Multivariate Function Research Articles

Research on controlled mining of end slope fire-burned area in open-pit mine

To solve the problem of controlling mining in the open-pit mine end slope fire-burned area, applying multivariate function fitting to the roof and floor modeling of multi-coal seam open pit mines, introducing the factor of coal quality changes in the fire-burned area, determining coal quality information at each location through proximate analysis of coal, to establish the net profit model of the mining area, it is determined the net profit of each mining position by numerical integration, the final mining position was determined without failure by calculating the slope stability based on the numerical simulation of strength reduction. Taking the Dananhu No. 2 open-pit mine in Hami, Xinjiang, China as the engineering background, the fire-burned area III within the southern end slope boundary of the first mining area is 240 m. It was finally determined that the optimal mining position is when the advancement degree is 182 m, the ultimate pit slope angle is 25°, the three-dimensional slope stability is 1.305, the profit is 671.96 million yuan, The deep boundary of the southern end slope fire-burned area of the slope is reduced by 58 m. This paper solves the problem of end slope mining in Dananhu No. 2 mine, and maximizes its net profit under the condition of ensuring safe production.

An optimized watermarking scheme based on genetic algorithm and elliptic curve

Digital watermarking serves as a crucial tool for tracing copyright infringements and ensuring the authenticity and integrity of sensitive information. The fundamental concept involves embedding a watermark in the host information, ensuring its undetectability by unauthorized parties. The efficacy of a watermarking scheme mainly depends on achieving high levels of imperceptibility, robustness, and embedding capacity. These attributes are intricately linked to both the selection of the host information segment and the embedding factor. Existing schemes often (i) employ the entire host information for embedding, incurring computational expenses, and (ii) optimize the embedding factor without considering imperceptibility, robustness, and embedding capacity simultaneously, resulting in less secure watermarks. To address these limitations, we introduce a novel watermarking scheme leveraging elliptic curves (ECs) and genetic algorithms (GA). We model the choice of the embedding part by generating pseudo-random numbers over ECs, taking advantage of their proven sensitivity, security, and low computational complexity. Due to parallel search and adaptability to non-linear relationships of GA, the scheme employs genetic optimization with a multivariate objective function to establish a balance between imperceptibility, robustness, and embedding capacity for optimal watermarked generation. Rigorous analysis and comparisons demonstrate that our proposed scheme attains significantly higher imperceptibility, robustness, and embedding capacity compared to existing optimized schemes. Furthermore, our scheme exhibits a speed advantage, being up to 278 and 21 times faster than optimized and non-optimized schemes, respectively, thereby affirming its practical applicability.

The effects of new product quality choices and customer behavior on the in-house refurbishing model versus the outsourcing model

Background Refurbishing encompasses the process of collecting used products, refurbishing them into new ones, and offering them to consumers. Within this framework, manufacturers can opt for either in-house refurbishing, managed internally, or outsource these operations to third parties and retailers. Methods Our study also takes into account the limited supply of remanufacturable products and incorporates customer behavior distinctions, differentiating between high-end and low-end customers in our models. The primary aim of this research is to formulate and analyze models for in-house refurbishing, refurbishing outsourcing to third parties, and refurbishing outsourcing to retailers. We seek to identify optimal solutions, conduct a comprehensive analysis, and implement strategies to maximize benefits. Additionally, we compare the in-house refurbishing model with the outsourcing refurbishing model. Leveraging multivariable function optimization techniques alongside the Karush-Kuhn-Tucker method, we endeavor to ascertain the optimal strategy, thereby facilitating a nuanced and methodical examination of the intricate dynamics governing profitability and sustainability within the refurbishing sector. Results Our study has applied rigorous mathematical modeling and optimization methodologies to identify optimal configurations for each model, aiming to enhance profitability across diverse parameters, including product quantities, quality selections, and pricing strategies. Conclusions Our analysis underscores that the in-house refurbishing approach tends to yield superior profits for profit-oriented enterprises, whereas outsourcing refurbishing to retailers is conducive to meeting environmental sustainability goals. Furthermore, our investigation has considered pivotal factors such as the constrained availability of remanufacturable products and consumer behavior, with a particular emphasis on discerning preferences among high-end and low-end clientele. We also perform sensitivity analyses to assess the impact of parameter variations on optimal outcomes. Our findings yield three distinct refurbishing models: the in-house refurbishing model, suitable for profit-driven companies under specific production cost parameters, and the outsourcing refurbishing to retailers model, tailored for environmentally-conscious companies, contingent upon particular production cost parameters.

Investigation and optimization heat and humidity environmental control model in construction tunnels: A multifactorial approach incorporating ventilation parameters and hot water quantity

The high rock temperature and hot water influx in the surrounding area of the tunnel construction cause an excessively humid and environment high temperature inside the excavated area of the tunnel, necessitating urgent investigations into the environmental characteristics coupled with these factors and optimizing the control scheme of the construction environment. The on-site data of the tunnel temperature, humidity fields, and external environmental parameters have been collected, along with statistical analysis of the amount of the heat in the resource at the excavation face during the construction period. Subsequently, a three-dimensional numerical model coupling with high rock temperature, hot water, and ventilation was established. Based on the site monitoring data comparison, it was found that the average errors of temperature and humidity field values in the numerical simulation were 4.7 % and 7.1 %, respectively. By introducing the Heat Index (HI), the tunnel environmental characteristics under different conditions were analyzed, showing a linear relationship between the maximum HI at the excavation face and the two following features which are: the temperature-humidity of the input wind, and the hot water quantity. At the same time the air volume exhibited a quadratic relationship. Furthermore, by utilizing multivariate functions to fit the research results, the results show that the maximum HI under various conditions can be predicted. In addition, ultimately establishing an environmental safety assessment model was effective in incorporating multiple factors like input wind temperature, humidity, air volume, and hot water quantity for construction tunnels.

Extreme eigenvalues of random matrices from Jacobi ensembles

Two-term asymptotic formulæ for the probability distribution functions for the smallest eigenvalue of the Jacobi β-Ensembles are derived for matrices of large size in the régime where β > 0 is arbitrary and one of the model parameters α1 is an integer. By a straightforward transformation this leads to corresponding results for the distribution of the largest eigenvalue. The explicit expressions are given in terms of multi-variable hypergeometric functions, and it is found that the first-order corrections are proportional to the derivative of the leading order limiting distribution function. In some special cases β = 2 and/or small values of α1, explicit formulæ involving more familiar functions, such as the modified Bessel function of the first kind, are presented.

Analytic continuations and numerical evaluation of the Appell F1, F3, Lauricella [formula omitted] and Lauricella-Saran [formula omitted] and their application to Feynman integrals

We present our investigation of the study of two variable hypergeometric series, namely Appell F1 and F3 series, and obtain a comprehensive list of its analytic continuations enough to cover the whole real (x,y) plane, except on their singular loci. We also derive analytic continuations of their 3-variable generalization, the Lauricella FD(3) series and the Lauricella-Saran FS(3) series, leveraging the analytic continuations of F1 and F3, which ensures that the whole real (x,y,z) space is covered, except on the singular loci of these functions. While these studies are motivated by the frequent occurrence of these multivariable hypergeometric functions in Feynman integral evaluation, they can also be used whenever they appear in other branches of mathematical physics. To facilitate their practical use, for analytical and numerical purposes, we provide four packages: AppellF1.wl, AppellF3.wl, LauricellaFD.wl, and LauricellaSaranFS.wl in Mathematica. These packages are applicable for generic as well as non-generic values of parameters, keeping in mind their utilities in the evaluation of the Feynman integrals. We explicitly present various physical applications of these packages in the context of Feynman integral evaluation and compare the results using other packages such as FIESTA. Upon applying the appropriate conventions for numerical evaluation, we find that the results obtained from our packages are consistent. Various Mathematica notebooks demonstrating different numerical results are also provided along with this paper.

Sex estimation from skull measurements of a contemporary Japanese population using three-dimensional computed tomography images.

In this study, we assessed the sexual dimorphism of the contemporary Japanese skull and established sex discriminant function equations based on cranial measurements using three-dimensional (3D) computed tomography (CT) images. The CT images of 263 corpses (142 males, 121 females) that underwent postmortem CT scanning and subsequent forensic autopsy were evaluated. Twenty-one cranial measurements were obtained from 3D CT reconstructed images, which extracted only bone data. We performed descriptive statistics and discriminant function analyses for the measurements. Nineteen measurements were significantly larger in males, suggesting sexual dimorphism of the Japanese skulls. Univariate discriminant function analyses using these measurements showed a sex classification accuracy of 57.8-88.2%, and bizygomatic breadth provided the highest correct prediction rate. Multivariate discriminant function analyses offered the most accurate model using seven variables with an estimation rate of 93.9%. Our results suggest that cranial measurements based on 3D CT images may help in the sex estimation of unidentified bodies in a contemporary Japanese population.

A generalized strain model for spectral rate-dependent constitutive equation of transversely isotropic electro-viscoelastic solids

We model the constitutive equation for nonlinear electro-viscoelastic transversely isotropic solids with short term memory via a generalized strain method, where the method is a change with respect to the methods that have been done in the last decades regarding mechanics of nonlinear solids. Our generalized strain model uses spectral invariants with a clear physical interpretation and hence they are attractive for use in experiments. The constitutive equation contains single-variable functions, which are easy to deal with when compared to multivariable functions. The effects of viscosity and electric fields are analyzed via the boundary value problem results. The efficacy the proposed prototype is scrutinized by comparing our theory with experimental data.

Analysis of a Class of Minimization Problems Lacking Lower Semicontinuity

The minimization of nonlower semicontinuous functions is a difficult topic that has been minimally studied. Among such functions is a Heaviside composite function that is the composition of a Heaviside function with a possibly nonsmooth multivariate function. Unifying a statistical estimation problem with hierarchical selection of variables and a sample average approximation of composite chance constrained stochastic programs, a Heaviside composite optimization problem is one whose objective and constraints are defined by sums of possibly nonlinear multiples of such composite functions. Via a pulled-out formulation, a pseudostationarity concept for a feasible point was introduced in an earlier work as a necessary condition for a local minimizer of a Heaviside composite optimization problem. The present paper extends this previous study in several directions: (a) showing that pseudostationarity is implied by (and thus, weaker than) a sharper subdifferential-based stationarity condition that we term epistationarity; (b) introducing a set-theoretic sufficient condition, which we term a local convexity-like property, under which an epistationary point of a possibly nonlower semicontinuous optimization problem is a local minimizer; (c) providing several classes of Heaviside composite functions satisfying this local convexity-like property; (d) extending the epigraphical formulation of a nonnegative multiple of a Heaviside composite function to a lifted formulation for arbitrarily signed multiples of the Heaviside composite function, based on which we show that an epistationary solution of the given Heaviside composite program with broad classes of B-differentiable component functions can in principle be approximately computed by surrogation methods. Funding: The work of Y. Cui was based on research supported by the National Science Foundation [Grants CCF-2153352, DMS-2309729, and CCF-2416172] and the National Institutes of Health [Grant 1R01CA287413-01]. The work of J.-S. Pang was based on research supported by the Air Force Office of Scientific Research [Grant FA9550-22-1-0045].

Research on Optimizing Human Resource Expenditure in the Allocation of Materials in Universities

This paper establishes a multivariate function model for natural human load-carrying walking in some typical scenarios such as college equipment and material relocation by students and a large amount of identical freight relocation in commercial activities. For classified material relocation needs and constraints, we obtain the relationship between walking speed and load weight for a single person, as well as the time cost for different round trips. By establishing an integer programming model with the minimum total transportation time cost and shelf life as the objective function and the requirements of negative weight and speed as the constraint conditions, we reach the optimal item allocation methods considering time cost and shelf life. We discover that there is an approximate linear relationship between the change in natural walking speed and travel time when the load is small, thus obtaining the time cost of student transportation under different round-trip situations. The Monte Carlo simulation algorithm, which is more efficient compared with other methods such as the integer programming method, is used to obtain the optimal allocation scheme that meets the efficiency and quality requirements. The analysis methods and results can be used as guidance for task scheduling optimization for material relocation in educational organizations as well as commercial agencies.

Detection of uniformity and spatial correlation of subgrade compaction quality based on intelligent compaction technology

Intelligent compaction uniformity and spatial correlation detection of subgrade is an important part of intelligent compaction technology. In this paper, a subgrade compaction uniformity detection method is established based on the univariate statistical normal distribution 3σ criterion, and further a subgrade compaction quality spatial correlation detection method is established by the multivariate statistical spatial semi-variance function, and finally the established subgrade compaction uniformity and spatial correlation methods are verified by field tests. The subgrade compaction uniformity and spatial correlation are detected based on field tests. The results obtained in this paper show that the 3σ criterion based on normal distribution can be used for the detection of intelligent compaction uniformity of subgrade, and the field test results are verified by this method, and all the intelligent compaction uniformity of subgrade meets the requirements; considering the construction efficiency and the influence between rollers, it is recommended that the rolling speed should be in the interval range of 2–3 m/s, and the construction spacing of rollers is maintained at 10 m and above.

Neural encoding of linguistic speech cues is unaffected by cognitive decline, but decreases with increasing hearing impairment

The multivariate temporal response function (mTRF) is an effective tool for investigating the neural encoding of acoustic and complex linguistic features in natural continuous speech. In this study, we investigated how neural representations of speech features derived from natural stimuli are related to early signs of cognitive decline in older adults, taking into account the effects of hearing. Participants without (n=25\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$n = 25$$\\end{document}) and with (n=19\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$n = 19$$\\end{document}) early signs of cognitive decline listened to an audiobook while their electroencephalography responses were recorded. Using the mTRF framework, we modeled the relationship between speech input and neural response via different acoustic, segmented and linguistic encoding models and examined the response functions in terms of encoding accuracy, signal power, peak amplitudes and latencies. Our results showed no significant effect of cognitive decline or hearing ability on the neural encoding of acoustic and linguistic speech features. However, we found a significant interaction between hearing ability and the word-level segmentation model, suggesting that hearing impairment specifically affects encoding accuracy for this model, while other features were not affected by hearing ability. These results suggest that while speech processing markers remain unaffected by cognitive decline and hearing loss per se, neural encoding of word-level segmented speech features in older adults is affected by hearing loss but not by cognitive decline. This study emphasises the effectiveness of mTRF analysis in studying the neural encoding of speech and argues for an extension of research to investigate its clinical impact on hearing loss and cognition.

High-order total differential solution of multivariate function based on Yang Hui triangle

High-order total differential solution of multivariate function based on Yang Hui triangle

Self-Organizing Optimization Based on Caputo’s Fractional Order Gradients

This paper analyses the condition necessary to guarantee no divergence for Caputo’s fractional order gradient descent (C-FOG) algorithm on multivariate functions. C-FOG is self-organizing, computationally efficient, simple, and understandable. It converges faster than the classical gradient-based optimization algorithms and converges to slightly different points when the order of the fractional derivative is different. The additional freedom of the order is very useful in situations where the diversity of convergence is required, and it also allows for more precise convergence. Comparative experiments on a typical poor conditioned function and adversarial sample generation frameworks demonstrate the convergence performance of C-FOG, showing that it outperforms currently popular algorithms in terms of convergence speed, and more excitingly, the diversity of convergence allows it to exhibit stronger and more stable attack capability in adversarial sample generation procedures (The code for experiments is available at: https://github.com/mulertan/self_optimizing/tree/main, accessed on 30 April 2024).

Verification of Taylor's theorem

Multivariate function calculus is an important part of mathematical analysis courses, and most conclusions can be found and generalized in univariate calculus. However, the biggest difficulty in teaching multivariate calculus lies in its abstraction, such as Taylors theorem, multiple integral regions drawing, and integral variable transformation. At the same time, ordinary differential equations are also one of the basic courses of the profession, and dynamic systems based on ordinary differential equations have extensive applications in mathematical models of continuity problems and optimal control problems. Software such as Mathematica, Python, Matlab, etc. can solve similar problems. Therefore, this article will use the visualization and computational capabilities of Mathematica to validate important definitions and conclusions in multivariate calculus, and compare the differences among the three software in solving approximate numerical solutions of dynamic systems of ordinary differential equations from different perspectives.

Utility of the metatarsal diaphyseal nutrient foramen in estimating sex in the South African Africans population

IntroductionSex estimation is challenging in cases where dismembered or non-intact skeletal remains are recovered. Therefore, the development of sex estimation standards using various bones that present with high recovery rates during forensic investigations, like the metatarsal bones, is needed. The usefulness of the dimensions around the metatarsal diaphyseal nutrient foramen in sex estimation has not been assessed in South African Africans (SAA), constituting the majority of the country's population. Materials and methodsFive measurements around the nutrient foramen were taken from 995 metatarsal bones (first to fifth) from 200 individual skeletons (100 males, 100 females). Measurements subjected to direct and stepwise discriminant function (DFA) and logistic regression (LRA) analyses included the total length, distance from proximal end to nutrient foramen, circumference, and mediolateral and dorsoplantar diameters at the level of the nutrient foramen. ResultsThe original classification accuracies for multivariable functions of the stepwise and direct DFA ranged from 75.1 to 80 % and 76–79.5 % respectively. The original classification accuracies for multivariable functions of the stepwise and direct LRA ranged from 76.3% to 79.5 % and 75%–80.5 % respectively. The cross-validation classifications showed a drop of 0–2% for DFA and 0.2–1.9 % for LRA. Overall breadth measurements showed better classification accuracies than length measurements and females were classified with higher accuracy rates than males. ConclusionThe dimensions around the nutrient foramen of the metatarsal bones show sexual dimorphism in the SAA. The generated DFA and LRA functions produced high average classification accuracies which can be appropriate for use in sex estimation in forensic settings, especially when an isolated foot is recovered.

Collaborative Optimization Algorithm Based on Chameleon Swarm Algorithm Designed for the Problem of Beer Production Scheduling

Aiming at finding a better way to solve the problem of beer production scheduling, a new collaborative optimization based on the Manhattan Distance and Chameleon Swarm Algorithm is proposed. Firstly, a dynamic relaxation factor is introduced to the constraints at the system level, which combines the changing trend of the inconsistency information and the optimal solution of the discipline level. Additionally, the Manhattan Distance is used to replace the square of the Euclidean Distance at the system level. Thirdly, the Chameleon Swarm Algorithm is used to improve the update rule during the process of iteration. As these improvements are applied to the collaborative optimization, the steps of this new algorithm are given. Through the test case of a multivariate function, it can be found that the algorithm has been improved compared to the original algorithm. Then, a model for beer production scheduling is proposed, and the results of the optimization show that the improved collaborative optimization has better optima effectiveness and fewer iterations and is not sensitive to initial points, which proves that the improved collaborative optimization has a better ability to solve the problem of beer production scheduling than normal collaborative optimization and collaborative optimization with fixed relaxation factors.

Algebraic formulas and geometric derivation of source identities

Source identities are fundamental identities between multivariable special functions. We give a geometric derivation of rational and trigonometric source identities. We also give a systematic derivation and extension of various determinant representations for source functions which appeared in previous literature as well as introducing the elliptic version of the determinants, and obtain identities between determinants. We also show several symmetrization formulas for the rational version.

Short-term load distribution model for cascade giant hydropower stations with complex hydraulic and electrical connections

The cascade giant hydropower station system (CGHSS) has been widely applied in countries with abundant water resources. However, the dense hydropower stations and power grids bring complex hydraulic and power connections, including the backwater effect, multiple hydroplants sharing one reservoir and one-reservoir multiple dispatch power grid, which leads to modelling difficulties and inefficient model solving. To address these challenges, a short-term load distribution (SLD) model considering complicated hydraulic and electricity constraints is developed. First, the discharge of each hydroplant and the downstream forebay water level are used as inputs to construct a multivariate nonlinear function, and the average tailwater levels are obtained. Second, a head constraint considering the shared water level is added to the model to calculate the head of each hydroplant, making the output more accurate. Finally, the nonlinear constraints are linearly reconstructed using the Big-M method, triangulation technique and special ordered sets to reduce the model solution time. The SLD model is applied to the Xiluodu-Xiangjiaba cascade hydropower system in the Jinsha River. The results show that the developed model reduces water consumption by 4.02 %, 4.70 %, 5.79 % and 5.28 % compared to the constant proportional distribution model and the real operation, generating a more executable SLD scheme.

On the approximation to fractional calculus operators with multivariate Mittag-Leffler function in the kernel

Several numerical techniques have been developed to approximate Riemann–Liouville (R - L) and Caputo fractional calculus operators. Recently linear positive operators have been started to use to approximate fractional calculus operators such as R - L, Caputo, Prabhakar and operators containing bivariate Mittag-Leffler functions. In the present paper, we first define and investigate the fractional calculus properties of Caputo derivative operator containing the multivariate Mittag-Leffler function in the kernel. Then we introduce approximating operators by using the modified Kantorovich operators for the approximation to fractional integral and Caputo derivative operators with multivariate Mittag-Leffler function in the kernel. We study the convergence properties of the operators and compute the degree of approximation by means of modulus of continuity and Hölder continuous functions. The obtained results corresponds to a large family of fractional calculus operators including R- L, Caputo and Prabhakar models.