Number Of Sign Changes Research Articles

On the sign changes of the coefficients attached to triple product L-functions

Let f and g be two distinct normalized primitive holomorphic cusp forms of even integral weights [Formula: see text] and [Formula: see text] for the full modular group SL[Formula: see text], respectively. In this paper, we investigate the sign changes of the sequence [Formula: see text] in short intervals, where [Formula: see text] denote the normalized coefficients of triple product L-functions [Formula: see text] attached to f and [Formula: see text], respectively. We provide quantitative results for the number of sign changes of the sequences mentioned above in short intervals.

High power sums of Fourier coefficients of holomorphic cusp forms and their applications

<p>Let $ \lambda_f(n) $ be the $ n $th normalized Fourier coefficient of a holomorphic cusp form $ f $ for the full modular group. In this paper, we will established asymptotic formulae for high power sums of Fourier coefficients of cusp forms and further improved previous results. Moreover, as an application, we studied the signs of the sequences $ \{\lambda_f(n)\} $ and $ \{\lambda_f(n)\lambda_g(n)\} $ in short intervals, and presented some quantitative results for the number of sign changes for $ n\leq x $.</p>

On the Simultaneous Sign Changes of Coefficients of Rankin–Selberg L-Functions over a Certain Integral Binary Quadratic Form

In this paper, we consider the simultaneous sign changes of coefficients of Rankin–Selberg L-functions associated to two distinct Hecke eigenforms supported at positive integers represented by some certain primitive reduced integral binary quadratic form with negative discriminant D. We provide a quantitative result for the number of sign changes of such sequence in the interval (x, 2x] for sufficiently large x.

Diesel Governor Tuning for Isolated Hybrid Power Systems

In recent decades, renewable energy sources, such as wind power, have extraordinarily increased their participation in the energy mix throughout the world. This progression has played an important role in lowering the usage of fossil fuels. In addition, it has reduced environmental hazards and increased the emergence of hybrid power systems, mainly in remote areas. In some of these areas, diesel power plants were the only previous source of energy. Irrespective of the benefits, hybrid power systems might face problems such as frequency deviations. To contribute to reducing these problems, this paper presents a methodology to tune diesel engine governors using the Student Psychology-Based Algorithm. This proposed methodology enhances some metrics of controller performance, such as the integral square error, integral absolute error, and number of sign changes in the frequency derivative. This approach has been tested against different perturbations (step, ramp and random). To validate the effectiveness of the proposed approach, it has been simulated in relation to the San Cristobal Island (Ecuador) hybrid wind–diesel power system. The simulation results show that the governor tuned with the proposed approach provides a better system response.

On sign changes of Fourier coefficients of Hermitian cusp forms of degree two

We prove a quantitative result for the number of sign changes of the Fourier coefficients of a Hermitian cusp form of degree 2. In addition, we prove a quantitative result for the number of sign changes of the primitive Fourier coefficients. We give an explicit upper bound for the first sign change of the Fourier coefficients of a Hermitian cusp form of degree 2 over certain imaginary quadratic extensions.

Internally Hankel $k$-Positive Systems

There has been an increased interest in the variation diminishing properties of controlled linear time-invariant (LTI) systems and time-varying linear systems without inputs. In controlled LTI systems, these properties have recently been studied from the external perspective of $k$-positive Hankel operators. Such systems have Hankel operators that diminish the number of sign changes (the variation) from past input to future output if the input variation is at most $k-1$. For $k=1$, this coincides with the classical class of externally positive systems. For linear systems without inputs, the focus has been on the internal perspective of $k$-positive state-transition matrices, which diminish the variation of the initial system state. In the LTI case and for $k=1$, this corresponds to the classical class of (unforced) positive systems. This paper bridges the gap between the internal and external perspectives of $k$-positivity by analyzing internally Hankel $k$-positive systems, which we define as state-space LTI systems where controllability and observability operators as well as the state-transition matrix are $k$-positive. We show that the existing notions of external Hankel and internal $k$-positivity are subsumed under internal Hankel $k$-positivity, and we derive tractable conditions for verifying this property in the form of internal positivity of the first $k$ compound systems. As such, this class provides new means to verify external Hankel $k$-positivity, and lays the foundation for future investigations of variation diminishing controlled linear systems. As an application, we use our framework to derive new bounds for the number of over- and undershoots in the step responses of LTI systems. Since our characterization defines a new positive realization problem, we also discuss geometric conditions for the existence of minimal internally Hankel $k$-positive realizations.

Variation diminishing linear time-invariant systems

This paper studies the variation diminishing property of k-positive linear time-invariant (LTI) systems, which diminish the number of sign changes (variation) from input to output, if the input variation is at most k−1. We characterize this property for the discrete-time Toeplitz and Hankel operators of finite-dimensional causal systems. Our main result is that these operators have a dominant approximation in the form of series or parallel interconnections of k first order positive systems. This is shown by expressing the k-positivity of a LTI system as the external positivity (that is, 1-positivity) of kcompound LTI systems. Our characterization generalizes well known properties of externally positive systems (k=1) and totally positive systems (k=∞; also known as relaxation systems in case of the Hankel operator). All results readily extend to continuous-time systems by considering sampled impulse responses.

Sign changes and nonvanishing of Fourier coefficients of holomorphic cusp forms

We focus on the problems of sign changes and nonvanishing of Fourier coefficients of holomorphic cusp forms. Let λf(n), λg(n) and λh(n) be the n-th Fourier coefficients of primitive holomorphic cusp forms f, g and h for SL(2,ℤ), respectively. Firstly, we study the behavior of the signs of the sequences {λf(nj)} with j≥3, {λf(ni)λg(nj)} with i≥1,j≥2, and {λf(n)λg(n)λh(n)} in short intervals, and present some quantitative results for the number of sign changes for n≤x. Here we improve and generalize previous results. Moreover, we investigate simultaneous nonvanishing of {λf(nj)λg(nj)λh(nj)} with j≥1 in short intervals.

Geometric Case Based Reasoning for Stock Market Prediction

Case based reasoning is a knowledge discovery technique that uses similar past problems to solve current new problems. It has been applied to many tasks, including the prediction of temporal variables as well as learning techniques such as neural networks, genetic algorithms, decision trees, etc. This paper presents a geometric criterion for selecting similar cases that serve as an exemplar for the target. The proposed technique, called geometric Case Based Reasoning, uses a shape distance method that uses the number of sign changes of features for the target case, especially when extracting nearest neighbors. Thus, this method overcomes the limitation of conventional case-based reasoning in that it uses Euclidean distance and does not consider how nearest neighbors are similar to the target case in terms of changes between previous and current features in a time series. These concepts are investigated against the backdrop of a practical application involving the prediction of a stock market index. The results show that the proposed technique is significantly better than the random walk model at p < 0.01. However, it was not significantly better than the conventional CBR model in the hit rate measure and did not surpass the conventional CBR in the mean absolute percentage error.

ROOTS OF TRIGONOMETRIC POLYNOMIALS AND THE ERDŐS–TURÁN THEOREM

We prove, informally put, that it is not a coincidence that cos ( n θ ) + 1 ≥ 0 and the roots of z n + 1 = 0 are uniformly distributed in angle—a version of the statement holds for all trigonometric polynomials with “few” real roots. The Erdős–Turán theorem states that if p ( z ) = ∑ k = 0 n a k z k is suitably normalized and not too large for | z | = 1 , then its roots are clustered around | z | = 1 and equidistribute in angle at scale ∼ n − 1 / 2 . We establish a connection between the rate of equidistribution of roots in angle and the number of sign changes of the corresponding trigonometric polynomial q ( θ ) = ℜ ∑ k = 0 n a k e i k θ . If q ( θ ) has ≲ n δ roots for some 0 < δ < 1 / 2 , then the roots of p ( z ) do not frequently cluster in angle at scale ∼ n − ( 1 − δ ) ≪ n − 1 / 2 .

Signs of Fourier coefficients of cusp form at sum of two squares

In this article, we investigate the sign changes of the sequence of coefficients at sum of two squares where the coefficients are the Fourier coefficients of the normalized Hecke eigen cusp form for the full modular group. We provide the quantitative result for the number of sign changes of the sequence in a small interval.

Spectrum of a Class of Difference Operators with Indefinite Weights

In this study, we use analytical methods and Sylvester inertia theorem to research a class of second order difference operators with indefinite weights and coupled boundary conditions. The eigenvalue problem with sign-changing weight has lasted a long time. The number of eigenvalues and the number of sign changes of the corresponding eigenfunctions of discrete equations under different boundary conditions are mainly studied. For the discrete Sturm-Liouville problems, similar conclusions about the properties of eigenvalues and the number of sign changes of the corresponding eigenfunctions are obtained under different boundary conditions, such as periodic boundary conditions, antiperiodic boundary conditions and separated boundary conditions etc. The purpose of this paper is to extend the similar conclusion to the coupled boundary conditions, which is of great significance to the perfection of the theory of the discrete Sturm-Liouville problems. We came to the following conclusions: first, the eigenvalues of the problem are real and single, the number of the positive eigenvalues is equal to the number of positive elements in the weight function, and the number of negative eigenvalues is equal to the number of negative elements in the weight function. Second, under some conditions, we obtain the sign change of the eigenfunction corresponding to the j-th positive/negative eigenvalue.

Asymmetric rectified electric fields between parallel electrodes: Numerical and scaling analyses.

Recent computational and experimental work has established the existence of asymmetric rectified electric fields (AREFs), a type of steady electric field that occurs in liquids in response to an applied oscillatory potential, provided the ions present have different mobilities [Hashemi Amrei et al., Phys. Rev. Lett. 121, 185504 (2018)PRLTAO0031-900710.1103/PhysRevLett.121.185504]. Here we use scaling analyses and numerical calculations to elaborate the nature of one-dimensional AREFs between parallel electrodes. The AREF magnitude is shown to increase quadratically with applied potential at low potentials, increase nonlinearly at intermediate potentials, then increase with a constant rate slower than quadratically at sufficiently high potentials, with no impact at any potential on the spatial structure of the AREF. In contrast, the AREF peak location increases linearly with a frequency-dependent diffusive length scale for all conditions tested, with corresponding decreases in both the magnitude and number of sign changes in the directionality of AREF. Furthermore, both the magnitude and spatial structure of the AREF depend sensitively on the ionic mobilities, valencies, and concentrations, with a potential-dependent peak AREF magnitude occurring at an ionic mobility ratio of D_{-}/D_{+}⪅5. The results are summarized with approximate scaling expressions that will facilitate interpretation of the steady component for oscillatory fields in liquid systems.

Routh-hurwitz criterion and bifurcation method for stability analysis of tuberculosis transmission model

Tuberculosis is an infectious disease; it caused by Mycobacterium tuberculosis. In this paper, we discuss how to use the Routh-Hurwitz stability criterion to analyze the stability of disease free of the tuberculosis transmission model. From this method, can be found the number of roots of the characteristic polynomial (eigenvalues) with positive real parts is equal to the number of changes in sign of the first column of the Routh array. If all of the eigenvalues are negative, then the model is stable. While the bifurcation method is used to analyze the stability of the endemic equilibrium point of the tuberculosis transmission, the endemic equilibrium point is locally asymptotically stable if reproduction number greater than one and additional parameters requirement that bifurcation met. Finally, numerical simulations are demonstrated to verify the used method.

Revisiting totally positive differential systems: A tutorial and new results

A matrix is called totally nonnegative (TN) [totally positive (TP)] if all its minors are nonnegative [positive]. Multiplying a vector by a TN matrix does not increase the number of sign variations in the vector. In a largely forgotten paper, Binyamin Schwarz (1970) considered matrices whose exponentials are TN or TP. He also analyzed the evolution of the number of sign changes in the vector solutions of the corresponding linear system. His work, however, considered only linear systems.In a seemingly different line of research, Smillie (1984), Smith (1991), and others analyzed the stability of nonlinear tridiagonal cooperative systems by using the number of sign variations in the derivative vector as an integer-valued Lyapunov function.We show that these two research topics are intimately related. This allows to derive important generalizations of the results by Smillie (1984) and Smith (1991) while simplifying the proofs. These generalizations are particularly relevant in the context of control systems. Also, the results by Smillie and Smith provide sufficient conditions for analyzing stability based on the number of sign changes in the vector of derivatives and the connection to the work of Schwarz allows to show in what sense these results are also necessary. We describe several new and interesting research directions arising from this new connection.

English

The classical Descartes’ rule of signs limits the number of positive roots of a real polynomial in one variable by the number of sign changes in the sequence of its coefficients. One can ask the question which pairs of nonnegative integers (p, n), chosen in accordance with this rule and with some other natural conditions, can be the pairs of numbers of positive and negative roots of a real polynomial with prescribed signs of the coefficients. The paper solves this problem for degree 8 polynomials.

Pressure-dipole solutions of the thin-film equation

We investigate self-similar sign-changing solutions to the thin-film equation, ht = −(|h|nhxxx)x, on the semi-infinite domain x ≥ 0 with zero-pressure-type boundary conditions h = hxx = 0 imposed at the origin. In particular, we identify classes of first- and second-kind compactly supported self-similar solutions (with a free-boundary x = s(t) = Ltβ) and consider how these solutions depend on the mobility exponent n; multiple solutions can exist with the same number of sign changes. For n = 0, we also construct first-kind self-similar solutions on the entire half-line x ≥ 0 and show that they act as limiting cases for sequences of compactly supported solutions in the limit of infinitely many sign changes. In addition, at n = 1, we highlight accumulation point-like behaviour of sign-changes local to the moving interface x = s(t). We conclude with a numerical investigation of solutions to the full time-dependent partial differential equation (based on a non-local regularisation of the mobility coefficient) and discuss the computational results in relation to the self-similar solutions.

Half-normal approximation for statistics of symmetric simple random walk

ABSTRACTIn 2013, Döbler used Stein’s method to obtain the uniform bounds in half-normal approximation for three statistics of a symmetric simple random walk; the maximum value, the number of returns to the origin and the number of sign changes up to a given time n. In this paper, we give the non-uniform bounds for these statistics by using Stein’s method and the concentration inequality approach.

Exact Values of the Best (𝛼, β)-Approximations for the Classes of Convolutions with Kernels that Do Not Increase the Number of Sign Changes

We obtain the exact values of the best (𝛼, β) -approximations for the classes K ∗ F of periodic functions K ∗ f such that f belongs to a given rearrangement-invariant set F and K is 2𝜋 -periodic kernel that does not increase the number of sign changes by the subspaces of generalized polynomial splines with nodes at the points 2kπ/n and 2kπ/n + h, n ∈ ℕ, k ∈ ℤ, h ∈ (0, 2π/n). It is shown that these subspaces are extremal for the Kolmogorov widths of the corresponding functional classes.

Sign changes of Fourier coefficients of Siegel cusp forms of degree two on Hecke congruence subgroups

In this paper, we give a lower bound on the number of sign changes of Fourier coefficients of a non-zero degree two Siegel cusp form of even integral weight on a Hecke congruence subgroup. We also provide an explicit upper bound for the first sign change of Fourier coefficients of such Siegel cusp forms. Explicit upper bound on the first sign change of Fourier coefficients of a non-zero Siegel cusp form of even integral weight on the Siegel modular group for arbitrary genus was dealt in an earlier work of Choie, the first author and Kohnen.