Random Polynomials Research Articles

Equilibrium measures on Julia sets of random quadratic polynomials

Abstract We consider sequences of compositions of quadratic polynomials f c n ( z ) = z 2 + c n . For such sequences one can naturally generalise the definitions of the Julia set and basin of infinity from the autonomous case. In this setting the Julia set depends on a sequence ω = ( c 0 , c 1 , … ) . We study the equilibrium (harmonic) measure on such Julia sets. In particular, we calculate the Hausdorff dimension of the equilibrium measure and study its dependence on the ‘scale of randomnes’.

PR-FCNN: a data-driven hybrid approach for predicting PM2.5 concentration

The atmosphere’s fine articulate Matter (PM2.5) poses various health-related risks. Even though multiple efforts have been made to lower the emissions of these substances, the mortality rate is continuously increasing, requiring immediate inclination of the scientific community towards the design and development of advanced predictive models. Conventional statistical approaches have become dormant due to their limitations in capturing the innate relationships between the pollutants, particularly for predicting PM2.5 concentrations. In contrast, machine and deep learning techniques have shown great potential for forecasting air quality, providing more accuracy than its predecessor techniques. The present study investigates the utilization of hybrid approaches by integrating machine learning models with deep learning models to improve the prediction capabilities of PM2.5 concentration. It uses datasets from the World Air Quality Index (WAQI) and the State of Global Air (SOGA) to analyze the performance of the models on both the daily and annual data, respectively. This ensures the model’s effectiveness on a diversified dataset. The present study implements Random Forest (RF), Polynomial Regression (PR), XGBoost, and Extra Tree Regressor (ETR) coupled with Fully Connected Neural Network (FCNN), Long Short-Term Memory (LSTM), and Bi-directional LSTM (Bi-LSTM) for obtaining optimized results. Finally, after a thorough investigation, the hybrid PR model coupled with FCNN (PR-FCNN) is found to be the best model with improved R-squared (R2) values, portraying its potential for predicting PM2.5 concentration accurately. Based on the experimentation, the preset study recommends implementing hybrid approaches, offering better predictive accuracy in forecasting air pollutants, especially PM2.5.

New Solutions to the Fractional Perturbed Chen Lee Liu Model with Time-Dependent Coefficients: Applications to Complex Phenomena in Optical Fibersns

In this paper, we consider the fractional perturbed Chen Lee Liu model with time-dependent coefficients (FPCLLM-TDCs). We apply the mapping method in order to get hyperbolic, elliptic, trigonometric and rational fractional solutions. These solutions are vital for understanding some fundamentally complicated phenomena. The obtained solutions will be very helpful for applications such as fiber optics and plasma physics. Finally, we show how the conformable fractional derivative order affect the exact solutions of the FPCLLM-TDCs. Furthermore, we examine the effects of time-dependent coefficients when these coefficients take on special cases such as random variables, polynomials, and hyperbolic functions.

Genetic Parameter Estimates for Growth of Hāpuku (Groper, Polyprion oxygeneios) in Land-Based Aquaculture Using Random Regression Models

Hāpuku (Polyprion oxygeneios) is a promising candidate for aquaculture production in New Zealand. Methods for spawning, juvenile production, and growout to harvest entirely on land, where water quality, pathogens, environmental impacts, and genetic “pollution” can be tightly controlled, have been developed, and genetic improvement to optimise land-based production is the obvious next step. However, estimates of genetic parameters are required to design a rigorous, disciplined, and effective selective breeding program. By using existing data consisting of irregularly spaced repeated measurements of fork length and live body weight collected on wild-collected founders and two generations of captively reared progeny, we evaluated the species’ genetic potential for improvement in growth. We first tested a range of univariate random regression models to identify the best-fitting models for these data. Subsequently, using a bivariate model, we estimated variance components for growth trajectories of fork length and whole body weight. With one to six records available per fish, the best-fitting univariate models included only a fixed effect for contemporary groups and fixed and random genetic third-order Legendre polynomials. More complex models that included full-sib family and/or permanent environmental effects produced unacceptable constrained and/or non-positive-definite solutions. Both traits are moderately heritable at all stages of the growout phase (~0.4–0.5), and the genetic correlation patterns between daily breeding values estimated via the covariance function are different for length and weight. Genetic correlations for length between all pairs of age-specific breeding values are positive and strong (>0.7) and change gradually and smoothly with increasing temporal separation. For weight, these correlations deteriorate more rapidly with increasing time lags between measurements and become negative for some age pairings. We conclude that random regression analyses are a valuable tool for extracting genetic information from irregularly spaced repeated measurements of fish size, speculate that emerging technologies for high-throughput genotyping and phenotyping will add to the value of this approach in the near future, and reason that a breeding strategy that rigorously takes into account the potentially unfavourable genetic correlations between breeding values for weight at some ages will further adapt hāpuku to land-based systems and enhance the profitability commercial-scale production.

On random polynomials with an intermediate number of real roots

For each α ∈ ( 0 , 1 ) \alpha \in (0, 1) , we construct a bounded monotone deterministic sequence ( c k ) k ⩾ 0 (c_k)_{k \geqslant 0} of real numbers so that the number of real roots of the random polynomial f n ( z ) = ∑ k = 0 n c k ε k z k f_n(z) = \sum _{k=0}^n c_k \varepsilon _k z^k is n α + o ( 1 ) n^{\alpha + o(1)} with probability tending to one as the degree n n tends to infinity, where ( ε k ) (\varepsilon _k) is a sequence of i.i.d. (real) random variables of finite mean satisfying a mild anti-concentration assumption. In particular, this includes the case when ( ε k ) (\varepsilon _k) is a sequence of i.i.d. standard Gaussian or Rademacher random variables. This result confirms a conjecture of O. Nguyen from 2019. More generally, our main results also describe several statistical properties for the number of real roots of f n f_n , including the asymptotic behavior of the variance and a central limit theorem.

Linearized fluctuating hydrodynamics via random polynomials

We argue that an ensemble of backgrounds best describes hydrodynamic dispersion relations in a medium with few degrees of freedom and is therefore subject to strong thermal fluctuations. In the linearized regime, dispersion relations become describable by polynomials with random coefficients. We give a short review of this topic and perform a numerical study of the distribution of the roots of polynomials whose coefficients are of the order of a Knudsen series but fluctuate in accordance with canonical fluctuations of temperature. We find that, remarkably, the analytic structure of the poles of fluctuating dispersion relations is very different from deterministic ones, particularly regarding the distribution of imaginary parts with respect to real components. We argue that this provides evidence that hydrodynamic behavior persists, and is enhanced, by nonperturbative background fluctuations. Published by the American Physical Society 2024

Application of computer vision and machine learning in morphological characterization of Adansonia digitata fruits

Measuring fruit mass and volume is a time-consuming and tedious task that can affect production planning. This study developed a computer vision system to estimate the volume and mass of baobab fruits from single-view images captured from inexpensive and readily available cameras such as those in smartphones. The baobab fruits were collected from two study fields within the savanna ecological zone. Their images were captured, and subsequently, they were detected and segmented with over 97 % accuracy. The segmented images were binarized, and two-dimensional (2D) features such as the segmented area, centroid, bounding box, equivalent diameter, and major diameter were extracted from them. The volumes of the fruits were estimated from the 2D features using random forest, linear, polynomial, and radial support vector machine models. All the models achieved high goodness of fit; however, the random forest model delivered the best performance, with an R2 value of 99.8 %. The relationship between mass and volume was a quadratic equation (mass = 38.23 + 0.25 × volume + 4.49e−05 × volume2) and had an R2 value of 92 %. Correlations were validated via plots and statistical tests, and credible intervals of point estimates were determined from the posterior distributions of their samples. This highlights the potential of artificial intelligence methods to be applied in a less constrained environmental setting for ecological research and agricultural management. Commercial companies producing baobab powder and seed oil should apply these models for effective production planning. To enhance the model, it would be beneficial to gain a better understanding of how climate gradients affect the morphological characteristics of baobab fruits.

Central Limit Theorem for the number of real roots of random orthogonal polynomials

Central Limit Theorem for the number of real roots of random orthogonal polynomials

Probabilistic Galois theory in function fields

We study the irreducibility and Galois group of random polynomials over function fields. We prove that a random polynomial f=yn+∑i=0n−1ai(x)yi∈Fq[x][y] with i.i.d. coefficients ai taking values in the set {a(x)∈Fq[x]:deg⁡a≤d} with uniform probability, is irreducible with probability tending to 1−1qd as n→∞, where d and q are fixed. We also prove that with the same probability, the Galois group of this random polynomial contains the alternating group An. Moreover, we prove that if we assume a version of the polynomial Chowla conjecture over Fq[x], then the Galois group of this polynomial is actually equal to the symmetric group Sn with probability tending to 1−1qd. We also study the other possible Galois groups occurring with positive limit probability. Finally, we study the same problems with n fixed and d→∞.

On Average Modulus of Random Polynomials over a Unit Circle and Disc

On Average Modulus of Random Polynomials over a Unit Circle and Disc

A predictive analytics approach for forecasting bike rental demand

The demand for rental bikes in urban areas fluctuates, leading to localized surpluses and shortages. To address this challenge, effective bike relocation strategies are essential for ensuring equitable distribution and maximizing customer satisfaction. This study aims to employ advanced machine learning techniques to forecast bike rental demand in urban areas, thereby enhancing the efficiency and accessibility of bike rental services and contributing to sustainable urban mobility. The study comprehensively analyzes various influencing factors using machine learning models, including Ordinary Least Squares regression, MLP Regression, Gradient Boosting Regression, Random Forest Regression, Polynomial Regression, and Decision Tree Regression. The primary objective is to identify the most accurate predictor by comparing key metrics such as R-squared (R2), Mean Squared Error (MSE), Mean Absolute Error (MAE), Root Mean Squared Error (RMSE), and Pearson Correlation Coefficient. Insights gained from this analysis aid in identifying influential variables and ensure the development of resource-efficient and adaptable models, leading to more informed decision-making for rental bike businesses. Additionally, future research directions involve the implementation of artificial intelligence technology to predict overall bike demand based on urban cities’ criteria, including the number of national and international tourists. By addressing these objectives, this study seeks to provide valuable insights and tools for rental bike businesses to optimize operations, make strategic decisions, and enhance customer experience in competitive urban markets.

Comparative Study of Predictive Classification Models on Data with Severely Imbalanced Predictors

Analysing pre-COVID-19 unemployment in West Java is vital for comprehending and tackling Indonesia’s economic challenges. This significance arises not only due to the region’s high unemployment rate, but also from the need to understand unemployment patterns before COVID-19, which has become more relevant now during the country’s post-pandemic recovery phase. This study evaluates four machine learning models (Random Forest, Linear SVM, RBF SVM, and Polynomial SVM) to classify employment status using demographic and job-related variables. The objective is to find the most suitable model, particularly considering the imbalanced nature of the study-case data. Data from the National Labor Force Survey (SAKERNAS) in August 2019 is utilized, comprising 54,429 respondents across districts in West Java. The four models are evaluated using holdout validation with a 70:30 stratified proportion, repeated for 100 times. Results indicate that the random forest model outperforms others in balanced accuracy, F1-score, and computational time. The random forest model also underscores the importance of gender and age in classifying employment status in West Java, suggesting a need for targeted intervention, especially for female citizens and individuals in productive age groups.

Probabilistic zero bounds of certain random polynomials

Abstract This paper introduces the notion of probabilistic zero bounds for random polynomials. It presents new results regarding the probabilistic bounds of random polynomials whose coefficients are independently and identically distributed as standard normal variates. Additionally, the paper provides a clear exposition of the developed methodology. To establish our results, we develop a novel approach utilizing the classical Cauchy’s bounds for the zeros of a deterministic polynomial with complex coefficients. We also corroborate our analytical results with extensive simulations. The methodology developed in the paper can potentially be applied to a broad class of problems regarding bounds and the distribution of zeros in the theory of random polynomials.

Exponential concentration for the number of roots of random trigonometric polynomials

Exponential concentration for the number of roots of random trigonometric polynomials

Zeros of a growing number of derivatives of random polynomials with independent roots

Let X 1 , X 2 , … X_1,X_2,\ldots be independent and identically distributed random variables in C {\mathbb {C}} chosen from a probability measure μ \mu and define the random polynomial P n ( z ) = ( z − X 1 ) … ( z − X n ) . \begin{align*} P_n(z)=(z-X_1)\ldots (z-X_n)\,. \end{align*} We show that for any sequence k = k ( n ) k = k(n) satisfying k ≤ log ⁡ n / ( 5 log ⁡ log ⁡ n ) k \leq \log n / (5 \log \log n) , the zeros of the k k th derivative of P n P_n are asymptotically distributed according to the same measure μ \mu . This extends work of Kabluchko, which proved the k = 1 k = 1 case, as well as Byun, Lee and Reddy [Trans. Amer. Math. Soc. 375, pp. 6311–6335] who proved the fixed k k case.

Probabilistic Galois Theory: The square discriminant case

AbstractThe paper studies the probability for a Galois group of a random polynomial to be . We focus on the so‐called large box model, where we choose the coefficients of the polynomial independently and uniformly from . The state‐of‐the‐art upper bound is , due to Bhargava. We conjecture a much stronger upper bound , and that this bound is essentially sharp. We prove strong lower bounds both on this probability and on the related probability of the discriminant being a square.

Connecting flow duration curve and precipitation duration curve based on the relationship deduced from machine learning in the watersheds of northern China

Obtaining reliable hydrological data for ungauged watersheds has always been a significant challenge in the water resources management. The flow duration curve (FDC) and precipitation duration curve (PDC) are classical methods to address this challenge. However, traditional FDC and PDC methods mostly rely on pre-assumed probability distribution function form, which may result in sub-optimal model parameters and consequently sub-optimal simulation results of hydrological data, and the common function forms are not applicable to deal with the data including zero-values, which often occurs in the discharge data of arid region and daily rainfall data. To address the issues in FDC and PDC , this study firstly determines the most suitable function form to describe FDC and PDC from various distribution functions, i.e., Log Normal, Generalized Pareto Distribution and H2018 function, from 1960 to 2010 in 7 watersheds in the north of China. The multiple methods including Random Forest, polynomial, Multilayer Perceptron and K-Nearest Neighbor are employed to simulate the parameters of the functions, aiming to deduce the optimal parameters of FDC functions. The results show that, H2018 function form has the best performance in simulating the PDC and FDC and in general, parameter a and b of H2018 function indicate an opposite trend, and parameter k declines in mostly condition at the annual scale from 1960 to 2010. The results of different methods indicate that the Random Forest method is the most efficient and accurate approach to deduce FDC parameters from measured precipitation and climate factors, compared with polynomial, Multilayer Perceptron and K-Nearest Neighbor. Based on the relationship between PDC parameters and FDC parameters, the FDC of ungauged basins can be deduced from the observed precipitation data. The research findings provide a new method to estimate FDC for the water resources management in the ungauged basins.

Expected Energy of Zeros of Elliptic Polynomials

AbstractIn 2011, Armentano, Beltrán and Shub obtained a closed expression for the expected logarithmic energy of the random point process on the sphere given by the roots of random elliptic polynomials. We consider a different approach which allows us to extend the study to the Riesz energies and to compute the expected separation distance.

Bergman kernel functions associated to measures supported on totally real submanifolds

Abstract We prove that the Bergman kernel function associated to a smooth measure supported on a piecewise-smooth maximally totally real submanifold 𝐾 in C n \mathbb{C}^{n} is of polynomial growth. For example, this holds in dimension one if 𝐾 is a finite union of transverse Jordan arcs in ℂ. Our bounds are sharp when 𝐾 is smooth. We give an application to the equidistribution of the zeros of random polynomials, which extends a result of Shiffman–Zelditch to the higher-dimensional setting.

The fractional free convolution of R-diagonal elements and random polynomials under repeated differentiation

Abstract We extend the free convolution of Brown measures of $R$-diagonal elements introduced by Kösters and Tikhomirov [ 28] to fractional powers. We then show how this fractional free convolution arises naturally when studying the roots of random polynomials with independent coefficients under repeated differentiation. When the proportion of derivatives to the degree approaches one, we establish central limit theorem-type behavior and discuss stable distributions.