Univariate Functions Research Articles

Stock structure analysis of Trichopodus trichopterus (Pallas, 1770) from Indonesia waters using truss morphometry

Most species identification inTrichopodus trichopterusis primarily based on phenotypic variation of the body, including color polymorphisms, specific markers, and size differences. These key attributes can sometimes be difficult to apply due to ontogeny development, environment interaction, and specimen quality. Therefore, this study aimed to determine intraspecific variation ofT. trichopterusbody conformation using truss morphometric analysis. The fish specimens used were collected from the museum in Sumatra (n = 31), Java (n = 30), Kalimantan (n = 30), and Sulawesi (n = 35). Truss measurement data were subjected to univariate and discriminant function analysis (DFA). The results showed that 5 measured characters were significant discriminators among the observed populations (P < 0.05). The initial classification accuracy was 54.2%, marginally decreasing to 46.6% after cross-validation. Based on geographic distributions, Kalimantan specimens had the highest classification at 96.7%, followed by Sulawesi, Java, and Sumatra at 94.3%, 94.1%, and 91.2%, respectively. Further investigation revealed that habitat differences had a significant impact on morphometric characteristics. In conclusion, differences were observed in the morphological characters of the T. trichopterus populations in Indonesian water.

Analogical proportions in monounary algebras

AbstractThis paper studies analogical proportions in monounary algebras consisting only of a universe and a single unary function, where we analyze the role of congruences, and we show that the analogical proportion relation is characterized in the infinite monounary algebra formed by the natural numbers together with the successor function via difference proportions.

Multivariate compactly supported C∞ functions by subdivision

This paper discusses the generation of multivariate C∞ functions with compact small supports by subdivision schemes. Following the construction of such a univariate function, called Up-function, by a non-stationary scheme based on masks of spline subdivision schemes of growing degrees, we term the multivariate functions we generate Up-like functions. We generate them by non-stationary schemes based on masks of three-directional box-splines of growing supports. To analyze the convergence and smoothness of these non-stationary schemes, we develop new tools which apply to a wider class of schemes than the class we study. With our method for achieving small compact supports, we obtain in the univariate case, Up-like functions with supports [0,1+ϵ] in comparison to the support [0,2] of the Up-function. Examples of univariate and bivariate Up-like functions are given. As in the univariate case, the construction of Up-like functions can motivate the generation of C∞ compactly supported wavelets of small support in any dimension.

Efficient continuous piecewise linear regression for linearising univariate non-linear functions

Due to their flexibility and ability to incorporate non-linear relationships, Mixed-Integer Non-Linear Programming (MINLP) approaches for optimization are commonly presented as a solution tool for real-world problems. Within this context, piecewise linear (PWL) approximations of non-linear continuous functions are useful, as opposed to non-linear machine learning-based approaches, since they enable the application of Mixed-Integer Linear Programming techniques in the MINLP framework, as well as retaining important features of the approximated non-linear functions, such as convexity. In this work, we extend upon fast algorithmic approaches for modeling discrete data using PWL regression by tuning them to allow the modeling of continuous functions. We show that if the input function is convex, then the convexity of the resulting PWL function is guaranteed. An analysis of the runtime of the presented algorithm shows which function characteristics affect the efficiency of the model, and which classes of functions can be modeled very quickly. Experimental results show that the presented approach is significantly faster than five existing approaches for modeling non-linear functions from the literature, at least 11 times faster on the tested functions, and up to a maximum speedup of more than 328,000. The presented approach also solves six benchmark problems for the first time.

Multivariate polynomial regression by an explainable sigma-pi neural network

&lt;p&gt;Over the years, data-driven regression on univariate functions has been extensively studied. However, fast, effective, and stable algorithms for multivariate function fitting are still lacking. Recently, Kolmogorov-Arnold networks have garnered significant attention among scholars due to their superior accuracy and interpretability compared to multi-layer perceptrons. In this paper, we have demonstrated that the sigma-pi neural network, a form of Kolmogorov-Arnold networks, can efficiently fit multivariate polynomial functions, including fractional-order multivariate polynomials. Three examples were employed to illustrate the regression performance of the designed neural networks. The explainable sigma-pi neural network will lay the groundwork for further development of general tools for multivariate nonlinear function regression problems.&lt;/p&gt;

Dynamic inequalities of Grüss, Ostrowski and Trapezoid type via diamond-$ \alpha $ integrals and Montgomery identity

&lt;abstract&gt;&lt;p&gt;In this article, the Montgomery identity and Ostrowski inequality are established for univariate first-order diamond-alpha differentiable functions. We also investigate the generalization of Ostrowski-type inequalities for bivariate functions with bounded second-order diamond-alpha derivatives by applying integration by parts for $ \diamondsuit_\alpha $-integrals. Moreover, some extensions of dynamic trapezoid- and Grüss-type inequalities are also obtained by using the Montgomery identity.&lt;/p&gt;&lt;/abstract&gt;

FRACTAL DIMENSIONS FOR THE MIXED (κ,s)-RIEMANN–LIOUVILLE FRACTIONAL INTEGRAL OF BIVARIATE FUNCTIONS

The research object of this paper is the mixed [Formula: see text]-Riemann–Liouville fractional integral of bivariate functions on rectangular regions, which is a natural generalization of the fractional integral of univariate functions. This paper first indicates that the mixed integral still maintains the validity of the classical properties, such as boundedness, continuity and bounded variation. Furthermore, we investigate fractal dimensions of bivariate functions under the mixed integral, including the Hausdorff dimension and the Box dimension. The main results indicate that fractal dimensions of the graph of the mixed [Formula: see text]-Riemann–Liouville integral of continuous functions with bounded variation are still two. The Box dimension of the mixed integral of two-dimensional continuous functions has also been calculated. Besides, we prove that the upper bound of the Box dimension of bivariate continuous functions under [Formula: see text] order of the mixed integral is [Formula: see text] where [Formula: see text].

Finiteness of One-Valued Function Classes in Many-Valued Logic

This paper addresses the theoretical issues in k-valued logic, which are crucial for developing solutions in various fields of science and technology. One of the fundamental issues is a complete description of the closed classes of functions of three-valued logic. The explicit description of closed classes in multivalued logic is an open problem. In this study, we consider a special case of the finite generation of all closed classes of three-valued logic through the operation of superposition. Previously, we considered the issue of the finite generation of classes containing a subset of single-variable functions. We have also provided a description of superlattices (lattices of lattices) containing a precomplete class of unary functions. The finite generation of these superlattices is proved. On the basis of these results, in this paper, we have proven that any class containing any of the precomplete classes from the set of single-valued functions is also finitely generated. The main result of this paper consists of three theorems on the finite generation of classes containing precomplete classes of single-valued functions and classes including all monotone unary functions. Thus, the obtained theoretical result provides easily verifiable criteria for the finiteness of classes of multivalued logic functions. It allows you to use simple procedures instead of cumbersome explicit constructs. The finite generation of overlattices allows the development of digital computing circuits that are crucial for practical applications. The proofs are based on an explicit description of these classes by an induction in the number of variables and essentially use the properties of functionally closed (Burle) classes of functions.

Evaluating the patella bone for sex estimation in Northwest Indian subjects: A radiological study

ObjectivesIn forensic identification, pelvis, post cranial bones especially long bones and skull are considered best for sex estimation. But there are circumstances where these bones cannot be recovered for establishing the correct biological profile of unknown human skeletal remains. In those cases, other skeletal bones should be investigated for sex estimation. The goal of the present study was to validate whether the patellar bone measurements through CT scan can be used for the sex estimation in Northwest Indian subjects. Materials and methodsThis study comprises of 344 subjects (179 males and 165 females) between the age group of 18- 80 years. Eight variables of patella were selected in the study for sex estimation using CT scan images of patella. ResultsAll the variables were statistically significant with p-value< 0.05 for indicating the sex. Univariate and multivariate discriminant function analysis (direct and stepwise method) were applied for sex estimation. The overall accuracy rate of classifying sex was 88.4% for multivariate stepwise discriminant function analysis and 87.8% for multivariate direct discriminant function analysis. Total perimeter of patella (TPP) was found to be the best variable for sex estimation with overall accuracy rate of 84.9%. ConclusionThe present study emphasized on the most dimorphic variables of patella in Northwest Indian subjects. New variables of patella namely Patellar Femoral Distance (PFD), Patellar Apex Angle (PAA), Frontal Surface Area of patella (FSA) and Total Perimeter of Patella (TPP) were added in the study of Indian subjects. This brought an overall accuracy rate of classification above 85%. The study confirmed that patellar bone's linear and angular measurements can be used for forensic identification casework.

Recruitment dynamics in a tropical karst seasonal rain forest: Revealing complex processes from spatial patterns

Spatial patterns can reveal many ecological processes in forest communities; thus, understanding the spatial patterns of trees and their interactions is key to exploring forest dynamics. However, few studies have explored the spatial patterns and interactions between adults and recruits over longer time spans in tropical karst forests. A forest dynamics census was conducted (2011–2021) in a 15-ha karst seasonal rainforest plot at the Nonggang National Nature Reserve, in Guangxi, China. The tree community structure, which included a total of 20 tree species (≥ 50 individuals) at various states of recruitment, survival, and mortality, were selected and investigated via detailed analyses. First, univariate pair correlation functions and bivariate mark correlation functions were utilized to analyze the spatial patterns of dead recruits, surviving recruits, and adults. Aggregation was the dominant intraspecific distribution pattern in the plot that decreased at larger scales, which indicated that dispersal was limited. The segregation of bivariate patterns in most species suggested that there was scramble competition between the surviving and dead recruits, which translated to their not clustering around adults. Second, it was revealed through testing that density dependence was not prevalent. However, negative density dependence was more common in species that exhibited density dependence, as various species respond differently to changes in population density. Third, most recruits had a low probability of survival, which for several dominant species was influenced by neighborhood effects. This study did not support the Janzen-Connell hypothesis, as the survival probability of most recruits was not enhanced with the distance from adults. This meant that the effects of distance from adults on the growth of recruits in Nonggang forests were negligible. These results suggested that dispersal limitations and habitat heterogeneity played more significant roles in the regulation of tree spatial patterns and seedling growth (or mortality) in tropical karst seasonal rain forests. This work will contribute to the restoration and conservation of fragile forest communities in Karst regions, while providing a theoretical basis for biodiversity research, forest management planning, and ecosystem services.

Approximation of curve-based sleeve functions in high dimensions

Sleeve functions are generalizations of the well-established ridge functions that play a major role in the theory of partial differential equation, medical imaging, statistics, and neural networks. Where ridge functions are non-linear, univariate functions of the distance to hyperplanes, sleeve functions are based on the squared distance to lower-dimensional manifolds. The present work is a first step to study general sleeve functions by starting with sleeve functions based on finite-length curves. To capture these curve-based sleeve functions, we propose and study a two-step method, where first the outer univariate function—the profile—is recovered, and second, the underlying curve is represented by a polygonal chain. Introducing a concept of well-separation, we ensure that the proposed method always terminates and approximates the true sleeve function with a certain quality. Investigating the local geometry, we study an inexact version of our method and show its success under certain conditions.

Asymptotic normality for the wavelet partially linear additive model components estimation

The focus of this article is on studying a partially linear additive model, which is defined using a measurable function ψ : R q → R . The model is given as follows: ψ ( Y i ) := Y i = Z i ⊤ β + ∑ ℓ = 1 d m ℓ ( X ℓ , i ) + ε i for 1 ≤ i ≤ n , where Z i = ( Z i , 1 , … , Z i p ) ⊤ and X i = ( X 1 , i , … , X i d ) ⊤ are vectors of explanatory variables, β = ( β 1 , … , β p ) ⊤ is a vector of unknown parameters, m 1 , … , m d are unknown univariate real functions, and ε 1 , … , ε n are independent random errors with mean zero, finite variances σ ε . Additionally, it is assumed that E ( ε | X , Z ) = 0 almost surely. The main contributions of this article are as follows. First, under certain mild conditions, we establish the asymptotic normality of the non linear additive components of the model. These components are estimated using the marginal integration device with the linear wavelet method. Second, we leverage our main result to construct confidence intervals for the estimated model.

Association of environmental polycyclic aromatic hydrocarbons exposure with periodontitis in NHANES 2009-2014: A mixtures approach.

Polycyclic aromatic hydrocarbons (PAHs) can invade and cause harm to the human body through various pathways, but there is currently little research on the relationship between mixed-PAHs exposure and periodontitis. The purpose of this study was to examine the effects of mixed-urinary PAHs exposure on periodontitis in adults in the United States. The cross-sectional study included 2749 subjects selected from the National Health and Nutrition Examination Survey (NHANES) 2009-2014 cycles. A professional examination of the periodontal status was conducted to distinguish between periodontitis and non-periodontitis based on the Centers for Disease Control and Prevention/American Academy of Periodontology (CDC/AAP) case definition. Laboratory testing of urine samples was performed to obtain the levels of urinary PAHs. Pearson correlation coefficients were utilized to determine the degree of correlation between urinary PAHs, while weighted binary logistic regression and Bayesian kernel machine regression (BKMR) were employed to evaluate the relationship between urinary PAHs and periodontitis. In a single-exposure model, 3-hydroxyfluorene (OH-3F), 2-hydroxyfluorene (OH-2F), 1-hydroxyphenanthrene (OH-1Ph), and 2-hydroxyphenanthrene and 3-hydroxyphenanthrene (OH-2,3Ph) were positively associated with periodontitis risk. In the mixed-exposure model, BKMR analysis demonstrated that mixed exposure to urinary PAHs was positively associated with periodontitis, with OH-2F being the most critical factor for the overall mixed effects (posterior inclusion probability [PIP]=0.98). Univariate exposure-response function and univariate effects analysis revealed a positive correlation between urinary OH-2F levels and periodontitis. The study reveals a significant positive correlation between exposure to mixed PAHs and periodontitis, with a particular emphasis on the pivotal role of OH-2F. Mitigating PAHs in the environment may serve as a preventive measure against periodontitis and alleviate its global public health burden.

Approximation Properties of Extended Beta-Type Szász–Mirakjan Operators

In this article, we introduce generalized beta extension of Szacute{a}sz-integral type operators and study their approximation properties. First, we calculate the some estimates for these operators. Further, we study the uniform convergence and order of approximation in terms of Korovkin-type theorem and modulus of continuity for the space of univariate continuous functions and bivariate continuous functions in their sections.. Moreover, numerical estimates and graphical representations for convergence of one- and two-dimensional sequences of operators are studied. In continuation, local and global approximation properties are studied in terms of the first- and second-order modulus of smoothness, Peetre’s K-functional and weight functions in various functional spaces.

Comparing perspective reformulations for piecewise-convex optimization

We aim at generalizing formulations for non-convex piecewise-linear problems to mathematical programs whose non-convexities are only expressed in terms of piecewise-convex univariate functions. This is motivated by solving Mixed-Integer Non-Linear Programming (MINLP) problems with separable non-convex functions via the Sequential Convex MINLP technique. We theoretically and computationally compare different formulations, showing that, unlike in the linear case, they are not equivalent when perspective reformulation is applied to strengthen the formulation of each single segment.

Building a digital twin of an EDFA for optical networks: a gray-box modeling approach

High-accuracy physical layer models enable intelligent, self-driving optical networks. The dynamic wavelength-dependent gain characteristics of erbium-doped fiber amplifiers (EDFAs) remain a crucial problem in terms of modeling. The gain model directly determines the power spectrum and is therefore important for estimating the optical signal-to-noise ratio as well as the magnitude of fiber nonlinearities. Black-box data-driven models have been widely studied, but they require a large size of data for training and suffer from poor generalizability. In this paper, we derive the gain spectra of EDFAs as a simple univariable linear function; then, based on it, we propose a gray-box EDFA gain modeling scheme. Experimental results show that, for automatic gain control (AGC) and automatic power control (APC) EDFAs, our model built with 8 data samples can achieve better performance than the neural network (NN) based model built with 900 data samples, which means the required data size for modeling can be reduced by at least 2 orders of magnitude. Moreover, in the experiment, the proposed model demonstrates superior generalizability to unseen scenarios since it is based on the underlying physics of EDFAs. With the proposed scheme, building a customized digital twin of each EDFA in optical networks becomes more feasible, which is essential, especially for next-generation multiband network operations.

EXAMPLES OF LINEAR ORDERS WITH A DEFINABLE UNARY FUNCTION AND THE INDEPENDENCE PROPERTY

After the appearance of the concept of o-minimality, which was introduced by L. van den Dries for expansions of the ordered field of real numbers and generalized to arbitrary linear orders by A. Pillay and C. Steinhorn, linearly ordered structures became firmly established in the circle of interests of specialists in model theory. Numerous generalizations of the concept of o-minimality have appeared in the works of various authors, such as weak o-minimality, quasi-o- minimality, weak quasi-o-minimality, dp-minimality, and o-stability. B. S. Baizhanov and V. V. Verbovskiy proved that o-stability generalizes all the above concepts for linearly ordered structures and that o-stability entails the absence of the independence property. They also proved that any linear order has an o-superstable theory. V. V. Verbovskiy studied o-stable ordered groups, in particular, he proved that they are commutative. In this paper, we begin the study of the question of how complex the theory of a linear order with one unary function can be. We construct an example of an expansion of a linearly ordered structure with one unary function, which has the independence property.

Intelligent Connectivity Through RIS-Assisted Wireless Communication: Exact Performance Analysis With Phase Errors and Mobility

The current research offers only approximations for evaluating the performance of wireless systems that use reconfigurable intelligent surfaces (RIS) over generalized fading channels with phase error caused by imperfect phase compensation at the RIS. This paper presents an exact analysis of RIS-assisted vehicular communication considering uniformly distributed phase error and generalized fading channels with a coherent combining of received signals reflected by RIS elements and direct transmissions from the source terminal. We use a generalized- <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$K$</tex-math></inline-formula> shadowed distribution for the direct link and asymmetrical channels for the RIS-assisted transmission with <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\kappa$</tex-math></inline-formula> - <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\mu$</tex-math></inline-formula> distribution for the first link and double generalized Gamma (dGG) distribution for the second link, combined with a statistical random waypoint (RWP) model for the moving vehicle. We employ a novel approach to represent the probability density function (PDF) and cumulative distribution function (CDF) of the resultant channel in terms of a single univariate Fox-H function and use the multivariate Fox-H approach to develop an exact statistical analysis of the end-to-end signal-to-noise ratio (SNR) for the RIS-assisted system. We also use the inequality between the arithmetic and geometric means to simplify the statistical results of the considered system in terms of the univariate Fox-H function. Our analysis also provides exact, upper bound, and asymptotic expressions of the outage probability and average bit-error-rate (BER) performance using the derived density and distribution functions. We conduct computer simulations in various practically relevant scenarios to assert that mitigating phase errors is achievable by augmenting the number of elements in the RIS module and employing a higher quantization level.

Separable Gaussian Neural Networks: Structure, Analysis, and Function Approximations

The Gaussian-radial-basis function neural network (GRBFNN) has been a popular choice for interpolation and classification. However, it is computationally intensive when the dimension of the input vector is high. To address this issue, we propose a new feedforward network-separable Gaussian neural network (SGNN) by taking advantage of the separable property of Gaussian-radial-basis functions, which splits input data into multiple columns and sequentially feeds them into parallel layers formed by uni-variate Gaussian functions. This structure reduces the number of neurons from O(Nd) of GRBFNN to O(dN), which exponentially improves the computational speed of SGNN and makes it scale linearly as the input dimension increases. In addition, SGNN can preserve the dominant subspace of the Hessian matrix of GRBFNN in gradient descent training, leading to a similar level of accuracy to GRBFNN. It is experimentally demonstrated that SGNN can achieve an acceleration of 100 times with a similar level of accuracy over GRBFNN on tri-variate function approximations. The SGNN also has better trainability and is more tuning-friendly than DNNs with RuLU and Sigmoid functions. For approximating functions with a complex geometry, SGNN can lead to results that are three orders of magnitude more accurate than those of a RuLU-DNN with twice the number of layers and the number of neurons per layer.

Higher order Bernstein–Bézier and Nédélec finite elements for the relaxed micromorphic model

The relaxed micromorphic model is a generalized continuum model that is well-posed in the space X=[H1]3×[H(curl)]3. Consequently, finite element formulations of the model rely on H1-conforming subspaces and Nédélec elements for discrete solutions of the corresponding variational problem. This work applies the recently introduced polytopal template methodology for the construction of Nédélec elements. This is done in conjunction with Bernstein–Bézier polynomials and dual numbers in order to compute hp-FEM solutions of the model. Bernstein–Bézier polynomials allow for optimal complexity in the assembly procedure due to their natural factorization into univariate Bernstein base functions. In this work, this characteristic is further augmented by the use of dual numbers in order to compute their values and their derivatives simultaneously. The application of the polytopal template methodology for the construction of the Nédélec base functions allows them to directly inherit the optimal complexity of the underlying Bernstein–Bézier basis. We introduce the Bernstein–Bézier basis along with its factorization to univariate Bernstein base functions, the principle of automatic differentiation via dual numbers and a detailed construction of Nédélec elements based on Bernstein–Bézier polynomials with the polytopal template methodology. This is complemented with a corresponding technique to embed Dirichlet boundary conditions, with emphasis on the consistent coupling condition. The performance of the elements is shown in examples of the relaxed micromorphic model.