A Comparison of Two Mixed-Integer Linear Programs for Piecewise Linear Function Fitting

The problem of fitting continuous piecewise linear (PWL) functions to discrete data has applications in pattern recognition and engineering, amongst many other fields. To find an optimal PWL function, the positioning of the breakpoints connecting adjacent linear segments must not be constrained and should be allowed to be placed freely. Although the univariate PWL fitting problem has often been approached from a global optimisation perspective, recently, two mixed-integer linear programming approaches have been presented that solve for optimal PWL functions. In this paper, we compare the two approaches: the first was presented by Rebennack and Krasko [Rebennack S, Krasko V (2020) Piecewise linear function fitting via mixed-integer linear programming. INFORMS J. Comput. 32(2):507–530] and the second by Kong and Maravelias [Kong L, Maravelias CT (2020) On the derivation of continuous piecewise linear approximating functions. INFORMS J. Comput. 32(3):531–546]. Both formulations are similar in that they use binary variables and logical implications modelled by big-[Formula: see text] constructs to ensure the continuity of the PWL function, yet the former model uses fewer binary variables. We present experimental results comparing the time taken to find optimal PWL functions with differing numbers of breakpoints across 10 data sets for three different objective functions. Although neither of the two formulations is superior on all data sets, the presented computational results suggest that the formulation presented by Rebennack and Krasko is faster. This might be explained by the fact that it contains fewer complicating binary variables and sparser constraints. Summary of Contribution: This paper presents a comparison of the mixed-integer linear programming models presented in two recent studies published in the INFORMS Journal on Computing. Because of the similarity of the formulations of the two models, it is not clear which one is preferable. We present a detailed comparison of the two formulations, including a series of comparative experimental results across 10 data sets that appeared across both papers. We hope that our results will allow readers to take an objective view as to which implementation they should use.

Temperature Sensor Based on Periodically Tapered Optical Fibers

In this paper, the fabrication and characterization of a temperature sensor based on periodically tapered optical fibers (PTOF) are presented. The relation between the geometry of the sensors and sensing ability was investigated in order to find the relatively simple structure of a sensor. Four types of PTOF structures with two, four, six and eight waists were manufactured with the fusion splicer. For each PTOF type, the theoretical free spectral range (FSR) was calculated and compared with measurements. The experiments were conducted for a temperature range of 20–70 °C. The results proved that the number of the tapered regions in PTOF is crucial, because some of the investigated structures did not exhibit the temperature response. The interference occurring inside the structures with two and four waists was found be too weak and, therefore, the transmission dip was hardly visible. We proved that sensors with a low number of tapered regions cannot be considered as a temperature sensor. Sufficiently more valuable results were obtained for the last two types of PTOF, where the sensor’s sensitivity was equal to 0.07 dB/°C with an excellent linear fitting (R2 > 0.99). The transmission dip shift can be described by a linear function (R2 > 0.97) with a slope α > 0.39 nm/°C.

A Note on Confusion between Linear and Affine Functions and the Generalized Forms of Gradient

Background: Arguably the most frequently used term in science, particularly in mathematics and statistics, is linear. However, confusion arises from the various meanings of linearity instructed in different levels of mathematical courses. The definition of linearity taught in high school is less correct than the one learned in a linear algebra class. The correlation coefficient of two quantitative variables is a numerical measure of the affinity, not only linearity, of two variables. However, every statistics book loosely says it is a measure of linear relationship. This clearly show that there is some confusion between use of the terms the linear function and affine function. Objective: This article aims at clarifying the confusion between use of the terms linear function and affine function. It also provides more generalized forms of the gradient in different branches of mathematics and show their equivalency. Materials and Methods: We have used the pure analytical deductive methods to proof the statements. Results: We have clearly presented that gradient is the measure of affinity, not just linearity. It becomes a special case of the derivative in calculus, of the least-squares estimate of the regression coefficient in statistics and matrix theory. The gradient can ­­­­be seen in terms of the inverse of the informative matrix in the most general setting of the linear model estimation. Conclusion: The article has been clearly written to show the distinction between the linear and affine functions in a concise and unambiguous manner. We hope that readers will clearly see various generalizations of the gradient and article itself would be a simple exposition, enlightening, and fun to read.

Route planning on orienteering maps with least-cost path analysis

The feature categories of an orienteering map are prepared to allow the map reader to estimate the travel time between any two points on the map with a good approximation. This requires not only an accurate map, but also a key that adapts to the speed of travel. Such map key is developed and maintained by the IOF (International Orienteering Federation), and technically all the orienteering maps are compiled by using it. Estimated time also plays an important role in planning the courses of orienteering races. The course setter estimates time based on a route he thinks is ideal, but the speed of travel is basically a non-linear function of terrain, road network and land cover. Because of this, the easiest (ideal) route between the two points and its time cost can be calculated using the least-cost path (LCP) GIS method, which can be prepared to take into account these three map feature categories. This method is based on the calculation of a cost surface, then the analysis of the ideal path from a given point to the destination. The automation can be adapted to any orienteering map due to the similarities of the map keys. This study shows that if the weight corresponding to the different feature categories is given properly, the ideal path between two points on orienteering maps can be calculated. The ideal path, however is still a subjective category, which may depend on the capabilities and preferences of the orienteer. In this study the routes calculated with the LCP method were compared with the suggestions of the ideal routes by orienteering runners of different ages. The results show that the routes given by sportsmen can be simulated with the LCP method and even the time cost of the calculated routes can be calculated. This study can lay the groundwork for a GIS tool helping the course setting process on standard orienteering maps.

LAMOST medium-resolution spectral survey of Galactic nebulae (LAMOST-MRS-N): subtraction of geocoronal Hα emission

We introduce a method of subtracting geocoronal Hα emissions from the spectra of LAMOST medium-resolution spectral survey of Galactic nebulae (LAMOST-MRS-N). The flux ratios of the Hα sky line to the adjacent OH λ6554 single line do not show a pattern or gradient distribution in a plate. More interestingly, the ratio is well correlated to solar altitude, which is the angle of the Sun relative to the Earthʼs horizon. It is found that the ratio decreases from 0.8 to 0.2 with the decreasing solar altitude from –17 to –73 degree. Based on this relation, which is described by a linear function, we can construct the Hα sky component and subtract it from the science spectrum. This method has been applied to the LAMOST-MRS-N data, and the contamination level of the Hα sky to nebula is reduced from 40% to less than 10%. The new generated spectra will significantly improve the accuracy of the classifications and the measurements of physical parameters of Galactic nebulae.

On the Difference between the Radii of Gluons and Quarks

In this paper, in the scope of a non-extensive statistical model for the nucleon’s structure function, the volume of the gluons in the nucleons and the relations among the temperature, T, the parameter “q” of Tsallis statistics, and the scattering energies, Q2, are studied. A system of equations with the usual sum rules are solved for the valence quarks, the experimental results for the polarized structure function, and the estimated carried moments for gluons and quarks. Each state of T and q leads to a set of chemical potentials and different radii for gluons and quarks. We conclude that gluons must occupy a larger volume than the quarks to fit the fraction of the total momentum. A linear function of the temperature with Q2 is obtained as an approach. The obtained range of temperatures is different from the previous models.

ASSESSMENT OF A NEW ISOTROPIC HYPERELASTIC CONSTITUTIVE MODEL FOR A RANGE OF RUBBERLIKE MATERIALS AND DEFORMATIONS

ABSTRACT The choice of an appropriate strain energy function W is key to accurate modeling and computational finite element analysis of the mechanical behavior of unfilled non-crystalizing rubberlike materials. Despite the existing variety of models, finding a suitable model that can capture many deformation modes of a rubber specimen with a single set of parameter values and satisfy the a priori mathematical and structural requirements remains a formidable task. Previous work proposed a new generalized neo-Hookean W (I1) function, showing a promising fitting capability and enjoying a structural basis. We now use two extended forms of that model that include an I1 term adjunct, W (I1, I2), for application to various boundary value problems commonly encountered in rubber mechanics applications. Specifically, two functional forms of the I2 invariant are considered: a linear function and a logarithmic function. The boundary value problems of interest include the in-plane uniaxial, equi-biaxial, and pure shear deformations and simple shear, inflation, and nonhomogeneous deformations such as torsion. By simultaneous fitting of each model to various deformation modes of rubber specimens, it is demonstrated that a single set of model parameter values favorably captures the mechanical response for all the considered deformations of each specimen. It is further shown that the model with a logarithmic I2 function provides better fits than the linear function. Given the functional simplicity of the considered W (I1, I2) models, the low number of model parameters (three in total), the structurally motivated bases of the models, and their capability to capture the mechanical response for various deformations of rubber specimens, the considered models are recommended as a powerful tool for practical applications and analysis of rubber elasticity.

Efficient protocols for oblivious linear function evaluation from ring-LWE1

An oblivious linear function evaluation protocol, or OLE, is a two-party protocol for the function f ( x ) = a x + b, where a sender inputs the field elements a, b, and a receiver inputs x and learns f ( x ). OLE can be used to build secret-shared multiplication, and is an essential component of many secure computation applications including general-purpose multi-party computation, private set intersection and more. In this work, we present several efficient OLE protocols from the ring learning with errors (RLWE) assumption. Technically, we build two new passively secure protocols, which build upon recent advances in homomorphic secret sharing from (R)LWE (Boyle et al. in: EUROCRYPT 2019, Part II (2019) 3–33 Springer), with optimizations tailored to the setting of OLE. We upgrade these to active security using efficient amortized zero-knowledge techniques for lattice relations (Baum et al. in: CRYPTO 2018, Part II (2018) 669–699 Springer), and design new variants of zero-knowledge arguments that are necessary for some of our constructions. Our protocols offer several advantages over existing constructions. Firstly, they have the lowest communication complexity amongst previous, practical protocols from RLWE and other assumptions; secondly, they are conceptually very simple, and have just one round of interaction for the case of OLE where b is randomly chosen. We demonstrate this with an implementation of one of our passively secure protocols, which can perform more than 1 million OLEs per second over the ring Z m , for a 120-bit modulus m, on standard hardware.

Prediction of Lambs Growth From Milk Production and Its Composition in Awassi Sheep and Description of Growth Curve of Non-Linear Function

This study was undertaken at the Ruminant’s Research Station, Office of Agricultural Research/Ministry of Agriculture (20 km west of Baghdad). 102 ewes were used in this study from 1/1/2020- 1/10/2021 to study prediction of lambs growth from milk production and its composition in Awassi sheep and description of growth cure of non–linear function. Simple regression was used to examine relationship between growth traits such as gain, weaning weight, weight at 6 months of age and milk production and its composition. Regression of weaning weight was positive and highly significant (P≤ 0.01) with daily milk yield (DMY) and significant (P≤0.05) with dam’s milk protein percentage (MPP), Analysis of regression also showed significant positive increasing (p≤ 0.05) in lambs' weight at six months of age with increasing of DMY and MPP, where coefficient of regression (CR) amount to 5.006 kg/kg and 0.663 kg/% respectively. Results showed that CR of gain from birth to weaning was highly significant with dam’s DMY, whereas was significant with MPP. Both DMY and MPP had significant and positive relationship with gain between birth weight and weight at six months, where amount of CR was 4.859 kg/kg and 0.619 kg/% with determination coefficient (R2) amount to 0.31 and 0.19 sequentially. Also, positive and significant regression was noticed for gain from birth to 6 months with MPP and milk Solids Not-Fat(SNF) percentages. Exponential functions also were used according to analysis of non-linear regression with calculated R2 for growth in Awassi lambs. Results showed that relationship which derives from regression weight at 6 months on weight at weaning according to equation WT6^=a+b WWT2 is considered the best because increasing value of R2 which was 0.69. we conclude, possibility prediction in lamb’s growth from birth to six months depend on DMY and MPP, also dependence non-linear functions gave an important results to describe growth curves between weaning to 6 months.