Univariate Functions Research Articles

On a Generalization of the Young Inequality Involving Intermediary Functions and Its Applications

In this article, we present new results on specific cases of a general Young integral inequality established by Páles in 1990. Our initial focus is on a bivariate function, defined as the product of two univariate and separable functions. Based on this, some new results are established, including particular Young integral-type inequalities and some upper bounds on the corresponding absolute errors. The precise role of the functions involved in this context is investigated. Several applications are presented, including one in the field of probability theory. We also introduce and study reverse variants of our inequalities. Another important contribution is to link the setting of the general Young integral inequality established by Páles to a probabilistic framework called copula theory. We show that this theory provides a wide range of functions, often dependent on adjustable parameters, that can be effectively applied to this inequality. Some illustrative graphics are provided. Overall, this article broadens the scope of bivariate inequalities and can serve related purposes in analysis, probability and statistics, among others.

NONPARAMETRIC IDENTIFICATION AND ESTIMATION OF A GENERALIZED ADDITIVE MODEL WITH A FLEXIBLE ADDITIVE STRUCTURE AND UNKNOWN LINK

This paper proposes a nonparametric approach to identify and estimate the generalized additive model with a flexible additive structure and with possibly discrete variables when the link function is unknown. Our approach allows for a flexible additive structure which provides applied researchers the flexibility to specify their model according to economic theory or practical experience. Motivated by the concerns from empirical research, our method also allows for multiple discrete variables in the covariates. By transforming our model into a generalized additive model with univariate component functions, our identification and estimation thereby follows a procedure adapted from the case with univariate components. The estimators converge to normal distributions in large sample with a one-dimensional convergence rate for the link function and a $d_k$ -dimensional convergence rate for the component function $f_k(\cdot )$ defined on ${\mathbb R}^{d_k}$ for all k.

A gradient-enhanced univariate dimension reduction method for uncertainty propagation

The univariate dimension reduction (UDR) method stands as a way to estimate the statistical moments of the output that is effective in a large class of uncertainty quantification (UQ) problems. UDR's fundamental strategy is to approximate the original function using univariate functions so that the UQ cost only scales linearly with the dimension of the problem. Nonetheless, UDR's effectiveness can diminish when uncertain inputs have high variance, particularly when assessing the output's second and higher-order statistical moments. This paper proposes a new method, gradient-enhanced univariate dimension reduction (GUDR), that enhances the accuracy of UDR by incorporating univariate gradient function terms into the UDR approximation function. Theoretical results indicate that the GUDR approximation is expected to be one order more accurate than UDR in approximating the original function, and it is expected to generate more accurate results in computing the output's second and higher-order statistical moments. Our proposed method uses a computational graph transformation strategy to efficiently evaluate the GUDR approximation function on tensor-grid quadrature inputs, and use the tensor-grid input-output data to compute the statistical moments of the output. With an efficient automatic differentiation method to compute the gradients, our method preserves UDR's linear scaling of computation time with problem dimension. Numerical results show that the GUDR is more accurate than UDR in estimating the standard deviation of the output and has a performance comparable to the method of moments using a third-order Taylor series expansion.

Multivariate Moment Least-Squares Variance Estimators for Reversible Markov Chains

Markov chain Monte Carlo (MCMC) is a commonly used method for approximating expectations with respect to probability distributions. Uncertainty assessment for MCMC estimators is essential in practical applications. Moreover, for multivariate functions of a Markov chain, it is important to estimate not only the auto-correlation for each component but also to estimate cross-correlations, in order to better assess sample quality, improve estimates of effective sample size, and use more effective stopping rules. Berg and Song introduced the moment least squares (momentLS) estimator, a shape-constrained estimator for the autocovariance sequence from a reversible Markov chain, for univariate functions of the Markov chain. Based on this sequence estimator, they proposed an estimator of the asymptotic variance of the sample mean from MCMC samples. In this study, we propose novel autocovariance sequence and asymptotic variance estimators for Markov chain functions with multiple components, based on the univariate momentLS estimators from Berg and Song. We demonstrate strong consistency of the proposed auto(cross)-covariance sequence and asymptotic variance matrix estimators. We conduct empirical comparisons of our method with other state-of-the-art approaches on simulated and real-data examples, using popular samplers including the random-walk Metropolis sampler and the No-U-Turn sampler from STAN. Supplemental materials for this article are available online.

Multi-Intelligent Reflecting Surfaces and Artificial Noise-Assisted Cell-Free Massive MIMO Against Simultaneous Jamming and Eavesdropping.

In a cell-free massive multiple-input multiple-output (MIMO) system without cells, it is assumed that there are smart jammers and disrupters (SJDs) that attempt to interfere with and eavesdrop on the downlink communications of legitimate users. A secure transmission scheme based on multiple intelligent reflecting surfaces (IRSs) and artificial noise (AN) is proposed. First, an access point (AP) selection strategy based on user location information is designed, which aims to determine the set of APs serving users. Then, a joint optimization framework based on the block coordinate descent (BCD) algorithm is constructed, and a non-convex optimization solution based on the univariate function optimization and semi-definite relaxation (SDR) is proposed with the aim of maximising the minimum achievable secrecy rate for users. By solving the univariate function maximisation problem, the multi-variable coupled non-convex problem is transformed into a solvable convex problem, obtaining the optimal AP beamforming, AN matrix and IRS phase shift matrix. Specifically, in a single-user scenario, the scheme of multiple intelligent reflecting surfaces combined with artificial noise can improve the user's achievable secrecy rate by about 11% compared to the existing method (single intelligent reflective surface combined with artificial noise) and about 2% compared to the scheme assisted by multiple intelligent reflecting surfaces without artificial noise assistance.

Associations between urinary phytoestrogen mixed metabolites and osteoarthritis risk.

This study aims to explore the relationship between urinary phytoestrogen mixed metabolites and the risk of osteoarthritis (OA). Using data from the National Health and Nutrition Examination Survey (NHANES), a Weighted Quantile Sum (WQS) regression analysis was conducted to determine the dominant metabolites. Additionally, a Bayesian kernel machine regression (BKMR) model was utilized to explore the combined effects of phytoestrogen mixed metabolites on OA. Compared to the lowest quartile group, the highest quartile group of Enterodiol showed a 46% increased risk of OA (OR = 1.46, 95% CI: 1.09-1.96), while the highest quartile group of Enterlactone showed a 30% decreased risk of OA (OR = 0.70, 95% CI: 0.52-0.96). The WQS regression model analysis revealed a positive relationship between urinary phytoestrogen mixed metabolites and OA risk, with Enterodiol found to have the highest weight in this association. The BKMR model indicated that the association between urinary phytoestrogens and OA increased with concentration but did not reach statistical significance. The univariate exposure-response function demonstrated a positive association between Enterodiol and OA. There is a positive relationship between urinary phytoestrogen mixed metabolites and OA, with Enterodiol being an important factor influencing OA risk.

EPCO-03. METABOLIC INSIGHT INTO GLIOMA HETEROGENEITY: MAPPING WHOLE EXOME SEQUENCING TOIN VIVO IMAGING WITH STEREOTACTIC LOCALIZATION AND DEEP LEARNING

Abstract Intra-tumoral heterogeneity (ITH) complicates the diagnosis and treatment of glioma, partly due to the diverse metabolic profiles driven by genomic alterations. While multiparametric imaging captures spatial and functional variations enabling ITH characterization, it falls short in assessing the metabolic activities underpinning phenotypic differences. This gap stems from the challenge of integrating easily accessible, co-located pathology and genomic data with metabolic insights. This study presents a multi-faceted approach combining stereotactic biopsy with clinical open-craniotomy for sample collection, voxel-wise analysis of MR images, regression-based GAM, and whole-exome sequencing. The aim is to demonstrate the potential of machine learning algorithms to predict variations in cellular and molecular tumor characteristics. This retrospective study enrolled ten treatment-naïve patients with radiologically confirmed glioma who underwent multiparametric MR scans (T1W, T1W-CE, T2W, T2W-FLAIR, DWI) prior to surgery. During craniotomy, at least one stereotactic biopsy was collected from each patient, with the screenshots of sample locations saved for spatial registration to pre-surgical MR data. Whole-exome sequencing was performed on flash-frozen tumor samples, prioritizing five glioma-related genes: IDH1, TP53, EGFR, PIK3CA, and NF1. Regression was implemented with a GAM using a univariate shape function for each predictor. Standard ROC analyses were used to evaluate detection, with AUC calculated for each gene target and MR contrast combination. Mean AUC for five gene targets and 31 MR contrast combinations was 0.75±0.11, with individual AUCs as high as 0.96 for IDH1 and TP53 with T2W-FLAIR and ADC, and 0.99 for EGFR with T2W and ADC. An average AUC of 0.85 across the five mutations was achieved using the combination of T1W, T2W-FLAIR, and ADC. These results suggest the possibility of predicting exome-wide mutation events from non-invasive, in vivo imaging by combining stereotactic localization of glioma samples with a semi-parametric deep learning method. This approach holds potential for refining targeted therapy by better addressing the genomic heterogeneity of glioma tumors.

Separate but equal: Equality in belief propagation for single-cycle graphs

Belief propagation is a widely used, incomplete optimization algorithm whose main theoretical properties hold only under the assumption that beliefs are not equal. Nevertheless, there is substantial evidence to suggest that equality between beliefs does occur. A published method to overcome belief equality, which is based on the use of unary function-nodes, is commonly assumed to resolve the problem.In this study, we focus on min-sum, the version of belief propagation that is used to solve constraint optimization problems. We prove that for the case of a single-cycle graph, belief equality can only be avoided when the algorithm converges to the optimal solution. Under any other circumstances, the unary function method will not prevent equality, indicating that some of the existing results presented in the literature are in need of reassessment. We differentiate between belief equality, which refers to equal beliefs in a single message, and assignment equality, which prevents the coherent assignment of values to the variables, and we provide conditions for both.

A unified theory of risk

Abstract A novel theory of comparative risk is developed and defended. Extant theories are criticized for failing the tests of extensional and formal adequacy. A unified diagnosis is proposed: extant theories consider risk to be a univariable function, but risk is a multivariate function. According to the theory proposed, which we call the unified theory of risk, the riskiness of a proposition is a function of both the proportion and the modal closeness of the possible worlds at which the proposition holds. It is argued the theory passes the tests of extensional and formal adequacy.

A probabilistic framework for offshore aquaculture suitability assessment using bivariate copulas

Aquaculture at sea is gaining increasing importance, not only as a (local) food source but also due to its potential of being combined with other offshore activities such as wind parks. Nevertheless, experience of offshore aquaculture is limited. This study aims to provide a framework to evaluate offshore aquaculture suitability accounting for the probabilistic dependence between relevant variables. This framework is applied to obtain suitability maps of aquaculture for the North Sea for the blue mussel Mytilus edulis and the sugar kelp Saccharina latissima. For each of these species, three ecological variables are selected and the optimal growth and critical survival limits are defined. Here, suitability is defined as the probability of meeting these conditions. Data on the selected variables is extracted from a large-scale 3D hydrodynamic and ecological model of the northwest European Shelf, of which daily extremes are sampled. The probabilistic model is developed using bivariate copula models, which are fitted to each variable pair to describe their joint distribution function at each studied location. Empirical distribution functions are used to describe the univariate distribution function of each variable and location. Using Monte-Carlo simulations, the probability of meeting the optimal and critical limits is estimated and suitability maps accounting for the probabilistic dependence between the variables are generated. In addition, suitability maps disregarding the dependence are generated and compared to those accounting for the probabilistic dependence. It was found that considering the dependence between variables significantly improves the accuracy of the results for optimal and critical growth conditions for both species. The presented method allows to identify potential areas where blue mussel and sugar kelp cultivation is the most suitable. For instance, in this study, a north-south elongated area west of the German and Danish coast appears to be most suitable for blue mussels, while estuaries and rivers are found the most suitable for the sugar kelp.

On bivariate fractal interpolation for countable data and associated nonlinear fractal operator

Abstract Fractal interpolation has been conventionally treated as a method to construct a univariate continuous function interpolating a given finite data set with the distinguishing property that the graph of the interpolating function is the attractor of a suitable iterated function system. On the one hand, attempts have been made to extend the univariate fractal interpolation from a finite data set to a countably infinite set. On the other hand, fractal interpolation in higher dimensions, particularly the theory of fractal interpolation surfaces (FISs), has received increasing attention for more than a quarter century. This article targets a two-fold extension of the notion of fractal interpolation by providing a general framework to construct FISs for a prescribed set consisting of countably infinite data on a rectangular grid. By using this as a crucial tool, we obtain a parameterized family of bivariate fractal functions simultaneously interpolating and approximating a prescribed bivariate continuous function. Some elementary properties of the associated nonlinear (not necessarily linear) fractal operators are established, thereby allowing the interaction of the notion of fractal interpolation with the theory of nonlinear operators.

General Solutions to the Navier-Stokes Equations for Incompressible Flow

<i>Waves are mainly generated by wind via the transfer of wind energy to the water through friction. When the wind subsides, the waves transition into swells and eventually dissipate. Friction plays a crucial role in the generation and dissipation of waves. Numerous wave theories have been developed based on the assumption of inviscid flow, but these theories are inadequate in explaining the transformation of waves into swells. This study addressed these limitations by analytically deriving general solutions to the Navier–Stokes equations. By expressing the velocity field as the product of a solution to the Helmholtz equation and a time-dependent univariate function, the Navier–Stokes equations are decomposed into an ordinary differential equation and the Euler equations, which are solved using tensor calculus. This paper provides solutions for viscous flow with shear currents when applied to the water wave problem. These solutions were validated through their application to the vorticity equation. The decay modulus of water waves was compared with experimental data, showing a significant degree of concordance. In contrast to other wave theories, this study clarified the process through which waves evolve into swells.</i>

Why do refugees live near to each other? A utility-based model for spatial segregation of Syrian refugees in Türkiye: 2016–2023 period

PurposeThe main purpose of this research is to understand and explain the clustering and/or segregation patterns of the rapid, massive and unexpected flow of Syrian refugees to Türkiye.Design/methodology/approachThe analytical framework we use consists of three parts. In the first stage, we created a two-group model in which utility is assumed to be a function of consumption and the proportion of the population belonging to the same group living in their neighborhood. We show that equilibrium utility can be reduced to a univariate function that is the population proportion of the group to which one belongs. Then, with the help of a Python-based simulation, we analyze the redistribution dynamics of the groups and clustering patterns. In the final stage, we compute the dissimilarity index to determine the degree of clustering.FindingsAlthough the dilution policy has been implemented, a relatively high spatial segregation of refugees still exists. The dissimilarity index we compute using the latest data shows that almost half of Syrians need to be displaced for a homogeneous geographical distribution.Research limitations/implicationsBy obtaining neighborhood-based housing data, it will be possible to conduct a more detailed analysis at the city level. This will improve policymakers' understanding of refugee policy at both local and national levels.Originality/valueIt is the first study in which a dissimilarity index is computed for Syrian refugees in Türkiye.

ORDERED STABILITY AND EXPANSIONS OF A PURE LINEAR ORDER BY A UNARY FUNCTION

A linearly ordered structure is said to be o-stable if each of its Dedekind cut has a “small” number of extensions to complete 1-types. This concept, which was introduced by B.S. Baizhanov and V.V. Verbovsky, generalizes such widely known concepts among specialists in model theory as weak o-minimality, (weak) quasi-o-minimality and dp-minimality of ordered structures. It is based on a combination of the concepts of o-minimality and stability. As we know, the elementary theory of any pure linear order is o-superstable. Indeed, this follows from the fact, which Rubin proved in the late 70s of the 20th century, that any type of one variable is determined by its cut and definable subsets, distinguished by unary predicates or formulas with one free variable. In this paper, we explore the question of what happens if a pure linear order is expanded with a unary function. Two examples were constructed when o-stability is violated; in addition, sufficient conditions for preserving o-stability with such language expansion were found. Research work on this topic is not yet finished, ideally, it would be good to find a criterion for preserving ordered stability when enriching a structure with pure linear order with a new function of one variable.

Towards multi-variable tsunami damage modeling for coastal roads: Insights from the application of explainable machine learning to the 2011 Great East Japan Event

The accurate assessment of tsunami-induced damage to coastal roads is crucial for effective disaster risk management. Traditional approaches, reliant on univariate fragility functions, often fail to capture the complex interplay of variables influencing road damage during tsunami events. This study addresses this limitation by employing machine learning techniques on an extensive dataset compiled after the 2011 Great East Japan tsunami. The dataset, enriched with additional explicative variables accounting for the hydraulic features of the event and the physical characteristics at roads’ location, enables a comprehensive analysis of road damage mechanisms. Results indicate that while inundation depth remains a significant predictor, factors such as wave approach angle, road orientation and potential overflow from inland watercourses also play critical roles.

A new improved convex underestimator for univariate functions in global optimization

ABSTRACT In this paper, we consider convex underestimators for univariate nonconvex functions over an interval in R . We propose a new convex underestimator that can be used for computing the lower bound of the range of nonconvex functions, or for solving global optimization problems. We show that the new underestimator is tighter than other underestimators which are developed in the literature.

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.

D-algebraic functions

Differentially-algebraic (D-algebraic) functions are solutions of polynomial equations in the function, its derivatives, and the independent variables. We revisit closure properties of these functions by providing constructive proofs. We present algorithms to compute algebraic differential equations for compositions and arithmetic manipulations of univariate D-algebraic functions and derive bounds for the order of the resulting differential equations. We apply our methods to examples in the sciences.

Floor failure behavior and water disaster prevention system of ultra-wide opposite pulling working face mining on confined aquifer

The floor failure depth of the ultra-wide opposite pulling working face (OPWF) mining on the confined aquifer is larger, and the risk of water inrush is higher. Based on the engineering background of Binhu Coal Mine (BHCM) in North China Coalfield, this paper analyzes the distribution rule of floor support pressure in OPWF mining, carries out numerical simulation before and after roof cutting, and microseismic-electrical joint dynamic monitoring. The results show that the failure depth, vertical stress, maximum principal stress, maximum shear stress, and pore water pressure of floor after roof cutting are decreased by 28.6 %, 15.5 %, 12.1 %, 30.9 %, and 41.7 %, respectively, the stress concentration coefficient is also decreased by 19.5 %. Roof cutting blocks the propagation path of mining stress, reduces the mine pressure concentration and controls the floor failure depth. Using the optimal univariate function between the floor failure depth and mining width, mining depth, mining height, roof cutting height, working face stagger distance, a multi-factor prediction model for the floor failure depth of the OPWF mining is established, and the accuracy of the model is more than 90 %. Combined with the methods of exploration, treatment, and monitoring, the water disaster prevention technical guarantee system in the ultra-wide OPWF mining on the confined aquifer is constructed, which provides a new method for the prevention and control of floor water disasters during mining.

Piecewise nonlinear approximation for non-smooth functions

Piecewise affine or linear approximation has garnered significant attention as a technique for approximating piecewise-smooth functions. In this study, we propose a novel approach: piecewise non-linear approximation based on rational approximation, aimed at approximating non-smooth functions. We introduce a method termed piecewise Padé Chebyshev (PiPC) tailored for approximating univariate piecewise smooth functions. Our investigation focuses on assessing the effectiveness of PiPC in mitigating the Gibbs phenomenon during the approximation of piecewise smooth functions. Additionally, we provide error estimates and convergence results of PiPC for non-smooth functions. Notably, our technique excels in capturing singularities, if present, within the function with minimal Gibbs oscillations, without necessitating the explicit specification of singularity locations. To the best of our knowledge, prior research has not explored the use of piecewise non-linear approximation for approximating non-smooth functions. Finally, we validate the efficacy of our methods through numerical experiments, employing PiPC to reconstruct a non-trivial non-smooth function, thus demonstrating its capability to significantly alleviate the Gibbs phenomenon.