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This paper explores the least absolute deviation (LAD) estimator of the autoregressive model with heavy-tailed G-GARCH(1, 1) noise. When the tail index α ∈ ( 1 , 2 ] , it is shown that the LAD estimator asymptotically converges to a linear function of a series of α -stable random vectors with a rate of convergence n 1 − 1 ∕ α . The result is significantly different from that of the corresponding least square estimator which is not consistent, and partially solves the problem on the asymptoticity of the LAD estimator when the tail index is less than 2. A simulation study is carried out to assess the performance of the LAD estimator and a real example is given to illustrate this approach.

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.

The moving asymptotes function is widely applied as sequential explicit convex approximation in topology optimization, due to the controllable conservatism and convergence. In virtue of these advantages, it is supposed that the efficiency of optimality criteria method will become higher if constraint functions are approximated by moving asymptotes function instead of linear function. This work presents an optimality criteria method for topology optimization under multiple constraints where the constraint functions are approximated by moving asymptotes function. The dual feasibility condition is customarily adopted to establish explicit update scheme of topological variable, where hypothesis of active topological variable set is avoided, in the case that gradient of objective function is positive. The complementary slackness condition and primal feasibility condition are combined into a simplified dual programming to solve for the Lagrange multipliers, where hypothesis of active constraint set is avoided. Three benchmark examples under multiple displacement, stress, compliance or eigenfrequency constraints are solved by the presented optimality criteria method, the results are compared to the method of moving asymptotes and the optimality criteria method with linear constraint approximation.

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.

We studied the temperature-dependent behavior of dark solitons in superfluid Fermi gases along the BCS–BEC crossover by solving the finite-temperature Bogoliubov-de Gennes equations. The notch depth of the soliton density distribution decreases with the increase in temperature in the whole BCS–BEC crossover. In particular, the notch depth has an obvious different temperature-dependent behavior from the BCS regime to the BEC regime, i.e, it has a concave function in the BCS regime, linear function in the unitary regime and convex function in the BEC regime.

Analysis of transient recovery voltage is essential to develop high-voltage direct current circuit breakers with active current injection technology. This study presents a theoretical analysis of the transient recovery voltage using the Laplace transform, thereafter the value of the transient recovery voltage obtained using this analytical equation is compared to measured reference values. It is found that the analytical transient recovery voltage is given by the sum of a linear function and a damped oscillation function. The amplitude of the damped oscillation is reduced by the resistance of post-arc plasma; therefore, it is necessary to consider the resistance owing to post-arc plasma to evaluate the transient recovery voltage accurately.

In this paper, we study the generalized submodular maximization problem with a non-negative monotone submodular set function as the objective function and subject to a matroid constraint. The problem is generalized through the curvature parameter α ∈ [ 0 , 1 ] which measures how far a set function deviates from linearity to submodularity. We propose a deterministic approximation algorithm which uses the approximation algorithm proposed by Buchbinder et al. [2] as a building block and inherits the approximation guarantee for α = 1 . For general value of the curvature parameter α ∈ [ 0 , 1 ] , we present an approximation algorithm with a factor of 1 + h α ( y ) + Δ ⋅ [ 3 + α − ( 2 + α ) y − ( 1 + α ) h α ( y ) ] 2 + α + ( 1 + α ) ( 1 − y ) , where y ∈ [ 0 , 1 ] is a predefined parameter for tuning the ratio. In particular, when α = 1 we obtain a ratio 0.5008 when setting y = 0.9 , coinciding with the renowned state-of-art approximate ratio; when α = 0 that the object is a linear function, the approximation factor equals one and our algorithm is indeed an exact algorithm that always produces optimum solutions.

It is shown that the number of irreducible quartic factors of the form g ( x ) = x 4 + a x 3 + ( 11 a + 2 ) x 2 − a x + 1 which divide the Hasse invariant of the Tate normal form E 5 in characteristic l is a simple linear function of the class number h ( − 5 l ) of the field Q ( − 5 l ) , when l ≡ 2 , 3 modulo 5. A similar result holds for irreducible quadratic factors of g ( x ) , when l ≡ 1 , 4 modulo 5. This implies a formula for the number of linear factors over F p of the supersingular polynomial s s p ( 5 ⁎ ) ( x ) corresponding to the Fricke group Γ 0 ⁎ ( 5 ) .

This paper is devoted to solving inverse problems of simulation modeling, which are presented in the form of an optimization problem. The article discusses the use of direct search methods taking into account the specifics of the problem under consideration. Due to the fact that these methods require a lot of computational experiments, two algorithms based on approximation were proposed for solving the problem. The first algorithm consists in determining and evaluating the parameters of the function of dependence (which can be linear or non-linear) of the output variable on the input variables and solving the inverse problem by minimizing increments of the arguments. In the second algorithm a linear function of dependence is iteratively constructed using the data set generated by changing the input variables in given increments, and the inverse problem is solved by minimizing increments of the arguments. The classical inventory management model with a threshold strategy is considered as an example. The inverse problem was solved using direct search and approximation-based methods.

Although a variety of research works have been carried out to investigate the behavior of nailed slopes and nail walls, only a few studies are devoted to the optimum design of soil nail walls. In this study, the limit equilibrium-based approach presented in FHWA manual, which is the prevalent design method in conjunction with nonlinear programming, has been used. Based on this approach, the optimum nail inclination angle (η) leading to the maximum safety (FS) factor against overall failure was determined for typical soil nail walls. Effects of some primary contributing factors such as nail diameter, nail length, soil friction, slope angle, back slope angle, and layout of nails on the optimum design indices were investigated thoroughly and presented in dimensionless graphs. Results indicate that increase in the nail diameter gives rise to an increase in both FS and ηopt. In addition, increasing the length of nails up to 1.875H leads to the most significant improvement in slope stability. Furthermore, the increase in soil friction results in an improvement in FS and a limited rise in ηopt. Steeper nail walls are less stable and require higher ηopt, a linear function of slope orientation (α). A higher back slope angle was found to reduce FS. Furthermore, it was found that inserting the nails in the lower 1/3 part of the slope leads to the highest efficiency, and the nails have a minor influence on the stability of short walls or highly cohesive slopes.

By a given family of convex functions on the real axis that grow at infinity faster than any linear function and by a certain logarithmically convex sequence of positive numbers, we construct the space of infinitely differentiable functions on the real line. Under the condition of a logarithmic gap between weight functions, we prove the possibility of approximation by polynomials in this space.

By using region pixels rather than edge points, a novel method is proposed for extracting circular objects in an image. A two-dimensional accumulator array with less columns is voted by region pixels instead of contour points. A set of quadratic functions and a linear function are fitted, and the coefficients of the fitted functions are used to compute the circle parameters. The quadratic polynomial function, which is determined by the circle radius and mean voting distance, is deduced by analyzing the relationship between voting values and the voting distance. A linear function, which is determined by the circle center, is deduced by analyzing the relationship between mean voting distances and the voting angle. In each column of two-dimensional array, a quadratic function is fitted and validated. The circle radius is computed based on the coefficients of the validated quadratic functions. After all columns are processed, a linear function is fitted. The circle center happens to be the coefficients of the fitted linear function. The proposed method is tested and compared using synthetic images and real-world images. Experimental results show the proposed method is fast and accurate in the presence of noise disturbance.

The influence of slope gradient on interrill erosion processes is a key scientific problem in the decision-making process regarding soil erosion control in Loess Plateau. The relationship of time to runoff (RT), flow velocity (V), runoff rate (RR) and interrill erosion rate (IER) with slope gradient was investigated to derive accurate experimental model to evaluate and quantify the influence of slopes on interrill erosion processes. The experimental soil was collected from Ansai County of Shaanxi Province, China. The average diameter of the test soil was 0.041 mm. The experiment was conducted at slopes of 8.74%, 17.62%, 26.78%, 36.38%, 46.6%, 57.70% and 69.97% under I of 90, 120 and 150 mm h−1, respectively, using indoor simulated rainfall. Time to runoff, flow velocity, runoff rate and interrill erosion rate were measured for each combination. Results showed that the time to runoff decreased as a linear function with increasing slope gradient. Slope gradient was a good predictor of time to runoff for different rainfall intensities with NSE from 0.90 to 0.97 and MSE from 0.1 to 0.25 and R2 from 0.90 to 0.97. The flow velocity increased as a power function with increasing slope gradients. Slope gradient was a good predictor of flow velocity for different rainfall intensities with NSE from 0.91 to 0.93 and MSE from 0.01 to 0.015 and R2 from 0.95 to 0.98. The runoff rate increased as a power function with increasing slope gradients. Slope gradient was a good predictor of runoff rate for different rainfall intensities with NSE from 0.90 to 0.95 and MSE from 0.000000024 to 0.000000044 and R2 from 0.94 to 0.97. The interrill erosion rate increased as a power function with increasing slope gradients. Slope gradient was a good predictor of interrill erosion rate for different rainfall intensities with NSE from 0.98 to 0.99 and MSE from 0.00022 to 0.00055 and R2 from 0.98 to 0.99. By performing the controlled simulated rainfall experiments, this study showed that slopes strongly influenced interrill erosion processes for different rainfall intensities.

The speed of short-term memory scanning is thought to be captured in the slope of the linear function of mean reaction times (RTs) regressed on set size in the Sternberg memory scanning task (SMST). Individual differences in the slope parameter have been hypothesized to correlate with general intelligence ('g'). However, this correlation can usually not be found. This present study chose a fixed-links model (FLM) approach to re-evaluate the RT slope parameter on a latent level in a sample of 102 participants aged 18 to 61 years who completed the SMST with set sizes 1, 3, and 5. The same was tried for P3 latencies to investigate whether or not both parameters measure the same cognitive processes in the SMST, and to assess the usability of both slopes to predict 'g'. For RTs, a linear increase with set size was found. The slope of mean RTs correlated with 'g' on a manifest level already. The FLM approach could better reveal this relationship with the correlation between the slope and 'g' being substantially higher. For P3 latencies, we found no evidence for a linear increase, but a general increase from the smallest set size to the two larger ones. This indicates that RTs and P3 latencies do not measure the same cognitive processes in the SMST. The FLM proved suitable to investigate the association between the speed of short-term memory scanning and intelligence.

Non-linearity of a Boolean function indicates how far it is from any linear function. Despite there being several strong results about identifying a linear function and distinguishing one from a sufficiently non-linear function, we found a surprising lack of work on computing the non-linearity of a function. The non-linearity is related to the Walsh coefficient with the largest absolute value; however, the naive attempt of picking the maximum after constructing a Walsh spectrum requires $\Theta(2^n)$ queries to an $n$-bit function. We improve the scenario by designing highly efficient quantum and randomised algorithms to approximate the non-linearity allowing additive error, denoted $\lambda$, with query complexities that depend polynomially on $\lambda$. We prove lower bounds to show that these are not very far from the optimal ones. The number of queries made by our randomised algorithm is linear in $n$, already an exponential improvement, and the number of queries made by our quantum algorithm is surprisingly independent of $n$. Our randomised algorithm uses a Goldreich-Levin style of navigating all Walsh coefficients and our quantum algorithm uses a clever combination of Deutsch-Jozsa, amplitude amplification and amplitude estimation to improve upon the existing quantum versions of the Goldreich-Levin technique., Comment: Accepted in ACM Transactions on Quantum Computing

When convection is thermal, there is no insolation, and water is fresh, the upward heat flux on the surface of a natural reservoir can be presented as the product of the root-mean-squire deviation (RMSD) of water-surface temperature from its mean in the power of 4/3 and a polynomial function of the Prandtl number, where the latter corresponds to the bulk water temperature. The error of measuring the heat flux arising due to this representation is estimated. To be applicable in the sea, the proposed heat-flux representation is corrected, so, that to take into account water salinity, which becomes essential when thermal convection develops in cold water. In the Prandtl number range 4.4–7.2, the skewness of the surface temperature, which, along with its RMSD from the mean, can be measured by infrared imaging, is a linear function of the Prandtl number (Verevochkin [4], Int J Heat Mass Transfer 143:118426). It is shown that, in this case, the relative error of calculating the Prandtl number is less than occurring while measuring the skewness. A possibility of using the spatial-average water-surface temperature to calculate the Prandtl number is also considered, and the relative error arising here is estimated. Two heat-flux values corresponding to the same conditions, but obtained in different ways, are used as coordinates of a point. The first value measured calorimetrically is taken from literature sources, while the second one is calculated by the being proposed here formulae. The linear function fitted to the set of thirteen such points by the least square method has the slope 1.15 ± 0.11.

We consider the quantity P ( G ) associated with a graph G that is defined as the probability that a randomly chosen subtree of G is spanning. Motivated by conjectures due to Chin, Gordon, MacPhee and Vincent on the behaviour of this graph invariant depending on the edge density, we establish first that P ( G ) is bounded below by a positive constant provided that the minimum degree is bounded below by a linear function in the number of vertices. Thereafter, the focus is shifted to the classical Erdos-Renyi random graph model G ( n , p ). It is shown that P ( G ) converges in probability to e - 1 / ( e p infinity ) if p -> p infinity > 0 and to 0 if p -> 0. As side results, we find in the dense case that the total number of subtrees satisfies a log-normal limit law, and that the number of vertices missing in a random subtree asymptotically follows a Poisson distribution.

This study examined the generality of the result that linear or logarithmic functions best describe the relationship between numerical stimuli and people’s judgments about those stimuli. In Experiment 1, one group of participants was told that 2 was a perfect example of category x and 6 was a perfect example of category y; another group was told the same about 6 and 18. The participants then rated how well logarithmically spaced numbers matched the category x number. Exponential and logarithmic functions best fit the data of individual participants who did the “2 vs. 6” task; power functions did the same in the “6 vs. 18” task. When participants’ data were averaged, a logarithmic function was the best fit for the ratings produced by the “2 vs. 6” task; power and exponential functions were the best fits for the ratings produced by the “6 vs. 18” task. In Experiment 2, two groups of participants were given similar rules about two numbers (2 vs. 4 or 4 vs. 6) and then rated how well linearly spaced numbers matched the category x number. For both tasks, most individuals’ ratings best fit a linear function. When the data were averaged, though, a logarithmic function was the best fit for the ratings produced by both tasks. The results highlight the importance of presenting individuals’ data and suggest that global and local numerical contexts influence people’s judgments about the numbers. Stimulus generalization may be a mechanism by which the local influence occurs.

Coefficient of restitution is regarded as a dominating parameter in rockfall research. Generally, small-scale experiments were developed without considering interactions between boulder and slope. However, preimpact moving statuses are essential to evaluate rockfall behaviors. To reveal the effect of preimpact interactions on coefficient of restitution, energy dissipation considering initial velocity, surface type, and slope angle is executed based on medium-scale tests. The results show that (1) as the inclination of initial velocity, higher rebound height, and the declining normal coefficient of restitution occur, a determinable linear function could demonstrate relationships among energy dissipation and all coefficient of restitution; when initial velocity exceeds 5 m/s, the recovery ability shows and produces an increasing trend with respect to the variation of kinematic coefficient of restitution and kinetic energy coefficient of restitution. (2) As the surface material varies, slope hardness and rebound ability influence normal coefficient of restitution, and the surface roughness and rotation feature dominate tangential coefficient of restitution; considering preimpact slope and boulder interactions, four types of coefficient of restitution follow declining trend with different material sequence. (3) Slope angle affects normal coefficient of restitution, and tangential coefficient of restitution relatively descends 18% and inclines 10% when the angle ranges from 30° to 75°; regarding preimpact moving status, it differs from bounce times. The correlation between preimpact energy dissipation and four coefficients of restitution can be represented by the same decreasing linear function, when increasing the slope angle.

We propose a multithreshold change plane regression model which naturally partitions the observed subjects into subgroups with different covariate effects. The underlying grouping variable is a linear function of observed covariates and thus multiple thresholds produce change planes in the covariate space. We contribute a novel two-stage estimation approach to determine the number of subgroups, the location of thresholds, and all other regression parameters. In the first stage we adopt a group selection principle to consistently identify the number of subgroups, while in the second stage change point locations and model parameter estimates are refined by a penalized induced smoothing technique. Our procedure allows sparse solutions for relatively moderate- or high-dimensional covariates. We further establish the asymptotic properties of our proposed estimators under appropriate technical conditions. We evaluate the performance of the proposed methods by simulation studies and provide illustrations using two medical data examples. Our proposal for subgroup identification may lead to an immediate application in personalized medicine.

The article provides analytical expressions for the electromagnetic fields generated by a lightning return stroke characterized by a channel-base current with arbitrary time waveform, in presence of either a perfectly conducting or a lossy ground, assuming the transmission line model for the current along the channel. In this second case, a time-domain analytical expression for the Cooray–Rubinstein formula is presented. The main idea that leads to the derivation of analytical formulas consists of dividing the channel into intervals in which the distance between the field source point and the observation point can be approximated with a linear function of the time, and of the spatial coordinates of both points. In the companion aticle, a detailed comparison is proposed with the classical (numerical) approach highlighting an excellent agreement both at close and far distances, considering all the values of practical interest for the ground conductivity. Moreover, the method guarantees a meaningful improvement in computational performance.

In this paper, we study the optimal execution problem by considering the trading signal and the transaction risk simultaneously. We propose an optimal execution problem by taking into account the trading signal and the execution risk with the associated decay kernel function and the transient price impact function being of generalized forms. In particular, we solve the stochastic optimal control problems under the assumptions that the decay kernel function is the Dirac function and the transient price function is a linear function. We give the optimal executing strategies in state-feedback form and the Hamilton‐Jacobi‐Bellman equations that the corresponding value functions satisfy in the cases of a constant execution risk and a linear execution risk. We also demonstrate that our results can recover previous results when the process of the trading signal degenerates.

The Khrapchenko method of finding a lower bound for the complexity of binary formulas is extended to formulas in $$k$$ -ary bases. The resulting extension makes it possible to evaluate the complexity of linear Boolean functions and a majority function of $$n$$ variables when realized by formulas in the basis of all $$k$$ -ary monotone functions and negation as $$\Omega(n^{g(k)})$$ , where $$g (k)=1+\Theta(1/\ln k)$$ . For a linear function, the complexity bound in this form is unimprovable. For $$k=3$$ , the sharper lower bound $$\Omega(n^{1.53})$$ is proved.

The production function has always been an active field in the research and application of economics. Researchers generally use the production function of constant elasticity of substitution, i.e., the C–D (Cobb–Douglas) production function and the CES (Constant Elasticity of Substitution) production function, to explore economic growth, but rarely use the VES (Variable Elasticity of Substitution) production function. The paper gives an extended VES production function model in the case the elasticity of substitution is a time linear function. With regard to the parameter estimation, the paper uses a modern intelligent algorithm, i.e., the PSO (Particle Swarm Optimization) algorithm. The conventional PSO algorithm has a low rate and the poor accuracy of convergence, so the paper proposes an improved PSO algorithm. In the final part, the paper calculates the contribution rates of influencing factors to Chinese economic growth, and the results are consistent with the actual situation.

We analyze the chirality of dispersion relations and propagation lengths of two fundamental modes (m=0 and 1) in single silver nanowire coated by chiral TDBC. The systems of both Ag nanowires only coated by left TDBC (ALM) and right TDBC (ARM) are considered, respectively. Finite-difference time-domain method is used to do the mode analysis. The results show that the dispersion relations curves of both ALM and ARM are similar to magnetic hysteresis loops. There is a dip in the propagation length of ALM or ARM versus the incident frequency at the resonant frequency of left TDBC or right TDBC. The propagation length is reduced shapely in chiral TDBC due to the dissipation of TDBC. The difference of the propagation length $$|\Delta L|$$ becomes larger with the incident frequency far away from the resonant frequency of chiral TDBC for these two modes. The maximum difference of the real part of the effective index between ALM and ARM ( $$\Delta n'$$ ) is the linear function of m and the radius of Ag nanowire. With the radius decreasing from 100nm to 10nm, the propagation length for m=0 decreases and the propagation length for m=1 increases. Also, the maximum difference of the propagation length for m=1 increases. The maximum normalized difference between Ag nanowires coated by left TDBC and right TDBC in both the dispersion relation for $$m=0$$ (0.024) and the propagation length for $$m=1$$ (0.16) are quite larger than the one of only chiral TDBC (1/3000). According to our analysis, chiral molecules can be distinguished in either the dispersion relation or the propagation length, and our proposed structure has potential applications in chiral molecules detecting.

We consider a system of $$n$$ th-order ordinary differential equations whose right-hand side is the sum of a linear function of the solution with a constant matrix, an essential nonlinearity of the relay type with hysteresis, and a perturbing continuous periodic function. The matrix of the linear function has only real simple nonzero eigenvalues, of which at least one is positive. We study the question of whether such systems have continuous solutions with two switching points in the state space (two-point oscillatory solutions) such that the time in which the solution returns to each of these points coincides with the period of the perturbing function or is an integer fraction of the latter. A sufficient condition for the nonexistence of such solutions is established, and a theorem is proved that gives sufficient conditions for the existence of a two-point oscillatory solution with return time equal to the period of the perturbing function. A corroborating example is given.

The generalized variable step-size diffusion continuous mixed p-norm (GVSS-DCMPN) algorithm is proposed in this paper, which is derived based on the improved continuous mixed p-norm (CMPN) strategy. In detail, a linear function is designed for the CMPN strategy. The proposed GVSS-DCMPN algorithm capable of exploiting various error norms to obtain performance improvement in non-Gaussian noise environment can be viewed as the generalization of the traditional p-norm-based algorithms in the sense of continuous errors. In particular, the GVSS-DCMPN algorithm transforms into the VSS-DCMPN algorithm when the slope of the linear function is set to 0. The computational complexity as well as the mean convergence is analyzed in the paper. Simulation results over the diffusion network show that the proposed algorithm achieves performance gain over some existing diffusion algorithms.

In this paper, we consider data driven control of Hammerstein systems. For such systems a common control structure is a transfer function followed by a static output nonlinearity that tries to cancel the input nonlinearity of the system, which is modeled as a polynomial or piece-wise linear function. The linear part of the controller is used to achieve desired disturbance rejection and tracking properties. To design a linear part of the controller, we propose a weighted average risk criterion with the risk being the average of the squared L2 tracking error. Here the average is with respect to the observations used in the controller and the weighting is with respect to how important it is to have good control for different impulse responses. This criterion corresponds to the average risk criterion leading to the Bayes estimator and we therefore call this approach Bayes control. By parametrizing the weighting function and estimating the corresponding hyperparameters we tune the weighting function to the information regarding the true impulse response contained in the data set available to the user for the control design. The numerical results show that the proposed methods result in stable controllers with performance comparable to the optimal controller, designed using the true input nonlinearity and true plant.

Semidefinite programming (SDP) is the task of optimizing a linear function over the common solution set of finitely many linear matrix inequalities (LMIs). For the running time of SDP solvers, the ...

In this work, we prove that the ratio of torsion and curvature of any dual rectifying curve is a non-constant linear function of its dual arc length parameter. Thereafter, a dual differential equation of third order is constructed for every dual curve. Then, several well-known characterizations of dual spherical, normal and rectifying curves are consequences of this differential equation. Finally, we prove a simple new characterization of dual spherical curves in terms of the Darboux vector.

When using the Cobb-Douglas (C-D) production function to measure the contribution rate of agricultural technological progress (ATP), it is necessary to estimate the output elasticity coefficient (OEC) of each input factor in C-D production function. For this purpose, it is usually necessary to take logarithm at both sides of C-D production function and convert it into a linear function, and then use regression analysis method to estimate the OECs of input factors. However, there are some problems in this method remains unsolved: first, the OECs estimated by taking logarithm of C-D production function are not the optimal estimation of the original C-D production function; second, the regression results sometimes fail to pass statistical test; third, some OECs cannot be guaranteed to be non-negative. Aiming at resolving these problems, a method for estimating OECs in C-D production function based on the Hybrid Improved Bat Optimization Algorithm (HIBA) was proposed. This method solves the problems existing in the OEC estimation methods in the existing literatures. To verify the effectiveness of the proposed method in this study, the OECs of input factors in China’ Sichuan Province from 1996 to 2018 was estimated. The estimation results show that, compared with other estimation methods in the existing literatures, the proposed method can not only guarantee that it is the optimal estimation of the OECs in the original C-D production function, but also ensure that the OECs are non-negative and with high precision and good fitting effect. Finally, based on the estimation results, this study measured and analyzed the contribution rate of agricultural input factors and ATP of Sichuan Province and puts forward corresponding suggestions for the agricultural development in this region.

Problems of differential equations construction of equilibrium of a geometrically and physically nonlinear continuous medium under conditions of one-dimensional plane deformation are considered, when the diagrams of volumetric and shear deformation are approximated by quadratic functions. The construction of physical dependencies is based on calculating the secant moduli of volumetric and shear deformation. When approximating the graphs of the volumetric and shear deformation diagrams using two segments of parabolas, the secant shear modulus in the first segment is a linear function of the intensity of shear deformations, the secant modulus of volumetric expansion - contraction is a linear function of the first invariant of the strain tensor. In the second section of the diagrams of both volumetric and shear deformation, the secant shear modulus is a fractional (rational) function of the shear strain intensity, the secant modulus of volumetric expansion - compression is a fractional (rational) function of the first invariant of the strain tensor. Based on the assumption of independence, generally speaking, from each other of the volumetric and shear deformation diagrams, six main cases of physical dependences are considered, depending on the relative position of the break points of the graphs of the diagrams volumetric and shear deformation, each approximated by two parabolas. The differential equations of equilibrium in displacements constructed in the article can be applied in determining the stressed and deformed state of a continuous medium under conditions of one-dimensional plane deformation, the closing equations of physical relations for which, constructed on the basis of experimental data, are approximated by biquadratic functions.

This paper is concerned with global stabilization of discrete-time multiple integrators with bounded controls by utilizing the energy function based approach. First, a discrete-time double integrators system subject to input additive disturbances is stabilized by a bounded feedback whose formula involves a linear function of the state and a saturation function only. Next, this result is used to stabilize a discrete-time multiple integrators of arbitrary length by bounded control with the aid of a special canonical form. Compared with the existing results, the proposed controllers require fewer saturation functions, which allow a better use of the control energy. Moreover, some free parameters that are introduced into these controllers can help improve the transient performance of the closed-loop systems significantly. Finally, a numerical example is given to illustrate the effectiveness of the proposed method.

A study has been made of the pressure field in filtration of a weakly compressible fluid in a homogeneous isotropic compressible medium at high-amplitude perturbations. The equation used in formulating the problem is nonlinear: the densities of a porous medium and of a saturating fluid are assumed to be dependent on the function sought. Use is made of the fact that the dependences of the densities of the fluid and the skeleton material on pressure are approximated with a high degree of accuracy by the linear function. Consideration is given to one-dimensional plane horizontal filtration. The porosity and permeability of a porous medium, and also the viscosity of a filtering medium, are considered to be constant. A solution to the problem has been found with asymptotic expansions where a cofactor of the compressibility factor of the liquid acts as a small parameter. Analytical expressions have been found for the zero and first expansion factors. It has been shown that the zero expansion factor may be used to investigate the evolution of the pressure field of an incompressible liquid, whereas the expression for the first factor contains information on the contribution of nonlinearity due to the fluid’s compressibility. Values of the zero and first residual terms have been determined. It has been proved that the zero and first residual terms contain terms of only higher orders as far as the asymptotic-expansion parameter is concerned, i.e., the corresponding requirement of asymptoticity of expressions of the zero and first factors is met. On the basis of a computational experiment, the regularities of the dependence of the contribution of the nonlinearity under study on time and on the spatial coordinate have been established.

In this paper, we investigate different activation functions (AFs) on convergence performance of a gradient-based neural network (GNN) for solving linear matrix equation, AXB + X = C . It is observed that, by employing different AFs, i.e., linear, power-sigmoid, sign-power, and general sign-bi-power functions, the presented GNN model can achieve different convergence performance. More specifically, if linear function is employed, the GNN model can achieve exponential convergence; if the power-sigmoid function is employed, superior convergence can be achieved as compared to the linear case; while if the sign-power and general sign-bi-power functions are employed, the GNN model can achieve finite- and fixed-time convergence, respectively. Detailed theoretical proofs are offered to demonstrate these facts. Besides, the exponential convergence rate and the upper bounds of finite and fixed convergence time are also theoretically estimated. Finally, two illustrative examples are performed to further substantiate the aforementioned theoretical results and the effectiveness of the presented GNN model for solving the linear matrix equation.

Ping’ou hybrid hazelnut is produced by cross cultivation and is widely cultivated in northern China with good development prospects. Based on a field experiment of fertilizer efficiency, the leaf spectral reflectance and leaf potassium (K) concentration were measured with different quantities of K fertilizer applied at four fruit growth stages (fruit setting stage, fruit rapid growth stage, fruit fat-change stage, and fruit near-maturity stage) of Ping’ou hybrid hazelnut in 2019. Spectral parameters that were significantly correlated with leaf K concentration were selected using Pearson correlation analysis, and spectral parameter estimation models of leaf K concentration were established by employing six different modelling methods (exponential function, power function, logarithmic function, linear function, quadratic function, and cubic function). The results indicated that at the fruit setting period, leaf K concentration was significantly correlated with Dy (spectra slope of yellow edge), Rg (reflectance of the green peak position), λo (red valley position), SDb (blue edge area), SDr/SDb (where SDr represents red edge area), and (SDr−SDb)/(SDr+SDb) (P

In this paper, weighted sum energy efficiency (WSEE) in uplink and downlink of a multi-user non-orthogonal multiple access (NOMA) system is considered. we adopt a more realistic power consumption model where signal processing power is modeled as a linear function of transmit power. Rather than the well-known network-centric global energy efficiency (GEE) metric, which is a pseudo-concave (PC) function, the weighted sum energy efficiency metric is considered, which is not PC in general. To find the optimum user power allocation Dinkelbach-like algorithm is adopted, by which each individual fractional EE function is converted to a parametric function where under some conditions on the weights falls into a class of convex optimization problems and it is solved in the dual domain. The dual variables are updated using a sub-gradient and cutting plane-based algorithm, which here ellipsoid method is used. Since the optimum solution restricts user weights, a low complexity suboptimum algorithm that does not consider any condition on user's weights is proposed. The problem is non-convex in general; hence, epigraph form followed by successive convex approximation (SCA) is used to deal with that problem. Results demonstrate that with the user-oriented metric, one can provide different priorities to users, and by choosing proper weights entail fairness among users.

In this paper, we consider some unusual features of dynamical regimes in the non-smooth potential $$V(x)=|x|$$ which is a piece-wise linear function. Also, we consider the dynamics in more complicated potential $$V(x)=\left| |x|-a\right| $$ which is quite similar to the well-known double-well potential within the Duffing model. Numerical results for Poincare sections, bifurcation diagrams, and Lyapunov spectra together with dependencies of the largest Lyapunov characteristic exponent on the parameters of the excitation force are also obtained and analyzed. A comparison of the proposed systems and the Duffing model with the same fixed points is also done. Our numerical results show that such a relatively simple oscillatory system has rich nonlinear dynamics and exhibits a conservative character of chaos. This makes it possible to consider these systems as promising sources of chaotic signals in the field of modern chaos-based information technologies and digital communications.

Surgical invasion activates nociception, while anesthesia suppresses it. Under general anesthesia, stimulation, which is the balance between nociception and anti-nociception, induces responses, including activation of the autonomic nervous system. To evaluate the associations between stimulation (S) and the resultant responses (R), we examined R values, which were calculated using mathematical models of Stevens’ power law, Gompertz function and logistic function. The previously developed Nociceptive Response (NR) formula was applied as a modified logistic model. S values were calculated using a linear function in the NR formula. In a retrospective study, we developed an exponential model of Stevens’ power law and a sigmoidal model of Gompertz function using differential equations, by adjusting R values to correspond to NR values, in consecutive patients undergoing surgery under general anesthesia (n = 4,395). In a subsequent prospective study, we validated the superiority of R values of Gompertz function and the NR formula in an exponential model in adult patients undergoing tympanoplasty (n = 141) and laparoscopic cholecystectomy (n = 86). In conclusion, both modified logistic function and Gompertz function are likely appropriate mathematical models for representing responses to stimulation resulting from the balance between nociception/anti-nociception during surgical procedures under general anesthesia.

The paper deals with estimating the r -colorability threshold for a random k -uniform hypergraph in the binomial model H ( n , k , p ) . We consider the sparse case, when the expected number of edges is a linear function of n , p n k = c n , and c > 0 , k ⩾ 4 , r ⩾ 3 are some fixed constants. We show that if c r k − 1 ln r − 1 2 ln r − r − 1 r + O ( k 2 r 1 − k ∕ 3 ln r ) then H ( n , k , c n ∕ n k ) is r -colorable with probability tending to 1 as n → ∞ .

Combinatorial optimization problems and methods of their solution have been a subject of numerous studies, since a large number of practical problems are described by combinatorial optimization models. Many studies consider approaches to and describe methods of solution for combinatorial optimization problems with linear or fractionally linear target functions on combinatorial sets such as permutations and arrangements. Studies consider solving combinatorial problems by means of well-known methods, as well as developing new methods and algorithms of searching a solution. We describe a method of solving a problem of a linear target function localization on a permutation set. The task is to find those locally admissible permutations on the permutation set, for which the linear function possesses a given value. In a general case, this problem may have no solutions at all. In the article, we propose a newly developed method that allows us to obtain a solution of such a problem (in the case that such solution exists) by the goal-oriented seeking for locally admissible permutations with a minimal enumeration that is much less than the number of all possible variants. Searching for the solution comes down to generating various permutations and evaluating them. Evaluation of each permutation includes two steps. The first step consists of function decreasing by transposing the numbers in the first n – 3 positions, and the second step is evaluation of the permutations for the remaining three numbers. Then we analyze the correlation (which is called balance) to define whether the considered permutation is the solution or not. In our article, we illustrate the localization method by solving the problem for n = 5. Keywords: localization, linear function, permutation, transposition, balance, position.

New calibration method of the capacitance and frequency domain reflectometry (ECH2O EC-5) for soil water monitoring was developed by introducing the slope-intercept relationship derived from the re...

We consider n-bit permutations with differential uniformity of 4 and null nonlinearity. We first show that the inverses of Gold functions have the interesting property that one component can be replaced by a linear function such that it still remains a permutation. This directly yields a construction of 4-uniform permutations with trivial nonlinearity in odd dimension. We further show their existence for all n = 3 and n ≥ 5 based on a construction in Alsalami (Cryptogr. Commun. 10(4): 611–628, 2018). In this context, we also show that 4-uniform 2-1 functions obtained from admissible sequences, as defined by Idrisova in (Cryptogr. Commun. 11(1): 21–39, 2019), exist in every dimension n = 3 and n ≥ 5. Such functions fulfill some necessary properties for being subfunctions of APN permutations. Finally, we use the 4-uniform permutations with null nonlinearity to construct some 4-uniform 2-1 functions from $\mathbb {F}_{2}^{n}$ F 2 n to $\mathbb {F}_{2}^{n-1}$ F 2 n − 1 which are not obtained from admissible sequences. This disproves a conjecture raised by Idrisova.

Based on the method for establishing a global plasmaspheric model using observations from COSMIC and MetOp-A orbit determination GNSS receivers, Chen et al. (2017) obtained a global plasmaspheric total electron content product with a spatial resolution of 2.5° × 5° and a time resolution of 4 h. In this paper, we use those global plasmaspheric electron content product in 2008, 2010, 2011, 2012, 2014 for 1446 days to establish a global plasmaspheric empirical model based on empirical orthogonal function (EOF). The model can well characterize the spatiotemporal variation of plasmaspheric electron content (PEC) and the influence of solar radiation on it. Only the first four orders of EOF sequences can characterize the 98.43% features of the original PEC dataset. The principal component coefficient Pk is decomposed twice during modeling, and the combination of trigonometric function and linear function is used to model Pk to characterize the solar cycle, annual cycle, semi-annual cycle and quarter-cycle variation. We compare the PEC model values with the actual observation data, the results show that the empirical PEC model values are highly correlated with the actual observations. The correlation between the two is above 0.96, and the RMS maximum of the difference between the PEC model values and the observed values are 0.70 TECU, and the average of the difference between the PEC model values and the observed values are −0.18 TECU, respectively. In addition, we validate the reliability of the global plasmaspheric model established by two empirical orthogonal function decomposition method using actual observation data, according to the global distribution of the differences between the PEC model values and the observed values in low solar activity and high solar activity, it can be seen that under low solar activity and high solar activity conditions, the model has good adaptability.

Comparative analyze of the methods of interpolation of dependence of electron beam radius on position of cutting plane by the longitudinal coordinate with using three basic point have been provided in the article. It is shown, that main particularity of analytical functions, which are described such dependences, is that they have one global minimum, and outside the minimum region the character of such dependences is similar to linear. Two different approaches to interpolation of such dependences are considered. First method is based on the assumption, that in the region of beam focus the parabolical interpolation is used, and outside the focal region used the linear function. For describing of such dependences, the apparat of mathematical logic and arithmetic-logic expressions have been used. The main disadvantage of such describing of boundary trajectories of electron beams, propagated in the ionized gas, is that it is usually difficult to find the basic point, where the transfer from linear to parabolical dependence have to be realized. Another method of interpolation is based on the use of class of special functions, which are defined as the roots of n power from the polynomials of the same power. It is shown with using the method of computer simulation, that the behavior of such functions is generally corresponded to the ravine dependence, which described the trajectories of electron beams, propagated in the ionized gas. It is shown, that with using of such interpolation functions the difference between the simulation results, which are interpolated, and the values of interpolation function, always isn’t grater, than few percent, but in the most cases it consist the part of percent. Therefore, it is proved on the base of testing computer experiments, that using as the interpolation functions the roots of n power from the polynomial’s of the same power is the effective instrument for describing the boundary trajectories of short-focus electron beams in ionized gas. The obtained results are very important for the solution the practical task of defining the focal parameters of electron beams, generated by the electron sources, based on the high voltage glow discharge.

This research aims to determine the mathematical connection ability of junior high school students in Bandung. This research uses descriptive qualitative method with data triangulation. The instrument used in the form is test description as many as three questions that have been consulted with experts and refers to indicators of mathematical connection ability. The results of this research describe 1) the ability of students to connect between mathematical concepts with a percentage of 58.33%, 2) the ability of students to connect and apply mathematical concepts in other disciplines of sciences with a percentage of 50%, 3) the ability of students to apply mathematical concepts to solving problems in daily life with a percentage of 66.67%. The results showed that the mathematical connection ability of junior high school students in Bandung on the topic of linear function was good enough.

A linear control problem is considered in the presence of an uncontrolled disturbance. It is only known that the values of the disturbance belong to a given connected compact set. The terminal time of the control process is fixed. The terminal component of the payoff depends on the modulus of a linear function of the phase variables, and the integral component is given by an integral of a power of the control. We admit the possibility of one breakdown leading to a change in the dynamics of the control process. The time of the breakdown is not known in advance. The construction of the control is based on the principle of minimization of the guaranteed result. The opponents are the disturbance and the time of the breakdown. Necessary and sufficient conditions for the optimality of an admissible control are found.

One of the most widely verified empirical regularities of ecology is Taylor’s power law of fluctuation scaling, or simply Taylor’s law (TL). TL says that the logarithm of the variances of a set of random variables or a set of random samples is (exactly or approximately) a linear function of logarithm of the means of the corresponding random variables or random samples: logvariance = log a + b log mean, a > 0. Ecologists have argued about the interpretation of the intercept log a and slope b of TL and about what the values of these parameters reveal about the underlying probability distributions of the random samples. We show here that the form and the values of the parameters of TL and of any other variance function (relationship of variance to mean in a set of samples or a family of random variables) say nothing whatsoever about the underlying probability distributions of the random samples (or random variables) other than that they have finite mean and variance. Specifically, given any real-valued random variable with a finite mean and a finite variance, and given any variance function (e.g., TL with specified intercept log a and slope b), we construct a family of random variables with probability distributions of the same shape as the probability distribution of the given random variable (i.e., that are the same up to location and scale, or in the same “location-scale family”) and that obeys the given variance function exactly (e.g., TL exactly with the given intercept log a and slope b). Every variance function can be produced by the location-scale family of any random variable with finite positive mean and finite positive variance. We illustrate some consequences of these findings by examples (e.g., for presence-absence sampling in agricultural pest control).

We consider the the implementation of Boolean functions by circuits from unreliable functional elements in a complete finite basis, which contains a linear function essentially dependent on at least two variables. We assume that all elements of a circuit are exposed to the faults of type 0 at the outputs with probability e ∈ (0, 1/2) independently of each other. We prove that almost any Boolean function can be implemented by an asymptotically optimal in reliability circuit functioning with the unreliability which is asymptotically equal to e with e → 0.

Under consideration is the possibility of constructing an almost periodic solution for one inhomogeneous system of linear differential equations with delay that is a linear function of the argument (time) under some assumptions about the right-hand side of the system. This solution is proved to be asymptotically stable. Also, we study the existence of an almost periodic solution to another system without neutral terms; in this event the solution is stable.