Polynomial Approximation Research Articles

On the density hypothesis for $L$-functions associated with holomorphic cusp forms

We study the range of validity of the density hypothesis for the zeros of L -functions associated with cusp Hecke eigenforms f of even integral weight, and prove that N_{f}(\sigma, T) \ll T^{2(1-\sigma)+\varepsilon} holds for \sigma \geq 1407/1601 . This improves upon a result of Ivić, who had previously shown the zero-density estimate in the narrower range \sigma\geq 53/60 . Our result relies on an improvement of the large value estimates for Dirichlet polynomials based on mixed moment estimates for the Riemann zeta function. The main ingredients in our proof are the Halász–Montgomery inequality, Ivić’s mixed moment bounds for the zeta function, Huxley’s subdivision argument, Bourgain’s dichotomy approach, and Heath-Brown’s bound for double zeta sums.

Conversion map from quantitative parameter mapping to myelin water fraction: comparison with R1·R2* and myelin water fraction in white matter.

To clarify the relationship between myelin water fraction (MWF) and R1⋅R2* and to develop a method to calculate MWF directly from parameters derived from QPM, i.e., MWF converted from QPM (MWFQPM). Subjects were 12 healthy volunteers. On a 3T MR scanner, dataset was acquired using spoiled gradient-echo sequence for QPM. MWF and R1⋅R2* maps were derived from the multi-gradient-echo (mGRE) dataset. Volume-of-interest (VOI) analysis using the JHU-white matter (WM) atlas was performed. All the data in the 48 WM regions measured VOI were plotted, and quadratic polynomial approximations of each region were derived from the relationship between R1·R2* and the two-pool model-MWF. The R1·R2* map was converted to MWFQPM map. MWF atlas template was generated using converted to MWF from R1·R2* per WM region. The mean MWF and R1·R2* values for the 48 WM regions were 11.96 ± 6.63%, and 19.94 ± 4.59s-2, respectively. A non-linear relationship in 48 regions of the WM between MWF and R1·R2* values was observed by quadratic polynomial approximation (R2 ≥ 0.963, P < 0.0001). MWFQPM map improved image quality compared to the mGRE-MWF map. Myelin water atlas template derived from MWFQPM may be generated with combined multiple WM regions.

Asymptotic Methods for Transaction Costs

We propose a general approximation method for the determination of optimal trading strategies in markets with proportional transaction costs, with a polynomial approximation of the residual value function. The method is exemplified by several problems, from optimally tracking benchmarks and hedging the log contract to maximizing utility from terminal wealth. Strategies are also approximated by practically executable, discrete trades. We identify the necessary trade-off between the trading frequency and trade size to ensure satisfactory agreement with the theoretically optimal, continuous strategies of infinite activity.

USE OF POLYNOMIAL APPROXIMATION OF FORCES IN THE METHOD OF CORRECTION FUNCTIONS

The paper considers the problem of polynomial approximation of the "force - time" dependencies in the elements of corroding hinged-rod structures from the point of view of the influence of the degree of the polynomial on the error in calculating their durability. A method for determining the coefficients of approximating polynomials is proposed, which is based on the use of a numerical-analytical algorithm for solving a system of differential equations describing the corrosion process. The results of numerical experiments are presented, illustrating for various constructions the dependence of the error in solving the problem on the degree of approximating polynomials.

Promotion Time Cure Model with Local Polynomial Estimation

Promotion Time Cure Model with Local Polynomial Estimation

Sixth-kind Chebyshev polynomials technique to numerically treat the dissipative viscoelastic fluid flow in the rheology of Cattaneo–Christov model

Abstract This study investigates the complex dynamics of a viscoelastic fluid subjected to magneto-hydrodynamics over a stretching sheet, incorporating the Cattaneo–Christov heat flux model. This model is especially advantageous for explaining heat transfer in materials possessing significant thermal conductivity, where the conventional Fourier’s law might not be precise. The investigation revolves around evaluating how the thermal relaxation time affects the boundary layer and how both thermal radiation and viscous dissipation influence the thermal field. The significance of this research lies in its contribution to understanding the intricate behavior of such fluids in the presence of magnetic fields and non-Fourier heat conduction. The primary objective is to analyze the impact of viscoelasticity, magnetohydrodynamics, and Cattaneo–Christov heat flux on the flow and heat transfer characteristics over the stretching sheet. The research methodology involves the application of mathematical models and numerical techniques, particularly the use of the shifted Chebyshev polynomials of the sixth-order approximation and spectral collocation technique. The major conclusion of the study underscores the significant influence of viscoelasticity, magnetic field, and Cattaneo–Christov heat flux on the transport properties of the fluid, providing valuable insights for applications in various engineering and industrial contexts. Certain notable results arising from the current issue indicate that heat transfer is more pronounced for the viscoelastic factor and magnetic parameter, whereas the thermal relaxation parameter exhibits the opposite trend. In addition, the inclusion of the Cattaneo–Christov term enhances our comprehension of thermal behavior.

On the Continuity of Functions and Their Application in Polynomial Approximation

On the Continuity of Functions and Their Application in Polynomial Approximation

Investigation of an optimization method for time constants to calibrate reaction calorimeters

In exothermic reactions, scale-up sometimes endangers the chemical process because of the risk of runaway reactions. Reaction calorimeters are used in a laboratory to design a safe chemical process by collecting thermal and kinetic parameters for scale-up. Because scale-up can magnify the errors obtained in laboratory experiments, the cooling capacity can be improperly calculated based on the underestimated heat generation due to thermal lag. Therefore, reaction calorimeters are required to provide accurate heat flow despite thermal delays inherent in the reaction calorimeters.This study validates the time-constant search method for a reaction calorimeter that employs a function of smoothness S(τ) in the presence of noise. The degree of noise was determined by measuring the actual value using a reaction calorimeter. The simulation conducted using spreadsheets demonstrated that even in the presence of noise, it is possible to achieve high accuracy in the search for time-constants when there is a small thermal lag and a large intensity of calibration heat. Moreover, the use of a third-order polynomial approximation to calculate the derivative value proved to be suitable for determining the time constant accurately.

Enhanced Estimation of Quantum Properties with Common Randomized Measurements

We present a technique for enhancing the estimation of quantum state properties by incorporating approximate prior knowledge about the quantum state. This consists in performing randomized measurements on a quantum processor and comparing the results with those obtained from a classical computer that stores an approximation of the quantum state. We provide unbiased estimators for expectation values of multicopy observables and present performance guarantees in terms of variance bounds that depend on the prior knowledge accuracy. We demonstrate the effectiveness of our approach through experimental and numerical examples detecting mixed-state entanglement, and estimating polynomial approximations of the von Neumann entropy and state fidelities. Published by the American Physical Society 2024

Robust thalamic nuclei segmentation from T1-weighted MRI using polynomial intensity transformation.

Accurate segmentation of thalamic nuclei, crucial for understanding their role in healthy cognition and in pathologies, is challenging to achieve on standard T1-weighted (T1w) magnetic resonance imaging (MRI) due to poor image contrast. White-matter-nulled (WMn) MRI sequences improve intrathalamic contrast but are not part of clinical protocols or extant databases. In this study, we introduce histogram-based polynomial synthesis (HIPS), a fast preprocessing transform step that synthesizes WMn-like image contrast from standard T1w MRI using a polynomial approximation for intensity transformation. HIPS was incorporated into THalamus Optimized Multi-Atlas Segmentation (THOMAS) pipeline, a method developed and optimized for WMn MRI. HIPS-THOMAS was compared to a convolutional neural network (CNN)-based segmentation method and THOMAS modified for the use of T1w images (T1w-THOMAS). The robustness and accuracy of the three methods were tested across different image contrasts (MPRAGE, SPGR, and MP2RAGE), scanner manufacturers (PHILIPS, GE, and Siemens), and field strengths (3T and 7T). HIPS-transformed images improved intra-thalamic contrast and thalamic boundaries, and HIPS-THOMAS yielded significantly higher mean Dice coefficients and reduced volume errors compared to both the CNN method and T1w-THOMAS. Finally, all three methods were compared using the frequently travelling human phantom MRI dataset for inter- and intra-scanner variability, with HIPS displaying the least inter-scanner variability and performing comparably with T1w-THOMAS for intra-scanner variability. In conclusion, our findings highlight the efficacy and robustness of HIPS in enhancing thalamic nuclei segmentation from standard T1w MRI.

New algorithms for fair k-center problem with outliers and capacity constraints

The fair k-center problem has been paid lots of attention recently. In the fair k-center problem, we are given a set X of points in a metric space and a parameter k∈Z+, where the points in X are divided into several groups, and each point is assigned a color to denote which group it is in. The goal is to partition X into k clusters such that the number of cluster centers with each color is equal to a given value, and the k-center problem objective is minimized. In this paper, we consider the fair k-center problem with outliers and capacity constraints, denoted as the fair k-center with outliers (FkCO) problem and the capacitated fair k-center (CFkC) problem, respectively. The outliers constraints allow up to z outliers to be discarded when computing the objective function, while the capacity constraints require that each cluster has size no more than L. In this paper, we design an Fixed-Parameter Tractability (FPT) approximation algorithm and a polynomial approximation algorithm for the above two problems. In particular, our algorithms give (1+ϵ)-approximations with FPT time for the FkCO and CFkC problems in doubling metric space. Moreover, we also propose a 3-approximation algorithm in polynomial time for the FkCO problem with some reasonable assumptions.

BiPFT: Binary Pre-trained Foundation Transformer with Low-Rank Estimation of Binarization Residual Polynomials

Pretrained foundation models offer substantial benefits for a wide range of downstream tasks, which can be one of the most potential techniques to access artificial general intelligence. However, scaling up foundation transformers for maximal task-agnostic knowledge has brought about computational challenges, especially on resource-limited devices such as mobiles. This work proposes the first Binary Pretrained Foundation Transformer (BiPFT) for natural language understanding (NLU) tasks, which remarkably saves 56 times operations and 28 times memory. In contrast to previous task-specific binary transformers, BiPFT exhibits a substantial enhancement in the learning capabilities of binary neural networks (BNNs), promoting BNNs into the era of pre-training. Benefiting from extensive pretraining data, we further propose a data-driven binarization method. Specifically, we first analyze the binarization error in self-attention operations and derive the polynomials of binarization error. To simulate full-precision self-attention, we define binarization error as binarization residual polynomials, and then introduce low-rank estimators to model these polynomials. Extensive experiments validate the effectiveness of BiPFTs, surpassing task-specific baseline by 15.4% average performance on the GLUE benchmark. BiPFT also demonstrates improved robustness to hyperparameter changes, improved optimization efficiency, and reduced reliance on downstream distillation, which consequently generalize on various NLU tasks and simplify the downstream pipeline of BNNs. Our code and pretrained models are publicly available at https://github.com/Xingrun-Xing/BiPFT.

Principal-Agent Reward Shaping in MDPs

Principal-agent problems arise when one party acts on behalf of another, leading to conflicts of interest. The economic literature has extensively studied principal-agent problems, and recent work has extended this to more complex scenarios such as Markov Decision Processes (MDPs). In this paper, we further explore this line of research by investigating how reward shaping under budget constraints can improve the principal's utility. We study a two-player Stackelberg game where the principal and the agent have different reward functions, and the agent chooses an MDP policy for both players. The principal offers an additional reward to the agent, and the agent picks their policy selfishly to maximize their reward, which is the sum of the original and the offered reward. Our results establish the NP-hardness of the problem and offer polynomial approximation algorithms for two classes of instances: Stochastic trees and deterministic decision processes with a finite horizon.

Vibration Suppression of Piecewise-Linear Stiffness Nes System Under Random Excitation

Nonlinear energy sinks (NESs) are critically important for structural vibration suppression. They can absorb vibrational energy across a broad frequency spectrum, possess strong robustness, and have a relatively small mass. This study addresses the vibration suppression in a piecewise linear stiffness NES system under random excitation. Initially, a theoretical model of the piecewise linear stiffness NES system is developed. The piecewise linear stiffness function is approximated using Legendre polynomial approximation. Following this, the steady-state Fokker–Planck–Kolmogorov (FPK) equation of the system is formulated via the Generalized Harmonic Function Method. The FPK equation is solved using the fourth-order central difference method, and the effectiveness of this approach is validated by comparing the FDM results with numerical simulations. Lastly, the influence of varying system parameters on the stability of the piecewise linear stiffness NES system is analyzed.

Near-Circularity in Capacity and Maximally Convergent Polynomials

AbstractIf f is a power series with radius R of convergence, $$R &gt; 1$$ R &gt; 1 , it is well-known that the method of Carathéodory–Fejér constructs polynomial approximations of f on the closed unit disk which show the typical phenomenon of near-circularity on the unit circle. Let E be compact and connected and let f be holomorphic on E. If $$\left\{ p_n\right\} _{n\in \mathbb {N}}$$ p n n ∈ N is a sequence of polynomials converging maximally to f on E, it is shown that the modulus of the error functions $$f-p_n$$ f - p n is asymptotically constant in capacity on level lines of the Green’s function $$g_\Omega (z,\infty )$$ g Ω ( z , ∞ ) of the complement $$\Omega $$ Ω of E in $$\overline{\mathbb {C}}$$ C ¯ with pole at infinity, thereby reflecting a type of near-circularity, but without gaining knowledge of the winding numbers of the error curves with respect to the point 0.

Higher order investigation on modulated waves in the Peyrard–Bishop–Dauxois DNA model

Despite the widespread use of transcendental functions in the modeling of the dynamics of DNA, most research efforts are limited in their analytical studies of this enthralling system to cubic order polynomial approximations of the corresponding equations of motion. In this paper, we present an investigation of waves in the Peyrard–Bishop–Dauxois model of DNA; while extending the polynomial approximation of the Morse potential up to the sixth order. We show that, within a generalized version of the reductive perturbation method that we have adopted, the equations governing the envelop consist of the standard cubic nonlinear Schrödinger equation and its non homogeneous linearizations. Exact and explicit analytical solutions that correspond to bright solitary waves are obtained for these coupled amplitude equations. A notable qualitative feature of these solutions is the dependence of their propagation speeds and frequencies on their amplitudes. Our approach additionally unveils that these solutions contain some harmonic terms; which are missed in existing works. A very good agreement is found between our analytical analysis and the numerical simulations of the full discrete nonlinear equation of the lattice which use these solutions as initial conditions.

Hazard rate estimation when the measurement error has a normal or logistic distribution

Statistical data analysis available in most scientific fields is often recorded with measurement error. The modeling of these statistical data by ignoring the measurement errors, leads to estimators of the parameters of the distributions, whose use does not achieve sufficient accuracy in the goodness of fit. In reliability criteria, one of the important issues is hazard rate function. It prompted us to investigate the hazard rate criterion in the presence of measurement error generated from the normal or logistic distribution. Now, while providing the estimator for the density function using local time polynomial estimator methods, the risk rate function is estimated according to the contamination degree of 15 or 30%. Finally, we present the numerical analysis.

Minimax Regret Robust Screening with Moment Information

Problem definition: We study a robust screening problem where a seller attempts to sell a product to a buyer knowing only the moment and support information of the buyer’s valuation distribution. The objective is to maximize the competitive ratio relative to an optimal hindsight policy equipped with full valuation information. Methodology/results: We formulate the robust screening problem as a linear programming problem, which can be solved efficiently if the support of the buyer’s valuation is finite. When the support of the buyer’s valuation is continuous and the seller knows the mean and the upper and lower bounds of the support for the buyer’s valuation, we show that the optimal payment is a piecewise polynomial function of the valuation with a degree of at most two. Moreover, we derive the closed-form competitive ratio corresponding to the optimal mechanism. The optimal mechanism can be implemented by a randomized pricing mechanism whose price density function is a piecewise inverse function adjusted by a constant. When the mean and variance are known to the seller, we propose a feasible piecewise polynomial approximation of the optimal payment function with a degree of at most three. We also demonstrate that the optimal competitive ratio exhibits a logarithmic decay with respect to the coefficient of variation of the buyer’s valuation distribution. Managerial implications: Our general framework provides an approach to investigating the value of moment information in the robust screening problem. We establish that even a loose upper bound of support or a large variance can guarantee a good competitive ratio. Funding: The research of S. Wang is partially supported by the National Natural Science Foundation of China [Grant 72394395]. The research of S. Liu is partly supported by the National Natural Science Foundation of China [Grant NSFC-72072117]. Supplemental Material: The online appendix is available at https://doi.org/10.1287/msom.2023.0072 .

Effect of Concurrent Metabolic Dysfunction-Associated Steatotic Liver Disease on Serial Non-invasive Fibrosis Markers in Chronic Hepatitis B.

Concurrent hepatic steatosis has diverse effects on chronic hepatitis B (CHB), however the combined effects of metabolic dysfunction-associated steatotic liver disease (MASLD) and CHB on liver fibrosis progression remains unclear. The primary aim of this study was to utilize serial fibrosis measurements to compare the dynamic change in fibrosis in CHB patients with/without concurrent MASLD. The secondary aim was to investigate factors associated with steatosis development and regression in CHB patients. This was a retrospective cohort study of all non-cirrhotic CHB patients identified from 1/1/2011 to 31/12/2016. Hepatic steatosis was diagnosed by ultrasound. Fibrosis markers included liver stiffness (LSM) by transient elastography, APRI and FIB-4. General linear mixed effects modelling was used to fit polynomial and linear estimates. Of 810 CHB patients (n = 2,373 LSM measurements; median age 44.4y; 48% male; 24% HBeAg positive), 14% had concurrent MASLD. LSM was higher at baseline but decreased in MASLD patients over time, while LSM remained stable in non-MASLD patients, such that all patients had similar LSM beyond 4-5 years. MASLD patients had lower APRI compared to non-MASLD patients, which was predominately due to a higher platelet count and higher ALT over time. There was substantial discordance between LSM, APRI and FIB-4. Baseline BMI was the only factor that predicted steatosis development and regression. We found no evidence of an association between concurrent MASLD and fibrosis progression amongst CHB patients without baseline advanced liver disease. APRI and FIB-4 may have reduced accuracy in MASLD patients.

A new graph theoretic analytical method for nonlinear distributed order fractional ordinary differential equations by clique polynomial of cocktail party graph

In this paper, we presented a new analytical method for one of the rapidly emerging branches of fractional calculus, the distributed order fractional differential equations (DFDE). Due to its significant applications in modeling complex physical systems, researchers have shown profound interest in developing various analytical and numerical methods to study DFDEs. With this motivation, we proposed an easy computational technique with the help of graph theoretic polynomials from algebraic graph theory for nonlinear distributed order fractional ordinary differential equations (NDFODE). In the method, we used clique polynomials of the cocktail party graph as an approximation solution. With operational integration and fractional differentiation in the Caputo sense, the NDFODEs transformed into a system of algebraic equations and then solved by Newton–Raphson's method to determine the unknowns in the Clique polynomial approximation. The proficiency of the proposed Clique polynomial collocation method (CCM) is illustrated with four numerical examples. The convergence and error analysis are discussed in tabular and graphical depictions by comparing the CCM results with the results of existing numerical methods.