Quadratic Forms Research Articles

Average estimates and sign change of Fourier coefficients of cusp forms at integers represented by binary quadratic form of fixed discriminant

Average estimates and sign change of Fourier coefficients of cusp forms at integers represented by binary quadratic form of fixed discriminant

Global stability analysis and modelling onchocerciasis transmission dynamics with control measures

ABSTRACT Background: Onchocerciasis infection is one of the neglected tropical diseases targeted for eradication by 2030. The disease is usually transmitted to humans through the bites of black flies. These black flies mostly breed near well-oxygenated fast-running water bodies. The disease is common in mostly remote agricultural villages near rivers and streams. Objective: In this study, a deterministic model describing the infection dynamics of human onchocerciasis disease with control measures is presented. Methods: We derived the model’s reproductive number and used a stability theorem of a Metzler matrix to show that disease-free equilibrium is both locally and globally asymptotically stable whenever the reproductive number is less than one. Parameter contribution was conducted using sensitivity analysis. The model endemic equation is shown to be a cubic polynomial in the presence of infected immigrants and a quadratic form in their absence. Results: When the inflow of infected immigrants is null, the model endemic equation may admit a unique equilibrium if the reproductive number is greater than one, or admits multiple endemic equilibria if the reproductive number is less than unity. We carried out a sensitivity analysis to identify the significant parameters that contribute to onchocerciasis spread. Conclusion: Onchocerciasis disease can be eradicated if the importation of infected immigrants is properly monitored. The integration of the One Health concept in the public health system is key in tackling the emergence and spread of diseases.

An Empirical Investigation of GDP, Industrialization, Population, Renewable Energy and CO2 Emission in Bangladesh: Bridging EKC-STIRPAT Models

This research paper explores the presence of the Environmental Kuznets Curve (EKC) in Bangladesh, delving into the intricate relationships among GDP, industrialization, renewable energy utilization, and CO2 emissions. Utilizing the Stochastic Impacts by Regression on Population, Affluence, and Technology (STIRPAT) model, our investigation spans the 1992 to 2021 timeframe, offering a comprehensive analysis of the interplay between these crucial factors in the context of Bangladesh's environmental sustainability. In this study, the F-bound test is utilized to establish the cointegration relationship among variables. And the short-run and long-run elasticity of explanatory variables is investigated by employing the Autoregressive Distributed Lag (ARDL) approach. This research also employed a pairwise granger causality test to explore the direction of causality between them. Empirical results prove the existence of a cointegration relationship among CO2 emission, GDP per capita, a quadratic form of GDP per capita, renewable energy, industrialization, and population. The findings provide strong support for the presence of the EKC in the case of Bangladesh. Besides that, significant inverse relationship was also obtained between CO2 emission and renewable energy consumption. Additionally, the study found the detrimental long-term effects of industrialization on the environment. The EKC hypothesis argues that Bangladesh can accomplish sustainable economic development. To attain the goal country should adopt appropriate government policies and ensure their implication. This study strongly advocates using sustainable energy sources and implementing regulations on pollutant industries.

Investigating the impact of storage duration and temperature on vitamin C in various citrus genotypes: A meta-analysis method

The present work disseminates a solid scientific meta-analysis method to investigate the impact of storage duration and temperature on vitamin C of citrus. This work is initiated by designing of the PICO framework, collecting, and organizing the articles, creating selection criteria, sorting articles, identifying factors influencing moderation effects and sources of diversity, tabulating data, and employing analysis in the form of a linear mixed model. Using this method, we identified 54 distinct studies from a pool of 289 eligible peer-reviewed publications, focusing on variations of vitamin C in citrus. The method provides mean values in both quadratic and linear regression forms.•This method provides a detailed description starting from topic selection to statistical methodologies intended for performing meta-analysis.•All guidelines for conducting this method have been approved by all authors and adhere to the standard PRISMA-P guidelines.•Disseminating this method in a peer-reviewed publication aims to facilitate scholarly discussions and promote transparency, ultimately improving the standard for performing meta-analysis on vitamin C levels in citrus concerning various genotypes, storage temperatures, and durations.

Cycle integrals of meromorphic modular forms and Siegel theta functions

AbstractWe study meromorphic modular forms associated with positive definite binary quadratic forms and their cycle integrals along closed geodesics in the modular curve. We show that suitable linear combinations of these meromorphic modular forms have rational cycle integrals. Along the way we evaluate the cycle integrals of the Siegel theta function associated with an even lattice of signature (1, 2) in terms of Hecke’s indefinite theta functions.

Isotropy of quadratic forms over function fields in characteristic 2

We extend to characteristic two recent results about isotropy of quadratic forms over function fields. In particular, we provide a characterization of function fields not only of quadratic forms but also more generally of polynomials in several variables, over which a quadratic form becomes isotropic. As an application of these results, we obtain criteria for stable birational equivalence of quadratic forms.

Simultaneous robust model predictive control and state estimation for nonlinear systems with state‐ and input‐dependent uncertainties

AbstractThe convergence and stability of uncertain nonlinear systems is a challenging problem in the nonlinear control area. Besides, in many practical cases, all states are not measurable and are affected by measurement noise. Based on this motivation, the first objective of this paper is to design a novel output feedback robust model predictive control approach for nonlinear systems with state‐ and input‐dependent uncertainties and measurement noise. This approach combines state estimation and robust model predictive control (MPC) into one min–max optimization and by solving the optimization, these two tasks are performed simultaneously. The studied nonlinear system comprises a linear part, a nonlinear part, and a function that denotes the state‐ and input‐dependent uncertainties. Therefore, the other objective is to reduce the computational complexity; thus, the system's nonlinear term and the aforementioned uncertainties are converted into additional disturbances. Subsequently, the optimization problem becomes a quadratic form, which leads to global convergence with the appropriate selection of objective function weights. Besides, this paper explores the convergence of the closed‐loop system states and the sufficient synthesis conditions to guarantee input‐to‐state stability. The implementation on a numerical example and a CSTR process demonstrate the applicability and reliability of the proposed approach.

Energy estimated frequencies of standing-wave solutions to nonlinear Klein-Gordon systems in higher dimensions

A system of nonlinear elliptic equations is considered, where the nonlinear term is the sum of a quadratic form and a Sobolev sub-critical term. An extra assumption is introduced on the nonlinear term, which is minimal among the ones which guarantee the existence of standing-waves obtained by estimating frequencies of minimizing sequences with the energy functional.

Strengthening stability with centralized event-triggered control system with the disturbances and artificial time delay in wireless connected vehicle platooning (CVSs)

This paper addresses the difficulties with connected vehicle systems (CVSs), particularly with vehicle platooning, are examined in this paper. For leader and follower-connected vehicles, the control protocol (which includes artificial delays, disturbances and proportional gains) is implemented. With tracking error systems, system dynamics are modelled while taking outside influences into consideration. Using creative thinking, a centralized event-triggered control system is implemented to maximize fleet wide communication updates. System stability is guaranteed by this centralized method in combination with quadratic form Lyapunov stability analysis. The risk of zeno behaviour is reduced by an event-triggered communication condition that is activated when a threshold is exceeded. The effectiveness of the centralized event-triggered system in improving stability and resilience in connected vehicle platooning scenarios is evaluated numerically through simulations.

Quadratic descent of generalized quadratic forms

Quadratic descent of generalized quadratic forms over a division algebra with involution of the first kind in characteristic two is investigated. Using the notion of transfer, it is shown that a system of quadratic forms, associated to such a generalized quadratic form, can be used to characterize its descent properties.

Near-miss identities and spinor genus classification of ternary quadratic forms with congruence conditions

In this paper, near-miss identities for the number of representations of some integral ternary quadratic forms with congruence conditions are found and proven. The genus and spinor genus of the corresponding lattice cosets are then classified. Finally, a complete genus and spinor genus classification for all conductor 2 lattice cosets of 2-adically unimodular lattices is given.

Explicit evaluation of triple convolution sums of the divisor functions

In this paper, we use the theory of modular forms and give a general method to obtain the convolution sums [Formula: see text] for odd integers [Formula: see text] and [Formula: see text], where [Formula: see text] is the sum of the [Formula: see text]th powers of the positive divisors of [Formula: see text]. We consider four cases, namely (i) [Formula: see text], (ii) [Formula: see text]; [Formula: see text] (iii) [Formula: see text]; [Formula: see text] and (iv) [Formula: see text], and give explicit expressions for the respective convolution sums. We provide several examples of these convolution sums in each case and further use these formulas to obtain explicit formulas for the number of representations of a positive integer [Formula: see text] by certain positive definite quadratic forms. The existing formulas for [Formula: see text] (in [20]), [Formula: see text] (in [7]), [Formula: see text] (in [35]), [Formula: see text], [Formula: see text] (in [30]) and [Formula: see text] (in [31]), which were all obtained by using the theory of quasimodular forms, follow from our method.

Multivariate unified skew-[formula omitted] distributions and their properties

The unified skew-t (SUT) is a flexible parametric multivariate distribution that accounts for skewness and heavy tails in the data. A few of its properties can be found scattered in the literature or in a parameterization that does not follow the original one for unified skew-normal (SUN) distributions, yet a systematic study is lacking. In this work, explicit properties of the multivariate SUT distribution are presented, such as its stochastic representations, moments, SUN-scale mixture representation, linear transformation, additivity, marginal distribution, canonical form, quadratic form, conditional distribution, change of latent dimensions, Mardia measures of multivariate skewness and kurtosis, and non-identifiability issue. These results are given in a parameterization that reduces to the original SUN distribution as a sub-model, hence facilitating the use of the SUT for applications. Several models based on the SUT distribution are provided for illustration.

Unraveling the Complexity of Inverting the Sturm–Liouville Boundary Value Problem to Its Canonical Form

The Sturm–Liouville boundary value problem (SLBVP) stands as a fundamental cornerstone in the realm of mathematical analysis and physical modeling. Also known as the Sturm–Liouville problem (SLP), this paper explores the intricacies of this classical problem, particularly the relationship between its canonical and Liouville normal (Schrödinger) forms. While the conversion from the canonical to Schrödinger form using Liouville’s transformation is well known in the literature, the inverse transformation seems to be neglected. Our study attempts to fill this gap by investigating the inverse of Liouville’s transformation, that is, given any SLP in the Schrödinger form with some invariant function, we seek the SLP in its canonical form. By closely examining the second Paine–de Hoog–Anderson (PdHA) problem, we argue that retrieving the SLP in its canonical form can be notoriously difficult and can even be impossible to achieve in its exact form. Finding the inverse relationship between the two independent variables seems to be the main obstacle. We confirm this claim by considering four different scenarios, depending on the potential and density functions that appear in the corresponding invariant function. In the second PdHA problem, this invariant function takes a reciprocal quadratic binomial form. In some cases, the inverse Liouville transformation produces an exact expression for the SLP in its canonical form. In other situations, however, while an exact canonical form is not possible to obtain, we successfully derived the SLP in its canonical form asymptotically.

Universal Quadratic forms and Dedekind Zeta functions

Universal Quadratic forms and Dedekind Zeta functions

Robust sparse concept factorization with graph regularization for subspace learning

Concept factorization (CF) is a powerful tool in subspace learning. Recently graph-based CF and local coordinate CF have been proposed to exploit the intrinsic geometrical structure of data, and have been shown quite successful in improving performance. However, these methods have limited robustness and might be sensitive to noises and disturbances in practical applications. In this paper, we propose a novel robust sparse CF framework (RSCF) for subspace learning. Specifically, we present a robust loss function to effectively eliminate the impact of the large outliers. The local coordinate constraint and the graph regularization term are incorporated into RSCF to simultaneously guarantee the sparsity of the coefficient matrix and maintain the local structure of the data. We prove that the local coordinate constraint implies the orthogonality of the coefficient matrix. By using the half-quadratic optimization technique, we transform the objective function of RSCF into a quadratic form and provide the iterative updating rules and the convergence analysis. Extensive experiments demonstrate the robustness and superiority of the proposed RSCF in comparison to state-of-the-art CF methods.

Distribution theories for genetic line of least resistance and evolvability measures.

Quantitative genetic theory on multivariate character evolution predicts that a population's response to directional selection is biased toward the major axis of the genetic covariance matrix G-the so-called genetic line of least resistance. Inferences on the genetic constraints in this sense have traditionally been made by measuring the angle of deviation of evolutionary trajectories from the major axis, or more recently by calculating the amount of genetic variance-the Hansen-Houle evolvability-available along the trajectories. However, there have not been clear practical guidelines on how these quantities can be interpreted, especially in a high-dimensional space. This study summarizes pertinent distribution theories for relevant quantities, pointing out that they can be written as ratios of quadratic forms in evolutionary trajectory vectors by taking G as a parameter. For example, a beta distribution with appropriate parameters can be used as a null distribution for squared cosine of the angle of deviation from a major axis or subspace. More general cases can be handled with the probability distribution of ratios of quadratic forms in normal variables. Apart from its use in hypothesis-testing, this latter approach could potentially be used as a heuristic tool for looking into various selection scenarios like directional and/or correlated selection as parameterized with mean and covariance of selection gradients.

The circle method and shifted convolution sums involving the divisor function

Let Q(x) be a positive definite integral quadratic form with the determinant D being squarefree, and r(n,Q) denote the number of representations of n by the quadratic form Q. In this paper, we apply the Hardy-Littlewood-Kloosterman circle method to derive the asymptotic formula for the shifted convolution sum of the divisor function d(n) and Fourier coefficients r(n,Q). With more efforts, our method should have a number of applications for other multiplicative functions.

On the properties of the anisotropic multivariate Hermite-Gauss functions

The Hermite-Gauss basis functions have been extensively employed in classical and quantum optics due to their convenient analytic properties. A class of multivariate Hermite-Gauss functions, the anisotropic Hermite-Gauss functions, arise by endowing the standard univariate Hermite-Gauss functions with a positive definite quadratic form. These multivariate functions admit useful applications in optics, signal analysis and probability theory, however they have received little attention in literature. In this paper, we examine the properties of these functions, with an emphasis on applications in computational optics.

About the Kyiv school mathematicians contribution to the extrema of functions of many variables theory

The subject of this article is the study of Kyiv school mathematicians’ contribution to the many variable extrema function theory. The purpose of this article is to study the work of Kyiv mathematician Professor M. Ye. Vashchenko-Zakharchenko. Task: for the first time, to carry out a detailed analysis of the results obtained in the article by M. Ye. Vashchenko-Zakharchenko «Signs of the highest and lowest value of functions». The research method is a historical and scientific analysis of the original source, which allows the scientific results obtained in the late 19th and early 20th centuries to be estimated from the viewpoint of modern mathematical analysis. The following results were obtained. Thanks to the analysis of this article, it was found out that here M. Ye. Vashchenko-Zakharchenko first used the methods of linear algebra and applied D. Sylvester’s criterion of positive (negative) definiteness of quadratic form to obtain sufficient conditions for the existence of an extremum of functions of many variables. The problem is considered in general for functions of many variables and for special cases of functions of two and three variables. Conclusions. Comparing the results obtained from the Kyiv mathematician, Professor M. Ye. Vashchenko-Zakharchenko with the presentation of this topic in modern textbooks on higher mathematics and mathematical analysis, we can conclude that they are included in these textbooks in virtually the same form. This is what determines the relevance of this topic: methods for solving problems for the extremum of functions of many variables, obtained at the turn of the 19th and 20th centuries, are used in modern optimization problems. The breadth of the scientific erudition of M. Ye. Vashchenko-Zakharchenko allowed him to immediately perceive new methods obtained by foreign scientists and immediately find their application to obtain sufficient conditions for the existence of an extremum of functions of many variables. This shows the high scientific level of the state of mathematics at Kyiv University in the second half of the 19th century. Further research should focus on the contribution of other mathematicians of the Kyiv school to the development of the theory of extrema of functions of many variables.