Exponential Sums Research Articles

On squares of Fourier coefficients twist exponential functions with applications

Let f be a Hecke–Maass cusp form of weight zero for [Formula: see text] and [Formula: see text] be the nth Fourier coefficient. For almost all [Formula: see text] we have [Formula: see text] which improves the result of Acharya [Exponential sums of squares of Fourier coefficients of cusp forms, Proc. Indian Acad. Sci. Math. Sci. 130 (2020) 24], who showed an upper bound larger than [Formula: see text] For all [Formula: see text] of type [Formula: see text] we also show that [Formula: see text] where [Formula: see text] This result relies heavily on a generalized double sum (see Theorem 1.3).

Low-rank tensor structure preservation in fractional operators by means of exponential sums

The use of fractional differential equations is a key tool in modeling non-local phenomena. Often, an efficient scheme for solving a linear system involving the discretization of a fractional operator is computing inverse fractional powers of the standard discretized Laplace operator. In this work, an exponential sum approximation for such fractional powers is derived. It is accurate over all positive real numbers larger than one, and allows to efficiently approximate the action of such operators on tensors stored in a variety of low-rank formats (CP, TT, Tucker). The results are relevant from a practical and theoretical perspective, as they predict the low-rank approximability of the solutions of these linear systems in low-rank tensor formats.

Small Fractional Parts of Polynomials and Mean Values of Exponential Sums

Abstract Let $k_i\ (i=1,2,\ldots ,t)$ be natural numbers with $k_1>k_2>\cdots >k_t>0$, $k_1\geq 2$ and $t<k_1.$ Given real numbers $\alpha _{ji}\ (1\leq j\leq t,\ 1\leq i\leq s)$, we consider polynomials of the shape $$\begin{align*} &\varphi_i(x)=\alpha_{1i}x^{k_1}+\alpha_{2i}x^{k_2}+\cdots+\alpha_{ti}x^{k_t},\end{align*}$$and derive upper bounds for fractional parts of polynomials in the shape $$\begin{align*} &\varphi_1(x_1)+\varphi_2(x_2)+\cdots+\varphi_s(x_s),\end{align*}$$by applying novel mean value estimates related to Vinogradov’s mean value theorem. Our results improve on earlier Theorems of Baker (2017).

On the Balog–Ruzsa theorem in short intervals

Abstract In this paper we give a short interval version of the Balog–Ruzsa theorem concerning bounds for the L1 norm of the exponential sum over r-free numbers. In particular, when r = 2, for $H \geq N^{\frac{9}{17}+\epsilon}$, we have the lower bound result $$ \int_{\mathbb T}\left|\sum_{|n-N| \lt H} \mu^2(n)e(n \alpha)\right| d \alpha \gg H^{\frac{1}{3}}, $$ and for $H \geq N^{\frac{18}{29}+\epsilon}$, we have the upper bound result $$ \int_{\mathbb T}\left|\sum_{|n-N| \lt H} \mu^2(n)e(n \alpha)\right| d \alpha \ll H^{\frac{1}{3}}.$$ As an application, we show that the L1 norm of the exponential sum $\sum_{|n-N| \lt H} \mu(n)e(n \alpha)$ has the lower bound $\gg H^{\frac{1}{6}}$ when $H \geq N^{\frac{9}{17} + \varepsilon}$.

Three-dimensional exponential sums under constant perturbation

By generalizing Bombieri and Iwaniec’s double large sieve inequality, we obtain an estimation on three-dimensional exponential sums with constant perturbation. As an application of our estimation on the three-dimensional exponential sums, we get that for any ϵ>0, ∑n⩽xΛ([xn])=x∑d⩾1Λ(d)d(d+1)+Oϵ(x1736+ϵ) as x→∞. This improves a recent result of Liu–Wu–Yang, which requires 9/19 in place of 17/36.

Large Deviations for Hawkes Processes with Randomized Baseline Intensity

The Hawkes process, which is generally defined for the continuous-time setting, can be described as a self-exciting simple point process with a clustering effect, whose jump rate depends on its entire history. Due to past events determining future developments of self-exciting point processes, the Hawkes model is generally not Markovian. In certain special circumstances, it can be Markovian with a generator of the model if the exciting function is an exponential function or the sum of exponential functions. In the case of non-Markovian processes, difficulties arise when the exciting function is not an exponential function or a sum of exponential functions. The intensity of the Hawkes process is given by the sum of a baseline intensity and other terms that depend on the entire history of the point process, as compared to a standard Poisson process. It is one of the main methods used for studying the dynamical properties of general point processes, and is highly important for credit risk studies. The baseline intensity, which is instrumental in the Hawkes model, is usually defined for deterministic cases. In this paper, we consider a linear Hawkes model where the baseline intensity is randomly defined, and investigate the asymptotic results of the large deviations principle for the newly defined model. The Hawkes processes with randomized baseline intensity, dealt with in this paper, have wide applications in insurance, finance, queue theory, and statistics.

Time evolution of electron distributions to bimodal steady states for electrons dilutely dispersed in theinert gases Ar, Kr, and Xe with deep Ramsauer Townsend minima in themomentum transfer cross section

The current paper considers the thermalization of an ensemble of electrons under the influence of an external electric field and dilutely dispersed in one of the inert gas moderators, Argon, Krypton or Xenon for which the electron momentum transfer cross sections have deep Ramsauer-Townsend minima. As a consequence, the steady state electron distribution functions are bimodal over a small range of external electric field strengths. The current work is directed towards the time evolution of the electron distribution function determined from the numerical solution of the Fokker-Planck equation. The kinetic theory of electrons dilutely dispersed in a heat bath of atoms at temperature T b has a very long history. The solution of the Fokker-Planck equation can be expressed as a sum of exponentials of the form where λ n are the eigenvalues of the Fokker-Planck operator. Alternatively, a finite difference algorithm is used to solve the time dependent Fokker-Planck equation to give the time dependent electron energy distribution function. We demonstrate the evolution of the initial Maxwellian into a nonequilibrium bimodal distribution which cannot be rationalized with either the Gibbs-Boltzmann entropy or the Tsallis nonextensive entropy. Instead, the time dependent approach of an initial Maxwellian to the bimodal distribution is described in terms of the Kullback-Leibler entropy. We also demonstrate the inapplicability of the Boltzmann entropy nor the Tsallis entropy for a model system with a power law momentum transfer cross section of the form, σ(x) = σ 0/x p , where is the reduced speed. This model with p = 2 is also employed to demonstrate a steady-state Kappa distribution which features prominently in space physics and other fields. For p > 2, we show distribution functions that increase without bound analogous to runaway electrons. The steady nonequilibrium distributions are interpreted as solutions of a Pearson ordinary differential equation.

Improved tail estimates for the distribution of quadratic Weyl sums

Improved tail estimates for the distribution of quadratic Weyl sums

A new two-term exponential sums and its fourth power mean

A new two-term exponential sums and its fourth power mean

Growth of Fourier coefficients of vector-valued automorphic forms

In this article, we establish polynomial-growth bound for the sequence of Fourier coefficients associated with even integer weight vector-valued automorphic forms of non-cocompact Fuchsian groups of the first kind. In the end, their L-functions and exponential sums have been discussed.

Exponential sums with Piatetski-Shapiro sequences and their application to modular hyperbolas

The Piatetski-Shapiro sequence associated with c>1 and c∉N is given by Nc=(⌊nc⌋)n∈N. Let p be an odd prime and let g(x),h(x)∈Fp[x] be coprime polynomials. We estimate exponential sums with Piatetski-Shapiro sequences of the shape∑m≤Mm∈Nc,(h(m),p)=1ep(g(m)h(m)), where ep(y)=exp⁡(2πiy/p). In particular, when g(x)/h(x)=rx+sx¯ and 1<c<6/5, our results improve the bound for Kloosterman sums with Piatetski-Shapiro sequences which was given by Shparlinski and Technau. Furthermore, bounds for max⁡{m,m˜} subject to m,m˜∈Z∩[1,p), z indivisible by p, g(m)m˜≡z(modp) and m belonging to some fixed Piatetski-Shapiro sequence are also obtained.

Mathematical Reflections on Locality

Starting from the principle of locality in quantum field theory, which states that an object is influenced directly only by its immediate surroundings, we review some features of the notion of locality arising in physics and mathematics. We encode these in locality relations, given by symmetric binary relations, and locality morphisms, namely maps that factorise on products of pairs in the graph of such locality relations. This factorisation is a key property in the context of renormalisation, as illustrated on the factorisation of an exponential sum on convex cones, discussed at the end of the paper. The subject of locality is so vast and the issues it raises are so subtle, that this brief and modest presentation can only offer a small glimpse into this fascinating topic.

Analytic twists of GL2×GL2$\rm GL_2\times \rm GL_2$ automorphic forms

AbstractLet f and g be holomorphic or Maass cusp forms for with normalized Fourier coefficients and , respectively. In this paper, we prove nontrivial estimates for the sum where , , is a large parameter and is some nonlinear real‐valued smooth function. Applications of these estimates include a subconvex bound for the Rankin–Selberg L‐function in the t‐aspect, an improved estimate for a nonlinear exponential twisted sum and the following asymptotic formula for the sum of the Fourier coefficients of certain Eisenstein series for any .

Bounds for theta sums in higher rank. I

Theta sums are finite exponential sums with a quadratic form in the oscillatory phase. This paper establishes new upper bounds for theta sums in the case of smooth and box truncations. This generalises a classic 1977 result of Fiedler, Jurkat and Körner for one-variable theta sums and, in the multi-variable case, improves previous estimates obtained by Cosentino and Flaminio in 2015. Key steps in our approach are the automorphic representation of theta functions and their growth in the cusps of the underlying homogeneous space.

A 4-Term Exponential-Quadratic Approximation for Gaussian Q or Error Functions Accurate to $1.65\times 10^{-4}$

Integrals on [0, ∞) where the integrand is of the form Qn(a√x) p(x), where Q is the Gaussian Q function, p(·) a Gamma PDF, n a positive integer and a &gt; 0; or of the form erfn(ax + b) xr exp(-c2x2 + 2dx), where erf(x) is the error function, with integers r ≥ 0, n &gt; 0, arise in performance modelling of communication and machine learning systems. Such integrals cannot be evaluated analytically in general, but they are reducible to a set of key integrals whose integrand is erfn(ax + b) N(x; m, s) where N() is a Gaussian PDF with mean m and variance s. Seeking an efficient and accurate evaluation method, we develop a new 4-term exponential quadratic approximator (EQA) for the error function that includes both linear and quadratic terms in its exponents. The EQA minimises a sum-of-squares cost function with two “spline-type” constraints, i.e., constraints on the function value and its first derivative. This constrained optimisation problem is reduced to an unconstrained one by inverting a 4-D linear system, then solved by gradient descent. The resulting approximator has a maximum absolute error of 1.65 × 10-4 on the real line, and outperforms many other exponential sum approximators for erf(x) on x ∈ [0, 1.5] and for Q(x) on x ∈ [0, 2]. Moreover, due to its functional form, the EQA leads to an analytical formula for the set of key integrals, which, in the n = 1 case, is accurate to 3 to 4 significant figures while being orders of magnitude more efficient than Monte Carlo integration. The EQA can equally be used to obtain closed forms for the average symbol error probability of various modulation schemes on Rayleigh fading channels.

Physical assets life cycle analysis in a food industry company - case study

Monitoring the life cycle of physical assets (PA) implies addressing issues, such as PA’s energy efficiency and its replacement and definition of the most proper moment to renew. The goals of this article are: to present a characterization of energy sources and analyze the PA life cycle in a food sector company. First, it will be characterized the costs and the expenses of the organization’s energy sources, then, a study about the replacement of PA is presented; Traditional methods were used, such as economic life; The models that underlie it are discussed throughout the article, using actual data, for validation. Three methods for depreciation of PA are used: Linear Depreciation; Sum of Digits and Exponential. Other methods were used to determine the Economic Cycle for replacing PA: Uniform Annual Expenditure (MRAU); Minimizing the Average Total Cost (MCMT); and the MCMT-Reduced to Present Value (MCMT-RVP). The equipment of the study was bakery ovens (gas and electrical). Results and conclusions from the application of the methods used in the evaluation of these PAs are presented.

On Stability of an Approximate Solution of the Cauchy Problem for Some First-Order Integrodifferential Equations

The Cauchy problem for a first-order evolutionary equation with memory with the time derivative of the Volterra integral term and difference kernel in the finite-dimensional Banach space is considered. The fundamental difficulties of the approximate solution of such problems are caused by nonlocality with respect to time when the solution at the current time depends on the entire history. Transformation of the first-order integrodifferential equation to a system of evolutionary local equations with the approximation of the difference kernel by a sum of exponential functions is used. For the weakly coupled system of local equations with additional ordinary differential equations, estimates of stability of solution with respect to initial data and right-hand side are obtained using the concept of logarithmic norm. Similar estimates are obtained for the approximate solution using two-level time approximations.

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.

Recurrences in Terms of Special Polynomials for Exponential Sums of Elementary Symmetric Polynomials over Finite Fields

Recurrences in Terms of Special Polynomials for Exponential Sums of Elementary Symmetric Polynomials over Finite Fields

Partial zeta functions, partial exponential sums, and p-adic estimates

Partial zeta functions of algebraic varieties over finite fields generalize the classical zeta function by allowing each variable to be defined over a possibly different extension field of a fixed finite field. Due to this extra variation their rationality is surprising, and even simple examples are delicate to compute. For instance, we give a detailed description of the partial zeta function of an affine curve where the number of unit poles varies, a property different from classical zeta functions. On the other hand, they do retain some properties similar to the classical case. To this end, we give Chevalley-Warning type bounds for partial zeta functions and L-functions associated to partial exponential sums.