , in analogy with Vinogradov's conjecture on quadratic non‐residues. We give a new and simple proof of this fact. We show, furthermore, that if is squarefree then for any th root of unity the number of such that is whenever . Consequently, when has sufficiently large order the sequence cannot cluster near for any . Our proof relies on a second moment estimate for short sums of the characters , averaged over , that is non‐trivial whenever has no small prime factors. In particular, given any we show that for all but powers , the partial sums of exhibit cancellation in intervals as long as is prime, going beyond Burgess' theorem. Our argument blends together results from pretentious number theory and additive combinatorics. Finally, we show that, uniformly over prime , the Pólya–Vinogradov inequality may be improved for on average over , extending work of Granville and Soundararajan.

Fourth power mean of the general s-dimensional Kloosterman sum mod p

In this article, we prove an asymptotic formula for the fourth power mean of a general s-dimensional hyper-Kloosterman sum. We find the number of solutions of certain congruence equations mod p which play an integral part to prove our main result. We use estimates for character sums and analytic methods to prove our theorem.

CocoSketch: High-Performance Sketch-Based Measurement Over Arbitrary Partial Key Query

Sketch-based measurement has emerged as a promising solutions due to its high accuracy and resource efficiency. Prior sketches focus on measuring single flow keys and cannot support measurement on multiple keys. This work takes a significant step towards supporting <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">arbitrary partial key queries</i> , which aims to provide information for any key in the predefined range of possible flow keys. The designed system, casts arbitrary partial key queries to the subset sum estimation problem and makes the theoretical tools for subset sum estimation practical. utilizes two techniques: (1) stochastic variance minimization to significantly reduce per-packet update delay, and (2) removing circular dependencies in the per-packet update logic to make the implementation hardware-friendly. This paper extends the conference version by discussing how adapts to new measurement requirements, including: (1) collecting the exact information of specified flow keys, and (2) distributed measurement. is implemented on five popular platforms (CPU, Open vSwitch, Redis, P4, and FPGA). Experiment results show that compared to baselines that use traditional single-key sketches, improves average packet processing throughput by <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$27.2\times$</tex-math> </inline-formula> and accuracy by <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$10.4\times$</tex-math> </inline-formula> when measuring six flow keys.

Three conjectures about character sums

We establish that three well-known and rather different looking conjectures about Dirichlet characters and their (weighted) sums, (concerning the Pólya–Vinogradov theorem for maximal character sums, the maximal admissible range in Burgess’ estimate for short character sums, and upper bounds for L(1,chi ) and L(1+it,chi )) are more-or-less “equivalent”. We also obtain a new mean value theorem for logarithmically weighted sums of 1-bounded multiplicative functions.

Suzuki type estimates for exponentiated sums and generalized Lie-Trotter formulas in JB-algebras

Lie-Trotter-Suzuki product formulas are ubiquitous in quantum mechanics, computing, and simulations. Approximating exponentiated sums with such formulas are investigated in the JB-algebraic setting. We show that the Suzuki type approximation for exponentiated sums holds in JB-algebras, we give explicit estimation formulas, and we deduce three generalizations of Lie-Trotter formulas for arbitrary number elements in such algebras. We also extended the Lie-Trotter formulas in a Jordan Banach algebra from three elements to an arbitrary number of elements.

Multi-Slot Over-the-Air Computation in Fading Channels

IoT systems typically involve separate data collection and processing, and face the scalability issue when the number of nodes increases. For some tasks, only the result of data fusion is needed. Then, the whole process can be realized in an efficient way, integrating the data collection and fusion in one step by over-the-air computation (AirComp). Its shortcoming, however, is signal distortion when channel gains of nodes are different, which cannot be well solved by transmission power control alone in times of deep fading. To address this issue, in this paper, we propose a multi-slot over-the-air computation (MS-AirComp) framework for the sum estimation in fading channels. Compared with conventional data collection (one slot for each node) and AirComp (one slot for all nodes), MS-AirComp is an alternative policy that lies between them, exploiting multiple slots to improve channel gains so as to facilitate power control. Avoiding to obtain instantaneous channel gains of all nodes at the sink is a key point. Specifically, the transmissions are distributed over multiple slots and a threshold of channel gain is set for distributed transmission scheduling. Each node transmits its signal only once, in the slot when its channel gain first gets above the threshold, or in the last slot when its channel gain remains below the threshold. Theoretical analysis gives the closed-form of the computation error in fading channels, based on which the optimal parameters are found. Noticing that computation error tends to be reduced at the cost of more transmission power, a method is suggested to control the increase of transmission power. Simulations confirm that the proposed method can effectively reduce computation error, compared with state-of-the-art methods.

Measuring child and adolescent well-being in Denmark: Validation and norming of the Danish KIDSCREEN-10 child/adolescent version in a national representative sample of school pupils in grades five through eight.

KIDSCREEN-10 is a generic instrument for measuring global health-related quality of life among 8-18-year-old children and adolescents. This study examines the criterion-related construct validity and psychometric properties of the Danish language version of the KIDSCREEN-10 using Rasch models. A further aim was to construct Danish norms based on the resulting person parameter estimates from the Rasch models. Data consists of a nationally representative cross-sectional survey of 8171 children in the 5th to 8th grade of primary school in Denmark. No adequate fit to the Rasch model or a graphical loglinear Rasch model could be established for the KIDSCREEN-10 in the full sample of children (n = 8171). Results based on analyses with increasing samples sizes showed that even with the smallest sample item 3 (Kid3) of the KIDSCREEN-10 did not fit the Rasch model. After elimination of Kid3, substantial local dependence and differential item functioning relative to gender and grade level was still present. Already with a sample size of 630 fit to the Rasch model or a graphical loglinear Rasch model adjusting for local dependence and differential item functioning was not established. Therefore, generation of Danish norms was not realizable, as this requires valid sum scores and estimates of the person parameters for an adequate number of cases. Thus, the Danish language version of the child/adolescent self-report KIDSCREEN-10 questionnaire cannot be recommended for use in population-level studies. Neither can use in small sample be recommended as adjustment for differential item functioning and local dependence is ambiguous.

Global estimates for sums of absolutely convergent Dirichlet series in a half-plane

Let $(\lambda_n)_{n=0}^{+\infty}$ be a nonnegative sequence increasing to $+\infty$, $F(s)=\sum_{n=0}^{+\infty} a_ne^{s\lambda_n}$ be an absolutely convergent Dirichlet series in the half-plane $\{s\in\mathbb{C}\colon \operatorname{Re} s&lt;0\}$, and let, for every $\sigma&lt;0$, $\mathfrak{M}(\sigma,F)=\sum_{n=0}^{+\infty} |a_n|e^{\sigma\lambda_n}$.&#x0D; Suppose that $\Phi\colon (-\infty,0)\to\overline{\mathbb{R}}$ is a function, and let $\widetilde{\Phi}(x)$ be the Young-conjugate function of $\Phi(\sigma)$, i.e.$\widetilde{\Phi}(x)=\sup\{x\sigma-\alpha(\sigma)\colon \sigma&lt;0\}$ for all $x\in\mathbb{R}$. In the article, the following two statements are proved:&#x0D; (i) There exist constants $\theta\in(0,1)$ and $C\in\mathbb{R}$ such that$\ln\mathfrak{M}(\sigma,F)\le\Phi(\theta\sigma)+C$ for all $\sigma&lt;0$ if and only if there exist constants $\delta\in(0, 1)$ and $c\in\mathbb{R}$ such that $\ln\sum_{m=0}^n|a_m|\le-\widetilde{\Phi}(\lambda_n/\delta)+c$ for all integers $n\ge0$ (Theorem 2);&#x0D; (ii) For every $\theta\in(0,1)$ there exists a real constant $C=C(\delta)$ such that $\ln\mathfrak{M}(\sigma,F)\le\Phi( \theta\sigma)+C$ for all $\sigma&lt;0$ if and only if for every $\delta\in(0,1)$ there exists a real constant $c=c(\delta)$ such that $\ln\sum_{m=0}^n|a_m|\le-\widetilde{\Phi}(\lambda_n/\delta)+c$ for all integers $n\ge0$ (Theorem 3).iii) Let $\Phi$ be a continuous positive increasing function on $\mathbb{R}$ such that $\Phi(\sigma)/\sigma\to+\infty$, $\sigma\to+ \infty$ and $F$ be a entire Dirichlet series.&#x0D; For every $q&gt;1$ there exists a constant $C=C(q)\in\mathbb{R}$ such that $\ln\mathfrak{M}(\sigma,F)\le \Phi(q\sigma)+C,\quad \sigma\in\mathbb{R},$ holds if and only if for every $\delta \in(0,1)$ there exist constants $c=c(\delta)\in\mathbb{R}$ and $n_0=n_0(\delta)\in\mathbb{N}_0$ such that $\ln \sum_{m=n}^{+\infty}|a_m|\le-\widetilde{\Phi}(\delta\lambda_n)+c,\quad n\ge n_0$ Theorem 5.&#x0D; These results are analogous to some results previously obtained by M.M. Sheremeta for entire Dirichlet series.

Decoupling inequalities for quadratic forms

We prove sharp $\ell^q L^p$ decoupling inequalities for $p,q \in [2,\infty)$ and arbitrary tuples of quadratic forms. Connections to prior results on decoupling inequalities for quadratic forms are also explained. We also include some applications of our results to exponential sum estimates and to Fourier restriction estimates. The proof of our main result is based on scale-dependent Brascamp--Lieb inequalities.