Multivariate Function Research Articles

Histopathology language-image representation learning for fine-grained digital pathology cross-modal retrieval

Large-scale digital whole slide image (WSI) datasets analysis have gained significant attention in computer-aided cancer diagnosis. Content-based histopathological image retrieval (CBHIR) is a technique that searches a large database for data samples matching input objects in both details and semantics, offering relevant diagnostic information to pathologists. However, the current methods are limited by the difficulty of gigapixels, the variable size of WSIs, and the dependence on manual annotations. In this work, we propose a novel histopathology language-image representation learning framework for fine-grained digital pathology cross-modal retrieval, which utilizes paired diagnosis reports to learn fine-grained semantics from the WSI. An anchor-based WSI encoder is built to extract hierarchical region features and a prompt-based text encoder is introduced to learn fine-grained semantics from the diagnosis reports. The proposed framework is trained with a multivariate cross-modal loss function to learn semantic information from the diagnosis report at both the instance level and region level. After training, it can perform four types of retrieval tasks based on the multi-modal database to support diagnostic requirements. We conducted experiments on an in-house dataset and a public dataset to evaluate the proposed method. Extensive experiments have demonstrated the effectiveness of the proposed method and its advantages to the present histopathology retrieval methods. The code is available at https://github.com/hudingyi/FGCR.

Fixed-time adaptive fuzzy control with prescribed tracking performances for flexible-joint manipulators

In this paper, we study the tracking control problem of flexible-joint manipulators with uncertain unstructured dynamics, and develop a prescribed fixed-time solution scheme by combining direct adaptive fuzzy control design with a class of smooth functions. With our scheme, only one adaptive law needs to be calculated online, and also there are no complicated terms, e.g., the partial derivatives of some multivariable functions, involved in virtual controllers and the actual control input, which show that the raised scheme is computationally attractive. Moreover, the boundedness of all closed-loop signals, and the fixed-time convergence of tracking error to a prescribed interval can be established through a rigorous proof, and the obtained results are further confirmed by simulation tests.

Investigation of Integral Transformation Associated with Extended Generalized Srivastava’s Hypergeometric Multi Variable Special Function

In recent year study on multivariate special functions and Integral transformation have been booming. In this work, we have focused on Srivastava hypergeometric function , , and  with triple variable. We have discussed the literature study and motivation from the recent works on the extension of Srivastava’s multivariable hypergeometric function , , and . In this paper, the extension of , , and  is studied based on the generalized beta  function  and the generalized Pochhammer’s symbol . Furthermore, the Mellin integral transformation and Inverse Mellin integral transformation have been studied for the  based extension of the functions , , and . A few of the most recent uses of these transformations in various scientific and engineering fields are also highlighted in this paper. In general, this work seeks to offer a thorough overview of recent breakthroughs in the importance and applications of several integral transforms of Multivariable functions.

Multispectral Thermometry Method Based on Optimisation Ideas.

Multispectral thermometry is based on the law of blackbody radiation and is widely used in engineering practice today. Temperature values can be inferred from radiation intensity and multiple sets of wavelengths. Multispectral thermometry eliminates the requirements for single-spectral and spectral similarity, which are associated with two-colour thermometry. In the process of multispectral temperature inversion, the solution of spectral emissivity and multispectral data processing can be seen as the keys to accurate thermometry. At present, spectral emissivity is most commonly estimated using assumption models. When an assumption model closely matches an actual situation, the inversion of the temperature and the accuracy of spectral emissivity are both very high; however, when the two are not closely matched, the inversion result is very different from the actual situation. Assumption models of spectral emissivity exhibit drawbacks when used for thermometry of a complex material, or any material whose properties dynamically change during a combustion process. To address the above problems, in the present study, we developed a multispectral thermometry method based on optimisation ideas. This method involves analysing connections between measured temperatures of each channel in a multispectral temperature inversion process; it also makes use of correlations between multispectral signals at different temperatures. In short, we established a multivariate temperature difference correlation function based on the principles of multispectral radiometric thermometry, using information correlations between data for each channel in a temperature inversion process. We then established a high-precision thermometry model by optimising the correlation function and correcting any measurement errors. This method simplifies the modelling process so that it becomes an optimisation problem of the temperature difference function. This also removes the need to assume the relationships between spectral emissivity and other physical quantities, simplifying the process of multispectral thermometry. Finally, this involves correction of the spectral data so that any impact of measurement error on the thermometry is reduced. In order to verify the feasibility and reliability of the method, a simple eight-channel multispectral thermometry device was used for experimental validation, in which the temperature emitted from a blackbody furnace was identified as the standard value. In addition, spectral data from the 468-603 nm band were calibrated within a temperature range of 1923.15-2273.15 K, resulting in multispectral thermometry based on optimisation principles with an error rate of around 0.3% and a temperature calculation time of less than 3 s. The achieved level of inversion accuracy was better than that obtained using either a secondary measurement method (SMM) or a neural network method, and the calculation speed achieved was considerably faster than that obtained using the SMM method.

Collision dynamics in AUV docking with conical hood type dock: influencing factors and performance analysis

ABSTRACT To enhance docking safety, the collision dynamics during autonomous underwater vehicle (AUV) docking is investigated. Various factors such as dock structures, motion conditions, and related parameters are considered. The relationship between docking collision force and key parameters is unveiled by the mathematical model based on the restoration coefficient method and equivalent spring damping method. Multivariate nonlinear fitting functions are proposed to evaluate collision force and docking time. Furthermore, the deductions made in the research are supported by a preliminary test and the groundwork for further analysis is laid. The study demonstrates the impacts of various factors on docking, including different docking cone hood profiles, structures, velocities, accelerations, eccentricities, traction forces, angles, and force indices. By clarifying relevant mechanism, docking parameters can be achieved to meet diverse docking requirements and enhance the device's lifespan. The research offers valuable insights for achieving smoother AUV docking with conical hood type docks.

Efficient quadratures for high-dimensional Bayesian data assimilation

Bayesian update is a common strategy used to combine (uncertain) model predictions and (noisy) observational data. A computational bottleneck in this data assimilation technique is the evaluation of high-dimensional quadratures involving multivariate probability density functions (PDFs) of system states. We explore “designed quadratures” as a means to reduce the computational cost of Bayesian update of multivariate joint PDFs. A series of numerical experiments demonstrate that our method outperforms stochastic collocation on sparse grids, a popular technique used to perform high-dimensional integration in the context of uncertainty quantification, in terms of both accuracy and computational efficiency.

Analytic Functions in a Complete Reinhardt Domain Having Bounded L-Index in Joint Variables

The manuscript is an initiative to construct a full and exhaustive theory of analytical multivariate functions in any complete Reinhardt domain by introducing the concept of L-index in joint variables for these functions for a given continuous, non-negative, non-vanishing, vector-valued mapping L defined in an interior of the domain with some behavior restrictions. The complete Reinhardt domain is an example of a domain having a circular symmetry in each complex dimension. Our results are based on the results obtained for such classes of holomorphic functions: entire multivariate functions, as well as functions which are analytical in the unit ball, in the unit polydisc, and in the Cartesian product of the complex plane and the unit disc. For a full exhaustion of the domain, polydiscs with some radii and centers are used. Estimates of the maximum modulus for partial derivatives of the functions belonging to the class are presented. The maximum is evaluated at the skeleton of some polydiscs with any center and with some radii depending on the center and the function L and, at most, it equals a some constant multiplied by the partial derivative modulus at the center of the polydisc. Other obtained statements are similar to the described one.

Structured interpolation for multivariate transfer functions of quadratic-bilinear systems

High-dimensional/high-fidelity nonlinear dynamical systems appear naturally when the goal is to accurately model real-world phenomena. Many physical properties are thereby encoded in the internal differential structure of these resulting large-scale nonlinear systems. The high dimensionality of the dynamics causes computational bottlenecks, especially when these large-scale systems need to be simulated for a variety of situations such as different forcing terms. This motivates model reduction where the goal is to replace the full-order dynamics with accurate reduced-order surrogates. Interpolation-based model reduction has been proven to be an effective tool for the construction of cheap-to-evaluate surrogate models that preserve the internal structure in the case of weak nonlinearities. In this paper, we consider the construction of multivariate interpolants in frequency domain for structured quadratic-bilinear systems. We propose definitions for structured variants of the symmetric subsystem and generalized transfer functions of quadratic-bilinear systems and provide conditions for structure-preserving interpolation by projection. The theoretical results are illustrated using two numerical examples including the simulation of molecular dynamics in crystal structures.

Olsson.wl & ROC2.wl: Mathematica packages for transformations of multivariable hypergeometric functions & regions of convergence for their series representations in the two variables case

We present the Olsson.wlMathematica package which aims to find transformations for some classes of multivariable hypergeometric functions. It is based on a well-known method developed by P. O. M. Olsson [1] (1964) to derive the analytic continuations of the Appell F1 double hypergeometric series using the linear transformations of the Gauss F12 hypergeometric function. We provide a brief description of the method of Olsson and demonstrate the use of the commands of the Olsson.wl package using some examples that are presented in the text and in some ancillary Mathematica notebooks. In particular, we reproduce various results of the literature on multivariable hypergeometric functions and show practical applications of this package in the derivation of novel formulas. In the context of high energy physics, we also demonstrate how it can be used to disentangle some known results about the analytic continuation of some series representations of the one-loop pentagon in multi-Regge kinematics and D=6−2ϵ. We also provide a companion package, called ROC2.wl, which is dedicated to the derivation of the regions of convergence of double hypergeometric series. This package can be used independently of Olsson.wl. Program summaryProgram Title:Olsson.wlCPC Library link to program files:https://doi.org/10.17632/gc63xzwzz5.1Licensing provisions: GNU General Public License v3.0.Programming language: Wolfram Mathematica version 11.3 and beyond.Nature of problem: To find the transformation formulas of multivariable hypergeometric series appearing in Feynman integral calculus.Solution method: Mathematica implementation of the method of Olsson [1]. The method uses the transformation theory of lower variable hypergeometric functions to find the transformation formulas of higher variable hypergeometric functions.Companion package:ROC2.wl, which is a Mathematica package used inside Olsson.wl and can also be used as a standalone package. The package is used to find the region of convergence of double hypergeometric functions, using Horn's theorem.

Low dimensional approximation and generalization of multivariate functions on smooth manifolds using deep ReLU neural networks

The expressive power of deep neural networks is manifested by their remarkable ability to approximate multivariate functions in a way that appears to overcome the curse of dimensionality. This ability is exemplified by their success in solving high-dimensional problems where traditional numerical solvers fail due to their limitations in accurately representing high-dimensional structures. To provide a theoretical framework for explaining this phenomenon, we analyze the approximation of Hölder functions defined on a d-dimensional smooth manifold M embedded in RD, with d≪D, using deep neural networks. We prove that the uniform convergence estimates of the approximation and generalization errors by deep neural networks with ReLU activation functions do not depend on the ambient dimension D of the function but only on its lower manifold dimension d, in a precise sense. Our result improves existing results from the literature where approximation and generalization errors were shown to depend weakly on D.

Univariate Flux Partition Functions for Planetary Boundary Layer Schemes at Gray Zone Resolutions

Abstract When the horizontal grid spacing of a numerical weather prediction model approaches kilometer scale, the so-called gray zone range, turbulent fluxes in the convective boundary layer (CBL) are partially resolved and partially subgrid scale (SGS). Knowledge of the partition between resolved and SGS turbulent fluxes is key to building scale-adaptive planetary boundary layer (PBL) schemes that are capable of regulating the SGS fluxes with varying grid spacing. However, flux partition depends not only on horizontal grid spacing, but also on local height, bulk stability of the boundary layer, and the particular turbulent flux. Such multivariate functions are difficult to construct analytically, so their implementations in scale-adaptive PBL schemes always involve certain levels of approximation that can lead to inaccuracies. This study introduces a physically based perspective for the flux partition functions that greatly simplifies their implementation with high accuracy. By introducing an appropriate scaling length λ that accounts for both height and bulk stability dependencies, the dimensionality of the partition functions is reduced to a single dimensionless group. Based on the analysis of a comprehensive large-eddy simulation dataset of the CBL, it is further shown that λ’s height and bulk stability dependencies can be separately represented by a similarity length scale and a stability coefficient. The resulting univariate partition functions are incorporated into a traditional first-order PBL scheme as a proof of concept. Our results show that the augmented scheme well-reproduces the SGS fluxes at gray zone resolutions. Significance Statement Flux partition functions are a key component in most scale-adaptive planetary boundary layer (PBL) schemes developed for kilometer- and subkilometer-resolution numerical weather prediction models. They regulate the parameterized turbulent fluxes as a function of horizontal grid spacing, while they also depend on height and atmospheric stability. Such multivariate dependencies forbid simple analytical expressions, and as a result, partition functions implemented in scale-adaptive PBL schemes are generally simplified at the cost of accuracy in previous works. This study investigates the possibility of constructing partition functions that are both accurate and easy to parameterize. Utilizing a physically based length scale, univariate partition functions are built, evaluated, and put into a conventional PBL scheme to improve the gray zone turbulence parameterization.

Fundamental period equations for plan irregular moment-resisting frame buildings

The fundamental natural period of oscillation is a critical parameter in evaluating the design base shear of buildings. Worldwide seismic design codes typically employ height-based empirical formulas to estimate this period for various building categories, without distinguishing between regular and irregular buildings. This study proposes a formula specifically for reinforced concrete (RC) moment-resisting frame (MRF) buildings with dominant re-entrant corner type plan irregularity. A total of 190 re-entrant corner dominant building models with different shapes (C-, L-, T-, and PLUS-type), heights, and floor configurations were prepared, and eigenvalue analysis (EVA) was conducted. The fundamental natural period of oscillation for each model was evaluated and compared with the height-based formulas from seismic design codes and the period–height relationship proposed in existing literature. A nonlinear regression model, using a multi-variable power function, is proposed to estimate the fundamental natural period for these re-entrant corner dominant building models. This model considers the A/L ratio in both directions of the building, along with its height. Both unconstrained and constrained regression analyses were performed to derive a formula that best fits the fundamental natural period data. The study recommends that the unconstrained best-fit minus one standard deviation curve can conservatively define the fundamental natural period of oscillation for re-entrant corner dominant RC building models. The equation defining this curve has the potential to replace the existing seismic design code-based period-height formula.

Optimal energy management system for grid-connected hybrid power plant and battery integrated into multilevel configuration

A novel optimal energy management system (EMS) using a nonlinear constrained multivariable function to optimize the operation of battery energy storages (BESs) used in a hybrid power plant with wind turbine (WT) and photovoltaic (PV) power plants is proposed in this work. The hybrid power plant uses a configuration based on a battery-stored impedance-based cascaded multilevel inverter to integrate renewable energy sources (PV power plants and WT) and BESs into the grid. The new optimal EMS seeks for satisfying the demanded power while dispatching power between BESs to optimize their efficiency. A grid-connected configuration is implemented to assess the efficiency of the suggested supervisory control under changes in renewable energy (changes in wind speed and irradiation), and in a varying active and reactive powers’ request. The BES efficiency obtained from the suggested EMS is set side by side to the BES efficiency got from a conventional EMS and a model predictive control (MPC), both working based on the state-of-charge (SOC) of the BES and balancing power EMS. The results from MATLAB simulation and the experimental results with the real-time OPAL-RT simulator (OP4510, OPAL-RT) and dSPACE MicroLabBox show the effectiveness of the suggested approach and the improvement in long-term BES efficiency.

Optimal Quadrature Formula of Hermite Type in the Space of Differentiable Functions

In this research work, a new derived optimal quadrature formula is discussed, which includes the sum of the values of the function and its first and second order derivatives at the points located at the same distance on the interval [0,1] in the L2(m)(0,1) space. we first obtain an analytical representation of the error function norm, and a system of equations of the Wiener-Hopf type construct using the method of Lagrange unknown multipliers for finding the conditional extremum of multivariable functions. Optimal coefficients found by solving the system. Using the exact form of the optimal coefficients, the norm of the error functional of the optimal quadrature formula for m=3 and m=4 calculate and the order of approximation was shown to be O(hm). The obtain theoretical conclusions confirmed by numerical experiments.

The latest steps of human evolution: What the hard evidence has to say about it?

The latest periods of human evolution are a heated topic of debate and have been at the center of paleoanthropological discussions since the beginning of the field. In the last twenty years, new excavations increased the geographic range of paleoanthropological data, new fossil hominins of the last third of the Pleistocene were found, and old key fossils were redated by modern and more accurate methods. The new picture emerging from this research changed the discussion about the neandertal-sapiens interaction, the finding of their common ancestor, and the understanding of how our lineage evolved. To integrate these new discoveries and to better contextualize how they fit in the hominin framework of the Chibanian, we compared the cranial morphology of 86 well-preserved fossils from Europe, Africa, and Asia of the last 500 thousand years. 25 linear measurements were used to describe the cranial morphology of the fossils, and their biological affinities were explored though multivariate discriminant function analyses. These analyses allow us to assess affinities and possible phylogenetic relationships of Archaic Homo species, including the expressive morphological variability of the specimens included under the umbrella-term “Homo heidelbergensis” from Africa, Asia, and Europe. Our results support that cannot be accommodated in one same species. Additionally, we contribute to the disputable discussion about the origin of H. sapiens, adding support to an African origin for our species.

Displacement and damage analysis of earth dams during the 2023 Turkiye earthquake sequence

The earthquake sequence that occurred on 6 February 2023 in Turkiye caused significant damage to various infrastructures including geostructures such as dams. A total of 17 earth dams within a 200-km radius of the earthquake epicenter experienced varying degrees of damage, ranging from minor (∼2 cm) to major (up to ∼150 cm) deformations. As study of these reveals that the damaged dams are located within the closest distance to the fault of less than 30 km, with an average value of ∼12 km. This study specifically focuses on the seismic displacement analysis of the 17 damaged dams, utilizing the sliding block methods. The recorded motion data was analyzed using the kriging technique to estimate the spectral response at the dam sites. Moreover, the recorded ground motions were scaled to the resonant period of the dam site to estimate acceleration time history. The findings reveal that the rigid block analysis can provide an average estimation of seismic displacement with a relative error of less than 44%. The results of the damage analysis indicate that seven dams reached the ultimate limit state and two dams experienced the serviceability limit state. Moreover, the univariate and multivariate fragility functions are developed to estimate seismic probabilistic analysis of earth dams based on the observed data and the limit states. The results show that the selection of a single intensity measure (IM) and a combination of IMs can affect the predicted probability of failure. The findings provide an insight into the resilience assessment of dams and other geosystems during this strong earthquake.

Likelihood ratio combination of multiple biomarkers viasmoothing spline estimated densities.

The diagnostic accuracy of multiple biomarkers in medical research is crucial for detecting diseases and predicting patient outcomes. An optimal method for combining these biomarkers is essential to maximize the Area Under the Receiver Operating Characteristic (ROC) Curve (AUC). Although the optimality of the likelihood ratio has been proven by Neyman and Pearson, challenges persist in estimating the likelihood ratio, primarily due to the estimation of multivariate density functions. In this study, we propose a non-parametric approach for estimating multivariate density functions by utilizing Smoothing Spline density estimation to approximate the full likelihood function for both diseased and non-diseased groups, which compose the likelihood ratio. Simulation results demonstrate the efficiency of our method compared to other biomarker combination techniques under various settings for generated biomarker values. Additionally, we apply the proposed method to a real-world study aimed at detecting childhood autism spectrum disorder (ASD), showcasing its practical relevance and potential for future applications in medical research.

Quantics Tensor Cross Interpolation for High-Resolution Parsimonious Representations of Multivariate Functions.

Multivariate functions of continuous variables arise in countless branches of science. Numerical computations with such functions typically involve a compromise between two contrary desiderata: accurate resolution of the functional dependence, versus parsimonious memory usage. Recently, two promising strategies have emerged for satisfying both requirements: (i)The quantics representation, which expresses functions as multi-index tensors, with each index representing one bit of a binary encoding of one of the variables; and (ii)tensor cross interpolation (TCI), which, if applicable, yields parsimonious interpolations for multi-index tensors. Here, we present a strategy, quantics TCI, which combines the advantages of both schemes. We illustrate its potential with an application from condensed matter physics: the computation of Brillouin zone integrals.

Trakya Havzasının Güneybatısında Fay Parametrelerinin Gravite Gradyent Verilerinden Kestirimi

Characterization of fault zones plays an important role for earthquake studies, geothermal and mineral resource exploration. This problem can be achieved by optimizing parameters of a defined synthetic model iteratively using gravity, gravity gradients or magnetic data. Curvature gradients indicate deviations of an equipotential surface from a spherical surface, therefore, reflecting deviations of mass distributions from a source point. In this study, differential curvature gradient is used as it identifies linear structures such as dip-slip faults and geological contacts. Curvature gradients are computed from gravity gradient observations which are derived from the high-resolution Earth Gravitational Model 2008. Estimating the fault parameters from gravity gradients is a complex nonlinear geophysical inverse problem that requires finding the minimum of a multi-variable cost function. A global optimization technique called simulated annealing (SA) is used to estimate location, dip angle and some depth parameters of the faults located in southwest side of the Thrace basin in Türkiye. In the optimization technique, a straight solution is inevitably applied to calculate the theoretical anomaly to be compared with the observed anomaly in each search step. In this case, the problem requires constructing a mathematical or geophysical interpretation model. Such a model is designed as a dip-slip fault model. The results show that dip angle estimates indicate high-angle faults which are found to be consistent with the seismic studies.

Gaining or losing perspective for convex multivariate functions on a simplex

Gaining or losing perspective for convex multivariate functions on a simplex