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# Multivariate Exponential Distribution

I'd like to generate random vectors according to a multivariate exponential distribution, that is with a pdf f: R^n->R given by for appropriate normalisation constant c_e. : Inference procedures for bivariate exponential distribution model of Gumbel based on life test of component and system. mvprpb computes the orthant probability of the multivariate Gaussian distribution. The intrinsic nature of the interval variables leads to special structures of the variance--covariance matrix, which is represented by five different possible configurations. It is procedure followed by statisticans to check the potential difference between scale-level dependent variable by a nominal-level variable having two or more categories. For p = 1; the pdf (1) reduces to that of a double exponential (or Laplace) distribution. the linear or exponential kernel) will not admit such efficiencies. In this paper, we consider the four-parameter bivariate generalized exponential distribution proposed by Kundu and Gupta [Bivariate generalized exponential distribution, J. Skip navigation. As βtends to 1, this distribution tends to a multivariate uniform distribution. to the Weibull distribution, generation of simulated complex multivariate Weibull data with prescribed covariance structure has been a challenging problem. Once the new ideas are mastered, it is often easier to work within the general exponential family framework than with speciﬁc instances. We see the connection between the exponential distribution and the gamma distribution. Random Number Distributions¶. It covers all significant advances that have occurred in the field over the past quarter century in the theory, methodology, inferential procedures, computational and simulational aspects, and applications of continuous multivariate. 2010;7:464-477] introduced Block and Basu bivariate Weibull (BBBW) distribution, which is a generalization of the Block and Basu bivariate exponential distribution, and provided the maximum likelihood estimators using EM algorithm. in [1] introduced a Bayesian analysis for the bivariate generalized exponential distribution based on FGM copula in the presence of censored data. This chapter discusses the multivariate exponential distributions and their applications in reliability. If Tau=1, the distribution is identical to the chi-square distribution with n degrees of freedom. Multivariate Exponential Distribution. *FREE* shipping on qualifying offers. Probability Distributions. In probability theory and statistics, the exponential distribution is the probability distribution of the time between events in a Poisson point process, i. Set lambda = 0. Choose a distribution. For a general bivariate case we write this as P(X 1 = x 1, X 2 = x 2). The exponential distaibution plays a central role in several fields of probability and statistics, and ranks in overall importance next to the normal distribution. The multivariate normal distribution is said to be "non-degenerate" when the symmetric covariance matrix is positive definite. Multivariate Weibull distributions encompass distributions exhibiting Weibull characteristics and relate to multivariate exponential distributions of various types. Section 3 contains the main theoretical results of the paper. Statistics is used in almost every field of research: the discovery of the Higgs particle, social sciences, climate research, With this, and with its well established foundations,. Method of Moments: Exponential Distribution Given a collection of data that may fit the exponential distribution, we would like to estimate the parameter which best fits the data. Defined in edward The multivariate normal distribution on R^k. The multivariate normal distribution is said to be "non-degenerate" when the symmetric covariance matrix is positive definite. The members of this family have many important properties which merits discussing them in some general format. only when the shape parameter of the multivariate Power Exponential (MPE) distribution is equal to one which corresponds to the multivariate normal distribution. Two families have a representation like the Marshall--Olkin multivariate exponential distribution. The Wishart distribution is the probability distribution of the maximum-likelihood estimator (MLE) of the precision matrix of a multivariate normal distribution. "outside the main mass of the distribution". The relationship between the exponential distribution and the Laplace distribution allows for a simple method for simulating bivariate asymmetric Laplace variables (including for the case of = ). Example 2: Exponential Cumulative Distribution Function (pexp Function) We can also use the R programming language to return the corresponding values of the exponential cumulative distribution function for an input vector of quantiles. Abstract A number of multivariate exponential distributions are known, but they have not been obtained by methods that shed light on their applicability. Exponential family comprises a set of ﬂexible distribution ranging both continuous and discrete random variables. 2 A Single Population Mean using the Student t Distribution, 9. Its importance derives mainly from the multivariate central limit theorem. This is essentially a Metropolis-Hastings algorithm where at each iteration the proposal distribution is formed by performing one iteration, starting at the current point, of Iterative Weighted Least Squares (IWLS). Various bivariate geometric distributions analogous to important bivariate exponential distributions like, Marshall-Olkin’s bivariate exponential, Downton’s bivariate exponential and Hawkes’ bivariate exponential are presented. dirichlet Multivariate generalization of Beta distribution. The moment expression for multivariate order. Abstract: A number of multivariate exponential distributions are known, but they have not been obtained by methods that shed light on their applicability. The multivariate normal probability function is overloaded to allow the variate vector $$y$$ and location vector $$\mu$$ to be vectors or row vectors (or to mix the two types). A random variable X is said to have an exponential distribution with PDF:. The distribution of Ax is multivariate normal with mean And covariance matrix. Exports Reproducible Summary Tables to Multi-Tab Spreadsheet Files (. Multivariate Anal. In this paper a multivariate failure rate representation based on Cox's conditional failure rate is introduced, characterizations of the Freund–Block and the Marshall–Olkin multivariate exponential distributions are obtained, and generalizations of the Block–Basu and the Friday–Patil bivariate exponential distributions are proposed. , Landsman, Z. Infection Data with Generalized Exponential Baseline Distribution David D. Remember that no matter how x is distributed, E(AX +b) = AE(X)+b. Specify a mixed multivariate distribution: # Multivariate Mixed distribution OutcomeDist (outcome. 1): η = p, T(x) = x, g(p) = log p 1−p, B(p) = log(1−p), h(x) = 1. This gives rise to a new distribution that can be regarded as a generalized multivariate beta distribution. Multivariate exponential distributions with latent factor structure and related topics Jan-Frederik Mai email: janfrederik. MODELING DEPENDENT RISKS WITH MULTIVARIATE ERLANG MIXTURES 155 measures such as value at risk (VaR) and tail VaR. There are multiple forms of the multivariate Laplace distribution. You can also easily derive expressions for correlated random variables because you can use linear algebra to diagonalize the random variables and find a linear transformation between the correlated random variables and a new set of independent random variables. in [1] introduced a Bayesian analysis for the bivariate generalized exponential distribution based on FGM copula in the presence of censored data. The reason for the special status of the Exponential family is that a number of important and useful calculations in statistics can be done all at one stroke within the framework of the Exponential family. Multivariate normal distribution. Kotz, Balakrishna and Johnson (2000) discussed seven multivariate exponential distributions. Univariate Distribution Relationships About the chart: The list on the left-hand side displays the names of the 76 probability distributions (19 discrete distributions given by the rectangular boxes and 57 continuous distributions given by the rectangular boxes with the rounded corners) present in the chart. 6(d) are associated provided that M and N, and El(1) and E2(1) are associated by the same reasoning in Remark 2. JSTP Diawara, N. Convergence in distribution does not imply convergence in probability. For a general bivariate case we write this as P(X 1 = x 1, X 2 = x 2). Advances in sequencing technology continue to deliver increasingly large molecular sequence datasets that are often heavily partitioned in order to accurately model. # Or the exponential distribution expvars - qexp (pvars) cor (expvars, rawvars) # We can see that the correlations after the exponential tranformations are # significantly weaker (from. The di culties stem from the complicated correlation struc-ture arising through the estimation process of both the re-. Multivariate Exponential Distribution. Abstract A number of multivariate exponential distributions are known, but they have not been obtained by methods that shed light on their applicability. exponential(). The conjugate prior for J with known µ is a matrix analogue of the gamma distribution called the Wishart distribution. It may [5] be obtained as a gamma mixture of independent exponential random variables in the following way. This paper introduces a new multivariate approach for jointly modeling crash counts by severity data based on Multivariate Poisson-Lognormal models. The exponential distribution has only one parameter, theta. Exponential Regression An exponential regression is the process of finding the equation of the exponential function that fits best for a set of data. The exponential distaibution plays a central role in several fields of probability and statistics, and ranks in overall importance next to the normal distribution. Several properties of this distribution are reported and their usefulness in reliability theory indicated. Each univariate distribution is an instance of a subclass of rv_continuous An exponential continuous random variable. A new class of bivariate and multivariate exponential distributions Article (PDF Available) in Far East Journal of Mathematical Sciences 50(2) · March 2015 with 465 Reads How we measure 'reads'. L Hsu] on Amazon. In addition to the conventional areas, the volume also includes discussions on exponential regression, models and applications of mixtures which includes the especially relevant bootstrapping method, and. (1964) considered the estimation of R when X and Y are normally distributed. Random Number Distributions¶. Keskustelualoitteita #52 Joensuun yliopisto, Taloustieteet GARCH Modelling with Power Exponential Distribution - Applications to Value at Risk. Suppose two standardized random. A class of multivariate linear models under the longitudinal setting, in which unobserved heterogeneity may evolve over time, is introduced. Abstract A number of multivariate exponential distributions are known, but they have not been obtained by methods that shed light on their applicability. The Exponentially modified Gaussian distribution, a convolution of a normal distribution with an exponential distribution, and the Gaussian minus exponential distribution, a convolution of a normal distribution with the negative of an exponential distribution. , we must invert the CDF F Inverting F might be easy (exponential), or difficult (normal) in which case. of a multivariate negative binomial (MVNB) distribution on the basis of multivariate exponential (MVE) distributions and convolution. Although the crash frequency by severity data are multivariate in nature, they have often been analyzed by modeling each severity. that the BWGED density is a unimodal distribution. The univariate marginals of such a distribution will not necessarily be exponential. We present a formulation of subexponential and exponential tail behavior for multivariate distributions. Recently, Kundu and Gupta [A class of absolute continuous bivariate distributions. Abstract A number of multivariate exponential distributions are known, but they have not been obtained by methods that shed light on their applicability. , and Carpenter, M. A random variable which has a normal distribution with a mean m=0 and a standard deviation σ=1 is referred to as Standard Normal Distribution. Inference for a Multivariate Exponential Distribution with a Censored Sample Abstract: Maximum likelihood estimators for the parameters of a multivariate exponential Cdf are easily obtained from partial information about a random sample, censored or not. from the binomial(p;n) distribution, then you can expect the sample to have a sample mean that is close to np. Difference between two poisson variable is not a poission variable. with exponential scaled minima have exponential univariate marginals; this fact is also immediate from (2. distribution is in the exponential family is just the sum of each rv's su cient statistic. Apply the univariate normal CDF of variables to derive probabilities for each variable. To estimate the distribution of lifetimes nonparametrically, based on right censored observa- tions, we use the Kaplan-Meier estimator. Remember that no matter how x is distributed, E(AX +b) = AE(X)+b. Such extensions involve the standard gamma (/3 = 1, "y = 0), or the exponential (a = 1), see Johnson and Kotz. Graphical Models via Univariate Exponential Family Distributions els obtained by the following construction: suppose the node-conditional distributions of each node conditioned on the rest of the nodes follows a univariate exponential family. only when the shape parameter of the multivariate Power Exponential (MPE) distribution is equal to one which corresponds to the multivariate normal distribution. The corresponding values for the normal distribution are 1. In a previous paper (Marshall and Olkin (1966)) the authors have derived a multivariate exponential distribution from points of view designed to indicate the applicability of the distribution. s that already has a normal limit in distribution. One chart is based on raw observations and the other on the transformed data. Covariance matrix of the distribution default one. Use of markov chain monte carlo methods in a bayesian analysis of the block and basu bivariate exponential distribution. Suppose two standardized random. A random variable X is said to have an exponential distribution with PDF:. Chapter 3 Exponential Families 3. Exponential distribution is widely used for survival analysis. For these reasons, I will concentrate on ﬁnite mixtures of normals. The univariate exponential distribution is also (sort of) closed under convolution. Skip navigation. Abstract: A number of multivariate exponential distributions are known, but they have not been obtained by methods that shed light on their applicability. The bivariate exponential distribution and the multivariate extension of ex-. Note: Theorem 4. Probability distribution function (PDF) of the univariate case (scalar domain, scalar parameter) The natural exponential families (NEF) are a subset of the exponential families. multivariate çok değişkenli multivariate analysis çok değişkenli analizler. At the opposite extreme, for points which are very close compared to the kernel size, Kernels which are very "flat" (e. A multivariate uniform occurs as kappa -> infinity. A Look Ahead 00:36:59 Joseph Blitzstein. Multivariate normal distribution. We say that the exponential family is generated by any of the distributions in it. The univariate exponential distribution is well known as a model in reliability theory. The following are code examples for showing how to use numpy. A Multivariate Exponential Distribution Created Date: 20160731195810Z. 6 billion annually to Virginia's economy. We define a multivariate negative binomial distribution (MVNB) as a bivariate Poisson distribution function mixed with a multivariate exponential (MVE) distribution. , 2015 ) did not allow positive dependencies for the exponential and Poisson generalizations. A multivariate generalization of the Lomax (Pareto type 2) distribution is obtained by mixing exponential variables. , Landsman, Z. Kotz, Balakrishna and Johnson (2000) discussed seven multivariate exponential distributions. Theorem: (Conditional distributions for the Multivariate Normal distribution) have p-variate Normal distribution with mean vector and Covariance matrix Then the conditional distribution of given is qi-variate Normal distribution with mean vector and Covariance matrix. , rate) The Exponential distribution uses a rate parameter, or "inverse scale", which can be intuited as, X ~ Exponential(rate=1) Y = X / rate See also. Infection Data with Generalized Exponential Baseline Distribution David D. _____ Problem 3 Let be a random sample drawn from an exponential distribution with the density function where. Example 2: Exponential Cumulative Distribution Function (pexp Function) We can also use the R programming language to return the corresponding values of the exponential cumulative distribution function for an input vector of quantiles. Our group has used eddy correlation to make over 10,000 hours of measurements of the turbulent momentum and heat fluxes over snow-covered sea ice in both the Arctic and the Antarctic. We conclude that the quality of the resulting design (in terms of expected cost) is not substantially affected by mis-specification of the distribution of process failure. This distribution is discussed in the framework of the exponential family, and its statistical properties regarding independence of the nodes are demonstrated. 21SEP05 M566 HOMEWORK 4 DUE THURSDAY 29SEP05 Problem 1. In the case of the multivariate Gaussian density, the argument ofthe exponential function, −1 2 (x − µ)TΣ−1(x − µ), is a quadratic form in the vector variable x. I want to do it by proving that it belongs to the exponential family and then bring to canonical form. 1 Multiparameter exponential families can be written as a 2-dimensional exponential family with su cient statistic Notation for multivariate derivatives It is. Multivariate t-distribution, which is another widely used spherically symmetric multivariate distribution. laplace is based on generation of a point s on the d-dimensional sphere and utilizes the auxiliary function generate. The mean and covariance are given by E(x) = (2) Var(x) = v v 2 1 The multivariate Tapproaches a multivariate Normal for large degrees of free-dom, v, as shown in Figure 1. The models and characterization investigated in this paper lead to the multivariate distribution with exponential marginals, which in the. Analysis of Variance also termed as ANOVA. In the first distribution (2. Figure 2: Multivariate Random Numbers with Normal Distribution. Hence we may treat this distribution as a multivariate gener-alization of double exponential distribution. Probability distribution function (PDF) of the univariate case (scalar domain, scalar parameter) The natural exponential families (NEF) are a subset of the exponential families. approximate exponential distribution using (3). software failure follows an exponential distribution and the author used Marshall-Olkin Multivariate Exponential Distribution (MOMED) as a base model for their proposed derivations. The relationship between the exponential distribution and the Laplace distribution allows for a simple method for simulating bivariate asymmetric Laplace variables (including for the case of = ). , and Indika, S. We first present some basic definitions and notations. 3 The assumption is the same as in corollary 1 of Theorem 4. In this case the method proposed by Marshall and Olkin (1997) may not produce the bivariate distribution in such a tractable form. Recently, Kundu and Gupta [A class of absolute continuous bivariate distributions. İngilizce Türkçe online sözlük Tureng. For example, let's say a Poisson distribution models the number of births in a. The generation of random samples from a truncated multivariate normal distribution, that is, a multivariate normal distribution subject to multiple linear inequality restrictions, is a recurring problem in the evaluation of integrals by Monte Carlo methods in econometrics and statistics. 2 CHAPTER 13. You can vote up the examples you like or vote down the ones you don't like. This class is an intermediary between the Distribution class and distributions which belong to an exponential family mainly to check the correctness of the. Tail conditional expectation for the multivariate Pareto distribution allocation and divided differences. 3), distributions with exponential scaled. distribution. These distributions, under multiplication and division, are self-replicating. Each approach used to derive this distribution was chosen for its intuitive appeal, and it is significant that each leads to the same distribution. Acknowledgments The author thanks Prof. The Marshall-. For most of the exponential family distribu-tions this is possible (a. On the other hand, using (2),the sample size had to be reduced to n =100 to obtain the same 100,000 draws within the same 25 second time period. In this case the method proposed by Marshall and Olkin (1997) may not produce the bivariate distribution in such a tractable form. 3 Distribution Needed for Hypothesis Testing, 9. 6(d) are associated provided that M and N, and El(1) and E2(1) are associated by the same reasoning in Remark 2. The verification condition implies verification of internal information x and y, thereby effectively resisting key recovery attack generated by the existence of equivalence key. This article proposes two multivariate exponential weighted moving average (MEWMA) charts to monitor the mean vector of Gumbel's bivariate exponential TBE model. Description of multivariate distributions • Discrete Random vector. The input argument pd can be a fitted probability distribution object for beta, exponential, extreme value, lognormal, normal, and Weibull distributions. Parameter Estimation and Characterization in a Generalized Bivariate Exponential Distribution. Since Σ is positive. You can find tips for working with the functions, means and variances of different distributions, and more. With the proper setting, we show that if all one dimensional marginals of a d-dimensional distribution are subexponential, then the distribution is multivariate subexponential. A new class of bivariate and multivariate exponential distributions Article (PDF Available) in Far East Journal of Mathematical Sciences 50(2) · March 2015 with 465 Reads How we measure 'reads'. We should have pij ≥ 0 and X i X j pij = 1. The exponential distribution with mean 1/β (ED(β)) and the Rayleigh distribution with parameter γ (RD(γ )) can be obtained as special cases from the LFR distribution. natural exponential family) but there are cases like the LogNormal distribution which while it belongs to the exponential family it cannot be writen in the canonical form. Thus, F T(t) = ˆ 0 if t<0; 1 exp( t= ) if t 0: Then,. But in our textbook (Dobson, Generalized linear model) there are only formulas for f(y) where y is not a vector as here y=(y1,y1) I want to do it by proving that it belongs to the exponential. It is an open access peer-reviewed textbook intended for undergraduate as well as first-year graduate level courses on the subject. The average run length performance of the MEWMA charts is compared to several individual TBE chart pairs. In this paper a multivariate failure rate representation based on Cox's conditional failure rate is introduced, characterizations of the Freund–Block and the Marshall–Olkin multivariate exponential distributions are obtained, and generalizations of the Block–Basu and the Friday–Patil bivariate exponential distributions are proposed. 2 for all of the simulations. Systematic component. distribution are special cases of the multivariate power exponential distribution for βequaling 0. It is often the case that there's more than one multivariate choice that seems to correspond to some univariate density - there's not always a natural one; hence we have papers with titles like "A multivariate exponential distribution", rather than "The multivariate exponential distribution". Keskustelualoitteita #52 Joensuun yliopisto, Taloustieteet GARCH Modelling with Power Exponential Distribution - Applications to Value at Risk. Also in particular, it addresses the common CRF problem of specifying "feature" functions determining the interactions between response variables and covariates. Conversely, if a member of the Exponential Family is specified, the. for all x ≥ x m, where x m is the (necessarily positive) minimum possible value of X, and k is a positive parameter. This article proposes two multivariate exponential weighted moving average (MEWMA) charts to monitor the mean vector of Gumbel's bivariate exponential TBE model. The next result shows that if we sample from the distribution of an exponential family, then the distribution of the random sample is itself an exponential family. Multivariate exponential distributions with latent factor structure and related topics Jan-Frederik Mai email: janfrederik. Index of R packages and their compatability with Renjin. A continuous multivariate exponential distribution is introduced which can model a full range of correlation structures and attains the Fréchet bounds in the bivariate case, is easy to simulate, arises as a model for reliability and failure due to shocks, and is analogous to the multivariate normal. The Wishart distribution is the probability distribution of the maximum-likelihood estimator (MLE) of the precision matrix of a multivariate normal distribution. I To check model ﬁt, we can generate samples from the posterior predictive distribution (letting X∗ = the observed. Use our online normal distribution calculator to find the area above, below or between the bell curve with the known values of mean and standard deviation. Just as the univariate normal distribution tends to be the most important statistical distribution in univariate statistics, the multivariate normal distribution is the most important distribution in multivariate statistics. Continuous Multivariate Distributions, Volume 1, Second Edition provides a remarkably comprehensive, self-contained resource for this critical statistical area. In the present paper we derive a family of bivariate exponential distributions based on an extended lack of memory property of a class of univariate distributions. Chapter 3 Exponential Families 3. We can write x e T§¡1x e = (x e2 ¡m e. For p = 1; the pdf (1) reduces to that of a double exponential (or Laplace) distribution. Y is also normal, and its. When both µ and J are unknown, the conjugate prior is a Gaussian-Wishart distribution. Bivariate Random Variables. = P(X = xi,Y = yj). It is procedure followed by statisticans to check the potential difference between scale-level dependent variable by a nominal-level variable having two or more categories. Such distributions can be represented by their probability density functions. 2015] did not allow positive dependencies for the exponential and Poisson generalizations. This deviation is not an indication that something is wrong. To learn about multivariate analysis, I would highly recommend the book "Multivariate analysis" (product code M249/03) by the Open University, available from the Open University Shop. The vector exponential distribution is defined over a subset of R^k, and parameterized by a (batch of) length- k loc vector and a (batch of) k x k scale matrix: covariance = scale @ scale. Probability Distributions. that is composed of a continuous distribution and a singular distribution, which seems unusual and unlikely for general real-world situations. Let X ˘N(0;1). 5 Exponential Approxmation 16. We consider methods for constructing multivariate dispersion models, illustrated by examples. That is, for any constant vector a Rk , the random variable Y = ax has a univariate normal distribution, where a univariate normal distribution with zero variance is a point mass on its mean. ") The covariance structure is specified by using a matrix whose ( i, j )th element specifies the covariance between the i th and j th variables. Each univariate distribution is an instance of a subclass of rv_continuous An exponential continuous random variable. 3), distributions with exponential scaled. For this distribution, the moment generating function is obtained, comparison is made with the case of independence, the distribution of the minimum is discussed, and various other properties are investigated. Autoregressive models with marginals as bivariate geometric distribution are developed. As βtends to 1, this distribution tends to a multivariate uniform distribution. These are the values of t that you use in a confidence interval. 6 ng/dL (SD±1130 ng/dL). In its simplest form, which is called the "standard" MV-N distribution, it describes the joint distribution of a random vector whose entries are mutually independent. Some of the justiﬁcations for this choice are as follows. case the situation is far from clear-cut. Figure 2 illustrates the output of the R code of Example 2. In this case the distribution has density[2] where is the determinant of. Description of multivariate distributions • Discrete Random vector. Such extensions involve the s t a n d a r d g a m m a (/3 = 1, "y = 0), or the exponential (a = 1), see Johnson and Kotz (1972). It is expressed in terms of equality of residual survival function with the survival function of the original distribution. In some cases, a distribution may be important because it is connected with other special distributions in interesting ways (via transformations, limits, conditioning, etc. This article proposes two multivariate exponential weighted moving average (MEWMA) charts to monitor the mean vector of Gumbel's bivariate exponential TBE model. The standardized trivariate normal distribution takes unit variances and. Autoregressive models with marginals as bivariate geometric distribution are developed. This distribution is discussed in the framework of the exponential family, and its statistical properties regarding independence of the nodes are demonstrated. In multivariate analysis controlling for transfusion burden and splenectomy, this difference was not significant for all genetic models considered (p = 0. 1) the conditional expectation of one variable decreases to zero with increasing values of the other one. The probit is a transformation with a mathematical connection to the normal (Gaussian) distribution, which is not only very similar in behaviour to the logit, but also more awkward to work with. This compendium contains a list of some of the most frequently encountered probability distributions. From the multivariate normal pdf in equation (1), we can re-express the term in the exponent as x e T§¡1x e = x e T 1 V11x e1 +x e T 1 V12x e2 +x e T 2 V21x e1 +x e T 2 V22x e2: (6) In order to compute the marginal and conditional distributions, we must complete the square in x e2 in this expression. Two bivariate distributions with exponential margins are analyzed and another is briefly mentioned. In this paper, a bivariate distribution with a two-parameter exponential conditional is obtained. Exponential distribution example Univariate Multivariate The score statistic Likelihoods are typically easier to work with on the log scale (where products become sums); furthermore, since it is only relative comparisons that matter with likelihoods, it is more meaningful to work with derivatives than the likelihood itself. continuous bivariate generalized exponential distribution with generalized exponential distribution as marginal using Clayton copula. Such extensions involve the standard gamma (/3 = 1, "y = 0), or the exponential (a = 1), see Johnson and Kotz. Then x has a multivariate normal distribution if and only if a’x follows a normal distribution for any. alization of a random variable having a Dirichlet distribution. Section 3 contains the main theoretical results of the paper. A class of multivariate linear models under the longitudinal setting, in which unobserved heterogeneity may evolve over time, is introduced. Multivariate extensions of gamma distributions such that all the marginals are again gamma are the most common in the literature. Poisson and Over-Dispersed Poisson Distributions. Meanwhile, the interest of this research is to define another version of the weighted exponential distribution following the content of Nasiru. This results in a new multivariate distribution for this exponential case. Note: Theorem 4. We illustrate the method of moments approach on this webpage. of this chapter is the simplicity and elegance of exponential family. We consider methods for constructing multivariate dispersion models, illustrated by examples. English Turkish online dictionary Tureng, translate words and terms with different pronunciation options. This deviation is not an indication that something is wrong. The moment expression for multivariate order. It crops up in many different applications including, for example, mean variance analysis and asset allocation, as well as geometric Brownian motion and the Black-Scholes[UNKNOWN]. normal distribution) for a continuous and diﬀerentiable function of a sequence of r. Sampling Exponential Distributions. Figure 4: Markov random eld POV on su cient statistics. A random variable which has a normal distribution with a mean m=0 and a standard deviation σ=1 is referred to as Standard Normal Distribution. Assume the distributions of the sample. ASTIN Bulletin 30 (1), 57-67. Discussions include exponential regression, models and applications of mixtures, and applications to survival analysis. In this paper, a bivariate distribution with a two-parameter exponential conditional is obtained. The Fisher-Tippett, extreme value, or log-Weibull distribution; Fisher's z-distribution. Multivariate normal distribution. Further, we have found it’s Marginal, multivariate conditional distributions,. This paper presents some meaningful derivations of a multivariate exponential distribution that serves to indicate conditions under which the distribution is appropriate. Find the probability. The mixed exponential distribution plays an important part in lifetime data analysis but we may suffer from extreme difficulties to get the maximal likelihood estimate due to the complexity of likelihood function,ZHU Liping etc. Multivariate extensions of gamma distributions such that all the marginals are again gamma are the most common in the literature. Statistical Distributions Fourth Edition Multivariate Distributions 24 17. The definitions are necessarily in terms of vague convergence of Radon measures rather than of ratios of distribution tails. 2 A Single Population Mean using the Student t Distribution, 9. The linear transform of a gaussian r. The Joint Distribution of Bivariate Exponential Under Linearly Related Model Pak. Statistics - Area of applied mathematics concerned with the data collection, analysis, interpretation and presentation. The simplest case of a Gaussian distribution is known as the standard normal probability distribution. The multivariate Gaussian distribution Gaussian likelihoods The Wishart distribution Multivariate Gaussian Analysis Ste en Lauritzen, University of Oxford BS2 Statistical Inference, Lecture 7, Hilary Term 2009 February 13, 2009 Ste en Lauritzen, University of Oxford Multivariate Gaussian Analysis. In canonical exponential families the log-likelihood function has at most one local maximum within Θ. Internal Report SUF-PFY/96-01 Stockholm, 11 December 1996 1st revision, 31 October 1998 last modiﬁcation 10 September 2007 Hand-book on STATISTICAL. (x) = exp(t(x) ( )) a proper probability density. Marshall-Olkin Bivariate Exponential Distribution: Generalisations and Applications 6. The univariate exponential distribution is also (sort of) closed under convolution. On Tue, Jun 14, 2011 at 08:40:00AM -0700, xuyongdeng wrote: > Any one know is there any package or function to generate bivariate > exponential distribution? I gusee there should be three parameters, two rate > parameters and one correlation parameter. I wondered if there was built in functionality to do this, if not how would I go about doing this manually?. , treatment arm in a trial) in a data model. via the multivariate sampling technique described in Gamerman (1997). In this paper, we consider the multivariate Bernoulli distribution as a model to estimate the structure of graphs with binary nodes. 1 Univariate Normal (Gaussian) Distribution Let Y be a random variable with mean (expectation) and variance ˙2 >0. Briefly, given univariate marginal densities and the corresponding distribution functions (here, the exponential distribution), the standard normal quantiles of the values of the distribution functions follow a multivariate standard normal. The quadrant probability in this special case is then given analytically by The quadrant probability in this special case is then given analytically by. An important property of PH-distributions is that they are closed under some operations. Some readers might be concerned that the distribution of the sample shown in Figure 2. In addition to satisfying (1. The linear transform of a gaussian r. approximate exponential distribution using (3). Probability Distributions. Achcar et al. Recently, probably due to the observation formulated by equa- is said to follow the multivariate Pareto distribution of the second tion (1. In this article, we de-velop a multivariate version of this distribution, which we call the conjugate multivariate (CM) distribution. We prove a characterization which states that a distribution is an MVME if, and only if, all positive linear combinations of the coordi-nates have a univariate matrix–exponential distribution. Besag (1974) developed this multivariate distribution for pairwise dependencies, and Yang et al. However, unlike the normal distribution, there is no natural extension available in a unique way.


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## Model Theory and Category TheoryJuly 29, 2008

Posted by dcorfield in Uncategorized.

David Kazhdan has some interesting things to say about model theory, and in particular its relationship to category theory, in his Lecture notes in Motivic Integration.

In spite of it successes, the Model theory did not enter into a “tool box” of mathematicians and even many of mathematicians working on “Motivic integrations” are content to use the results of logicians without understanding the details of the proofs.

I don’t know any mathematician who did not start as a logician and for whom it was “easy and natural” to learn the Model theory. Often the experience of learning of the Model theory is similar to the one of learning of Physics: for a [short] while everything is so simple and so easily reformulated in familiar terms that “there is nothing to learn” but suddenly one find himself in a place when Model theoreticians “jump from a tussock to a hummock” while we mathematicians don’t see where to “put a foot” and are at a complete loss.

## Same but multifacetedJuly 11, 2008

Posted by Alexandre Borovik in Uncategorized.

Continuing the topic of “sameness”, it is interesting to compare behaviour of two familiar objects: the field of real numbers $\mathbb{R}$ and the field of complex numbers $\mathbb{C}$.

$\mathbb{C}$ is uncountably categorical, that is, it is uniquely described in a language of first order logic among the fields of the same cardinality.

In case of $\mathbb{R}$, its elementary theory, that is, the set of all closed first order formulae that are true in $\mathbb{R}$, has infinitely many models of cardinality continuum $2^{\aleph_0}$.

In naive terms, $\mathbb{C}$ is rigid, while $\mathbb{R}$ is soft and spongy and shape-shifting. However, $\mathbb{R}$ has only trivial automorphisms (an easy exercise), while $\mathbb{C}$ has huge automorphism group, of cardinality $2^{2^{\aleph_0}}$ (this also follows with relative ease from basic properties of algebraically closed fields). In naive terms, this means that there is only one way to look at $\mathbb{R}$, while $\mathbb{C}$ can be viewed from an incomprehensible variety of different point of view, most of them absolutely transcendental. Actually, there are just two comprehensible automorphisms of $\mathbb{C}$: the identity automorphism and complex conjugation. It looks like construction of all other automorphisms involves the Axiom of Choice. When one looks at what happens at model-theoretic level, it appears that “uniqueness” and “canonicity” of a uncountable structure is directly linked to its multifacetedness. I am still hunting appropriate references for this fact. Meanwhile, I got the following e-mail from a model theorist colleague, Zoe Chatzidakis:

Models of uncountably categorical theories behave really like vector spaces: if inside a model $M$ you take a maximal independent set $X$ of elements realizing the generic type, and take any permutation of $X$, it extends to an automorphism of the model. So, if $M$ is of size $\kappa > \aleph_0$, then any basis has size $\kappa$, and its automorphism group has size $2^\kappa$.

I don’t know a reference, but it should be in any model theory book which talks about strongly minimal sets. Or maybe in the paper by ??? Morley ??? which shows that you have a notion of dimension and so on? I.e., that $\aleph_1$ categorical theories and strongly minimal sets are the same.

It is really a well-known result, so you probably don’t need a reference if you cite it in a paper.

## Facing EternityJuly 4, 2008

Posted by Alexandre Borovik in Uncategorized.

Mikhail Zlatkovsky. Facing Eternity.

Under a changed title “Coat Star” the cartoon won first prize at Ken Sprague Fund competition at  Earthworks 2008.

## Ultraproducts of fields, IIJune 28, 2008

Posted by Alexandre Borovik in Uncategorized.
1 comment so far

I continue my post on ultraproducts. So, we want to understand in what sense an ultraproduct of finite fields $F_i$ of unbounded order is a limit at infinity of finite fields. The answer now should be obvious: since ultraproducts are residue fields for maximal ideals in the cartesian product

$R = \prod F_i,$

the topology in question should be the canonical topology (Zariski topology) of the spectrum of the ring $\mbox{Spec}(R)$. It instantly follows from the description of ideals and maximal ideals in $R$ that this is the Stone topology on the set of ultrafilters on $\mathbb{N}$, or, what is th same, the Cech-Stone compactification $\beta\mathbb{N}$ of the set $\mathbb{N}$ with descrete topology. Therefore the answer is: an ultraproduct is the limit in the Cech-Stone compactification of a discrete countable set.

I have to admit that at this point I reached limits of my knowledge of set-theoretic topology and had to dip into Wikipedia. It happened that $\beta\mathbb{N}$, and even more so its non-principal part ${\mathbb{N}}^* = \beta{\mathbb{N}} \setminus {\mathbb{N}}$ is characterised by some unique properties: If the continuum hypothesis holds then ${\mathbb{N}}^*$ is the unique Parovicenko space, up to isomorphism.

According to Wikipedia, a Parovicenko space is a topological space X satisfying the following conditions:

• X is compact Hausdorff
• X has no isolated points
• X has weight c, the cardinality of the continuum (this is the smallest cardinality of a base for the topology).
• Every two disjoint open Fσ subsets of X have disjoint closures
• Every nonempty Gδ of X has non-empty interior.

As you can see, ${\mathbb{N}}^*$ is uniquely characterised by very natural properties.

It is yet another manifestation of of one of the most pardoxical properties of mathematical infinity: canonicity of workable constructions in the infinite domain.

## NHS invented a new type of infinity…June 21, 2008

Posted by Alexandre Borovik in Uncategorized.

My posts are likely to become shorter and sparser — as a result of a work trauma, I have developed a medical condition (see a photo) which makes typing very difficult, and I depend on the kind help of my wife with all my typing needs. On the bright side, my experience gave me a new understanding of infinity. From a mathematical viewpoint, indefinitely long waiting lists for treatment on NHS (National Health Service) were not something new, they were just a special case of potentially infinite natural series

Week 1, Week 2, Week 3, … etc.

going on and on in a very old-fashioned way. The real contribution of NHS to mathematics of infinity is that they make patients to wait (indefinitely again) to be included on a waiting list. It is an infinity of natural numbers enhanced by an additional constraint: you are not allowed to start counting.

## A letter from a studentJune 17, 2008

Posted by Alexandre Borovik in Uncategorized.

Hello Professor,

I hope everything is good and that you recovered from the accident in your
finger.

I just wanted to share some personal thoughts. Suppose we have a system that is discrete and finite. We have the natural numbers {1,2,3,…,T} where T is the symbol for the ”biggest natural number”. We will never have a value for T but we accept that it is a fixed natural number. We can also include 0 in our system. How much mathematics can we do?

We could define the usual addition and multiplication. But we will have a problem when the result is ”greater than T”. But nothing ”greater than T” exist…

Suppose for example that T=10. Then we have 2+3=5, 4+5=9, 5+5=10. But what about 5+7? We could just define 5+7= err (error). just like the small calculators do when they reach their limit. But err is not in our system… We could work modulo T OR we could just say that 5+7=10=T. and 8+9=T. and 7+8=10=T etc.

(If we choose the last option then obviously 1+T=T and T+T=T. So T has a similar behaviour as the infinite that we learned at school. But the big difference is that in our system T is a fixed natural number!)

I just can not see why we NEED infinite to make mathematics. Is it a matter of convenience? Is it just to make things simpler? If this is the case then we should accept that infinite and continuous entities are just tools, ideas, symbols that make our life easier. But we should remember that IDEAS themselves are FINITE since they live in the finite world of our brains!

I think the whole problem is more about aesthetics. I think that someone can accept that only discrete and finite things exist and, at the same time, that this belief does not destroy all the beautifull mathematics that have been created until now.

While I was searching the Internet I found this PDF document which I attached and I found amusing. I also found other people expressing similar views (against infinite) like for example Alexander Yessenin-Volpin.

## Ultraproducts of fields, IJune 16, 2008

Posted by Alexandre Borovik in Uncategorized.

My immediate research interests more and more focus on an interplay between finite and infinite in algebra (at least this is where my chats with my PhD student drift to). In particular, I have to use frequently a specific construction, an ultrafilter product of fields. It is pretty sublime in the sense of David Corfield and leads to appearance of very interesting canonical objects.

We start with a family of fields $F_i$, $i \in I$ . For simplicity assume that all fields are finite of unbounded order and that the index set $I$ is just the set of natural numbers $\mathbb{N}$ (actually this is the case most interesting to me). We form the Cartesian product $R$ of $F_i$: this is just a set of infinite strings

$\{ (f_1, f_2, \dots ) \mid f_i \in F_i \}$

with componentwise operations of addition and multiplication. In what follows, we shall make frequent use of zero set of a string $f = (f_1, f_2, \dots )$, that is, the set of indices where the string components are zeroes:

$zero(f) = \{ i \in {\mathbb{N}} \mid f_i =0\}$.

Obviously, $R$ is a commutative ring with unity. Let us look at its ideals. One can easily see that an ideal $I$ in $R$ is uniquely determined by the set

$zero(I) = \{zero(f) \mid f \in I\}$;

indeed, non-zero components of a string $f \in I$ can be arbitrarily changed without moving $f$ outside of $I$ by multiplying by appropriate string of invertible elements. One can also instantly see that $zero(I)$ is a filter on $\mathbb{N}$, that is, it is a collection of non-empty subsets of $\mathbb{N}$ closed under taking finite intersections and supersets, and, moreover, that the correspondence

$I \mapsto zero(I)$

is a one-to-one correspondence between proper ideals in $R$ and filters of $\mathbb{N}$ which preserves embedding of ideals and embedding of filters. Therefore, maximal ideals in $R$ correspond to maximal filters on $\mathbb{N}$; the former and the latter exist by the Zorn Lemma, one of the equivalent formulations of the Choice Axiom. If now $I$ is a maximal ideal in $R$, the fact that the factor ring $R/I$ is a field and, in particular, has no zero divisors, translates in the fact that a maximal filter $\mathcal{F}$ is an ultrafilter: it has the property that, for any subset $X \subseteq \mathbb{N}$ either $X$ or its complement ${\mathbb{N}} \smallsetminus X$ belongs to $\mathcal{F}$.

Given an ultrafilter $\mathcal{F}$ on $\mathbb{N}$, the ultraproduct $F = \prod F_i/\mathcal{F}$ is nothing more than the corresponding residue field $R/I$. There are obvious principal (ultra)filters on $\mathbb{N}$, they consist of all subsets containing a given element $i \in \mathbb{N}$; obviously, the corresponding ultraproduct is just the original field $F_i$.

Non-principal filters do exists. One very interesting non-principal filter on $\mathbb{N}$ is the Frechet filter consisting of all subsets with finite complements.

But if we take an ultrafilter containing a non-principal filter (it exists by the Zorn Lemma), the corresponding ultraproduct $F$ has many marvelous properties. In particular, if all $F_i$ have different characteristics, $F$ turns out to be a field of characteristic zero (I leave the proof of this fact as an exercise to the reader).

In the next instalment of this post, I will discuss the meaning of a frequently used assertion that an ultraproduct $F$ is a limit at infinity of finite fields $F_i$.

## The SublimeJune 11, 2008

Posted by dcorfield in Uncategorized.

There’s an interesting post – Whatever Happened to Sublimity? – at the blog Siris. It includes a quotation from Edmund Burke

But let it be considered that hardly any thing can strike the mind with its greatness which does not make some sort of approach toward infinity; which nothing can do while we are able to perceive its bounds; but to see an object distinctly, and to perceive its bounds, are one and the same thing. A clear idea is, therefore, another name for a little idea. (A Philosophical Inquiry into the Origin of Our Ideas of the Sublime and the Beautiful, Part II, Section V.)

A natural question to ask, then, is where do we encounter the sublime in mathematics? And an obvious answer, you might think, would be the mathematical infinite.

Joseph Dauben has an interesting section in his book – Georg Cantor: his mathematics and philosophy of the infinite, Harvard University Press, 1979 – on how Cantor, receiving such discouragement from his mathematical colleagues, found an audience in certain thinkers within the Catholic church. Where earlier in the nineteenth century any attempt to describe a completed infinity was viewed as a sacrilegious attempt to circumscribe God, some theologians were open to Cantor’s new hierarchy of infinities, with its unreachable Absolute Infinite leaving room for the divine.

Personally, set theory has rarely invoked in me a sense of the sublime. On the other hand, the following comment by Daniel Davis does:

Behrens and Lawson use stacks, the theory of buildings, homotopy fixed points, the above model category, and other tools to make it possible to use the arithmetic of Shimura varieties to help with understanding the stable homotopy groups of spheres.

There’s plenty about the infinite in that statement, it’s true, but there’s so much more to it than that.

Brandon Watson, the blogger at Siris, writes

…what often strikes me when I look around at the philosophical scene today is how foreign this has all become. There are a few exceptions, but sublimity has vanished as a serious concern.

With regard to the philosophy of mathematics in particular I couldn’t agree more.

## Axiom of Choice, quotesJune 1, 2008

Posted by Alexandre Borovik in Uncategorized.
• The Axiom of Choice is obviously true, the Well-ordering theorem obviously false, and who can tell about Zorn’s lemma ?” ::— Jerry Bona -(This is a joke: although the axiom of choice, the well-ordering principle, and Zorn’s lemma are all mathematically equivalent, most mathematicians find the axiom of choice to be intuitive, the well-ordering principle to be counterintuitive, and Zorn’s lemma to be too complex for any intuition.)
• “The Axiom of Choice is necessary to select a set from an infinite number of socks, but not an infinite number of shoes.” ::— Bertrand Russell -(The observation here is that one can define a function to select from an infinite number of pairs of shoes by stating for example, to choose the left shoe. Without the axiom of choice, one cannot assert that such a function exists for pairs of socks, because left and right socks are (presumably) identical to each other.)
• “The axiom gets its name not because mathematicians prefer it to other axioms.” ::— A. K. Dewdney -( This quote comes from the famous April Fool’s Day article in the computer recreations column of the Scientific American , April 1989.)

[Source]

## Induction and recursion IIMay 28, 2008

Posted by David Pierce in Uncategorized.

Formal logic provides examples of structures that allow proof by induction, but not necessarily definition by recursion. Let us consider, for example, the propositional logic that is apparently due to Łukasiewicz. We first define the (well-formed) formulas of the logic. We start with some set of so-called propositional variables. Then the set of formulas is the smallest set of strings that contains these variables and is closed under certain formation rules, namely:

1. the singulary operation converting a string S to its negation, ~S;
2. the binary operation converting a pair (S, T) of strings to the implication, (S → T).

The set of theorems of the logic is the smallest set of formulas that contains all of the axioms and is closed under a certain rule of inference. The axioms are the formulas of any of three forms:

1. (F → (G → F))
2. ((F → (G → H)) → ((F → G) → (F → H)))
3. ((~F → ~G) → (G → F))

Here F and G stand for formulas. The rule of inference is detachment (or modus ponens), the binary operation that converts a pair (F, (F → G)) of formulas into the formula G. As stated, this is only a partial operation; we can make it total by having it convert (F, H) to F whenever H does not have the form (F → G) for some formula G.
The “inductive” definition of theorems is thus similar to the inductive definition of formulas. In each case, we start with a set and close under one or more operations. Immediately, proof by induction is possible on the set so defined. For example, we can prove inductively that each formula features as many left as right brackets. A feature of theorems that is proved by induction will be given in a moment.
However, unlike the formation rules for formulas, our rule of inference for theorems is not obviously well-defined. To see that it is well-defined, we must establish “unique readability” of formulas, namely:

1. each formula is exactly one of the following: a variable, a negation, or an implication;
2. the formation rules are injective.

Here injectivity of the formation rules corresponds to injectivity of the successor-operation on the set of natural numbers. The partition of the set of formulas into variables, negations, and implications corresponds to the partition of the set of natural numbers into the successors and the initial number (zero or one, as one prefers).
Unique readability of formulas allows recursive definitions. Indeed, the definition of the rule of detachment can be understood as recursive: If we denote this operation by D, then we have

1. D(F, P) = F for all propositional variables P;
2. D(F, ~G) = F;
3. D(F, (F → G)) = G;
4. D(F, (H → G)) = F if H is not F.

Another example of a recursively defined function on formulas is a truth-assignment. Consider the set {true, false} as the two-element field {1, 0}. Given a function φ from the set of propositional variables into the this field, we have a unique function Φ on the set of all formulas that agrees with φ on the variables, that takes ~F to 1 + Φ(F), and that takes (F → G) to 1 + Φ(F) + Φ(F)Φ(G).
By induction, every theorem takes the value 1 under every truth-assignment. But the truth-assignment is not recursively defined on the set of theorems: it is recursively defined on the set of formulas. Since the rule of detachment is not injective, we do not automatically have recursively defined functions on the set of theorems.
As far as I know, we do not have an efficient algorithm to determine whether a given formula is indeed a theorem: this lack of an algorithm would appear to be connected to the non-injectivity of the rule of detachment. A formula carries within itself the method of its construction; but a theorem does not carry within itself its proof, and indeed it will have infinitely many proofs.

format: plain


text:
# Two slabs A and B of different materials but of the same thicknesss are joined end to end to form a composite slab. The thermal conductivities of A an

185 views
in Physics
closed
Two slabs A and B of different materials but of the same thicknesss are joined end to end to form a composite slab. The thermal conductivities of A and B are K_(1) and K_(2) respectively. A steady temperature difference of 12^(@)C is maintained across the composite slab. If K_(1) = K_(2)/2, the temperature difference across slabs A is
A. 4^(@)C
B. 6^(@)C
C. 8^(@)C
D. 10^(@)C

by (90.7k points)
selected

Correct Answer - C
The given equation can be shown as

Rate of flow of heat will be equal in both the slabs
therefore (12-x)K_(1)= K_(2) (x-0)
12-x = 2x rArr x=4^(@)C (therefore K_(1) = (K_(2)/2)
The temperature difference across slab
A = (12-x) = (12-4) = 8^(@)C


text:
transform() is a function used for modifying columns in SoilProfileCollections.

It allows the user to specify an arbitrary number of expressions that resolve to the (re-)calculation of one or more site or horizon level attributes. For instance: mutate(spc, thickness = hzdepb - hzdept). The expressions may depend on one another, and are evaluated from left to right.

# S4 method for SoilProfileCollection
transform(_data, ...)

Arguments

_data

A SoilProfileCollection

...

Comma-separated set of R expressions e.g. thickness = hzdepb - hzdept, hzdepm = hzdept + round(thk / 2)

Value

A SoilProfileCollection

Andrew G. Brown.


text:
Integer type:  int32  int64  nag_int  show int32  show int32  show int64  show int64  show nag_int  show nag_int

PDF version (NAG web site, 64-bit version, 64-bit version)
Chapter Contents
Chapter Introduction
NAG Toolbox

# NAG Toolbox: nag_rand_field_1d_generate (g05zp)

## Purpose

nag_rand_field_1d_generate (g05zp) produces realizations of a stationary Gaussian random field in one dimension, using the circulant embedding method. The square roots of the eigenvalues of the extended covariance matrix (or embedding matrix) need to be input, and can be calculated using nag_rand_field_1d_user_setup (g05zm) or nag_rand_field_1d_predef_setup (g05zn).

## Syntax

[state, z, ifail] = g05zp(ns, s, lam, rho, state, 'm', m)
[state, z, ifail] = nag_rand_field_1d_generate(ns, s, lam, rho, state, 'm', m)

## Description

A one-dimensional random field $Z\left(x\right)$ in $ℝ$ is a function which is random at every point $x\in ℝ$, so $Z\left(x\right)$ is a random variable for each $x$. The random field has a mean function $\mu \left(x\right)=𝔼\left[Z\left(x\right)\right]$ and a symmetric non-negative definite covariance function $C\left(x,y\right)=𝔼\left[\left(Z\left(x\right)-\mu \left(x\right)\right)\left(Z\left(y\right)-\mu \left(y\right)\right)\right]$. $Z\left(x\right)$ is a Gaussian random field if for any choice of $n\in ℕ$ and ${x}_{1},\dots ,{x}_{n}\in ℝ$, the random vector ${\left[Z\left({x}_{1}\right),\dots ,Z\left({x}_{n}\right)\right]}^{\mathrm{T}}$ follows a multivariate Normal distribution, which would have a mean vector $\stackrel{~}{\mathbf{\mu }}$ with entries ${\stackrel{~}{\mu }}_{i}=\mu \left({x}_{i}\right)$ and a covariance matrix $\stackrel{~}{C}$ with entries ${\stackrel{~}{C}}_{ij}=C\left({x}_{i},{x}_{j}\right)$. A Gaussian random field $Z\left(x\right)$ is stationary if $\mu \left(x\right)$ is constant for all $x\in ℝ$ and $C\left(x,y\right)=C\left(x+a,y+a\right)$ for all $x,y,a\in ℝ$ and hence we can express the covariance function $C\left(x,y\right)$ as a function $\gamma$ of one variable: $C\left(x,y\right)=\gamma \left(x-y\right)$. $\gamma$ is known as a variogram (or more correctly, a semivariogram) and includes the multiplicative factor ${\sigma }^{2}$ representing the variance such that $\gamma \left(0\right)={\sigma }^{2}$.
The functions nag_rand_field_1d_user_setup (g05zm) or nag_rand_field_1d_predef_setup (g05zn), along with nag_rand_field_1d_generate (g05zp), are used to simulate a one-dimensional stationary Gaussian random field, with mean function zero and variogram $\gamma \left(x\right)$, over an interval $\left[{x}_{\mathrm{min}},{x}_{\mathrm{max}}\right]$, using an equally spaced set of $N$ points. The problem reduces to sampling a Normal random vector $\mathbf{X}$ of size $N$, with mean vector zero and a symmetric Toeplitz covariance matrix $A$. Since $A$ is in general expensive to factorize, a technique known as the circulant embedding method is used. $A$ is embedded into a larger, symmetric circulant matrix $B$ of size $M\ge 2\left(N-1\right)$, which can now be factorized as $B=W\Lambda {W}^{*}={R}^{*}R$, where $W$ is the Fourier matrix (${W}^{*}$ is the complex conjugate of $W$), $\Lambda$ is the diagonal matrix containing the eigenvalues of $B$ and $R={\Lambda }^{\frac{1}{2}}{W}^{*}$. $B$ is known as the embedding matrix. The eigenvalues can be calculated by performing a discrete Fourier transform of the first row (or column) of $B$ and multiplying by $M$, and so only the first row (or column) of $B$ is needed – the whole matrix does not need to be formed.
As long as all of the values of $\Lambda$ are non-negative (i.e., $B$ is non-negative definite), $B$ is a covariance matrix for a random vector $\mathbf{Y}$, two samples of which can now be simulated from the real and imaginary parts of ${R}^{*}\left(\mathbf{U}+i\mathbf{V}\right)$, where $\mathbf{U}$ and $\mathbf{V}$ have elements from the standard Normal distribution. Since ${R}^{*}\left(\mathbf{U}+i\mathbf{V}\right)=W{\Lambda }^{\frac{1}{2}}\left(\mathbf{U}+i\mathbf{V}\right)$, this calculation can be done using a discrete Fourier transform of the vector ${\Lambda }^{\frac{1}{2}}\left(\mathbf{U}+i\mathbf{V}\right)$. Two samples of the random vector $\mathbf{X}$ can now be recovered by taking the first $N$ elements of each sample of $\mathbf{Y}$ – because the original covariance matrix $A$ is embedded in $B$, $\mathbf{X}$ will have the correct distribution.
If $B$ is not non-negative definite, larger embedding matrices $B$ can be tried; however if the size of the matrix would have to be larger than maxm, an approximation procedure is used. See the documentation of nag_rand_field_1d_user_setup (g05zm) or nag_rand_field_1d_predef_setup (g05zn) for details of the approximation procedure.
nag_rand_field_1d_generate (g05zp) takes the square roots of the eigenvalues of the embedding matrix $B$, and its size $M$, as input and outputs $S$ realizations of the random field in $Z$.
One of the initialization functions nag_rand_init_repeat (g05kf) (for a repeatable sequence if computed sequentially) or nag_rand_init_nonrepeat (g05kg) (for a non-repeatable sequence) must be called prior to the first call to nag_rand_field_1d_generate (g05zp).

## References

Dietrich C R and Newsam G N (1997) Fast and exact simulation of stationary Gaussian processes through circulant embedding of the covariance matrix SIAM J. Sci. Comput. 18 1088–1107
Schlather M (1999) Introduction to positive definite functions and to unconditional simulation of random fields Technical Report ST 99–10 Lancaster University
Wood A T A and Chan G (1994) Simulation of stationary Gaussian processes in ${\left[0,1\right]}^{d}$ Journal of Computational and Graphical Statistics 3(4) 409–432

## Parameters

### Compulsory Input Parameters

1:     $\mathrm{ns}$int64int32nag_int scalar
The number of sample points to be generated in realizations of the random field. This must be the same value as supplied to nag_rand_field_1d_user_setup (g05zm) or nag_rand_field_1d_predef_setup (g05zn) when calculating the eigenvalues of the embedding matrix.
Constraint: ${\mathbf{ns}}\ge 1$.
2:     $\mathrm{s}$int64int32nag_int scalar
$S$, the number of realizations of the random field to simulate.
Constraint: ${\mathbf{s}}\ge 1$.
3:     $\mathrm{lam}\left({\mathbf{m}}\right)$ – double array
Must contain the square roots of the eigenvalues of the embedding matrix, as returned by nag_rand_field_1d_user_setup (g05zm) or nag_rand_field_1d_predef_setup (g05zn).
Constraint: ${\mathbf{lam}}\left(i\right)\ge 0,i=1,2,\dots ,{\mathbf{m}}$.
4:     $\mathrm{rho}$ – double scalar
Indicates the scaling of the covariance matrix, as returned by nag_rand_field_1d_user_setup (g05zm) or nag_rand_field_1d_predef_setup (g05zn).
Constraint: $0.0<{\mathbf{rho}}\le 1.0$.
5:     $\mathrm{state}\left(:\right)$int64int32nag_int array
Note: the actual argument supplied must be the array state supplied to the initialization routines nag_rand_init_repeat (g05kf) or nag_rand_init_nonrepeat (g05kg).
Contains information on the selected base generator and its current state.

### Optional Input Parameters

1:     $\mathrm{m}$int64int32nag_int scalar
Default: the dimension of the array lam.
$M$, the size of the embedding matrix, as returned by nag_rand_field_1d_user_setup (g05zm) or nag_rand_field_1d_predef_setup (g05zn).
Constraint: ${\mathbf{m}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,2\left({\mathbf{ns}}-1\right)\right)$.

### Output Parameters

1:     $\mathrm{state}\left(:\right)$int64int32nag_int array
Contains updated information on the state of the generator.
2:     $\mathrm{z}\left({\mathbf{ns}},{\mathbf{s}}\right)$ – double array
Contains the realizations of the random field. The $j$th realization, for the ns sample points, is stored in ${\mathbf{z}}\left(i,j\right)$, for $i=1,2,\dots ,{\mathbf{ns}}$. The sample points are as returned in ${\mathbf{xx}}$ by nag_rand_field_1d_user_setup (g05zm) or nag_rand_field_1d_predef_setup (g05zn).
3:     $\mathrm{ifail}$int64int32nag_int scalar
${\mathbf{ifail}}={\mathbf{0}}$ unless the function detects an error (see Error Indicators and Warnings).

## Error Indicators and Warnings

Errors or warnings detected by the function:
${\mathbf{ifail}}=1$
Constraint: ${\mathbf{ns}}\ge 1$.
${\mathbf{ifail}}=2$
Constraint: ${\mathbf{s}}\ge 1$.
${\mathbf{ifail}}=3$
Constraint: ${\mathbf{m}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,2×\left({\mathbf{ns}}-1\right)\right)$.
${\mathbf{ifail}}=4$
On entry, at least one element of lam was negative.
Constraint: all elements of lam must be non-negative.
${\mathbf{ifail}}=5$
Constraint: $0.0\le {\mathbf{rho}}\le 1.0$.
${\mathbf{ifail}}=6$
On entry, state vector has been corrupted or not initialized.
${\mathbf{ifail}}=-99$
An unexpected error has been triggered by this routine. Please contact NAG.
${\mathbf{ifail}}=-399$
Your licence key may have expired or may not have been installed correctly.
${\mathbf{ifail}}=-999$
Dynamic memory allocation failed.

## Accuracy

Not applicable.

Because samples are generated in pairs, calling this function $k$ times, with ${\mathbf{s}}=s$, say, will generate a different sequence of numbers than calling the function once with ${\mathbf{s}}=ks$, unless $s$ is even.

## Example

This example calls nag_rand_field_1d_generate (g05zp) to generate $5$ realizations of a random field on $8$ sample points using eigenvalues calculated by nag_rand_field_1d_predef_setup (g05zn) for a symmetric stable variogram.
```function g05zp_example

fprintf('g05zp example results\n\n');

% Choose the Symmetric stable variogram
icov1 = int64(1);
params = [0.1; 1.2];

% Random Field variance
var = 0.5;
% Domain endpoints
xmin = -1;
xmax = 1;
% Number of sample points
ns = int64(8);
% Scaling factor, rho = 1.
icorr = int64(2);

% Get square roots of the eigenvalues of the embedding matrix
[lam, xx, m, approx, rho, icount, eig, ifail] = ...
g05zn( ...
ns, xmin, xmax, var, icov1, params, 'icorr', icorr);

fprintf('\nSize of embedding matrix = %d\n\n', m);

% Display approximation information if approximation used
if approx == 1
fprintf('Approximation required\n\n');
fprintf('rho = %10.5f\n', rho);
fprintf('eig = %10.5f%10.5f%10.5f\n', eig(1:3));
fprintf('icount = %d\n', icount);
else
fprintf('Approximation not required\n\n');
end

state = initialize_state();

% Compute s random field realisations.
s = int64(5);
[state, z, ifail] = g05zp( ...
ns, s, lam(1:m), rho, state);

% Display realizations
fprintf('Random field realisations:\n     ');
fprintf('%10d',[1:5]);
fprintf('\n');
disp([xx, z]);

function state = initialize_state()
genid = int64(1);
subid = int64(1);
seed  = [int64(14965)];
[state, ifail] = g05kf( ...
genid, subid, seed);
```
```g05zp example results

Size of embedding matrix = 16

Approximation not required

Random field realisations:
1         2         3         4         5
-0.8750   -0.4166   -0.8185   -0.9769    0.6741   -0.6762
-0.6250    0.0146    1.4538    0.0248    0.5218    1.9466
-0.3750   -0.5556    0.2913   -0.0853    0.4214   -0.1389
-0.1250   -0.5568    0.3199   -0.6094    0.2019    0.9085
0.1250   -0.0423    0.0486    1.4590    0.3608   -0.5288
0.3750   -0.2806   -0.7969    0.2330    0.1335    0.4012
0.6250    0.9298   -0.3956   -0.8455   -0.2749    0.5270
0.8750    0.3222    1.5227   -2.1645    0.1794    1.1937

```

PDF version (NAG web site, 64-bit version, 64-bit version)
Chapter Contents
Chapter Introduction
NAG Toolbox

© The Numerical Algorithms Group Ltd, Oxford, UK. 2009–2015


text:
# 11.18. Easy Multiple Choice Questions¶

These problems are easier than most of those that you will usually see on the AP CS A exam.

10-12-1: What best describes the purpose of a class’s constructor?
• Initialize the fields in the object.
• A constructor is often used to initialize the fields to their default values or in the case of a parameterized constructor, to the values passed in to the constructor.
• Determines the amount of space needed for an object and creates the object.
• The object is already created before the constructor is called.
• Names the new object.
• Constructors do not name the object.
10-12-2: Under which of these conditions is it appropriate to overload a method (ie: the class will contain two methods with the same name)?
• The methods do different things.
• Methods that do different things should be named differently.
• The methods have different parameter names.
• There is no reason the parameter names ought to be different if the two methods are performing the same action.
• The methods have different post-conditions.
• If the methods have different post-conditions, they are performing different functions, and should be named differently.
• Two methods with the same name can never be included in the same class.
• If two methods perform the same function, they can be named the same. However, the number of parameters, type of parameters, or order of parameter types must be different.
• The methods have different numbers of parameters
• Overloading occurs when two methods perform the same essential operation, but take a different number and/or type of parameters.

10-12-3: Which of the following statements about a class that contains an abstract method is (are) true?

I. You can't have any constructors in this class.
II.  This class must be declared as abstract.
III.  You can't declare any fields in this class.

• I and II only
• An abstract class can have constructors. A class with an abstract method must also be declared as abstract.
• II only
• A class with an abstract method must also be declared abstract. You can have constructors and fields in an abstract class.
• I, II and III
• A class with an abstract method must also be abstract. You can have constructors and fields in an abstract class.
• I only
• Only II is true. You can have constructors in an abstract class. A class with an abstract method must also be declared abstract.
• III only
• Only II is true. You can have fields in an abstract class. A class with an abstract method must also be declared abstract.
10-12-4: Which of the following is true about abstract classes?
• Abstract classes cannot be instantiated, but they can be sub-classed.
• Sub-classes must implement the abstract methods declared in the abstract class or also be declared abstract.
• Abstract classes can be instantiated, but they cannot be sub-classed.
• You can not create an object of an abstract class type. You can only create objects from concrete (not abstract) classes.
• Abstract classes can only contain abstract methods. They can be sub-classed.
• Abstract classes can contain fields and non-abstract methods.
• Abstract classes can only contain abstract methods. They cannot be sub-classed.
• Abstract classes can contain fields and non-abstract methods. They can also be sub-classed.
10-12-5: A car dealership needs a program to store information about the cars for sale. For each car, they want to keep track of the following information: number of doors (2 or 4), whether the car has air conditioning, and its average number of miles per gallon. Which of the following is the best design?
• Use four unrelated classes: Car, Doors, AirConditioning, and MilesPerGallon.
• Only Car should be a class. The number of doors, flag if it has air conditioning, and the average number of miles per gallon are attributes of a car so they belong in a Car class.
• Use a class Car with three subclasses: Doors, AirConditioning, and MilesPerGallon.
• Doors, air conditioning, and miles per gallon are not a kind of car. Child classes need to be able to be substituted for the parent class.
• Use a class Car, with fields: numDoors, hasAir, and milesPerGallon.
• The number of doors, flag if it has air conditioning, and the average number of miles per gallon are attributes of a car. Each of these is a simple value so they can just be fields of a Car class.
• Use a class Car, with subclasses of Doors, AirConditioning, and MilesPerGallon.
• A door is not a type of car. A flag for air conditioning is not a type of door, and a miles per gallon is not a type of air conditioning flag. Child classes need to be able to be substituted for the parent class.
• Use classes: Doors, AirConditioning, and MilesPerGallon, each with a subclass Car.
• A class Car can't be a subclass of three different classes. Each class can only have one parent class. Also a car is not a type of door, air conditioning flag, or miles per gallon. Child classes need to be able to be substituted for the parent class.
10-12-6: A program is being written by a team of programmers. One programmer is implementing a class called Employee; another programmer is writing code that will use the Employee class. Which of the following aspects of the public methods and fields of the Employee class does not need to be known by both programmers?
• How the methods are implemented.
• Only the programmer of the Employee class must know how the public methods work. The programmer that is using the Employee class can just use the public methods and not worry about how they are implemented.
• The method names.
• The programmer who writes the methods will need to know what the names are. The programmer who will use the public methods will also need to know the names of the methods in order to invoke them.
• The method return types.
• In order to use the public methods of the Employee class, a programmer must know the method return types.
• Constants
• Constants are public fields and are meant to be used by people using a class.
• The number and types of the method parameters.
• In order to use the public methods of the Employee class, a programmer must know the number of parameters and the type for each parameter.
10-12-7: A bookstore is working on an on-line ordering system. For each type of published material (books, movies, audio tapes) they need to track the id, title, author(s), date published, and price. Which of the following would be the best design?
• Create one class PublishedMaterial with the requested fields plus type.
• This will complicate the process of retrieving objects based on their type. Also if we need to add information that is specific to Book or Movie or AudioTape it would be best if these were subclasses of PublishedMaterial.
• Create classes Book, Movie, and AudioTape with the requested fields.
• This involves writing more code than is necessary (usually people copy and paste the shared code) and makes it harder to fix errors. It would be better to put common fields and methods in the superclass PublishedMaterial and have Book, Movie, and AudioTape be subclasses.
• Create one class BookStore with the requested fields plus type.
• The class name, BookStore, seems to imply the thing that keeps track of the store. This would be an appropriate class name for an object that handles the items in the Bookstore. However, for the published material, it would be better to use a superclass PublishedMaterial and subclasses for Book, Movie and AudioTape.
• Create classes for each.
• This is more classes than is necessary. Items such as Title, Price, ID, Author and DatePublished are simple variables that do not need a class of their own but should be fields in a PublishedMaterial superclass, with Movie, AudioTape and Book as subclasses.
• Create the class PublishedMaterial with children classes of Book, Movie, and AudioTape.
• We will need to get objects based on their type so we should create classes for Book, Movie, and AudioTape. They have common fields so we should put these in a common superclass PublishedMaterial.
Next Section - 11.19. Medium Multiple Choice Questions


text: If $a=\sqrt[3]{2+\sqrt{6}}$ then $a^3 - 2 = \sqrt{6}$ so $a^6 - 2a^3 - 2 = 0$. Thus, $a$ is root of polynomial $x^6 - 2x^3 - 2$. This polynomial is irreducible so $\mathbb{Q}(a)$ are all elements of form $a_0 + a_1 a+ ... + a_5a^5$ for $a_i \in \mathbb{Q}$.


text:
# American Institute of Mathematical Sciences

January  2016, 36(1): 469-479. doi: 10.3934/dcds.2016.36.469

## Center specification property and entropy for partially hyperbolic diffeomorphisms

 1 College of Mathematics and Information Science, and Hebei Key Laboratory of Computational Mathematics and Applications, Hebei Normal University, Shijiazhuang, 050024, China

Received  September 2014 Revised  April 2015 Published  June 2015

Let $f$ be a partially hyperbolic diffeomorphism on a closed (i.e., compact and boundaryless) Riemannian manifold $M$ with a uniformly compact center foliation $\mathcal{W}^{c}$. The relationship among topological entropy $h(f)$, entropy of the restriction of $f$ on the center foliation $h(f, \mathcal{W}^{c})$ and the growth rate of periodic center leaves $p^{c}(f)$ is investigated. It is first shown that if a compact locally maximal invariant center set $\Lambda$ is center topologically mixing then $f|_{\Lambda}$ has the center specification property, i.e., any specification with a large spacing can be center shadowed by a periodic center leaf with a fine precision. Applying the center spectral decomposition and the center specification property, we show that $h(f)\leq h(f,\mathcal{W}^{c})+p^{c}(f)$. Moreover, if the center foliation $\mathcal{W}^{c}$ is of dimension one, we obtain an equality $h(f)= p^{c}(f)$.
Citation: Lin Wang, Yujun Zhu. Center specification property and entropy for partially hyperbolic diffeomorphisms. Discrete & Continuous Dynamical Systems - A, 2016, 36 (1) : 469-479. doi: 10.3934/dcds.2016.36.469
##### References:
 [1] L. Barreira and Y. Pesin, Nonuniform Hyperbolicity,, Cambridge University Press, (2007).  doi: 10.1017/CBO9781107326026.  Google Scholar [2] D. Bohnet, Codimension-1 partially hyperbolic diffeomorphisms with a uniformly compact center foliation,, Journal of Modern Dynamics, 7 (2013), 565.  doi: 10.3934/jmd.2013.7.565.  Google Scholar [3] C. Bonatti and D. Bohnet, Partially hyperbolic diffeomorphisms with uniformly compact center foliations: the quotient dynamics,, to appear in Ergodic Theory Dynam. Systems, ().   Google Scholar [4] C. Bonatti, L. Diaz and M. Viana, Dynamics Beyond Uniform Hyperbolicity: A Global Geometric and Probabilistic Perspective,, Encyclopaedia Math. Sci., (2005).   Google Scholar [5] R. Bowen, Periodic points and measures for Axiom A diffeomorphisms,, Trans. Amer. Math., 154 (1971), 377.   Google Scholar [6] M. Brin and J. Pesin, Partially hyperbolic dynamical systems,, Izv. Akad. Nauk SSSR Ser. Mat., 38 (1974), 170.  doi: 10.1070/IM1974v008n01ABEH002101.  Google Scholar [7] P. D. Carrasco, Compact Dynamical Foliations,, Ph.D thesis, (2011).   Google Scholar [8] P. D. Carrasco, Compact dynamical foliations,, to appear in Ergodic Theory Dynam. Systems, ().   Google Scholar [9] F. Hertz, J. Hertz and R. Ures, Partially Hyperbolic Dynamics,, $28^0$ Coloquio Brasileiro de Matematica, (2011).   Google Scholar [10] M. Hirsch, C. Pugh and M. Shub, Invariant Manifolds,, Lect. Notes in Math. 583, (1977).   Google Scholar [11] H. Hu, Y. Zhou and Y. Zhu, Quasi-shadowing for partially hyperbolic diffeomorphisms,, Ergodic Theory Dynam. Systems, 35 (2015), 412.  doi: 10.1017/etds.2014.126.  Google Scholar [12] H. Hu, Y. Zhou and Y. Zhu, Quasi-shadowing and quasi-stability for dynamically coherent partially hyperbolic diffeomorphisms,, , ().   Google Scholar [13] A. Katok and B. Hasselblatt, Introduction to the Modern Theory of Dynamical Systems,, Cambridge University Press, (1995).  doi: 10.1017/CBO9780511809187.  Google Scholar [14] S. Kryzhevich and S. Tikhomirov, Partial hyperbolicity and central shadowing,, Discrete Continuous Dynam. Systems, 33 (2013), 2901.  doi: 10.3934/dcds.2013.33.2901.  Google Scholar [15] Y. Pesin, Lectures on Partial Hyperbolicity and Stable Ergodicity,, Zurich Lectures in Advanced Mathematics. European Math. Soc., (2004).  doi: 10.4171/003.  Google Scholar [16] C. Pugh, M. Shub and A. Wilkinson, Holder foliations, revisited,, J. Modern Dyn., 6 (2012), 79.  doi: 10.3934/jmd.2012.6.79.  Google Scholar [17] P. Walters, An Introduction to Ergodic Theory,, Springer, (1982).   Google Scholar

show all references

##### References:
 [1] L. Barreira and Y. Pesin, Nonuniform Hyperbolicity,, Cambridge University Press, (2007).  doi: 10.1017/CBO9781107326026.  Google Scholar [2] D. Bohnet, Codimension-1 partially hyperbolic diffeomorphisms with a uniformly compact center foliation,, Journal of Modern Dynamics, 7 (2013), 565.  doi: 10.3934/jmd.2013.7.565.  Google Scholar [3] C. Bonatti and D. Bohnet, Partially hyperbolic diffeomorphisms with uniformly compact center foliations: the quotient dynamics,, to appear in Ergodic Theory Dynam. Systems, ().   Google Scholar [4] C. Bonatti, L. Diaz and M. Viana, Dynamics Beyond Uniform Hyperbolicity: A Global Geometric and Probabilistic Perspective,, Encyclopaedia Math. Sci., (2005).   Google Scholar [5] R. Bowen, Periodic points and measures for Axiom A diffeomorphisms,, Trans. Amer. Math., 154 (1971), 377.   Google Scholar [6] M. Brin and J. Pesin, Partially hyperbolic dynamical systems,, Izv. Akad. Nauk SSSR Ser. Mat., 38 (1974), 170.  doi: 10.1070/IM1974v008n01ABEH002101.  Google Scholar [7] P. D. Carrasco, Compact Dynamical Foliations,, Ph.D thesis, (2011).   Google Scholar [8] P. D. Carrasco, Compact dynamical foliations,, to appear in Ergodic Theory Dynam. Systems, ().   Google Scholar [9] F. Hertz, J. Hertz and R. Ures, Partially Hyperbolic Dynamics,, $28^0$ Coloquio Brasileiro de Matematica, (2011).   Google Scholar [10] M. Hirsch, C. Pugh and M. Shub, Invariant Manifolds,, Lect. Notes in Math. 583, (1977).   Google Scholar [11] H. Hu, Y. Zhou and Y. Zhu, Quasi-shadowing for partially hyperbolic diffeomorphisms,, Ergodic Theory Dynam. Systems, 35 (2015), 412.  doi: 10.1017/etds.2014.126.  Google Scholar [12] H. Hu, Y. Zhou and Y. Zhu, Quasi-shadowing and quasi-stability for dynamically coherent partially hyperbolic diffeomorphisms,, , ().   Google Scholar [13] A. Katok and B. Hasselblatt, Introduction to the Modern Theory of Dynamical Systems,, Cambridge University Press, (1995).  doi: 10.1017/CBO9780511809187.  Google Scholar [14] S. Kryzhevich and S. Tikhomirov, Partial hyperbolicity and central shadowing,, Discrete Continuous Dynam. Systems, 33 (2013), 2901.  doi: 10.3934/dcds.2013.33.2901.  Google Scholar [15] Y. Pesin, Lectures on Partial Hyperbolicity and Stable Ergodicity,, Zurich Lectures in Advanced Mathematics. European Math. Soc., (2004).  doi: 10.4171/003.  Google Scholar [16] C. Pugh, M. Shub and A. Wilkinson, Holder foliations, revisited,, J. Modern Dyn., 6 (2012), 79.  doi: 10.3934/jmd.2012.6.79.  Google Scholar [17] P. Walters, An Introduction to Ergodic Theory,, Springer, (1982).   Google Scholar
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text:
## Elementary Algebra

Published by Cengage Learning

# Chapter 1 - Some Basic Concepts of Arithmetic and Algebra - 1.4 - Integers: Multiplication and Division - Problem Set 1.4: 45

69

#### Work Step by Step

Because multiplication is before addition in order of operations, we first multiply and then we add. We see that we are multiplying a negative number by a negative number in this problem. Recall, when we multiply two negative numbers, the negatives will cancel out, so the answer will always be positive. Thus, we obtain: $(-3)(-8)+(-9)(-5)=24+45=69$

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