Exponential Estimates Research Articles

Global well-posedness and exponential decay estimates for semilinear Newell–Whitehead–Segel equation

Abstract This article presents the application of the Faedo–Galerkin compactness method to establish the local well-posedness of the Newell–Whitehead–Segel equation. By analyzing a finite-dimensional approximate problem, the existence and uniqueness of a local solution were demonstrated. A priori estimates were derived, enabling the transition to the limit and the recovery of the original problem’s local solution. The study further proves the uniqueness and continuous dependence of the solution on initial data. Additionally, under certain conditions, it is shown that the energy norm of the solution decays exponentially over time, and the L 2 {L}^{2} -norm of the time derivative of the solution approaches zero asymptotically.

Modal volatility function

We in this article propose a novel non‐parametric estimator for the volatility function within a broad context that encompasses nonlinear time series models as a special case. The new estimator, built on the mode value, is designed to complement existing mean volatility measures to reveal distinct data features. We demonstrate that the suggested modal volatility estimator can be obtained asymptotically as well as if the conditional mean regression function were known, assuming observations are from a strictly stationary and absolutely regular process. Under mild regularity conditions, we establish that the asymptotic distributions of the resulting estimator align with those derived from independent observations, albeit with a slower convergence rate compared to non‐parametric mean regression. The theory and practice of bandwidth selection are discussed. Moreover, we put forward a variance reduction technique for the modal volatility estimator to attain asymptotic relative efficiency while maintaining the asymptotic bias unchanged. We numerically solve the modal regression model with the use of a modified modal‐expectation‐maximization algorithm. Monte Carlo simulations are conducted to assess the finite sample performance of the developed estimation procedure. Two real data analyses are presented to further illustrate the newly proposed model in practical applications. To potentially enhance the accuracy of the bias term, we in the end discuss the extension of the method to local exponential modal estimation. We showcase that the suggested exponential modal volatility estimator shares the same asymptotic variance as the non‐parametric modal volatility estimator but may exhibit a smaller bias.

Variance estimation using memory type estimators based on EWMA statistic for time scaled surveys in stratified sampling

In this article, we have proposed memory-type exponential and non-exponential estimators for population variance based on exponentially weighted moving average (EWMA) statistic in stratified sampling. We drive mathematical expressions for mean square errors of the proposed estimators using Taylor and exponential expansions. Our analysis demonstrates that the proposed memory-type estimators outperform the conventional estimators under stratification, particularly, when the information of previous sample is utilized. The performances of the proposed estimators are evaluated mathematically by deriving the conditions in which the memory-type estimators would perform better than their corresponding conventional estimators under stratification. Through an extensive simulation, we evaluated the performance of the proposed estimators across various population parameters, revealing their enhanced efficiency in time-scaled survey. Additionally, a real data application is also used to support the mathematical findings, confirming the practical utility of the proposed estimators. The results of numerical study underscore the importance of the use of previous sampled information which significantly improves the accuracy and reliability of the proposed estimator for variance estimation for time-scaled surveys.

A new modified estimator of population variance in calibrated survey sampling

In survey statistics, estimating and reducing population variation is crucial. These variations can occur in any sampling design, including stratified random sampling, where stratum weights may increase the variance of estimators. Calibration techniques, which use additional auxiliary information, can help mitigate this issue. This paper examines three calibration-based estimators—calibration variance, calibration ratio, and calibration exponential ratio estimators—within the framework of stratified random sampling. The study generates data from normal, gamma, and exponential distributions to test these estimators. Results demonstrate that the proposed calibration estimators offer more accurate estimates of population variance and outperform existing methods in estimating population variance under stratified random sampling, providing more accurate and reliable estimates.

State of Charge Estimation of Lithium-Ion Batteries for Electric Vehicle Application Using Gaussian Process Regression Approach

For the purpose of ensuring a secure, dependable and affordable performancealong with clean energy in electric vehicles, the estimation of the precise state of charge of LIB is very important. In this article, Gaussian Process Regression with different kernel functions-based SOC prediction is proposed and their performance with good health and well-beingare evaluated and analyzed. A useful benefit of employing GPR is the ability to quantify and estimate uncertainties, allowing for the evaluation of the SOC estimate's dependability. The kernel function serves as a crucial hyperparameter that improves GPR performance. GPR considers the temperature and voltage of the battery, which are independent of one another, as their respective input parametersthat relates Industry, innovation and infrastructure where target-dependent variable is battery SOC. Initially, the training process involves determining the ideal hyperparameters of a kernel function to accurately represent the characteristics of the data. The accuracy of predicting SOC of the battery is evaluated using test data. According to the simulation outcomes, the squared exponential kernel function-based GPR estimates SOC with high accuracy and lower RMSE and MAE which ensures energy efficiency and quality education.

ADCMDES: Design of an augmented cross-domain collaborative recommendation model using novel distance metric and ensemble stratification

Collaborative recommendation involves leveraging information from one entity to suggest attributes of other entities through pattern analysis. These models are invaluable for understanding data behaviour’s impact on related entities. Researchers propose various pattern recognition and correlation models, employing distance metrics like Jaccard, Cosine, etc., to estimate correlations between user queries and recommendation datasets. However, these models become less efficient as dataset size increases due to exponential correlation estimation delays. To enhance scalability while retaining recommendation quality, a new approach is introduced: ADCMDES. This novel augmented cross-domain collaborative model employs a hybrid distance metric and ensemble stratification for dataset pruning. It operates semi-supervisedly, requiring information about the entities being collaborated upon. This data is used to cluster similar collaborative entities, producing a condensed cluster with greater relevancy to user queries. Utilizing word2vec, records are transformed into features for an ensemble classification engine. The resultant model categorizes user input, directing it to the most relevant cluster. Entries within this cluster are ranked using a hybrid metric amalgamating 18 distance measures, enhancing correlation between input queries and recommendations. In testing, ADCMDES exhibited 15% better accuracy, 8% better precision, 9% better recall, and 3% lower RMSE compared to standard models across datasets. While some delay is introduced due to ensemble classification and augmented feature pooling, this doesn't considerably affect long-term recommendation performance and can be mitigated through parallel processing and redundancy reduction techniques.

An optimal estimation approach in non-response under simple random sampling utilizing dual auxiliary variable for finite distribution function

Auxiliary data needs to be incorporated into survey sampling in order to create a precise population parameter estimator. This study investigates improving the efficiency of these estimators, the researchers use study variable [cumulative distribution function, (CDF)] and auxiliary variable (CDF, mean and ranks) to achieve this goal in the condition of non-response. The researchers suggest two ideal families of exponential estimators to obtain an enhanced population (CDF) estimators in simple random sampling under non-response, by utilizing the data from dual auxiliary variable. The first-order approximation is used for the bias, mean squared error (MSE), and minimal mean squared error of the new and existing estimators. Using dual auxiliary information, the new families of exponential estimators turn out to be precise and efficient while estimating the distribution function, whenever non-response is contained within three Situation such as (Situation-I when non-response occurs in both study and auxiliary variables, next to it Situation-II when non-response occurs only in study variable and Situation-III when non-response occurs only in auxiliary variable). We employ a single population to assess the effectiveness and suitability of the suggested family estimators; varying values of k = 2 and 3 produce varying mean square error and percentage relative efficiency values. The suggested families of estimators have lower (MSE) and higher (PRE) values for the population which showed in tables 2–13 The percentage relative efficiency (PRE) of proposed families of estimators when k = 2 for all the three situation of non-response is 183.52715, 189.78448, 184.82791, 185.49885, and 223.06144 and when k = 3 for all the three situation of non-response is 223.06144, 119.32556, 165.89993, 166.29598, 112.90767 and 113.25086.

An Effective Exponential Ratio and Product Type Estimator in Post-Stratification

In this paper we present a unique estimator for the population parameter under post-stratification. By segmenting the population into homogeneous subgroups, post-stratification is a widely used strategy in survey sampling that increases the accuracy of estimators. In the post-stratification scenario, this paper addressed the issue of estimating the mean of a finite population. For post- stratification, better separate ratio and product exponential type estimators are proposed. This study's primary goal is to evaluate our suggested estimators' performance against that of current estimators. We perform a thorough simulation research to assess the precision and effectiveness of our estimator. The mean squared errors and biases of the proposed estimators are obtained to the first degree of approximation. Theoretical and practical researches have demonstrated that the proposed estimators are more efficient than other estimators that were taken into consideration.

The ergodicity and uniform large deviations for the 1D stochastic Landau-Lifshitz-Bloch equation

In this article, we consider two aspects asymptotic behaviors for the 1D stochastic Landau-Lifshitz-Bloch equation driven by additive noise including the unique ergodicity and the uniform large deviations. Using a deterministic argument relying on the exponential moment and exponential stability estimates, we first establish the unique ergodicity of invariant measure in H 1. Then, the uniform large deviations is established in C ( [ 0 , T ] ; H 1 ) by the uniform contraction principle.

Singular extension of critical Sobolev mappings under an exponential weak-type estimate

Given m∈N∖{0} and a compact Riemannian manifold N, we construct for every map u in the critical Sobolev space Wm/(m+1),m+1(Sm,N), a map U:B1m+1→N whose trace is u and which satisfies an exponential weak-type Sobolev estimate. The result and its proof carry on to the extension to a half-space of maps on its boundary hyperplane and to the extension to the hyperbolic space of maps on its boundary sphere at infinity.

Improved Ratio-type Exponential Estimator for Estimating Population Mean in Ranked Set Sampling

In this paper we propose a ratio-type exponential estimator for estimating population mean of the study variable under Ranked set sampling when auxiliary information is known. The bias and Mean square error of the proposed estimator has been derived up to the first degree of approximation. A simulation study has been carried out to judge the performance of the newly proposed estimator along with existing estimators. It is obtained that the proposed estimator is more efficient as compared to the competing estimators.

Improved estimation of the mean through regressed exponential estimators based on sub-sampling non-respondents

The present study concerns the issue of estimating the population mean and presents novel and improved regressed exponential estimators using different parameters of an auxiliary character based on sub-sampling non-respondents. The bias and mean square error (MSE) of the proposed estimators for the most pragmatic simple random sampling without replacement (SRSWOR) scheme have been derived up to the first order of approximation (i.e. the expression containing errors up to the power of two so that the expectation comes only in terms of the mean, variance and covariance). The optimum value of the MSE of the estimators is found, along with the necessary conditions for optimising the MSE. The effectiveness of the suggested estimators, outperforming the existing ones in terms of their MSE, has been studied theoretically, while the empirical as well as the simulation studies have confirmed these findings.

Robust Estimation of Variance Using Supplementary Information

This study introduces robust exponential ratio-type estimators for finite population variance, incorporating the minimum covariance determinant (M.C.D.) and minimum volume ellipsoid (M.V.E.) robust covariance matrices in the context of simple random sampling. We utilize a first-order approximation to derive expressions for the bias and mean square error (M.S.E.) of these proposed estimators. Through a comprehensive review of existing literature, it is evident that these robust estimators outperform other alternatives in terms of efficiency. Both the M.C.D. and M.V.E. methods are known for their resilience to outliers, making them particularly effective when such anomalies are present in the data. Simulation studies and empirical results consistently demonstrate that the proposed robust exponential ratio-type estimators yield lower mean square errors compared to traditional estimators under simple random sampling. Additionally, the efficiency of these estimators is further validated through simulated studies and real-world data sets. Theoretical analysis and numerical evidence collectively affirm that the proposed class of estimators consistently outperforms competing methods across various scenarios.

Uniform attractors for the Bingham model

In this paper, on the base of the theory of attractors for non-invariant trajectory spaces, the limiting behavior of solutions to a periodic in spatial variables problem for the Bingham model is investigated. For the considered problem, necessary exponential estimates are established, the trajectory space is determined, and the existence of a minimal uniform trajectory and uniform global attractors is proved.

Improved growth estimate for one-dimensional sixth-order Boussinesq equation with logarithmic nonlinearity

This paper provides an improved exponential growth estimate, surpassing the growth rate given in the previous work. This finding elucidates the impact of the power index k in the logarithmic nonlinearity uln|u|k on the growth behavior of the solution to the initial boundary value problem for the one-dimensional sixth-order nonlinear Boussinesq equation with logarithmic nonlinearity.

Looking backward toward the future: A bibliometric analysis of the last 40 years of meningioma global outcomes.

This study is the first comprehensive bibliometric analysis about meningioma to date. The aim of this study is to identify the most influential publications in this field through citation and co-citation analysis, to examine international collaborations, to identify the conceptual framework of the subject and emerging trending topics through keyword analysis, and to identify the most productive countries, authors and journals. 9619 articles on meningioma published between 1980 and 2023 were downloaded from the Web of Science (WoS) database and statistically analyzed. In this study, various bibliometric techniques were utilized, including trend keyword analysis, thematic evolution analysis, factor analysis, conceptual structure analysis, citation and co-citation analyses. Bibliometric network visualization maps were created to identify trend topics, citation analysis and cross-country collaborations. The Exponential Smoothing estimator was used to predict article productivity in the coming years. The first 3 countries that contributed the most to the literature were respectively; USA (2664, 27.7%), Japan (972, 10.1%), Germany (943, 9.8%). The first three most productive journals were respectively; Journal of Neurosurgery (number of article = 496), World Neurosurgery (399), Acta Neurochirurgica (378). The most productive author was Mcdermott MW (number of article = 88) and the most active institution was the University of California System (number of article = 470). In addition to high-grade meningiomas, the most studied topics from past to present have been magnetic resonance imaging, recurrence, radiation therapy, and skull base. As a result of the analyses to determine trend topics, the subjects studied in recent years were diagnostic and imaging methods, surgical and treatment methods, prognosis and survival, epidemiology and quality of life, and with the advancement of technology, machine learning and prediction models. Scientific collaboration was seen primarily in articles from western countries, especially the USA, European countries, and Canada. However, there was also a not insignificant effect in developing countries such as China, India, and Turkey.

Equivalence of Sobolev Norms with Respect to Weighted Gaussian Measures

AbstractWe consider the spaces $${\text {L}}^p(X,\nu ;V)$$ L p ( X , ν ; V ) , where X is a separable Banach space, $$\mu $$ μ is a centred non-degenerate Gaussian measure, $$\nu :=Ke^{-U}\mu $$ ν : = K e - U μ with normalizing factor K and V is a separable Hilbert space. In this paper we prove a vector-valued Poincaré inequality for functions $$F\in W^{1,p}(X,\nu ;V)$$ F ∈ W 1 , p ( X , ν ; V ) , which allows us to show that for every $$p\in (1,\infty )$$ p ∈ ( 1 , ∞ ) and every $$k\in \mathbb {N}$$ k ∈ N the norm in $$W^{k,p}(X,\nu )$$ W k , p ( X , ν ) is equivalent to the graph norm of $$D_H^{k}$$ D H k (the k-th Malliavin derivative) in $${\text {L}}^p(X,\nu )$$ L p ( X , ν ) . To conclude, we show exponential decay estimates for the V-valued perturbed Ornstein-Uhlenbeck semigroup $$(T^V(t))_{t\ge 0}$$ ( T V ( t ) ) t ≥ 0 , defined in Section 2.6, as t goes to infinity. Useful tools are the study of the asymptotic behaviour of the scalar perturbed Ornstein-Uhlenbeck $$(T(t))_{t\ge 0}$$ ( T ( t ) ) t ≥ 0 , and pointwise estimates for $$|D_HT(t)f|_H^p$$ | D H T ( t ) f | H p by means of both $$T(t)|D_Hf|^p_H$$ T ( t ) | D H f | H p and $$T(t)|f|^p$$ T ( t ) | f | p .

On Artin’s Primitive Root Conjecture for Function Fields over 𝔽q

ABSTRACT In 1927, E. Artin proposed a conjecture for the natural density of primes p for which g generates $(\mathbb{Z}/p\mathbb{Z})^\times$. By carefully observing numerical deviations from Artin’s originally predicted asymptotic, Derrick and Emma Lehmer (1957) identified the need for an additional correction factor, leading to a modified conjecture which was eventually proved to be correct by Hooley (1967) under the assumption of the generalized Riemann hypothesis. An appropriate analogue of Artin’s primitive root conjecture may moreover be formulated for an algebraic function field K of r variables over $\mathbb{F}_{q}$. Relying on a soon to be established theorem of Weil (1948), Bilharz (1937) provided a proof in the particular case that K is a global function field (that is, r = 1), which is correct under the assumption that $g \in K$ is a geometric element. Under the same assumptions, Pappalardi and Shparlinski (1995) established a quantitative version of Bilharz’s result. In this paper we build upon these works by both generalizing to function fields in r variables over $\mathbb{F}_{q}$ and removing the assumption that $g \in K$ is geometric, thereby completing a proof of Artin’s primitive root conjecture for function fields over $\mathbb{F}_{q}$. In doing so, we moreover identify an interesting correction factor which emerges when g is not geometric. A crucial feature of our work is an exponential sum estimate over varieties that we derive from Weil’s Theorem.

Understanding stunting and its determinant among children in the most populous state of India using estimation of population mean under ranked set sampling in the presence of missing data

ABSTRACT In real-life scenarios, encountering data with missing values is common, and if not managed carefully from the outset of a study, it can lead to significant biases in survey estimates. Various methods exist for imputing missing values in sampling procedures. Ranked set sampling (RSS) is widely recognized for its superior efficiency compared to simple random sampling. However, limited research has been conducted on ranked set sampling in the presence of missing data. This article introduces novel imputation methods designed to estimate population means in the context of missing data under RSS. These innovative estimators are developed by integrating ratio, exponential, and logarithmic estimators judiciously. Expressions for the bias and mean squared error of the proposed estimators are derived up to the first-order approximation. Through simulation studies and an application to stunting and its determinants among children in Uttar Pradesh, India’s most populous state, the effectiveness of the suggested estimators in handling missing data is demonstrated. Numerical examples involving stunting in Uttar Pradesh, as well as simulated data generated using R software, confirm the superior performance of the proposed estimators over existing methods, as evidenced by comparisons of percentage relative efficiency and mean squared error. The results are promising, indicating improvement over all existing imputation methods. Additionally, pertinent recommendations are provided for survey professionals regarding future research.

Exponential Decay Estimates and Smoothness of the Moduli Space of Pseudoholomorphic Curves

The compacified moduli space of bordered stable maps carries a Kuranishi structure with boundary. Smoothness of Kuranishi structure along the boundary requires smoothness of coordinate changes along the boundary. The proof of smoothness is written by the authors in [AMS/IP Stud. Adv. Math., 46.1, 2009, xii+396 pp.], [AMS/IP Stud. Adv. math., 46.2, 2009, pp. i–xii and 397–805], and [Surv. Differ. Geom., vol. 22, pp. 133–190, 2018] based on some uniform exponential decay estimates of the stable maps with respect to a parameter T T , the length of the gluing cylindrical neck, near the boundary of the moduli space. In this paper we establish this exponential decay estimates in a precise manner by carefully examining the dependence on the parameter T T of the gluing construction in the setting of bordered Riemann surfaces with boundary punctures. We also show that the smoothness of the collar follows from the aforementioned exponential estimates by taking s = 1 / T s = 1/T as the radial coordinate of the collar.