Formula Complexity of a Linear Function in a $$k$$-ary Basis

CHAPTER 1. AN INTRODUCTION TO DATA AND FUNCTIONS. 1.1 Describing Single-Variable Data. 1.2 Describing Relationships between Two Variables. 1.3 An Introduction to Functions. 1.4 The Language of Functions. 1.5 Visualizing Functions. CHAPTER 2. RATES OF CHANGE AND LINEAR FUNCTIONS. 2.1 Average Rates of Change. 2.2 Change in the Average Rate of Change. 2.3 The Average Rate of Change Is a Slope. 2.4 Putting a Slant on Data. 2.5 Linear Functions: When Rates of Change Are Constant. 2.6 Visualizing Linear Functions. 2.7 Constructing Graphs and Equations of Linear Functions. 2.8 Special Cases. 2.9 Breaking the Line: Piecewise Linear Functions. 2.10 Constructing Linear Models of Data. 2.11 Looking for Links between Education and Earnings: A Case Study on Using Regression Lines. CHAPTER 3. WHEN LINES MEET: LINEAR SYSTEMS. 3.1 Interpreting Intersection Points: Linear and Nonlinear Systems. 3.2 Visualizing and Solving Linear Systems. 3.3 Reading between the Lines: Linear Inequalities. 3.4 Systems with Piecewise Linear Functions: Tax Plans. CHAPTER 4. THE LAWS OF EXPONENTS AND LOGARITHMS: MEASURING THE UNIVERSE. 4.1 The Numbers of Science: Measuring Time and Space. 4.2 Positive Integer Exponents. 4.3 Zero, Negative, and Fractional Exponents. 4.4 Converting Units. 4.5 Orders of Magnitude. 4.6 Logarithms as Numbers. CHAPTER 5. GROWTH AND DECAY: AN INTRODUCTION TO EXPONENTIAL FUNCTIONS. 5.1 Exponential Growth. 5.2 Exponential Decay. 5.3 Comparing Linear and Exponential Functions. 5.4 Visualizing Exponential Functions. 5.5 Exponential Functions: A Constant Percent Change. 5.6 More Examples of Exponential Growth and Decay. 5.7 Compound Interest and the Number e. 5.8 Semi-Log Plots of Exponential Functions. CHAPTER 6. LOGARITHMIC LINKS: LOGARITHMIC AND EXPONENTIAL FUNCTIONS. 6.1 Using Logarithms to Solve Exponential Equations. 6.2 Using Natural Logarithms to Solve Exponential Equations Base e. 6.3 Visualizing and Applying Logarithmic Functions. 6.4 Using Semi-Log Plots to Construct Exponential Models for Data. C H A P T E R 7. POWER FUNCTIONS. 7.1 The Tension between Surface Area and Volume. 7.2 Direct Proportionality: Power Functions with Positive Powers. 7.3 Visualizing Positive Integer Power Functions. 7.4 Comparing Power and Exponential Functions. 7.5 Inverse Proportionality: Power Functions with Negative Powers. 7.6 Visualizing Negative Integer Power Functions. 7.7 Using Logarithmic Scales to Find the Best Functional Model. CHAPTER 8. QUADRATICS AND THE MATHEMATICS OF MOTION. 8.1 An Introduction to Quadratic Functions: The Standard Form. 8.2 Visualizing Quadratics: The Vertex Form. 8.3 The Standard Form vs. the Vertex Form. 8.4 Finding the Horizontal Intercepts: The Factored Form. 8.5 The Average Rate of Change of a Quadratic Function. 8.6 The Mathematics of Motion. CHAPTER 9. NEW FUNCTIONS FROM OLD. 9.1 Transformations. 9.2 The Algebra of Functions. 9.3 Polynomials: The Sum of Power Functions. 9.4 Rational Functions: The Quotient of Polynomials. 9.5 Composition and Inverse Functions. 9.6 Exploring, Extending & Expanding. APPENDIX Student Data Tables for Exploration 2.1. Data Dictionary for FAM1000 Data. SOLUTIONS For all Algebra Aerobics and Check Your Understanding problems for odd-numbered problems in the Exercises and Chapter Reviews. All solutions are grouped by chapter ANS-1. INDEX.

A two-phase procedure for calculation of multiple linear classification functions has been developed. Phase I approximates the distribution of samples in the measurement space by a series of linear functions and preselects only those samples lying in the critical region. Phase II utilizes the reduced number of samples for calculation by a linear programming algorithm, the multiple linear classification functions that can separate given samples strictly. The procedure was applied to the problem of spoken-word recognition. About 3300 samples of Japanese spoken digits, uttered by 55 male speakers and represented as 56-dimensional measurement vectors by a vocoder-type frequency analyzer are collected. During Phase I, the number was reduced to 16 linear functions on the average. Precise classification functions were then calculated for each digit in Phase II. Most of the digits were separated by a single linear function while others were separated by two functions. An error rate of less than 0.2% was achieved for 550 samples newly uttered by the same 55 speakers.

Whilst widely recognised as an excellent means for solving problems and for designing software, functional programming languages have suffered from their inefficient implementations on conventional computers. A route to improved runtime performance is to transform recursively defined functions into programs which execute more quickly and/or consume less space. We derive equivalent imperative programming language loops for a large class of linear recursive functions of which the tail-recursive functions form a very small subset. We first identify a small set of primitive function defining expressions for which we determine the corresponding loop-expressions. We then determine the loop-expressions for linear functions defined by any expressions which are formed from those primitives. In this way, a very general class of linear functions can be transformed automatically into loops in the parsing phase of a compiler, since the parser has in any case to determine the hierarchical structure of function definitions. Further transformation may involve specific properties of particular defining expressions, and adopt previous schemes. In addition, equivalent linear functions can be found for many non-linear ones which can therefore also be transformed into loops.

Evolutionary algorithms are randomized search heuristics that are often described as robust general purpose problem solvers. It is known, however, that the performance of an evolutionary algorithm may be very sensitive to the setting of some of its parameters. A different perspective is to investigate changes in the expected optimization time due to small changes in the fitness landscape. A class of fitness functions where the expected optimization time of the (1+1) evolutionary algorithm is of the same magnitude for almost all of its members is the set of linear fitness functions. Using linear functions as a starting point, a model of a fitness landscape is devised that incorporates important properties of linear functions. Unexpectedly, the expected optimization time of the (1+1) evolutionary algorithm is clearly larger for this fitness model than on linear functions.

Consider the problem of estimating unbiasedly a linear function, $\lambda?' \omega, \omega' = (\omega_1, \cdots, \omega_p)$, of $p$ weights $\omega_1, \cdots, \omega_p (0 < \omega_i < \infty$ for all $i = 1, \cdots, p)$ when the number of possible weighing operations, $n$, is smaller than $p$. Assume further that the weighing design is of the chemical type (each entry of the design matrix can assume one of the values $-1, 0, +1)$. A weighing design in which the number of weighing operations is smaller than the number of objects, $n < p$, will be called a fractional weighing design. A fractional weighing design is singular. As is well known, if a non-randomized weighing design is singular, one cannot estimate unbiasedly each of the $p$ weights. Some linear functions $\lambda?'\omega$ can be estimated, however, unbiasedly. For example, the sum of all $p$ weights can be estimated unbiasedly by a singular weighing design in which, at each weighing operation, all the objects are placed on one pan. Such a singular weighing design is in fact optimal. One can not attain the same precision in estimating the sum of all the $p$ weights by any non-singular design (see K. S. Banerjee [1]). Some linear functions of the weights can thus be estimated, with a sufficient degree of precision, on the basis of a number of weighing operations, $n$, smaller than the number of weights, $p$. A. M. Mood [4] has shown that if the design matrix of a non-singular weighing design is a Hadamard matrix (if it exists) then the design is optimal. We shall therefore study fractional weighing designs based on Hadamard matrices. If $p$ is such that no Hadamard matrix of order $p$ exists, we shall consider the smallest integer $p'$ larger than $p$, for which a Hadamard matrix exists and add $(p' - p)$ dummy weights. The linear functions $\lambda?'\omega$ can be extended in a way that assigns the dummy weights the coefficient zero. We shall then consider randomized fractional weighing designs which are constructed by choosing at random $n$ rows, independently and with replacement, from the given Hadamard matrix according to a probability vector $\xi$ of order $p$. Every such row specifies a weighing operation to be performed. Non-randomized fractional designs are special cases of the randomized designs under consideration. It is well known that if the weighing design is fractional $(n < p)$ then not all the linear functions $\lambda?'\omega$ are estimable under non-randomized designs. It is shown in the present paper that any linear function $\lambda?'\omega$ can be estimated unbiasedly under a proper randomization scheme. Optimal randomization procedures depend on the functional $\lambda?$. Complete class of unbiased estimation procedures is given for any linear function $\lambda?'\omega$. It is shown that every functional $\lambda?$ specifies a subset of, say $r (1 \leqq r \leqq p)$, admissible weighing operations (rows of the Hadamard matrix); in the sense that if other weighing operations are chosen then, either the estimation procedure is biased, or has a variance larger than the one that can be attained under the admissible weighing operations. It is also proven that if each of the $r$ admissible weighing operations $(1 \leqq r \leqq p)$ is chosen with probability equal to $1/r$, then the corresponding unbiased estimator has a uniformly (in $\omega$ and $\sigma^2$) minimum variance, relative to all the above specified randomization procedures. The formulae of the unbiased estimator of $\lambda?'\omega$ and its variance are extended in Section 4 to the case of random choice without replacement, of $n$ rows out of the $r$ admissible ones. This randomization procedure improves the results that can be obtained under the best randomization procedure with replacement. Finally, the relationship of the present study to the studies of S. Ehrenfeld and S. Zacks [2], [3] is discussed in Section 5.

Stacked generalization is a flexible method for multiple classifier combination; however, it tends to overfit unless the combiner function is sufficiently smooth. Previous studies attempt to avoid overfitting by using a linear function at the combiner level. This paper demonstrates experimentally that even with a linear combination function, regularization is necessary to reduce overfitting and increase predictive accuracy. The standard linear least squares regression can be regularized with an L2 penalty (Ridge regression), an L1 penalty (lasso regression) or a combination of the two (elastic net regression). In multi-class classification, sparse linear models select and combine individual predicted probabilities instead of using complete probability distributions, allowing base classifiers to specialize in subproblems corresponding to different classes.

Ninety hybrid (mainly Large White × Landrace) pigs from 2 experimental replicates were used to study the potential use of computed tomography (CT) as a nondestructive technology for estimating the chemical body composition of growing pigs. Body tissue components (lean, fat, and bone) of 6 live pigs from each sex (boars, gilts, and barrows) were assessed by CT imaging before slaughter at approximately 30, 60, 90, 120, and 150 kg of BW. After slaughter, the empty body components were ground and frozen until analyzed for protein, lipid, ash, and moisture content. Several growth functions were evaluated and the allometric function (Y = aBW(b)), which was evaluated as log(10)chemical component weight = b(0) + b(1)log(10)BW, provided the best fit to the data. For each sex, the allometric coefficient (b(1)) for protein (0.92 to 0.99) was close to but less than 1; for ash (1.03 to 1.12), it was close to but greater than 1; for moisture (0.82 to 0.86), it was less than 1, and for lipid (1.61 to 1.71), it was greater than 1. Deposition rates (change in component weight per unit change in BW) for each chemical component were predicted using derivatives of the function. The mean deposition rates for protein and lipid were 0.141 and 0.286 kg/kg of BW gain, respectively. The deposition rate for protein was generally stable across different BW, whereas that for lipid increased as BW increased. In addition, linear, quadratic, exponential, and logistic functions were fitted to the data to study the relationship between the CT data and chemical components. The linear function was assessed to be the best equation, based on the Bayesian information criterion. The prediction equation for protein (kg) = -1.64 + 0.28 × CT lean (kg), and for lipid (kg) = -0.69 + 1.09 × CT fat (kg), had R(2) values of 0.924 and 0.987, respectively. Sex had no effect (P > 0.05) on the prediction of protein and lipid. The effect of BW was not significant (P > 0.05) for the prediction of lipid, but it was significant (P > 0 0.05) for the prediction of protein. However, the addition of BW to the base prediction equation for protein resulted in an increase of only 0.013 in the R(2) value. It was concluded from this study that CT scanning has great potential as a nondestructive technology for estimating the physical and chemical body composition of pigs. Additional research is required to validate the utility and accuracy of the prediction equations.

Chapter 4 - ELEMENT INTERPOLATION AND LOCAL COORDINATES

In this work, we introduce multiplicative drift analysis as a suitable way to analyze the runtime of randomized search heuristics such as evolutionary algorithms. We give a multiplicative version of the classical drift theorem. This allows easier analyses in those settings where the optimization progress is roughly proportional to the current distance to the optimum. To display the strength of this tool, we regard the classical problem how the (1+1) Evolutionary Algorithm optimizes an arbitrary linear pseudo-Boolean function. Here, we first give a relatively simple proof for the fact that any linear function is optimized in expected time $O(n \log n)$, where $n$ is the length of the bit string. Afterwards, we show that in fact any such function is optimized in expected time at most ${(1+o(1)) 1.39 \euler n\ln (n)}$, again using multiplicative drift analysis. We also prove a corresponding lower bound of ${(1-o(1))e n\ln(n)}$ which actually holds for all functions with a unique global optimum. We further demonstrate how our drift theorem immediately gives natural proofs (with better constants) for the best known runtime bounds for the (1+1) Evolutionary Algorithm on combinatorial problems like finding minimum spanning trees, shortest paths, or Euler tours.

In this chapter, we continue our study of Banach and Hilbert spaces. Here, we mainly consider linear functionals, i.e., additive, homogeneous, and continuous number functions given on such spaces. The problems under consideration are mostly grouped around two fundamental facts, namely, the Hahn-Banach theorem on extensions of linear functionals and the Banach-Steinhaus theorem (the principle of uniform boundedness). The general form of linear continuous functionals in many important spaces and some geometric problems in the theory of Hilbert spaces are also investigated.

In this paper we address a min–max problem of fractional quadratic (not necessarily convex) over linear functions on a feasible set described by linear and (not necessarily convex) quadratic functions. We propose a conic reformulation on the cone of completely positive matrices. By relaxation, a doubly nonnegative conic formulation is used to provide lower bounds with evidence of very small gaps. It is known that in many solvers using Branch and Bound the optimal solution is obtained in early stages and a heavy computational price is paid in the next iterations to obtain the optimality certificate. To reduce this effort tight lower bounds are crucial. We will show empirical evidence that lower bounds provided by the copositive relaxation are able to substantially speed up a well known solver in obtaining the optimality certificate.

Aimed at the phenomenon that there is few theories and applications about multi-layer ADALINE neural network (NN), in this paper, mathematic expressions of the multi-input/ single-output (MISO) and multi-input/ multi-output (MIMO) ADALINE NN with multi-layer had been deduced according to the theory of linear neuron transfer function. At result, expressions are also linear functions respect with input variables with more complex weights and thresholds. In order to verify the equivalence of multi-layer ADALINE NNand single-layer ADALINE NN, two examples had been simulated with MATLAB software for function approximation, respectively. The simulation results showed that the multi-layer ADALINE NN can approximate linear functions but not as well as the single-layer ADALINE NN does. In conclusion, the existence of multi-layer adaline NN is not necessary, and its function can be simply realized with the single-layer ADALINE NN completely.

Let $X$ be a finite-dimensional real normed space, and $K$ a&nbsp;closed additive subgroup of $X$. Let $a\in X\setminus K$ and let $d_X(a,K)$ be the distance from $a$ to&nbsp;$K$. We say that a linear functional $f\in X^*$ separates $a$ from $K$ if $d_\ma

Laser-induced breakdown spectroscopy (LIBS) has been applied to many fields for the quantitative analysis of diverse materials. Improving the prediction accuracy of LIBS regression models is still of great significance for the Mars exploration in the near future. In this study, we explored the quantitative analysis of LIBS for the one-dimensional ChemCam (an instrument containing a LIBS spectrometer and a Remote Micro-Imager) spectral data whose spectra are produced by the ChemCam team using LIBS under the Mars-like atmospheric conditions. We constructed a convolutional neural network (CNN) regression model with unified parameters for all oxides, which is efficient and concise. CNN that has the excellent capability of feature extraction can effectively overcome the chemical matrix effects that impede the prediction accuracy of regression models. Firstly, we explored the effects of four activation functions on the performance of the CNN model. The results show that the CNN model with the hyperbolic tangent (tanh) function outperforms the CNN models with the other activation functions (the rectified linear unit function, the linear function and the Sigmoid function). Secondly, we compared the performance among the CNN models using different optimization methods. The CNN model with the stochastic gradient descent optimization and the initial learning rate = 0.0005 achieves satisfactory performance compared to the other CNN models. Finally, we compared the performance of the CNN model, the model based on support vector regression (SVR) and the model based on partial least square regression (PLSR). The results exhibit the CNN model is superior to the SVR model and the PLSR model for all oxides. Based on the above analysis, we conclude the CNN regression model can effectively improve the prediction accuracy of LIBS.

As in the binary case, ternary bent functions are defined as most non-linear ternary functions meaning that they are at the largest possible Hamming distance from affine ternary functions. It is therefore interesting to observe that some ternary bent functions can be constructed as various combinations of linear ternary functions. By starting from the combination of linear functions corresponding to the basic ternary bent functions the construction of different other ternary bent functions can be performed by the application of different combinations of FFT-like permutation matrices for ternary functions.