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In this study, we conducted a basic investigation involving the discrimination of eye movement condition (peripheral and central vision) using deep learning techniques. The subjects were 6 males aged 21–23 years. They watched two three-minute videos for central vision and peripheral vision in a random order for a total of eight sessions (four sessions each). The subjects wore an eye movement measurement device, and their eye movements (viewing angles) during the viewing of each video were continuously. From the time series data for eye movement, with four different lengths (0.5 s, 1 s, 2 s, 3 s) and shift length of 0.5 s, short time series data for each 3 min was obtained in sets of 350, and the data were utilized for deep learning and its evaluation. For the deep learning model, input nodes according to data length were placed in the input layer. For the middle layer, seven to eight units were put in place that brought together the one-dimensional convolution layer, the batch-normalization layer, normalized linear function, and the max-pooling layer. The output layer consisted of the fully-connected layer, sigmoid function, and multi-class cross-entropy. As a result, the accuracy of the discrimination was improved as the data length increased, and it was possible to determine the condition with an accuracy of over 90% if the eye movement data was at least one second.

This paper addresses the minmax regret 1-sink location problem on dynamic flow path networks with parametric weights. We are given a dynamic flow network consisting of an undirected path with positive edge lengths, positive edge capacities, and nonnegative vertex weights. A path can be considered as a road, an edge length as the distance along the road and a vertex weight as the number of people at the site. An edge capacity limits the number of people that can enter the edge per unit time. We consider the problem of locating a sink in the network, to which all the people evacuate from the vertices as quickly as possible. In our model, each weight is represented by a linear function in a common parameter t, and the decision maker who determines the location of a sink does not know the value of t. We formulate the sink location problem under such uncertainty as the minmax regret problem. Given t and a sink location x, the cost of x under t is the sum of arrival times at x for all the people determined by t. The regret for x under t is the gap between the cost of x under t and the optimal cost under t. The task of the problem is formulated as the one to find a sink location that minimizes the maximum regret over all t. For the problem, we propose an \(O(n^4 2^{\alpha (n)} \alpha (n) \log n)\) time algorithm where n is the number of vertices in the network and \(\alpha (\cdot )\) is the inverse Ackermann function. Also for the special case in which every edge has the same capacity, we show that the complexity can be reduced to \(O(n^3 2^{\alpha (n)} \alpha (n) \log n)\).

The least squares method is widely used in statistics with common application to regression. The aim is to fit a linear function to set of points that minimizes the sum of the squares of the residuals. In some cases, fitting a linear function with a relatively small error is impossible. Keeping the linear character of approximation, the data points can be split into a sequence of segments, where to each of the segments the line given by a linear function is fitted. The optimization objectives for this problem is to minimize the total least squares error for all segments and to minimize the number of segments used. We refer to such a problem as segmented least squares.

Researchers have thoroughly studied decomposition in many different contexts, such as switching theory or data mining. In recent years, a renewed interest in this problem was caused by memory-based pattern matching circuits, particularly in the synthesis of Index Generation Functions. In this case, function is a composition of a linear function L and a general function G. The function L is implemented using ExOR gates, while G is usually realized using embedded memories.

An inventory model for a single deteriorating item under fuzzy environment has been presented in this paper. Here demand rate is considered to be constant for some time period, post which the same is a linear function of time. This situation is common during the time of a new product launch in the market. As the product becomes popular, its demand increases with time although it remains constant during the initial days. Cycle time is considered to be constant in most of the models. However, practically it has been observed that it is difficult to pro-actively predict the cycle time. Because of this problem, cycle time has been considered as uncertain and has been further described as Symmetric Triangular Fuzzy number. The Signed Distance method has been used for defuzzification of the total cost function. For illustration of the process for finding the total optimal cost and the cycle time, numerical examples have been considered. The effects of changing parameter values on the optimal solution of the system have been demonstrated through Sensitivity Analysis.

TD(\(\lambda \)) has become a crucial algorithm of modern reinforcement learning (RL). By introducing the trace decay parameter \(\lambda \), TD(\(\lambda \)) elegantly unifies Monte Carlo methods (\(\lambda =1\)) and one-step temporal difference prediction (\(\lambda =0\)), which can learn the optimal value significantly faster than extreme cases with an intermediate value of \(\lambda \). However, it is mainly used in tabular or linear function approximation cases, which limits its practicality in large-scale learning and prevents it from adapting to modern deep RL methods. The main challenge of combining TD(\(\lambda \)) with deep RL methods is the “deadly triad” problem between function approximation, bootstrapping and off-policy learning. To address this issue, we explore a new deep multi-step RL method, called SAC(\(\lambda \)), to relieve this dilemma. Firstly, our method uses a new version of Soft Actor-Critic algorithm which stabilizes the learning of non-linear function approximation. Secondly, we introduce truncated TD(\(\lambda \)) to reduce the impact of bootstrapping. Thirdly, we further use importance sampling as the off-policy correction. And the time complexity of the training process can be reduced via parallel updates and parameter sharing. Our experimental results show that SAC(\(\lambda \)) can improve the training efficiency and the stability of off-policy learning. Our ablation study also shows the impact of changes in trace decay parameter \(\lambda \) and emerges some insights on how to choose an appropriate \(\lambda \).

Consider a set-associative cache with $p^n$ sets and $p^n$ ways where $p$ is prime and $n>0$. Furthermore, assume that the cache may be shared among $p^n$ mutually distrusting principals that may use the Prime+Probe side-channel attack against one another; architecturally, these principals occupy separate security domains (for example, separate processes, virtual machines, sandboxes, etc.). This paper shows that there exists a linear skewing of cache sets over the Galois field $G_{p^n}$ that exhibits the following property: each cache set of each security domain intersects every cache set of every other security domain exactly once. Therefore, a random eviction from a single cache set in security domain $A$ may be observed via Prime+Probe in any of security domain $B$'s cache sets. This paper characterizes this linear skewing and describes how it can be implemented efficiently in hardware., 8 pages, 5 figures

We study evolutionary algorithms in a dynamic setting, where for each generation a different fitness function is chosen, and selection is performed with respect to the current fitness function. Specifically, we consider Dynamic BinVal, in which the fitness functions for each generation is given by the linear function BinVal, but in each generation the order of bits is randomly permuted. For the \((1 + 1)\)-EA it was known that there is an efficiency threshold \(c_0\) for the mutation parameter, at which the runtime switches from quasilinear to exponential. Previous empirical evidence suggested that for larger population size \(\mu \), the threshold may increase. We prove rigorously that this is at least the case in an \(\varepsilon \)-neighborhood around the optimum: the threshold of the \((\mu + 1)\)-EA becomes arbitrarily large if the \(\mu \) is chosen large enough.

The Ring Learning with Errors (RLWE) problem has become one of the most widely used cryptographic assumptions for the construction of modern cryptographic primitives. Most of these solutions make use of power-of-two cyclotomic rings mainly due to its simplicity and efficiency. This work explores the possibility of substituting them for multiquadratic rings and shows that the latter can bring about important efficiency improvements in reducing the cost of the underlying polynomial operations. We introduce a generalized version of the fast Walsh-Hadamard Transform which enables faster degree-n polynomial multiplications by reducing the required elemental products by a factor of $\mathcal{O}(\log n)$. Finally, we showcase how these rings find immediate application in the implementation of OLE (Oblivious Linear Function Evaluation) primitives, which are one of the main building blocks used inside Secure Multiparty Computation (MPC) protocols.

A novel Radon–Nikodym approach to machine learning (ML) is developed, implemented numerically, tested on various datasets, and it’s parallelization is discussed. It consists in building a “wavefunction” ψ(x), a linear function on input attributes x then constructing a classificator of the form ⟨ψ2f⟩/⟨ψ2⟩. The solution is obtained from a generalized eigenproblem |f|ψ[i]⟩ = λ[i]|ψ[i]⟩ with left– and right– hand side matrices calculated from the input dataset. A solution to classification problem (predict f on an unseen x) is found without using a norm testing how predicted f differs from the one observed. Possible outcomes and their probabilities are obtained separately using the Lebesgue quadrature technique, this makes the method robust to input data with outliers, spikes, etc. A remarkable feature is the knowledge of the invariant group (what input attributes transforms do not change the solution). Radon–Nikodym approach is typically slower than the other methods used, this is the cost of being a “norm–free”. A parallel implementation is expected to improve this deficiency.

Artificial neural networks (ANNs) constitute a promising modeling approach that may be used in control systems for postharvest preservation and storage processes. The study investigated the ability of multilayer perceptron and radial-basis function ANNs to predict fungal population levels in bulk stored rapeseeds with various temperatures (T = 12&ndash, 30 &deg, C) and water activity in seeds (aw = 0.75&ndash, 0.90). The neural network model input included aw, temperature, and time, whilst the fungal population level was the model output. During the model construction, networks with a different number of hidden layer neurons and different configurations of activation functions in neurons of the hidden and output layers were examined. The best architecture was the multilayer perceptron ANN, in which the hyperbolic tangent function acted as an activation function in the hidden layer neurons, while the linear function was the activation function in the output layer neuron. The developed structure exhibits high prediction accuracy and high generalization capability. The model provided in the research may be readily incorporated into control systems for postharvest rapeseed preservation and storage as a support tool, which based on easily measurable on-line parameters can estimate the risk of fungal development and thus mycotoxin accumulation.

Hick's law, one of the few law-like relationships involving human performance, expresses choice reaction time as a linear function of the mutual information between the stimulus and response events. However, since this law was first proposed in 1952, its validity has been challenged by the fact that it only holds for the overall reaction time (RT) across all the stimuli, and does not hold for the reaction time (RTi ) for each individual stimulus. This paper introduces a new formulation in which RTi is a linear function of (1) the mutual information between the event that stimulus i occurs and the set of all potential response events and (2) the overall mutual information for all stimuli and responses. Then Hick's law for RT follows as the weighted mean of each side of the RTi equation using the stimulus probabilities as the weights. The new RTi equation incorporates the important speed-accuracy trade-off characteristic. When the performance is error-free, RTi becomes a linear function of two entropies as measures of stimulus uncertainty or unexpectancy. Reanalysis of empirical data from a variety of sources provide support for the new law-like relationship.

This paper aims to disclose the effects of the long-term creep of an expressway slope on project safety. Firstly, the displacements at multiple points of the slope were monitored by multi-point displacement meters. The displacement data were decomposed through time series analysis into a trend term and a period term. A creep fitting model of the slope is proposed by mathematical modeling. Then, the linear function and periodic function were combined to fit the creep curves and predict the long-term creep of the slope. The results show that the combined fitting function has a high accuracy and applies well to the prediction of long-term creep of the slope; the target slope will continue to creep in future and should be reinforced after 40 years into operation. The research results shed new light on the slope safety of expressways.

An $L$-estimator is a form of linear function of order statistics. Xiang [8] investigated the bias of this estimator. In this investigation, we derive a formula for the mean square error of the smooth $L$-estimator.

The success of deep learning relies on the availability of large-scale annotated data sets, the acquisition of which can be costly requiring expert domain knowledge. Semi-supervised learning (SSL) mitigates this challenge by exploiting the behavior of the neural function on large unlabeled data. The smoothness of the neural function is a commonly used assumption exploited in SSL. A successful example is the adoption of mixup strategy in SSL that enforces the global smoothness of the neural function by encouraging it to behave linearly when interpolating between training examples. Despite its empirical success, however, the theoretical underpinning of how mixup regularizes the neural function has not been fully understood. In this paper, we offer a theoretically substantiated proposition that mixup improves the smoothness of the neural function by bounding the Lipschitz constant of the gradient function of the neural networks. We then propose that this can be strengthened by simultaneously constraining the Lipschitz constant of the neural function itself through adversarial Lipschitz regularization, encouraging the neural function to behave linearly while also constraining the slope of this linear function. On three benchmark data sets and one real-world biomedical data set, we demonstrate that this combined regularization results in improved generalization performance of SSL when learning from a small amount of labeled data. Our code is available at https://github.com/Prasanna1991/Mixup-LR.

Activation function is a crucial component in artificial neural networks for its capability of converting linear function of input to complex nonlinear expression. It also plays an important role generating enhancement nodes in broad learning system(BLS). In this paper, we perform the comparison of 20 popular activation functions on different datasets in classification and regression. Among all selected activation functions, sigmoid leads to faster training process and greater approximation capability than others in general tasks. Meanwhile, the statistical analysis demonstrates that the type of activation function does not affect the performance of BLS too much. Afterwards, we assemble some best-performing activation functions to form a combination within convex restriction, which achieves better performance than corresponding base activation functions in standard BLS.

We consider a nonautonomous system \[ \dot x=A(t)x+f(t,x,y),\quad \dot y = g(t,y)\] and give conditions under which there is a transformation of the form $H(t,x,y)=(x+h(t,x,y),y)$ taking its solutions onto the solutions of the partially linearized system \[ \dot x=A(t)x,\quad \dot y = g(t,y).\] Shi and Xiong \cite{SX} proved a special case where $g(t,y)$ was a linear function of $y$ and $\dot x=A(t)x$ had an exponential dichotomy. Our assumptions on $A$ and $f$ are of the general form considered by Reinfelds and Steinberga \cite{RS}, which include many of the generalizations of Palmer's theorem proved by other authors. Inspired by the work of Shi and Xiong, we also prove H\" older continuity of $H$ and its inverse in $x$ and $y$. Again the proofs are given in the context of Reinfelds and Steinberga but we show what the results reduce to when $\dot x=A(t)x$ is assumed to have an exponential dichotomy. The paper is concluded with the discrete version of the results., Revised version. Accepted for publication in the Journal of Differential Equations

The Holt–Klee theorem says that the graph of a d -polytope, with edges oriented by a linear function on P that is not constant on any edge, admits d independent monotone paths from the source to the sink. We prove that the digraphs obtained from oriented matroid programs of rank d + 1 on n + 2 elements, which include those from d -polytopes with n facets, admit d independent monotone paths from source to sink if d ≤ 4 . This was previously only known to hold for d ≤ 3 and n ≤ 6 .

Volt-Amp profiles are used to describe electrodialysis (ED) dynamics when the concentration of the diluate is equal to the concentration of the concentrate and both are at high concentration. A standard electronic model that treats the ED process as a series of resistors failed to describe the performance of a 10-cell pair, 200 cm2 per membrane pilot electrodialysis unit treating 0.5–6% NaCl. The standard electronic model resulted in a regression coefficient of R2 0.999. The current was plotted as a function of the conductivity of the salt at a constant conductivity of electrode rinse solution. Each curve was a function of the applied voltage. The value of the saturation coefficient for the system plots as {Dsat } ∼ 22.1 mS/cm (2.21 S/m). The kinetic coefficient iMAX plotted as a linear function of the effective voltage at 25 °C (298°K). More importantly, a series of tests with a constant feed conductivity and with varied electrode rinse solution conductivity also plotted as Michaelis-Menten type curves. The saturation coefficient for the electrode rinse solution also fitted well to {Esat } ∼ 22.1 mS/cm (2.21 S/m) and the values for i MAX were also linear with respect to applied voltage. Overall, the results from 12 data sets follow the formula i = αβi MAX where α and β are the ratios of the conductivity of the fluid divided by the sum of the conductivity plus the saturation, e.g., for electrode rinse solution α = {E}/({E} + {Esat}) and for the electrode rinse β = {D}/({D} + {Dsat}), indicating the co-dependence of the feedstock and electrode rinse in the overall process dynamics. This interpretation is in opposition to the standard modeling of ED where the process is viewed as a series of resistances and the electrode rinse resistance is usually determined to be negligible. These data suggest that the effects of saturated feedstock and saturated rinse solution, measured as current or ion transport, are inextricably co-dependent. Three full-batch test runs of between 7.2 and 9-hr duration using different initial feedstock and electrode rinse water conditions were analyzed to estimate the relative effects of diluate and concentrate concentration on the test results. A third saturation coefficient was introduced, γ = {C}/({C} + {Csat}) where the terms {C} and {C sat } represent the conductivity and the saturation coefficient of the concentrate. The model for long duration batch tests, including the different effects of the concentrate and diluate, is well represented by i = (αβ0.8 γ 0.2 i MAX). The empirical power coefficients, β0.8 and γ0.2, indicate that the diluate conductivity carries more influence on the rate of the process than does the conductivity of the concentrate. The model has the potential to allow for the scale-up of the ED process using at most one adjustable parameter (the saturation coefficient of the membrane measured as mS/cm or S/cm). Michaelis-Menten kinetics and the modifications by others represent a rich literature on the modeling of different types of inhibition, topics of concern for membrane transport phenomenon.

We investigated theoretically the time dependence of ultra-short laser pulse scattering by an atom at the high-frequency limit for the spectral and total probability of the process using new expression which we derived in this paper. We established that the time dependence of spectral scattering is presented by the curve with the maximum for sufficiently large detuning of scattering frequency from the carrier frequency of the pulse, while the total scattering probability is always the monotonically increasing function of time. We also studied the dependence of scattering probability on pulse duration at the long-time limit. It was shown that, at the long-pulse limit, the scattering probability is a linear function of pulse duration, while in the opposite case, it is a function with maximum. The position of this maximum is determined by the detuning of the scattering frequency from the carrier frequency of the pulse.

Early CoVID-19 growth often obeys: , with Ko = [(ln 2)/(tdbl)], where tdbl is the pandemic doubling time, prior to society-wide Social Distancing. Previously, we modeled Social Distancing with tdbl as a linear function of time, where N [t] 1 ≈ exp[+KAt/ (1+,γot)] is used here. Additional parameters besides {Ko, γo} are needed to better model different ρ[t] = dN [t]/dt shapes. Thus, a new Orthogonal Function Model [OFM] is developed here using these orthogonal function series: where N (Z) and Z[t] form an implicit N [t] N (Z[t]) function, giving: with Lm(Z) being the Laguerre Polynomials. At large MF values, nearly arbitrary functions for N [t] and ρ [t] = dN [t]/dt can be accommodated. How to determine {KA, γo} and the {gm; m = (0, +MF)} constants from any given N (Z) dataset is derived, with ρ [t] set by: The bing com USA CoVID-19 data was analyzed using MF = (0, 1, 2) in the OFM. All results agreed to within about 10 percent, showing model robustness. Averaging over all these predictions gives the following overall estimates for the number of USA CoVID-19 cases at the pandemic end: which compares the pre- and post-early May bing com revisions. The CoVID-19 pandemic in Italy was examined next. The MF = 2 limit was inadequate to model the Italy ρ [t] pandemic tail. Thus, regions with a quick CoVID-19 pandemic shutoff may have additional Social Distancing factors operating, beyond what can be easily modeled by just progressively lengthening pandemic doubling times (with 13 Figures).

Zebra patterns (zebras) play an important role in the plasma diagnostics during solar flares. Considering their double plasma resonance (DPR) model, we present an improved method for the determination of the gyro-harmonic numbers of the zebra stripes that are essential in determining the electron density and magnetic field strength in zebra sources. Furthermore, we present the magnetic field and density spatial scales in zebra sources. Compared to the previous method, we change the basic assumption of the method. Namely, the assumption that the ratio $R={L_{\mathrm{bh}}} / {L_{\mathrm{nh}}}$ ( $L_{\mathrm{bh}}$ and $L_{\mathrm{nh}}$ are the magnetic field and density scales) is constant in the whole zebra source is changed to its more generalized form, where the ratio $R$ is a linear function. Using this improved method, first, we determine the gyro-harmonic numbers of several observed zebras and variations of the spatial scales. Then, knowing the gyro-harmonic numbers of zebra stripes, we compute the electron plasma density and magnetic field strength in zebra sources. It is shown that in all cases the gyro-harmonic numbers of zebra stripes are quite high (> ≈50). This significantly reduces the magnetic field strength and thus increases the plasma beta parameter in zebra sources. The change in the ratio of the magnetic field and density scales along the axis of the radiating tube for the studied zebras is within ± 5 percent. For zebras at high frequencies, this ratio increases with the height, and for zebras at lower frequencies it decreases. The ratio of the magnetic field and density scales across the radiating tube is close to 1 and varies in the range 0.87–1.20.

This paper is devoted to the chemotaxis model with indirect production and general kinetic function $$\begin{aligned} \left\{ \begin{aligned}&u_t=\Delta u-\chi \nabla \cdot (u\nabla v)+f(u),\qquad&x\in \Omega ,\,t>0,\\&v_t=\Delta v-v+w,\qquad&x\in \Omega ,\,t>0,\\&\tau w_t+\lambda w=g(u),\qquad&x\in \Omega ,\,t>0, \end{aligned} \right. \end{aligned}$$ in a bounded domain $$\Omega \subset \mathbb {R}^n(n\le 3)$$ with smooth boundary $$\partial \Omega $$ , where $$\chi , \tau , \lambda $$ are given positive parameters, f and g are known functions. We find several explicit conditions involving the kinetic function f, g, the parameters $$\chi $$ , $$\lambda $$ , and the initial data $$\Vert u_0\Vert _{L^1(\Omega )}$$ to ensure the global-in-time existence and uniform boundedness for the corresponding 2D/3D Neumann initial-boundary value problem. Particularly, when $$f\equiv 0$$ , and g is a linear function, the global bounded classical solutions to the corresponding 2D Neumann initial-boundary value problem with arbitrarily large initial data and chemotactic sensitivity are established. Our results partially extend the results of Hu and Tao (Math Models Methods Appl Sci 26:2111–2128, 2016), Tao and Winkler (J Eur Math Soc 19:3641–3678, 2017), etc.

A classic result by Cook, Gerards, Schrijver, and Tardos provides an upper bound of $$n \Delta $$ on the proximity of optimal solutions of an Integer Linear Programming problem and its standard linear relaxation. In this bound, n is the number of variables and $$\Delta $$ denotes the maximum of the absolute values of the subdeterminants of the constraint matrix. Hochbaum and Shanthikumar, and Werman and Magagnosc showed that the same upper bound is valid if a more general convex function is minimized, instead of a linear function. No proximity result of this type is known when the objective function is nonconvex. In fact, if we minimize a concave quadratic, no upper bound can be given as a function of n and $$\Delta $$ . Our key observation is that, in this setting, proximity phenomena still occur, but only if we consider also approximate solutions instead of optimal solutions only. In our main result we provide upper bounds on the distance between approximate (resp., optimal) solutions to a Concave Integer Quadratic Programming problem and optimal (resp., approximate) solutions of its continuous relaxation. Our bounds are functions of $$n, \Delta $$ , and a parameter $$\epsilon $$ that controls the quality of the approximation. Furthermore, we discuss how far from optimal are our proximity bounds.

In this paper, the energy efficiency (EE) of full-duplex (FD) two-hop relay system with decode-and-forward (DF) and amplify-and-forward (AF) relaying protocols is studied. The residual self-interference (RSI) is modeled as a linear function of relay transmit power first. Then, with the constant transmission rate, the EE maximization problem is transformed into transmit power minimization problem and the minimum transmit power consumptions of the DF and AF protocols are obtained. Furthermore, the EE comparison of the DF and AF protocols is given with the minimum transmit power consumption. The simulation results are given to verify the theoretical EE analysis and comparison.

Aiming at the shortage of differential accuracy of rapid tangent sigmoid tracking differentiator in tracking input signal and the slow convergence of tangent sigmoid tracking differentiator near the equilibrium point and the poor ability to suppress noise, a rapid tangent sigmoid tracking differentiator is designed by choosing tangent sigmoid function, linear function and terminal attractor function. The frequency domain characteristics are analyzed by sweeping test, and its stability is proved. In the tracking differentiator of rapid tangent sigmoid function, the continuous function form of linear and non-linear combination effectively improves the tracking and differential ability of the system, and the terminal attractor function effectively suppresses the noise, so that the tracking input signal has high precision and small error. The linear function accelerates the convergence speed when the state is far from and near the equilibrium point, and the form is simple. The numerical simulation shows that the tracking differentiator with fast tangent sigmoid function has excellent performance.

This paper presents a method of polynomial transition curve application for making agricultural aggregate movement paths during headland turn drives as well as within the field. Four types of agricultural aggregate paths in five different variant designs are discussed. Each path is composed of only two curves, making the so-called transition bi-curve. The curvature described by the linear function as well as the third, fifth, seventh, and ninth degree polynomials was designated. Moreover, a trajectory planning algorithm in which the movement proceeds along two transition curves composing the so-called bi-curve was proposed. The simulation was carried out applying the MATLAB program in which the 4th order Runge&ndash, Kutta method was used. The results were presented by means of figures showing the proposed paths and kinematic quantity courses in the displacement function. The obtained trajectories were compared regarding the size and kinematic quantities. The trajectories, whose curvature is described by the 3&deg, polynomial, were found to possess the smallest absolute values of maximal acceleration and jerk and to lack jerk discontinuity. The proposed solutions can be applied for planning trajectory of not only agriculture machines and aggregates but also autonomous vehicles or mobile robots.

We examine available data on the number of individuals infected by the Covid-19 virus, across several different states in India, over the period January 30, 2020 to April 10, 2020. It is found that the growth of the number of infected individuals N(t) can be modeled across different states with a simple linear function N(t) = γ + αt beyond the date when reasonable number of individuals were tested (and when a countrywide lockdown was imposed). The slope α is different for different states. Following recent work by Notari (arxiv:2003.12417), we then consider the dependency of the α for different states on the average maximum and minimum temperatures, the average relative humidity and the population density in each state. It turns out that like other countries, the parameter α, which determines the rate of rise of the number of infected individuals, seems to have a weak correlation with the average maximum temperature of the state. In contrast, any significant variation of α with humidity or minimum temperature seems absent with almost no meaningful correlation. Expectedly, α increases (slightly) with increase in the population density of the states; however, the degree of correlation here too is negligible. These results seem to barely suggest that a natural cause like a hot summer (larger maximum temperatures) may contribute towards reducing the transmission of the virus, though the role of minimum temperature, humidity and population density remains somewhat obscure from the inferences which may be drawn from presently available data.

Regularized regression analysis is a mature analytic approach to identify weighted sums of variables that predict outcomes. Typically, the number of subjects (N) is smaller than the number of predictors (p). Here, we present a novel coarse approximation linear function (CALF) to frugally select important predictors, build linear models, and discover causality.CALF is a linear regression strategy applied to normalized data that employs only a few (such as 2 to 20) nonzero weights, each +1 or -1. Metrics can be Welch t-test p-value, area under curve (AUC) of receiver operating characteristic, or Pearson correlation, depending upon data type and user preferences. For quantitative approximations, a linear fit (adding an intercept value and rescaling ±1 weights by a common multiplier) can be added to optimize mean squared error (MSE).Real medical data of five types were used to generate examples with goals of binary classification or real variable approximation. Predictors considered were real, sets of integers, or ternary values of single nucleotide polymorphisms. When applied to real data, CALF approximations outperformed in p-value, AUC, correlation, or MSE a popular regularized linear regression algorithm, namely, basic LASSO. It appears that using LASSO without considering CALF might risk wasting resources.AvailabilityR version: Comprehensive R Archive (CRAN): https://cran.r-project.org/web/packages/CALF/index.htmlPython 3.x version: GitHub: https://github.com/jorufo/CALF_PythonContactclark_jeffries@med.unc.edu

Previous dosimetric studies during photodynamic therapy (PDT) of superficial lesions within a cavity such as the nasopharynx, demonstrated significant intra- and interpatient variations in fluence rate build-up as a result of tissue surface re-emitted and reflected photons, which depends on the optical properties. This scattering effect affects the response to PDT. Recently, a meta-tetra(hydroxyphenyl)chlorin-mediated PDT study of malignancies in the paranasal sinuses after salvage surgery was initiated. These geometries are complex in shape, with spatially varying optical properties. Therefore, preplanning and in vivo dosimetry is required to ensure an effective fluence delivered to the tumor. For this purpose, two 3D light distribution models were developed: first, a simple empirical model that directly calculates the fluence rate at the cavity surface using a simple linear function that includes the scatter contribution as function of the light source to surface distance. And second, an analytical model based on Lambert's cosine law assuming a global diffuse reflectance constant. The models were evaluated by means of three 3D printed optical phantoms and one porcine tissue phantom. Predictive fluence rate distributions of both models are within +/- 20% accurate and have the potential to determine the optimal source location and light source output power settings.

In a recent paper published in Environmental Earth Sciences, Jain et al. (Environ Earth Sci 77:253, 2018) proposed an interesting study about the $$\beta$$-factor (the inverse of the turbulent Schmidt number, i.e. the ratio of momentum diffusivity to mass diffusivity in a turbulent flow). They proposed an equation for the depth-averaged $$\beta$$-factor which was used in the Rouse equation to calculate concentration profiles of suspended sediments in open-channel turbulent flows. Despite the interest, the study shows a weakness related to some inconsistencies and contradictions in the method. The main result should be improved. In this note, the weakness in this study will be pointed out. Some used equations are in contradiction with the initial assumptions. The Rouse equation is a solution of the steady-state one-dimensional convection–diffusion equation with a parabolic eddy viscosity which is based on a logarithmic velocity profile. In the same study, Jain et al. (2018) used two different mixing length equations, parabolic and linear relations. The parabolic mixing length and a constant mixing velocity are in contradiction with the used exponential-type eddy viscosity profile. The used equation for coefficient $$\gamma$$ is based on the finite mixing length model and the related assumptions of a linear eddy viscosity and a logarithmic velocity distribution (Nielsen and Teakle in Phys Fluids 16(7):2342–2348, 2004) which are in contradiction with the used exponential-type eddy viscosity profile and velocity distribution. An improved approach and new equations for the depth-averaged $$\beta$$-factor are proposed. The first is based on the equation for the coefficient $$A$$ proposed by Jain et al. (2018) and a second based on a linear function for $$A$$ which gives an equation for $$\beta$$ similar to that of van Rijn (J Hydraul Eng ASCE 1104(11):1613–1641, 1984).

A competition process is a continuous time Markov chain that can be interpreted as a system of interacting birth-and-death processes, the components of which evolve subject to a competitive interaction. This paper is devoted to the study of the long-term behaviour of such a competition process, where a component of the process increases with a linear birth rate and decreases with a rate given by a linear function of other components. A zero is an absorbing state for each component, that is, when a component becomes zero, it stays zero forever (and we say that this component becomes extinct). We show that, with probability one, eventually only a random subset of non-interacting components of the process survives. A similar result also holds for the relevant generalized Polya urn model with removals.

It is well known in quantum mechanics that a large energy gap between a Hilbert subspace of specific interest and the remainder of the spectrum can suppress transitions from the quantum states inside the subspace to those outside due to additional couplings that mix these states, and thus approximately lead to a constrained dynamics within the subspace. While this statement has widely been used to approximate quantum dynamics in various contexts, a general and quantitative justification stays lacking. Here we establish an observable-based error bound for such a constrained-dynamics approximation in generic gapped quantum systems. This universal bound is a linear function of time that only involves the energy gap and coupling strength, provided that the latter is much smaller than the former. We demonstrate that either the intercept or the slope in the bound is asymptotically saturable by simple models. We generalize the result to quantum many-body systems with local interactions, for which the coupling strength diverges in the thermodynamic limit while the error is found to grow no faster than a power law $t^{d+1}$ in $d$ dimensions. Our work establishes a universal and rigorous result concerning nonequilibrium quantum dynamics., Comment: 5 pages, 3 figures. Accompanying paper arXiv:2001.03421. Here, we prove the asymptotic bound and provide general ideas for the many-body generalization, which is numerically demonstrated

We introduce a family of values for TU-games that offers a compromise between the proportional and equal division values. Each value, called an alpha-mollified value, is obtained in two steps. First, a linear function with respect to the worths of all coalitions is defined which associates a real number to every TU-game. Second, the weight assigned by this function is used to weigh proportionality and equality principles in allocating the worth of the grand coalition. We provide an axiomatic characterization of this family, and show that this family contains the affine combinations of the equal division value and the equal surplus division value as the only linear values. Further, we identify the proportional division value and the affine combinations of the equal division value and the equal surplus division value as those members of this family, that satisfy projection consistency. Besides, we provide a procedural implementation of each single value in this family.

A linear control problem is considered in the presence of exposure from an uncontrolled disturbance. It is known only that its values belong to the given connected compact. The payoff is the absolute value of linear function at the end time of the controlled process. It’s believed that one breakdown is possible, which leads to a change in the dynamics of the controlled process. The time of occurrence of breakdown is not known in advance. Control is constructed based on the principle of minimizing the guaranteed result. The opposite side is the disturbance and the moment of occurrence of a breakdown. The necessary and sufficient conditions are found under which an feasible control is optimal. As an example, the problem of controlling a rod, which is rigidly attached to an electric motor rotor, is considered. A rotary flywheel is attached to the other end of the rod. The voltages applied to the rotor and flywheel are control. As disturbance, we take the nonlinear addend in the equations of motion. The goal of choosing controls is to minimize the modulus of the deviation of the angle formed by the rod and the vertical axis from the given value at a fixed time moment.

In this paper, we investigate the team formation problem to balance the coverage gained and the cost incurred. This problem can be formulated as maximizing the difference of two set functions \(f-l\), where f is non-negative monotone approximately submodular function, and l is non-negative linear function. We propose three online bicriteria algorithms. The first two handle the cases where the function f is \(\gamma \)-weakly submodular, and strictly \(\gamma \)-weakly submodular, respectively. The last algorithm integrates the first two with more parameters introduced.

Our main point of focus is the set of closed geodesics on hyperbolic surfaces. For any fixed integer $k$, we are interested in the set of all closed geodesics with at least $k$ (but possibly more) self-intersections. Among these, we consider those of minimal length and investigate their self-intersection numbers. We prove that their intersection numbers are upper bounded by a universal linear function in $k$ (which holds for any hyperbolic surface). Moreover, in the presence of cusps, we get bounds which imply that the self-intersection numbers behave asymptotically like $k$ for growing $k$., 19 pages, 5 figures

We construct the first actively-secure Multi-Party Computation (MPC) protocols with an arbitrary number of parties in the dishonest majority setting, for an arbitrary field \(\mathbb {F}\) with constant communication overhead over the “passive-GMW” protocol (Goldreich, Micali and Wigderson, STOC ‘87). Our protocols rely on passive implementations of Oblivious Transfer (OT) in the boolean setting and Oblivious Linear function Evaluation (OLE) in the arithmetic setting. Previously, such protocols were only known over sufficiently large fields (Genkin et al. STOC ‘14) or a constant number of parties (Ishai et al. CRYPTO ‘08).

In this paper, we consider a class of boundary value problems of nonlinear fractional differential equation with integral boundary conditions. By applying the monotone iterative method and some inequalities associated with Green’s function, we obtain the existence of minimal and maximal positive solutions and establish two iterative sequences for approximating the solutions to the above problem. It is worth mentioning that these iterative sequences start off with zero function or linear function, which is useful and feasible for computational purpose. An example is also included to illustrate the main result of this paper.

Let $X$ be a complex affine variety in $\mathbb{C}^N$, and let $f:\mathbb{C}^N\to \mathbb{C}$ be a polynomial function whose restriction to $X$ is nonconstant. For $g:\mathbb{C}^N \to \mathbb{C}$ a general linear function, we study the limiting behavior of the critical points of the one-parameter family of $f_t: =f-tg$ as $t\to 0$. Our main result gives an expression of this limit in terms of critical sets of the restrictions of $g$ to the singular strata of $(X,f)$. We apply this result in the context of optimization problems. For example, we consider nearest point problems (e.g., Euclidean distance degrees) for affine varieties and a possibly nongeneric data point., Comment: 24 pages

The International GNSS Service (IGS) real-time service (RTS) provides access to real-time precise products. State-Space Representation (SSR) products are disseminated through the Internet using the Networked Transport of the RTCM (Radio Technical Commission for Maritime Services) via the Internet Protocol (NTRIP). However, communication outages caused by a loss of the communication link during ephemeris changes can occur. Unfortunately, any break in providing orbit and clock corrections affects the possibility to perform precise point positioning. To eliminate this problem, various methods have been developed and presented in the literature. The solution proposed by the authors is to directly predict geocentric corrections. This manuscript presents the results and analysis of geocentric correction predictions under two scenarios: the first between the IODE (issue of data ephemeris) value change and the second where prediction must be done for epochs containing a change in IODE ephemeris. In this case, the prediction uses data from a previous message. The Root Mean Square (RMS) values calculated based on the differences between the true correction values and the predicted geocentric corrections using a linear function, a second-degree polynomial and a constant value do not differ significantly. The numerical results show that, in most cases, maintaining the constant value of the last registered SSR correction is the best option.

In the paper, we propose a deterministic approximation algorithm for maximizing a generalized monotone submodular function subject to a matroid constraint. The function is generalized through a curvature parameter \(c\in [0,1]\), and essentially reduces to a submodular function when \(c=1\). Our algorithm employs the deterministic approximation devised by Buchbinder et al. [3] for the \(c=1\) case of the problem as a building block, and eventually attains an approximation ratio of \(\frac{1+g_c(x)+\Delta \cdot \left[ 3+c-(2+c)x-(1+c)g_c(x)\right] }{2+c+(1+c)(1-x)}\) for the curvature parameter \(c\in [0,1]\) and for a calibrating parameter that is any \(x\in [0,1]\). For \(c=1\), the ratio attains 0.5008 by setting \(x=0.9\), coinciding with the renowned performance guarantee of the problem. Moreover, when the submodular set function degenerates to a linear function, our generalized algorithm always produces optimum solutions and thus achieves an approximation ratio 1.

This chapter investigates the unknown-input state observation problem for hybrid dynamical systems with state jumps. The problem considered is that of deriving an asymptotic estimate of a linear function of the state of a given system in the presence of unknown inputs. The systems addressed are hybrid dynamical systems which exhibit a continuous-time linear behaviour except at isolated points of the time axis, where their state shows abrupt changes ruled by an algebraic linear equation. The systems belonging to this class are also known as linear impulsive systems. It will be assumed that the time interval between two consecutive state discontinuities is lower bounded by a positive real constant. The presence of unknown inputs precludes, in general, the possibility of asymptotically estimating the whole system state, but may permit the asymptotic estimation of a linear function of the state. Thus, the objective of this chapter is to state and prove necessary and sufficient conditions for the existence of solutions to this problem, in the context of linear impulsive systems. The methodology adopted is structural, based on the use of properly defined geometric objects and properties. A general necessary and sufficient condition is proven first. Then, under more restrictive, yet acceptable in contexts of practical interest, assumptions, a constructive necessary and sufficient condition is shown. The latter condition can be checked by an algorithmic procedure, since it is based on subspaces which can be easily computed and on properties which can be systematically ascertained. Special attention is paid to the synthesis of asymptotic observers whose state has the minimal possible dimension, briefly referred to as minimal-order observers.

Recall from Sect. 13.5 that E-BLUP, the conventional procedure for predicting a predictable linear function τ = cTβ + u under the prediction-extended general mixed linear model $$\displaystyle \left \{\left (\begin {array}{c}\mathbf {y} \\ u\end {array}\right ),\left (\begin {array}{c}\mathbf { X}\boldsymbol {\beta } \\ 0 \end {array}\right ),\left (\begin {array}{cc}{\mathbf {V}}_{yy}(\boldsymbol {\theta }) & {\mathbf {v}}_{yu}(\boldsymbol {\theta }) \\ {\mathbf {v}}_{yu}(\boldsymbol {\theta })^T & \mathbf { v}_{uu}(\boldsymbol {\theta }) \end {array}\right )\right \}, $$ is to first obtain an estimate \(\hat {\boldsymbol {\theta }}\) of θ and then proceed as though this estimate was the true θ.

Let $f$ be an $\mathbb{F}_q$-linear function over $\mathbb{F}_{q^n}$. If the $\mathbb{F}_q$-subspace $U= \{ (x^{q^t}, f(x)) : x\in \mathbb{F}_{q^n} \}$ defines a maximum scattered linear set, then we call $f$ a scattered polynomial of index $t$. As these polynomials appear to be very rare, it is natural to look for some classification of them. We say a function $f$ is an exceptional scattered polynomial of index $t$ if the subspace $U$ associated with $f$ defines a maximum scattered linear set in $\mathrm{PG}(1, q^{mn})$ for infinitely many $m$. Our main results are the complete classifications of exceptional scattered monic polynomials of index $0$ (for $q>5$) and of index $1$. The strategy applied here is to convert the original question into a special type of algebraic curves and then to use the intersection theory and the Hasse-Weil theorem to derive contradictions., 23 pages

An equation of state (EOS) denotes the relationship between the pressure and the volume, involving two quantities, the isothermal bulk modulus (B0) and its first pressure derivative (B′), both at zero pressure. An accurate representation of P-V relationship in solids is provided by the modified Murnaghan equation (MME) in which B′ is assumed to be a linear function of pressure. We have derived a new equation of state by assuming B′ to be constant plus a weakly dependent function p of P and solving the resulting differential equation by the perturbation method. If the weak pressure-dependent part is chosen to be proportional to P, a situation exactly analogous to that of the MME, our perturbation solution acquires an algebraic structure somewhat simpler to the MME. As far as numerical applications are concerned, the two equations do not differ much. Our treatment has the additional advantage that it can be easily generalized to include higher order terms if desired. We have reported the V/V0 versus P values of NaCl and the same have been compared with Freund–Ingalls three-parameter equation and the Birch equation.

In adaptive DE algorithms, the parameter p is used to strike a balance between the algorithm’s exploration and exploitation abilities. In this paper, an effective method for generating a suitable p value is proposed. The probability distribution function (PDF) of p is viewed as a linear function. At the start of the evolutionary process, the PDF of p is a linearly increasing function. Hence, there is a high probability of p having a high value. With successive generations, the PDF of p changes to a linearly decreasing function, and there is a high probability of p having a smaller value. Furthermore, a method for allowing a variable archive size is also proposed. After each generation, the archive size for the next generation is computed. If the archive size has to be reduced, the worst-ranking individuals are deleted from the archive. The accuracy of the proposed method is computationally compared with those of eight other advanced DE variants using benchmark functions from the IEEE Congress on Evolutionary Computation 2017. The results show that the proposed algorithm achieves the best performance among all the variants for most of the benchmark functions in the case of 30-dimensional problems.

We study the asymptotic value of a frequency-dependent zero-sum game with separable payoff following a differential approach. The stage payoffs in such games depend on the current actions and on a linear function of the frequency of actions played so far. We associate to the repeated game, in a natural way, a differential game and although the latter presents an irregularity at the origin, we prove that it has a value. We conclude, using appropriate approximations, that the asymptotic value of the original game exists in both the n-stage and the {\lambda}-discounted games and that it coincides with the value of the continuous time game., Comment: 22 pages, accepted in Dynamic Games and Applications

We classify all solutions to − u t det ⁡ D 2 u = f ( x ) in R − n + 1 , where f ∈ C α ( R n ) is a positive periodic function in x. More precisely, if u is a parabolically convex solution to above equation, then u is the sum of a convex quadratic polynomial in x, a periodic function in x and a linear function of t. It can be viewed as a generalization of the work of Gutierrez and Huang in 1998. And along the line of approach in this paper, we can treat other parabolic Monge–Ampere equations.