text:
Statistics, Optimization & Information Computing http://iapress.org/index.php/soic
Statistics, Optimization and Information Computing (SOIC) is an international refereed journal dedicated to the latest advancement of statistics, optimization and applications in information sciences.  Topics of interest are (but not limited to):

Statistical theory and applications

Statistical computing, Simulation and Monte Carlo methods, Bootstrap, Resampling methods, Spatial Statistics, Survival Analysis, Nonparametric and semiparametric methods, Asymptotics, Bayesian inference and Bayesian optimization

Stochastic processes, Probability, Statistics and applications

Statistical methods and modeling in life sciences including biomedical sciences, environmental sciences and agriculture

Decision Theory, Time series analysis,  High-dimensional  multivariate integrals, statistical analysis in market, business, finance, insurance, economic and social science, etc

 Optimization methods and applications

Linear and nonlinear optimization

Stochastic optimization, Statistical optimization and Markov-chain etc.

Game theory, Network optimization and combinatorial optimization

Variational analysis, Convex optimization and nonsmooth optimization

Global optimization and semidefinite programming

Complementarity problems and variational inequalities

Optimal control: theory and applications

Operations research, Optimization and applications in management science and engineering

Information computing and machine intelligence

Machine learning, Statistical learning, Deep learning

Artificial intelligence, Intelligence computation, Intelligent control and optimization

Data mining, Data analysis, Cluster computing, Classification

Pattern recognition, Computer vision

Compressive sensing and sparse reconstruction

Signal and image processing, Medical imaging and analysis, Inverse problem and imaging sciences

Genetic algorithm, Natural language processing, Expert systems, Robotics, Information retrieval and computing

Numerical analysis and algorithms with applications in computer science and engineering

 en-US Authors who publish with this journal agree to the following terms:

Authors retain copyright and grant the journal right of first publication with the work simultaneously licensed under a Creative Commons Attribution License that allows others to share the work with an acknowledgement of the work's authorship and initial publication in this journal.

Authors are able to enter into separate, additional contractual arrangements for the non-exclusive distribution of the journal's published version of the work (e.g., post it to an institutional repository or publish it in a book), with an acknowledgement of its initial publication in this journal.

Authors are permitted and encouraged to post their work online (e.g., in institutional repositories or on their website) prior to and during the submission process, as it can lead to productive exchanges, as well as earlier and greater citation of published work (See The Effect of Open Access).

 david.iapress@gmail.com (David G. Yu) nhma0004@gmail.com (IAPress technical support) Tue, 18 Feb 2020 20:06:36 +0800 OJS 3.1.2.1 http://blogs.law.harvard.edu/tech/rss 60 Integral stochastic ordering of the multivariate normal mean-variance and the skew-normal scale-shape mixture models http://iapress.org/index.php/soic/article/view/863
In this paper, we introduce integral stochastic ordering of two most important classes of distributions that are commonly used to fit data possessing high values of skewness and (or) kurtosis. The first one is based on the selection distributions started by the univariate skew-normal distribution. A broad, flexible and newest class in this area is the scale and shape mixture of multivariate skew-normal distributions. The second one is the general class of Normal Mean-Variance Mixture distributions. We then derive necessary and sufficient conditions for comparing the random vectors from these two classes of distributions. The integral orders considered here are the usual, concordance, supermodular, convex, increasing convex and directionally convex stochastic orders. Moreover, for bivariate random vectors, in the sense of stop-loss and bivariate concordance stochastic orders, the dependence strength of random portfolios is characterized in terms of order of correlations.
 Dariush Jamali, Mehdi Amiri, Ahad Jamalizadeh, N. Balakrishnan Copyright (c) 2020 Statistics, Optimization & Information Computing http://iapress.org/index.php/soic/article/view/863 Mon, 17 Feb 2020 17:27:55 +0800 The Ristic-Balakrishnan odd log-logistic family of distributions: Properties and Applications http://iapress.org/index.php/soic/article/view/715
This paper investigates general mathematical properties of a new generator of continuous distributions with two extra parameter called the Ristic-Balakrishnan odd log-logistic family of distributions. We present some special models and investigate the asymptotes. The new density function can be expressed as a linear combination of exponentiated densities based on the same baseline distribution. Explicit expressions for the ordinary and incomplete moments, generating functions and order statistics, which hold for any baseline model, are determined. Further, we discuss the estimation of the model parameters by maximum likelihood and present a simulation study based on maximum likelihood estimation. A regression model based on proposed model was introduced. Finally, three applications to real data were provided to illustrate the potentiality of the family of distributions.
 Hamid Esmaeili, Fazlollah Lak, Emrah Altun Copyright (c) 2020 Statistics, Optimization & Information Computing http://iapress.org/index.php/soic/article/view/715 Mon, 17 Feb 2020 17:30:52 +0800 A weighted transmuted exponential distributions with environmental applications http://iapress.org/index.php/soic/article/view/785
In this paper, we introduce a new three-parameter distribution. It is based on the combination of a re-parametrization of a general family of distributions (known as the EGNB2 distribution) and the so-called quadratic rank transmutation map defined with the exponential distribution as baseline. We explore some mathematical properties of this distribution including the hazard rate function, moments, the moment generating function, the quantile function, various entropy measures and (reversed) residual life functions. A statistical study investigates estimation of the parameters using the method of maximum likelihood. The distribution along with other existing distributions are fitted to two environmental data sets and its superior performance is assessed by using some goodness-of-fit tests. As a result, some environmental measures associated with these data are obtained such as the return level and mean deviation about this level.
 Christophe Chesneau, Hassan S Bakouch, Muhammad Nauman Khan Copyright (c) 2020 Statistics, Optimization & Information Computing http://iapress.org/index.php/soic/article/view/785 Mon, 17 Feb 2020 17:32:11 +0800 An Alternative Diagnostic Procedure for Meta-Regression http://iapress.org/index.php/soic/article/view/864
This paper proposes an alternative procedure for detecting outliers in meta-regression using the penalized maximum likelihood with smoothly clipped absolute deviation penalty function. The coordinate descent algorithm is implemented to estimate the parameters where the cross-validation criterion is used to determine the tuning parameter. Extensive simulation experiments demonstrate the usefulness of our proposed procedure as well as its improved power performance compared to previous procedures. Simulation results demonstrate that the performance has a direct relationship with the number of studies and an inverse relationship with the heterogeneity between studies. An illustrative application with real data, implementing the proposed procedure and others, is given.
 Ali Hassan Abuzaid, Enass Abed, Abdu Atta, Esam Mahdi Copyright (c) 2020 Statistics, Optimization & Information Computing http://iapress.org/index.php/soic/article/view/864 Mon, 17 Feb 2020 17:33:41 +0800 A Density-Based Empirical Likelihood Ratio Approach for Goodness-of-fit Tests in Decreasing Densities http://iapress.org/index.php/soic/article/view/707
In this paper, we propose a test for the null hypothesis that a decreasing density function belongs to a given
parametric family of distribution functions against the non-parametric alternative. This method, which is based on an empirical likelihood (EL) ratio statistic, is similar to the test introduced by Vexler and Gurevich [23]. The consistency of the test statistic proposed is derived under the null and alternative hypotheses. A simulation study is conducted to inspect the power of the proposed test under various decreasing alternatives. In each scenario, the critical region of the test is obtained using a Monte Carlo technique. The applicability of the proposed test in practice is demonstrated through a few real data examples.

 Vahid Fakoor, Masoud Ajami, Seyed Mahdi Amir Jahanshahi, Ali Shariati Copyright (c) 2020 Statistics, Optimization & Information Computing http://iapress.org/index.php/soic/article/view/707 Mon, 17 Feb 2020 17:35:47 +0800 Estimation of Parameters and Reliability Characteristics in Lindley Distribution Using Randomly Censored Data http://iapress.org/index.php/soic/article/view/692
This article deals with the estimation of parameters and reliability characteristics of Lindley distribution under
random censoring. Expected time on test based on randomly censored data is obtained. The maximum likelihood estimators of the unknown parameters and reliability characteristics are derived. The asymptotic, bootstrap p and bootstrap t confidence intervals of the parameters are constructed. The Bayes estimators of the parameters and reliability characteristics under squared error loss function using non-informative and gamma informative priors are obtained. For computing of Bayes estimates, Lindley approximation and MCMC methods are considered. Highest posterior density (HPD) credible intervals of the parameters are obtained using MCMC method. Various estimation procedures are compared using a Monte Carlo simulation study. Finally, a real data set is analyzed for illustration purposes.
 Renu Garg, Madhulika Dube, Hare Krishna Copyright (c) 2020 Statistics, Optimization & Information Computing http://iapress.org/index.php/soic/article/view/692 Mon, 17 Feb 2020 17:37:05 +0800 Improved Estimator of the Conditional Tail Expectation in the case of heavy-tailed losses http://iapress.org/index.php/soic/article/view/665
In this paper, we investigate the extreme-value methodology, to propose an improved estimator of the conditional tail expectation (CTE) for a loss distribution with a finite mean but infinite variance.
The present work introduces a new estimator of the CTE based on the bias-reduced estimators of high quantile for heavy-tailed distributions. The asymptotic normality of the proposed estimator is established and checked, in a simulation study. Moreover, we compare, in terms of bias and mean squared error, our estimator with the known old estimator.
 Mohamed Laidi, Abdelaziz Rassoul, Hamid Ould Rouis Copyright (c) 2020 Statistics, Optimization & Information Computing http://iapress.org/index.php/soic/article/view/665 Mon, 17 Feb 2020 00:00:00 +0800 Weighted Cumulative Residual (Past) Inaccuracy For Minimum (Maximum) of Order Statistics http://iapress.org/index.php/soic/article/view/695
In this paper, we propose a measure of weighted cumulative residual inaccuracy between survival function of the first-order statistic and parent survival function $\bar{F}$. We also consider weighted cumulative inaccuracy measure between distribution of the last- order statistic and parent distribution $F$. For these concepts, we obtain some reliability properties and characterization results  such as relationships with other functions, bounds, stochastic ordering and effect of linear transformation. Dynamic versions of these weighted measures are considered.
 Safeih Daneshi, Ahmad Nezakati, Saeid Tahmasebi Copyright (c) 2020 Statistics, Optimization & Information Computing http://iapress.org/index.php/soic/article/view/695 Mon, 17 Feb 2020 17:42:26 +0800 The accuracy of modeling of Gaussian stochastic process in some Orlicz spaces http://iapress.org/index.php/soic/article/view/670
The main purpose of this study is the construction of a model of a Gaussian stochastic process with given reliability and accuracy in some Orlicz spaces. In the paper, a suitable model is presented, conditions for the model parameters are derived, and some examples of their calculations are given.
 YU. V. Kozachenko, Antonina Tegza, N. V.Troshki Copyright (c) 2020 Statistics, Optimization & Information Computing http://iapress.org/index.php/soic/article/view/670 Mon, 17 Feb 2020 17:43:40 +0800 Stochastic models to estimate population dynamics http://iapress.org/index.php/soic/article/view/488
The growth dynamics that a population follows is mainly due to births, deaths or migrations, each of these
phenomena is affected by other factors such as public health, birth control, work sources, economy, safety and conditions of quality of life in neighboring countries, among many others. In this paper is proposed two statistical models based on a system of stochastic differential equations (SDE) that model the dynamics of population growth, and three computational algorithms that allow the generation of probability distribution samples in high dimensions, in models that have non-linear structures and that are useful for making inferences. The algorithms allow to estimate simultaneously states solutions and parameters in SDE models. The interpretation of the parameters is important because they are related to the variables of growth, mortality, migration, physical-chemical conditions of the environment, among other factors. The algorithms are illustrated using real data from a sector of the population of the Republic of Ecuador, and are compared with the results obtained with the models used by theWorld Bank for the same data, which shows that stochastic models Proposals based on an SDE more adequately and reliably adjust the dynamics of demographic randomness, sampling errors and environmental randomness in comparison with the deterministic models used by the World Bank. It is observed that the population grows year by year and seems to have a definite tendency; that is, a clearly growing behavior is seen. To measure the relative success of the algorithms, the relative error was estimated, obtaining small percentage errors.
 Saba Infante, luis sanchez, Aracelis Hernandez Copyright (c) 2020 Statistics, Optimization & Information Computing http://iapress.org/index.php/soic/article/view/488 Mon, 17 Feb 2020 17:44:34 +0800 On Distributions of One Class of Random Sums and their Applications http://iapress.org/index.php/soic/article/view/698
We propose results of the investigation of properties of the random sums of random variables. We consider the case, where the number of summands is the first moment of an event occurrence. An integral equation is presented that determines distributions of random sums. With the help of the obtained results we analyse the distribution function of the time during which the Geiger-Muller counter will not lose any particles, the distribution function of the busy period of a redundant system with renewal, and the distribution function of the sojourn times of a single-server queueing system.
 Ivan Matsak, Mikhail Moklyachuk Copyright (c) 2020 Statistics, Optimization & Information Computing http://iapress.org/index.php/soic/article/view/698 Tue, 18 Feb 2020 19:17:45 +0800 Inexact Double Step Length Method For Solving Systems Of Nonlinear Equations http://iapress.org/index.php/soic/article/view/532
In this paper, a single direction with double step length method for solving systems of nonlinear equations is presented. Main idea used in the algorithm is to approximate the Jacobian via acceleration parameter. Furthermore, the two step lengths are calculated using inexact line search procedure. This method is matrix-free, and so is advantageous when solving large-scale problems. The proposed method is proven to be globally convergent under appropriate conditions. The preliminary numerical results reported in this paper using a large-scale benchmark test problems show that the proposed method is practically quite effective.
 Abubakar Sani Halilu, Mohammed Yusuf Waziri, Yau Balarabe Musa Copyright (c) 2020 Statistics, Optimization & Information Computing http://iapress.org/index.php/soic/article/view/532 Tue, 18 Feb 2020 19:20:44 +0800 On the Distributed Order Fractional Multi-Strain Tuberculosis Model: a Numerical Study http://iapress.org/index.php/soic/article/view/621
In this paper, a novel mathematical distributed order fractional model of multistrain Tuberculosis is presented. The proposed model is governed by a system of distributed order fractional differential equations, where the distributed order fractional derivative is defined in the sense of the Grünwald-Letinkov definition. A nonstandard finite difference method is proposed to study the resulting system. The stability analysis of the proposed model is discussed. Numerical simulations show that the nonstandard finite difference method can be applied to solve such distributed order fractional differential equations simply and eectively.
 Nasser Sweilam, S. M. AL-Mekhlafi, A. O. Albalawi Copyright (c) 2020 Statistics, Optimization & Information Computing http://iapress.org/index.php/soic/article/view/621 Tue, 18 Feb 2020 19:25:24 +0800 Higher-order symmetric duality in nondifferentiable multiobjective fractional programming problem over cone contraints http://iapress.org/index.php/soic/article/view/601
In this paper, we introduce the definition of higher-order K-(C, α, ρ, d)-convexity/pseudoconvexity over cone and discuss a nontrivial numerical examples for existing such type of functions. The purpose of the paper is to study higher order fractional symmetric duality over arbitrary cones for nondifferentiable Mond-Weir type programs under higher- order K-(C, α, ρ, d)-convexity/pseudoconvexity assumptions. Next, we prove appropriate duality relations under aforesaid assumptions.
 Ramu Dubey, Deepmala, Vishnu Narayan Mishra Copyright (c) 2020 Statistics, Optimization & Information Computing http://iapress.org/index.php/soic/article/view/601 Tue, 18 Feb 2020 00:00:00 +0800 Vector-valued nonuniform multiresolution analysis related to Walsh function http://iapress.org/index.php/soic/article/view/681
In this paper, we introduce vector-valued nonuniform multiresolution analysis on positive half-line related to Walsh function. We obtain the necessary and sufficient condition for the existence of associated wavelets.
 Abdullah Copyright (c) 2020 Statistics, Optimization & Information Computing http://iapress.org/index.php/soic/article/view/681 Tue, 18 Feb 2020 19:31:09 +0800 Informational Energy and Entropy Applied to Testing Exponentiality http://iapress.org/index.php/soic/article/view/616
The exponential distribution is widely used in reliability and life testing analysis. In this paper, two tests of fit for the exponential distribution based on Informational Energy and entropy are constructed. Consistency and other properties of the tests are proved. Using a simulation study, critical values of the proposed tests are obtained and then power values of tests are computed and compared with each other against various alternatives. Finally, we apply the tests for time between failures of secondary reactor pumps and waiting times for fatal plane accidents in the USA from 1983 to 1998.
 Hadi Alizadeh Noughabi, Havva Alizadeh Noughabi, Jalil Jarrahiferiz Copyright (c) 2020 Statistics, Optimization & Information Computing http://iapress.org/index.php/soic/article/view/616 Tue, 18 Feb 2020 19:47:47 +0800 Improving firefly-based multi-objective optimization based on attraction law and crowding distance http://iapress.org/index.php/soic/article/view/382
Multi-objective optimization problems are so designed that they simultaneously minimize several objectives functions (which are sometimes contradictory). In most cases, the objectives are in conflict with each other such that optimization of one objective does not lead to the optimization of another ones. Therefore, we should achieve a certain balance of goals to solve these problems, which usually requires the application of an intelligent method. In this regard, use of meta-heuristic algorithms will be associated with resolved problems. In this paper, we propose a new multi-objective firefly optimization method which is designed based on the law of attraction and crowding distance. The proposed methods efficiency has been evaluated by three valid test functions containing convex, nonconvex and multi discontinuous convex Pareto fronts. Simulation results confirm the significant accuracy of proposed method in defining the Pareto front for all three test functions. In addition, the simulation results indicates that proposed algorithm has higher accuracy and greater convergence speed, compared to other well known multi-objective algorithms such as non-dominated sorting genetic algorithm, Bees algorithm, Differential Evolution algorithm and Strong Pareto Evolutionary Algorithm.
 Farid shayesteh, Reihaneh Kardehi Moghaddam Copyright (c) 2020 Statistics, Optimization & Information Computing http://iapress.org/index.php/soic/article/view/382 Tue, 18 Feb 2020 19:50:26 +0800 An itertive algorithm with error terms for solving a system of implicit n-variational inclusions http://iapress.org/index.php/soic/article/view/705
A new system of implicit n-variational inclusions is considered. We propose a new algorithm with error terms for computing the approximate solutions of our system. The convergence of the iterative sequences generated by the iterative algorithm is also discussed. Some special cases are also discussed.
 Zubair Khan, Syed Shakaib Irfan, M. Firdosh Khan, P. Shukla Copyright (c) 2020 Statistics, Optimization & Information Computing http://iapress.org/index.php/soic/article/view/705 Tue, 18 Feb 2020 19:51:26 +0800 A Consensus Clustering Method for Clustering Social Networks http://iapress.org/index.php/soic/article/view/716
Detecting Communities in networks is one of the appealing fields in computer science. A wide range of methods are proposed for this problem. These methods employ different strategies and optimization functions to detect communities (or clusters). Therefore, it seems a good idea to combine these strategies to take advantage of the strengths of the methods and overcome their problems. This is the idea behind consensus clustering technique which combines several clustering results into one. In this paper, we propose a very good-performing method based on consensus clustering to detect communities of a network. Our method, called “Azar”, employed several community detection methods as base methods. Then Azar generates a new compressed network based on the common views of the used base methods and, gives this new compressed network to the last community detection method to find the final partition. We evaluate our approach by employing real and artificial datasets. The implementation results compare the base methods with Azar according to accuracy measures such as modularity and Normalized Mutual Information (NMI). The results show the good-performing behavior of Azar even for the most difficult networks. The results show the brilliant power of Azar in comparison with all the other methods.
 Masoumeh Kheirkhahzadeh, Morteza Analoui Copyright (c) 2020 Statistics, Optimization & Information Computing http://iapress.org/index.php/soic/article/view/716 Tue, 18 Feb 2020 19:53:05 +0800 Business Analytics using Dynamic Pricing based on Customer Entry-Exit Rates Tradeoff http://iapress.org/index.php/soic/article/view/551
This paper concerns with an integrated business process to be applied as a decision support for market analysis and decision making. The proposed business intelligence and analytics system makes use of an extract, transform and load mechanism for data collection and purification. As a mathematical decision optimization, dynamic pricing is formulated based on customer entry-exit rates in a history-based pricing model. The optimal prices for products are obtained so that aggregated profit is maximized. A case study is reported to show the effectiveness of the approach. Also, analytical investigations on the impacts of the sensitive parameters of the pricing model are given.
 Hamed Fazlollahtabar, Minoo Talebi Ashoori Copyright (c) 2020 Statistics, Optimization & Information Computing http://iapress.org/index.php/soic/article/view/551 Tue, 18 Feb 2020 19:55:12 +0800 Optimal control of a rectilinear motion of a rocket http://iapress.org/index.php/soic/article/view/741
In this work, we have modelled the problem of maximizing the velocity of a rocket moving with a rectilinear motion by a linear optimal control problem, where the control represents the action of the pilot on the rocket. In order to solve the obtained model, we applied both analytical and numerical methods. The analytical solution is calculated using the Pontryagin maximum principle while the approximate solution of the problem is found using the shooting method as well as two techniques of discretization: the technique using the Cauchy formula and the one using the Euler formula. In order to compare the different methods, we developed an implementation with MATLAB and presented some simulation results.
 Mohamed Aliane, Nacima Moussouni, Mohand Bentobache Copyright (c) 2020 Statistics, Optimization & Information Computing http://iapress.org/index.php/soic/article/view/741 Tue, 18 Feb 2020 19:57:46 +0800 Probability Model Based on Cluster Analysis to Classify Sequences of Observations for Small Training Sets http://iapress.org/index.php/soic/article/view/690
The problem of recognizing patterns, when there are few training data available, is particularly relevant and arises in cases when collection of training data is expensive or essentially impossible. The work proposes a new probability model MC&CL (Markov Chain and Clusters) based on a combination of markov chain and algorithm of clustering (self-organizing map of Kohonen, k-means method), to solve a problem of classifying sequences of observations, when the amount of training dataset is low. An original experimental comparison is made between the developed model (MC&CL) and a number of the other popular models to classify sequences: HMM (Hidden Markov Model), HCRF (Hidden Conditional Random Fields),LSTM (Long Short-Term Memory), kNN+DTW (k-Nearest Neighbors algorithm + Dynamic Time Warping algorithm). A comparison is made using synthetic random sequences, generated from the hidden markov model, with noise added to training specimens. The best accuracy of classifying the suggested model is shown, as compared to those under review, when the amount of training data is low.
 Sergey S Yulin, Irina N Palamar Copyright (c) 2020 Statistics, Optimization & Information Computing http://iapress.org/index.php/soic/article/view/690 Tue, 18 Feb 2020 19:59:29 +0800 The Alpha-Beta Skew Logistic Distribution: Properties and Applications http://iapress.org/index.php/soic/article/view/706
A new family of skew distributions is introduced by extending the alpha skew logistic distribution proposed by Hazarika-Chakraborty [9]. This family of distributions is called the alpha-beta skew logistic (ABSLG) distribution.Density function, moments, skewness and kurtosis coefficients are derived. The parameters of the new family are estimated by maximum likelihood and moments methods. The performance of the obtained estimators examined via a Monte carlo simulation. Flexibility, usefulness and suitability of ABSLG is illustrated by analyzing two real data sets.
 Hamid Esmaeili, Fazlollah Lak, Morad Alizadeh, Mohammad esmail Dehghan monfared Copyright (c) 2020 Statistics, Optimization & Information Computing http://iapress.org/index.php/soic/article/view/706 Tue, 18 Feb 2020 20:00:33 +0800 A Robust Statistical method to Estimate the Intervention Effect with Longitudinal Data http://iapress.org/index.php/soic/article/view/811
Segmented regression is a standard statistical procedure used to estimate the effect of a policy intervention on time series outcomes. This statistical method assumes the normality of the outcome variable, a large sample size, no autocorrelation in the observations, and a linear trend over time. Also, segmented regression is very sensitive to outliers. In a small sample study, if the outcome variable does not follow a Gaussian distribution, then using segmented regression to estimate the intervention effect leads to incorrect inferences. To address the small sample problem and non-normality in the outcome variable, including outliers, we describe and develop a robust statistical method to estimate the policy intervention effect in a series of longitudinal data. A simulation study is conducted to demonstrate the effect of outliers and non-normality in the outcomes by calculating the power of the test statistics with the segmented regression and the proposed robust statistical methods. Moreover, since finding the sampling distribution of the proposed robust statistic is analytically difficult, we use a nonparametric bootstrap technique to study the properties of the sampling distribution and make statistical inferences. Simulation studies show that the proposed method has more power than the standard t-test used in segmented regression analysis under the non-normality error distribution. Finally, we use the developed technique to estimate the intervention effect of the Istanbul Declaration on illegal organ activities. The robust method detected more significant effects compared to the standard method and provided shorter confidence intervals.
 Mohammad M Islam, Erik L Heiny Copyright (c) 2020 Statistics, Optimization & Information Computing http://iapress.org/index.php/soic/article/view/811 Mon, 17 Feb 2020 00:00:00 +0800


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# Playback

A Bar Chart with a Playback component

Replays time-series data in Data Grid and charting components

## Basics

Data Source
A data source containing a time series (number, dates or timestamps), shared with the component being played back.
Selected Value
A View State Parameter for the time series
Selected Column
An Integer or temporal column from the Data Source, to be used in playback sequence
Time Column
A temporal column from the Data Source (can be the Selected Column)
Mode
Governs the rate of replay: in Event mode, each event is one unit; in Time mode, units are temporal
Speed Controls

When checked, offers a choice of replay speeds

Multi Controls

When checked, offers fast-forward and rewind

Max Overall Time
Upper limit in data points (of the Time Column, e.g. seconds, nanaoseconds) for replay based on Selected Value View State Parameter
Interval Time
Sampling interval of the Selected Value View State Parameter: values above 1 will skip data points; e.g. a value of 2 will use every second data point of the Selected Value View State Parameter
Speed
Define default speed for playback
Playing
Define for playback: 0 is static, 1 plays forward, -1 plays backwards. Multiples, e.g. 4 will play at x4 speed.
Display Warning Message
Support messaging enabled

## Margins, Style and Format

Style for common settings

format: plain


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Guest Apr 27, 2017
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$$\frac{1\times 10^{-14}}{4.6\times 10^{-4}}=\frac{1}{10^{14}}:\frac{4.6}{10^4}$$

One divides by a fraction, multiplying by the reciprocal value of the fraction.

A power with negative exponent is equal to its reciprocal value with positive exponent. $$10^{-14}=\frac{1}{10^{14}}$$

$$\frac{1}{10^{14}}:\frac{4.6}{10^4}=\frac{1}{10^{14}}\times\frac{10^4}{4.6}=\frac{1}{4.6\times 10^{10}}=\frac{0.2173..}{10^{10}}=\frac{2.173..}{10^{11}}$$

$$=2.17391304348\times 10^{-11}$$

!

asinus  Apr 27, 2017
edited by asinus  Apr 27, 2017

### 7 Online Users

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# 2.7: Box Plot Demo

Skills to Develop

• Understand what it means for a distribution to balance on a fulcrum
• Learn which measure of central tendency will balance a distribution

### Instructions

• The box plot in the simulation is based on the data shown to the left of the box plot.
• Notice that various aspects of the box plot such as the mean and median are labeled.
• These labels can be hidden by unchecking the "show labels on box plot" button.
• Beneath the data is a pair of buttons that let you specify whether you want to be able to enter data or to see statistics based on the data.
• When you use the simulation, try to modify the data in various ways and see how it affects the box plot.
• Try putting in some extreme values and see if they get labeled as outliers.
• Outside values are shown as "$$o's$$" beyond the inner fence.
• Far out values are shown as $$*'s$$ outside the outer fence.
• You can delete all the data (by pressing the "Clear All button") and enter your own data.
• Your data can be typed in or pasted in from another application.
• When pasting, use keyboard shortcut for pasting (Command-V for Mac, CTRL-V for Windows).
• When you change the data, the box plot will disappear.

After you have entered new data, click the "Draw box plot" button to redraw the box plot.

### Illustrated Instructions

The screenshot below shows the box plot simulation with its default data on the left.

Figure $$\PageIndex{1}$$: Box Plot simulation

You can change some or all of the data on the left by editing individual numbers or pasting in your own data. When pasting, you must use the keyboard shortcut for pasting (Command-V for Mac, CTRL-V for Windows).

### Contributor

• Online Statistics Education: A Multimedia Course of Study (http://onlinestatbook.com/). Project Leader: David M. Lane, Rice University.


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# pgfplots and final size of the externalized pdf image

I have the following piece of code:

% command: pdflatex --jobname=try-f1 try.tex
\documentclass[12pt]{book}
\usepackage{tikz,pgfplots}
\pgfrealjobname{try}
\begin{document}
\beginpgfgraphicnamed{try-f1}%
\begin{tikzpicture}
\pgfplotsset{minor grid style={color=black!20,line width=0.25pt},major grid style={color=black!30,line width=0.35pt}}
\begin{semilogyaxis}[grid=both,enlargelimits=false,scale only axis,width=6.5cm,height=4.5cm,tick style={draw=none},xticklabels={},yticklabels={},axis background/.style={fill=black!3}]
\addplot[solid] coordinates{600 points coded as (0.028,6.28386e-005) for instance}
\end{semilogyaxis}
\end{tikzpicture}
\endpgfgraphicnamed
\end{document}

the size of tex source as a text file is 12ko and 4 ko when zipped while the final pdf is 8ko. Is all this expected? I thought the final pdf would be lighter than the zipped file.

-
Why would you assume the final pdf to be smaller? It includes a lot more information than the zipped source code, so if anything I'm surprised it's not much larger. –  Jake Aug 3 '11 at 0:44
I thought pdf was somehow more efficient than zip because of destructive compression. –  pluton Aug 3 '11 at 0:55
No. For example, the PDF includes embedded fonts that is not included in the flat format TEX file. –  Werner Aug 3 '11 at 1:25
correct but in the provided example, fonts have been removed on purpose. –  pluton Aug 3 '11 at 5:03
There are macros to control the compression level of pdf. I do not have the reference at hand, but I believe the macro (for pdflatex) is \pdfcompresslevel=<integer>. Even further compression can be archived if \pdfminorversion=5 (or something like this) because a particular pdf version introduced not just stream data compression but also compression of pdf object definitions. The pgfplots manual (doc/latex/pgfplots/pgfplots.tex ) has these macros somewhere. \pdfcompresslevel=0 should result in a completely uncompressed pdf. –  Christian Feuersänger Aug 3 '11 at 7:44

There are macros to control the compression level of pdf. I do not have the reference at hand, but I believe the macro (for pdflatex) is \pdfcompresslevel=<integer>. Even further compression can be archived if \pdfminorversion=5 (or something like this) because a particular pdf version introduced not just stream data compression but also compression of pdf object definitions. The pgfplots manual (doc/latex/pgfplots/pgfplots.tex ) has these macros somewhere. \pdfcompresslevel=0 should result in a completely uncompressed pdf.


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## sounddevices 702, power up problems

3

Hi all Had to learn it the hard way yesterday. Ready to record, and the 702, just wouldnt power up. Had a fresh batt in it. Went home put in the ac, pulled the ac, and everything works again, even when power back on and off several times. Today i tried to reproduce the problem, and only needed 1 try. Same problem again, and same way of treating it.

Any os you know what this could be or, have heard of it before? The dealer told me I should put away somewhere in the region of $300-400 to get it checked. Thanks alot. Best wishes Mikkel very weird. are you on the latest firmware? – Shaun Farley – 2012-04-25T14:15:10.013 Thanks Shaun. Yes I am. Im really keeping my fingers crossed that Im not in for a$900 repair bill. Thats crazy . – Mikkel Nielsen – 2012-04-25T17:41:37.360

2

I had the same issue with our 722 last year. After sending it back to Sound Devices, this is what they found:

"The technician working on this unit has determined the fault lies in a A Ball Grid Array FPGA. Sound Devices does not have the facilities to reliably replace this component. I have confirmed the fault. Cooling this component allowed the unit to power up. The unit remained operational over the weekend and has now powered on after being powered down for three days. As this was not a technical repair, we must assume the fault may return at some time in the future. This leads me to recommend the option of replacing the main board set with a new production set. The new set carries a one year warranty against manufacture defect. Additional replacement of the CH1 input XLR connector (insert pushed in, separated from the body) and Headphone Volume Potentiometer (left side scratchy) are recommended. As the unit requires hardware updates to bring it to the V2.0 level, these updates will be applied at no charge and flat rate labor charge will not apply. Should you choose not to proceed with installation of the new boards set, we can, if you wish, continue with the updates and XLR and HP pot repairs with parts charges only. Or we can return the recorder in its current state. Of course the unit would not be warranted for the FPGA issue should it arise again." PN 1605 - 722 Tested PCB Assembly US$836.42 PN 113 - XLRF connector US$3.84 PN 3425 - Potentiometer US$0.96 We decided to have it repaired. Now I'm not sure that this is the problem that you are facing, but just letting you know what you could potentially be in for. Woow... Thanks Bill. Are you saying that you paid close to$900 for this repair? – Mikkel Nielsen – 2012-04-25T17:36:32.223

Yes. We went back and forth on trying to decide to pay the $900, or use that money to buy a new lesser expensive recorder. With the amount of money spent on the initial investment for the 722 of over$2500, we felt that it would be a mistake to mothball it. Really hope that your's is a cheaper repair! :) – Bill Mellow – 2012-04-25T20:15:31.943

Thanks Bill. Yes I can see what you mean. I dont hope Im in for that kind of reapair cost either. That would really suck. Thx again. – Mikkel Nielsen – 2012-04-26T06:49:26.260

0

Happened to me due to faulty contacts of battery or the 702´s battery connection. Even when it works for sometimes... Try different packs.

Good luck!


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# Thread: Volume of a Spatial Figure

1. ## Volume of a Spatial Figure

Find the volume of a spatial figure which is limited the implicate function:

$\left( {\frac{{x^2 }}{{a^2 }} + \frac{{y^2 }}{{b^2 }} + \frac{{z^2 }}{{c^2 }}} \right)^2 =$ $\frac{z}{h}\exp \left( {\frac{{{{x^2 } \mathord{\left/{\vphantom {{x^2 } {c^2 }}} \right.\kern-\nulldelimiterspace} {c^2 }}}}{{{{x^2 } \mathord{\left/{\vphantom {{x^2 } {a^2 }}} \right.\kern-\nulldelimiterspace} {a^2 }} + {{y^2 } \mathord{\left/{\vphantom {{y^2 } {b^2 }}} \right.\kern-\nulldelimiterspace} {b^2 }} + {{z^2 } \mathord{\left/{\vphantom {{z^2 } {c^2 }}} \right.\kern-\nulldelimiterspace} {c^2 }}}}} \right),$

$a,b,c,h{\text{ }} - {\text{ positive parameters}}{\text{.}}$

Please, say: which a system of coordinates I must use?
I tried to use the cylindrical system of coordinates, but a very difficult triple integral.

2. My guess would be spherical coordinates (slightly modified).
Try
$x = a\cdot r\cdot sin(\phi)cos(\theta)$
$y = b\cdot r\cdot sin(\phi)sin(\theta)$
$z = c\cdot r\cdot cos(\phi)$

3. I used also the generalized spherical coordinates.
There's my solution here.

So, we need find a volume $V$ of area $G \subset \mathbb{R}^3$, which is limited a surface $S$:

$S = \left\{ {\left. {x,y,z} \right|\left( {\frac{{x^2 }}{{a^2 }} + \frac{{y^2 }}{{b^2 }} + \frac{{z^2 }}{{c^2 }}} \right)^2 } \right.$ $\left. { = \frac{z}{h}\exp \left( {\frac{{{{x^2 } \mathord{\left/{\vphantom {{x^2 } {c^2 }}} \right. \kern-\nulldelimiterspace} {c^2 }}}}{{{{x^2 } \mathord{\left/{\vphantom {{x^2 } {a^2 }}} \right.\kern-\nulldelimiterspace} {a^2 }} + {{y^2 } \mathord{\left/{\vphantom {{y^2 } {b^2 }}} \right.\kern-\nulldelimiterspace} {b^2 }} + {{z^2 } \mathord{\left/{\vphantom {{z^2 } {c^2 }}} \right.\kern-\nulldelimiterspace} {c^2 }}}}} \right)} \right\}.$

Obviously, $V = \iiint\limits_G {dV}.$

There's a using of the generalized spherical coordinates:

$\left\{ \begin{gathered}x = ar\cos \phi \sin \theta , \hfill \\y = br\sin \phi \sin \theta , \hfill \\z = cr\cos \theta . \hfill \\ \end{gathered} \right.$

Where $r > 0$, $0 < \theta < \pi$, $0 < \phi < 2\pi$.

It's jacobian of the transformation:

$J = \left| {\begin{array}{*{20}c}{\frac{{\partial x}}{{\partial r}}} & {\frac{{\partial x}}{{\partial \phi }}} & {\frac{{\partial x}}{{\partial \theta }}} \\{\frac{{\partial y}}{{\partial r}}} & {\frac{{\partial y}}{{\partial \phi }}} & {\frac{{\partial y}}{{\partial \theta }}} \\{\frac{{\partial z}}{{\partial r}}} & {\frac{{\partial z}}{{\partial \phi }}} & {\frac{{\partial z}}{{\partial \theta }}} \\\end{array} } \right| =$ $^{} \left| {\begin{array}{*{20}c}{a\cos \phi \sin \theta } & { - ar\sin \phi \sin \theta } & {ar\cos \phi \cos \theta } \\{b\sin \phi \sin \theta } & {br\cos \phi \sin \theta } & {br\sin \phi \cos \theta } \\{c\cos \theta } & 0 & { - cr\sin \theta } \\\end{array} } \right| =$ $^{} abcr^2 \sin \theta {\text{ }}\left( {{\text{absolute value}}} \right).$

$\left( {\frac{{x^2 }}{{a^2 }} + \frac{{y^2 }}{{b^2 }} + \frac{{z^2 }}{{c^2 }}} \right)^2 = \left( {\frac{{\left( {ar\cos \phi \sin \theta } \right)^2 }}{{a^2 }} + \frac{{\left( {br\sin \phi \sin \theta } \right)^2 }}{{b^2 }} + \frac{{\left( {cr\cos \theta } \right)^2 }}{{c^2 }}} \right)^2 =$

$= \left( {r^2 \cos ^2 \phi \sin ^2 \theta + r^2 \sin ^2 \phi \sin ^2 \theta + r^2 \cos ^2 \theta } \right)^2$ $= r^4 \left( {\sin ^2 \theta \left( {\cos ^2 \phi + \sin ^2 \phi } \right) + \cos ^2 \theta } \right)^2 = r^4 .$

$r^4 = \frac{{cr\cos \theta }}{h}\exp \left( {\frac{{a^2 r^2 \cos ^2 \phi \sin ^2 \theta }}{{c^2 r^2 }}} \right) \Leftrightarrow r^3 = \frac{c}{h}\cos \left( \theta \right)e^{\left( {\frac{a}{c}} \right)^2 \sin ^2 \left( \theta \right)\cos ^2 \left( \phi \right)} .$

So, we have [ $0 < \theta < \frac{\pi }{2}$ because $r>0$]

$V = abc\int\limits_0^{2\pi } {d\phi \int\limits_0^{{\pi \mathord{\left/
{\vphantom {\pi 2}} \right.\kern-\nulldelimiterspace} 2}} {\sin \left( \theta \right)d\theta \int\limits_0^{R\left( {\phi ,\theta } \right)} {r^2 } dr{\text{ }}} }$
where $R\left( {\phi ,\theta } \right) = \sqrt[3]{{\frac{c}{h}\cos \left( \theta \right)e^{\left( {\frac{a}{c}} \right)^2 \sin ^2 \left( \theta \right)\cos ^2 \left( \phi \right)} }}.$

The calculation of the internal integrals.

$\int\limits_0^{R\left( {\phi ,\theta } \right)} {r^2 } dr = \left. {\frac{{r^3 }}{3}} \right|_0^{R\left( {\phi ,\theta } \right)} = \frac{c}{{3h}}\cos \left( \theta \right)e^{\left( {\frac{a}{c}} \right)^2 \sin ^2 \left( \theta \right)\cos ^2 \left( \phi \right)} .$

$\int\limits_0^{{\pi \mathord{\left/{\vphantom {\pi 2}} \right.\kern-\nulldelimiterspace} 2}} {\sin \theta d\theta } \int\limits_0^{R\left( {\phi ,\theta } \right)} {r^2 } dr = \frac{c}{{3h}}\int\limits_0^{{\pi \mathord{\left/{\vphantom {\pi 2}} \right.\kern-\nulldelimiterspace} 2}} {\sin \theta \cos \left( \theta \right)e^{\left( {\frac{a}{c}} \right)^2 \sin ^2 \left( \theta \right)\cos ^2 \left( \phi \right)}d\theta } =$ $\left\{ \begin{gathered}\sin ^2 \left( \theta \right) = t, \hfill \\2\sin \left( \theta \right)\cos \left( \theta \right)d\theta = dt \hfill \\ \end{gathered} \right\} =$

$= \frac{c}{{6h}}\int\limits_0^1 {e^{t\left( {\frac{a}{c}} \right)^2 \cos ^2 \left( \phi \right)} dt} = \left. {\frac{c}{{6h}}\left( {\frac{a}{c}\cos \phi } \right)^{ - 2} e^{t\left( {\frac{a}{c}\cos \phi } \right)^2 } } \right|_0^1 = \frac{c}{{6h}}\left[ {\frac{{e^{\left( {\frac{a}{c}\cos \phi } \right)^2 } - 1}}{{\left( {\frac{a}{c}\cos \phi } \right)^2 }}} \right] =$

$= \frac{c}{{6h}}\left( {\frac{a}{c}\cos \phi } \right)^{ - 2} \left[ {\left( {\frac{a}{c}\cos \phi } \right)^2 + \frac{1}{{2!}}\left( {\frac{a}{c}\cos \phi } \right)^4 + \ldots + \frac{1}{{n!}}\left( {\frac{a}{c}\cos \phi } \right)^{2n} + \ldots} \right] =$

$= \frac{c}{{6h}}\left[ {1 + \frac{1}{{2!}}\left( {\frac{a}{c}\cos \phi } \right)^2 + \ldots + \frac{1}{{n!}}\left( {\frac{a}{c}\cos \phi } \right)^{2n - 2} + \ldots } \right] = \frac{c}{{6h}}\sum\limits_{n = 1}^\infty {\frac{1}{{n!}}\left( {\frac{a}{c}\cos \phi } \right)^{2n - 2} .}$

So, we have

$V = abc\int\limits_0^{2\pi } {d\phi \int\limits_0^{{\pi \mathord{\left/
{\vphantom {\pi 2}} \right.\kern-\nulldelimiterspace} 2}} {\sin \left( \theta \right)d\theta \int\limits_0^{R\left( {\phi ,\theta } \right)} {r^2 } dr = } } \frac{{abc^2 }}{{6h}}\int\limits_0^{2\pi } {\sum\limits_{n = 1}^\infty {\frac{1}{{n!}}\left( {\frac{a}{c}\cos \phi } \right)^{2n - 2} } d\phi } =$
$\frac{{abc^2 }}{{6h}}\sum\limits_{n = 1}^\infty {\left[ {\frac{1}{{n!}}\left( {\frac{a}{c}} \right)^{2n - 2} \int\limits_0^{2\pi } {\left( {\cos \phi } \right)^{2n - 2} d\phi } } \right].}$

The integral $\int\limits_0^{2\pi } {\left( {\cos \phi } \right)^{2n - 2} d\phi }$, calculated by the methods of complex analysis, is equal $\frac{{2\pi \left( {2n - 2} \right)!}}{{2^{2n - 2} \left( {\left( {n - 1} \right)!} \right)^2 }}$.

Finally, we have

$V = \frac{{\pi abc^2 }}{{3h}}\sum\limits_{n = 1}^\infty {\frac{{\left( {2n - 2} \right)!}}{{2^{2n - 2} \left( {\left( {n - 1} \right)!} \right)^2 n!}}\left( {\frac{a}{c}} \right)^{2n - 2} } .$


text:
A Vandermonde quasimatrix can be made like this in Chebfun:

x = chebfun('x');
A = x.^(0:10);

This particular example has 11 columns corresponding to the functions $1, x, \dots, x^n$ with $n=10$, and it is quite ill-conditioned:

cond(A)
ans =
3.072959852624344e+03

In fact, the condition number grows exponentially as $n\to\infty$ at the rate $\rho_c^n$ with $\rho_c = 1+\sqrt 2$:

for n = 1:20
c(n) = cond(x.^(0:n));
end
rhoc = 1+sqrt(2);
semilogy([c; rhoc.^(1:20)]','.-'), grid on
xlabel n, ylabel('condition number')
legend('Vandermonde matrix','asymptotics','location','northwest')

So far as I am aware, this growth constant $\rho_c$ was first identified by Walter Gautschi in 1975 [2, eq. (6.5)], and a much more complete theory has been provided by Beckermann [1, Thm. 4.1]. (These authors were working not with quasimatrices, but with discrete square matrices sampled at Chebyshev points. The crucial property of $\rho_c$ is that if the circle $|z|=\rho$ in the complex $z$-plane is mapped to an ellipse in the complex $x$-plane by the Joukowski map $x = (z+z^{-1})/2$, $\rho_c$ is the value of $\rho$ for which the ellipse just touches the unit circle. This happens at $x = \pm i$, the Joukowski images of $z = \pm i \rho_c$.

Here is an explanation of why this property controls the condition number. (It amounts to a proof that the condition number can grow no faster than this.) $A$ represents the map from coefficient vectors $c = (c_0,\dots, c_n)^T$ to polynomials $p(x) = c_0 + \cdots + c_n x^n$. The essential question for the condition number is, how small can $p$ be relative to $c$? Specifically, working in the $\infty$-norm for simplicity, if $|p(x)| \le \varepsilon$ for all $x\in [-1,1]$ but $|c_k|\ge 1$, for some $0\le k \le n$, how small can $\varepsilon$ be?

The answer comes from combining Cauchy's estimate with Bernstein's inequality of 1912, which can be found on p. 60 of Approximation Theory and Approximation Practice. Bernstein tells us that if $|p(x)|\le \varepsilon$ on $[-1,1]$, then $|p(x)| \le \rho^n\varepsilon$ on the $\rho$-ellipse for any $\rho>1$. In particular, we have $|p(x)| \le \rho_c^n\varepsilon$ on the $\rho_c$-ellipse. But since the $\rho_c$-ellipse contains the unit circle, we also have $|p(x)| \le \rho_c^n\varepsilon$ on the unit circle. On the other hand by Cauchy's estimate, for each $0\le k \le n$, we must have $|p(x)|\ge |c_k|$ for some $x$ on the unit circle. And thus we conclude $$\varepsilon \ge \rho^{-n}.$$

[1] B. Beckermann, The condition number of real Vandermonde, Krylov and positive definite Hankel matrices, Numerische Mathematik 85 (2000), 553-577.

[2] W. Gautschi, Norm estimates for inverses of Vandermonde matrices, Numerische Mathematik 23 (1975), pp. 337--347.


text:
# Main Page

MediaWiki has been successfully installed.

Consult the User's Guide for information on using the wiki software.

## MathJax Works

Math can by written between dollar signs, like this: $\Gamma(x)=\int_0^\infty s^x e^s \frac{ds}{s}$. There's probably a lot of weird interaction with the MediaWiki editor.

How about $\Gamma(x)=\int_0^\infty s^x e^s \frac{ds}{s}$.


text:
# Category of categories and anafunctors.

1)Category of sets is a (1,0)-category, isn't it?

2)Anafunctors are the factorization of class of functors by equivalence relation(isomorphism) on its values, aren't they?

3)Which kind of category small categories and anafunctors forms? ((2,1)-category?) Let it be $Cat_{ana}$.

4)May I say that $Set$ is "anaisomorphic" to $Set^{op}$ (in $Cat_{ana})$? ( I can not, because of different truthness of proposition "all morphisms to the initial object are isomorphisms". Anafunctors are not related to this statement.)

• To see 'all possible values' of a functor $\mathcal A\to\mathcal B$ (such as a limit functor) at once, we can also use its induced profunctor $\mathcal A^{op}\times\mathcal B\to\mathcal{Set}$ with its cograph category which disjointly contain $\mathcal A$ and $\mathcal B$ and 'heteromorphisms' between their objects. Then, the original assignment is given by reflection to $\mathcal B$, so that all target objects are equally presented this way. – Berci Sep 4 '16 at 23:10
• @Berci, I am trying to understand what you want to say and which question you answer. Limit functor sends functors of type A->B, where A,B-categories to objects of B, am I right? It sends natural transformations to morphisms between limits in B, am I right? (I didn't understand the rest. I also edited my post by adding answer to 4th question, is it ok?) – First Last Sep 5 '16 at 18:13
• Every functor induces two profunctors. (Which are "cocontrabifunctor" and "contracobifunctor" :-) ) – First Last Sep 5 '16 at 18:39
• I didn't aim to answer your question, only wanted to reason about why - if not - the theory of anafunctors is not very well developed. Actually, you can choose either profunctor of any given functor. What I meant by cograph is written here: ncatlab.org/nlab/show/…. In the cograph category, the image of an object appears as a [co-]reflection of the object, and if we take all [co-]reflection arrows, we get sth very similar to a saturated anafunctor. – Berci Sep 5 '16 at 21:59
• A $(1,0)$-category is just a groupoid, isn't it? (Which sets aren't). – neth Sep 6 '16 at 23:54

1. As stated in the comments, $Set$ is not a groupoid (which is what we mean by (1,0)-category) unless we ask it to be.
2. I'm not sure what this means, but anafunctors are certain spans of functors, $C\xleftarrow{J} C' \xrightarrow{F} D$, and every functor $G$ gives rise to a canonical functor $C\xleftarrow{id} C \xrightarrow{G} D$. The functor $J$ is surjective on objects and fully faithful, and the set of objects $c'$ of $C'$ in the preimage of a given object $x$ of $C$ give rise to a set of objects $F(c')$ in $D$, all of which are isomorphic, and which Makkai calls the possible values of $F$ at $c$ (think of the limit of a diagram for instance: it is not unique, but all such limits are isomorphic, and any of them will do).
4. Even ignoring the size distinctions (which are not important, but you have been specifying small categories until now), $Set$ is not equivalent to $Set^{op}$ in the bicategory of categories, anafunctors and transformations, as (for instance) $Set$ is a(n elementary) topos and $Set^{op}$ is not, and the property of being a topos is invariant under equivalence (even in the anafunctor setting).


text:
Two–parameter scaling theory of transport near a spectral node

# Two–parameter scaling theory of transport near a spectral node

## Abstract

We investigate the finite–size scaling behavior of the conductivity in a two–dimensional Dirac electron gas within a chiral sigma model. Based on the fact that the conductivity is a function of system size times scattering rate, we obtain a two–parameter scaling flow toward a finite fixed point. The latter is the minimal conductivity of the infinite system. Depending on boundary conditions, we also observe unstable fixed points with conductivities much larger than the experimentally observed values, which may account for results found in some numerical simulations. By including a spectral gap we extend our scaling approach to describe a metal–insulator transition.

###### pacs:
05.60.Gg, 72.10.Bg, 73.22.Pr

Transport in a one–band metal is based on the dynamics of non–interacting electrons which are subject to random scattering. Physical quantities, such as the conductivity or the electronic diffusion coefficient, are obtained after averaging with respect to a random distribution of the scatterers. Then transport properties are controlled by large–scale correlations in the electronic system which occur due to spontaneous symmetry breaking. The order parameter of the latter is the average density of states (1), while the symmetry depends on the specific form of the Hamiltonian . Weak fluctuations on large scales around the symmetry breaking saddle point are obtained by a gradient expansion, which has the action of a nonlinear sigma model (NLSM)(1); (2); (3):

 S=1tTr(∂μQ∂μQ) (1)

with the nonlinear field . (A symmetry–breaking term is omitted here.) The latter is determined by the underlying symmetry of the two–particle Green’s function with , rather than by the symmetry of the Hamiltonian itself.

A particular class of metallic systems consists of two electronic bands with spectral nodes, where the Hamiltonian is expanded in terms of Pauli matrices . Prominent examples are graphene (4); (5), topological insulators (6); (7) and quasiparticles in D–wave superconductors (8); (9) with the generic Hamiltonian

 H=H0+V,  H0=hxτx+hyτy+hzτz , (2)

where is random with mean zero and variance . In the special case of graphene, we have for the vicinity of each node , with the Fermi velocity , the components of the momentum , and the gap parameter . All explicit calculations will use this specific case of .

A number of different nonlinear fields has been proposed for two bands (8); (11); (10); (9). The reason for this variety of symmetry groups is that there are actually two major approaches for studying the symmetry: Either the supersymmetry is enforced by construction (12) or spontaneous supersymmetry breaking is permitted (13).

Motivated by the accurate transport measurements in graphene  (4); (5); (14); (15); (16); (17), there has been much activity from the theoretical side to evaluate transport quantities such as the conductivity . In most calculations it is assumed that disorder is rather smooth, which implies the absence of inter-node scattering. Of particular interest is the size dependence, since typical graphene sample are rather small and vary in size from sample to sample. The behavior of under a change of the linear system size has been studied numerically (18); (19); (20). There are two characteristic observations, namely (i) that the increases logarithmically with and with the disorder strength, and (ii) that the –function is always positive but decreases monotonically without a finite fixed point. These results disagree substantially with earlier speculations on the shape of the –function, where two finite fixed points were proposed (11). Given the fact that there is a very robust minimal conductivity in the experiments, it is rather surprising that the numerical calculations do not indicate the existence of a finite fixed point for the conductivity. This might be a hint that the simulations have not reached the asymptotic regime.

In the following we assume weak and slowly varying disorder so that there is no scattering between different spectral nodes. Then we briefly discuss the realization of the chiral sigma model (CSM) with broken supersymmetry for a two-band system of Ref. (9) and evaluate the corresponding finite–size scaling of the conductivity. Although the –function is sensitive to the existence or absence of a zero mode in the finite system, it always describes a flow towards a finite attractive fixed point that agrees with the minimal conductivity at the Dirac node. This provides a surprisingly simple two–parameter scaling picture for transport in two–band metals with a spectral node.

There are several options to evaluate the transport properties at the Dirac node. One is based on the diffusion coefficient

 D0=limϵ→0ϵ2∑rr2kTr2⟨Gr0(iϵ)G0r(−iϵ)⟩d , (3)

another one is provided by the Kubo formula of the conductivity as

 σ(ω)=−e2ω22h∑rr2kTr2⟨Gr0(ω/2)G0r(−ω/2)⟩d (4)

for the response to an external electromagnetic field with frequency . is the trace with respect to the Pauli matrices of the two–band Hamiltonian. These expressions are connected by the analytic continuation . The correlation function

 Krr′=⟨Grr′(iϵ)Gr′r(−iϵ)⟩d , (5)

which appears in both expressions from the average with respect to random scatterers, is available from a field–theoretical calculation (24). This is based on the symmetry relation of the two–band Hamiltonian. A consequence is that the energy eigenfunction in the upper and the lower band are related as . It allows us to write

 G(−iϵ)=(H0+V−iϵ)−1=−τx(H∗0−V+iϵ)−1τx , (6)

which implies the chiral symmetry for

 ^H=(H0+V00H∗0−V),   ^S=(0φτxφ′τx0) . (7)

The symmetry group depends only on the single pair of Grassmann variables . Thus, the nonlinear field is and we can write

 Krr′=4η2g21N∫φrφ′r′e−S2D[φ], N=∫e−S2D[φ] (8)

with the bilinear CSM action

 S2=4ηg[ϵTr(φφ′)+DTr(∂μφ∂μφ′)] . (9)

It should be noticed that the bilinear form is characteristic for the Dirac node. There are also quartic terms away from the node (28). is the (renormalized) diffusion coefficient

 D=ηg2Tr∑rr2k¯Gr0(iη)¯G0r(−iη) (10)

with the effective Green’s function . The definition of the diffusion coefficient in Eq. (3) and the correlation function in Eqs. (5), (8) imply the relation . Moreover, by comparing the NLSM action of a one–band metal with the CSM action we get for their prefactors the relation

 t−1⟷4ηD/g=2D0, (11)

which is the conductivity due to the Einstein relation . This can be used now to calculate the –function from , in analogy with the treatment of a one–band metal. In order to determine the size dependence of we use a simple approximation for a first estimate and in a second step a more detailed numerical summation of in Eq. (10).

For a finite sample of size and no gap () the main effect on is an infrared cut–off in the Fourier integral, assuming that the largest wavelength is :

 σ≈2η2π∫∞L−1k dk[k2+η2]2∼1π(1−1η2L2) . (12)

for . This result indicates that the conductivity increases monotonically with the size and its –dependence scales with the scattering rate : . Moreover, the –function reads in this approximation

 β∼2π(1−πσ) , (13)

which has a fixed point in units of . This is the well–known minimal conductivity of Dirac fermions. Although this approximation is reliable near the fixed point, it may not be so good further away from the fixed point. The reason is that we have not considered (i) that the spectrum of a finite system is discrete and (ii) that the boundary conditions can be crucial. The effect of the latter is know to be important, for instance, in graphene ribbons, because the system may or may not have a gap (25); (26); (27).

The discrete spectrum of the gapless Dirac Hamiltonian in Eq. (2) is with wave numbers , . The parameter depends on the boundary condition (BC). In particular, we have for periodic BC and for BC with a phase shift of the wave function at opposite boundaries. Thus, only has a zero mode, whereas has a spectral gap that increases with increasing . This mimics the situation of the tight–binding model in the case of a graphene ribbon, where armchair (zigzag) boundaries provide a gapless (gapped) spectrum (25); (26); (27). With this discrete spectrum we calculate the conductivity in Eqs. (10) and (11) as a function of size with generic BC, characterized by the phase shift , at the Dirac point ():

 σ(η,L)=4η2L2∑n,m 1[k2n+k2m+η2]2 . (14)

The sum converges and gives us a conductivity that depends only on . is plotted in Fig. 1a, where for its value agrees with the minimal conductivity of Eq. (12). For intermediate values , on the other hand, the conductivity depends strongly on the parameter , though. In the case of periodic BC () the behavior is dominated by the zero energy mode. Its contribution to the conductivity decreases as with increasing sample size, and the conductivity represents a monotonically decreasing function of the length . For the zero mode is strongly suppressed. In this case the conductivity increases monotonically with (cf. Fig. 1a). In particular, there is a relatively broad regime where it grows logarithmically with , i.e., which agrees with known analytical (10); (21) and numerical (18); (19) results. Finally, there is an intermediate regime for , in which the conductivity increases up to a maximum and then approaches the asymptotic minimal conductivity from above.

The scattering rate , which so far appeared in the conductivity as a free parameter, can also be calculated as a function of system size and disorder strength, using the self–consistent Born approximation (22); (23)

 1g=1L2+L∑n,m=−L 1k2n+k2m+η2. (15)

The calculation for a finite sample is again a sum over the discrete wave numbers , in analogy to the calculation the conductivity, and gives a non–monotonous scattering rate with respect to that increases up to a certain length and approaches asymptotically a finite value. The asymptotic value depends on but is indifferent to . The way approaches this value depends significantly on , though: It decreases with increasing and decreasing , cf. Fig. 3.

Once the –dependence of the scattering rate is taken into account the –function for different values of and is calculated from Eq. (14). Plotting the curves for different values of together, the graphs collapse on a single curve, as depicted in Fig. 2a. Moreover, regardless of the parameters, all solutions are attracted to the fixed point with the value of the minimal conductivity. However, there are two types of –functions, one that approaches the fixed point from positive values (like the approximation in Eq. (13)) and another one from negative values. The positive branch of the –function coincide with , while the negative branch is associated with . The negative branch also starts for small with positive values of the –function and reaches an unstable fixed points at values much larger than the experimentally observed conductivity. However, the –function does not stop there but keeps flowing toward the only attractive fixed point at the observable value . Thus, the BC related parameter enforces the two–parameter scaling, whereas for fixed we obtain the one–parameter scaling. In particular, the positive branches resemble the numerically evaluated –functions found in (18) and (19). Moreover, the main part of these branches is fitted excellently with the double logarithm formula obtained in leading order of perturbation theory in Ref. (28), cf. Fig. 2b. However, the limitation of the one–loop approximation does not allow to approach analytically the quasi–fixed point at which the –function changes the sign.

Close to the fixed point, the –function exhibits a power law behavior , with an exponent that approaches unity for very small deviations from the fixed point, in agreement with the approximation in Eq. (13). For we can fit our curves with , as it is shown in Fig. 2c. This is a crossover to asymptotic power law with exponent 1, which might be important for comparison with numerical simulations and experimental measurement. For the latter we have typical values of meV (14); (29) and typical sizes  m (30); (31) such that we get , which matches well the parametric regime .

Metal–insulator transition in the gapped disordered 2D Dirac electron gas:– Returning to the Hamiltonian in Eq. (2) we include now a gap term . This would allow us to study a metal-insulator transition (MIT), as predicted earlier in the literature (24); (32). The possibility of tuning the gap experimentally in a sample with a particular disorder configuration, for instance, by controllable hydrogenation (15); (16), provides a transition at fixed disorder strength by varying the gap: For we have a metal and for a band insulator. On the level of practical implementation, the gap is built into Eq. (14) and in the self–consistent Born approximation by replacing with . Then both, the scattering rate and the conductivity, depend also on which describes the –function in a 3D parametric space, as illustrated in Fig. 4. The fixed point turns out be unstable with respect to the variable : Gradually increasing the gap from zero upward we observe a shift of the fixed point from toward zero. At zero a critical gap is reached, where the critical value depends on the disorder strength. For the system does not have any fixed points with finite conductivity but undergoes a transition to the insulating phase. For a broad range of disorder strengths it is verified that the asymptotical behavior of the –function on the critical trajectory is .

Conclusions:– A number of experimental investigations (4); (5); (14); (15); (16); (17) provides strong evidence for a universal, sample shape and disorder strength independent conductivity of a weakly disordered 2D Dirac electron gas. This apparently contradicts to claims of some numerical (18); (19) and analytical (10); (11); (21) work, which predict a supermetallic fixed point at infinite conductivity. In this work we have investigated the conductivity within the CSM approach (24) and found that the conductivity can indeed flow to (unstable) fixed points at values much larger than the experimentally observed conductivity. However, the –function does not stop there but keeps flowing back to smaller conductivities to reach eventually the attractive bulk fixed point at in units of . The details of this flow depend crucially on the boundary condition. The conductivity depends on the scattering rate and the system length as . A spectral gap shifts the fixed point to smaller values, indicating an unstable fixed point against gap opening. This leads eventually to a metal–insulator transition.

### References

1. F. Wegner, Z. Physik B 35, 207 (1979).
2. L. Schäfer and F. Wegner, Z. Physik B 38, 113 (1980).
3. S. Hikami, A. Larkin, and Y. Nagaoka, Prog. Theor. Phys. 63, 707 (1980).
4. K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, M. I. Katsnelson, I. V. Grigorieva, S. V. Dubonos, A. A. Firsov, Nature 438, 197 (2005).
5. Y. Zhang, Y.-W. Tan, H. L. Stormer, P. Kim, Nature 438, 201 (2005).
6. M. Z. Hasan and C. L. Kane, Rev. Mod. Phys. 82, 3045 (2010).
7. X.-L. Qi and S.-C. Zhang, Rev. Mod. Phys. 83, 1057 (2011).
8. A. A. Nersesyan, A. M. Tsvelik, and F. Wenger, Phys. Rev. Lett. 72, 2628 (1994).
9. K. Ziegler, M. H. Hettler, and P. J. Hirschfeld, Phys. Rev. Lett. 77, 3013 (1996).
10. M. Boquet, D. Serban, and M. R. Zirnbauer, Nucl. Phys. B 578, 628 (2000).
11. P. M. Ostrovsky, I. V. Gornyi, and A. D. Mirlin, Phys. Rev. Lett. 98, 256801 (2007).
12. K. Efetov, Supersymmetry in Disorder and Chaos, Cambridge University Press, (1997).
13. K. Ziegler, Phys. Rev. B 55, 10661 (1997); Phys. Rev. Lett. 80, 3113 (1998).
14. Y.-W. Tan, Y. Zhang, K. Bolotin, Y. Zhao, S. Adam, E. H. Hwang, S. Das Sarma, H. L. Stormer, and P. Kim, Phys. Rev. Lett. 99, 246803 (2007).
15. D. C. Elias, R. R. Nair, T. M. G. Mohiuddin, S. V. Morozov, P. Blake, M. P. Halsall, A. C. Ferrari, D. W. Boukhvalov, M. I. Katsnelson, A. K. Geim, and K. S. Novoselov, Science 323, 610 (2009).
16. A. Bostwick, J. L. McChesney, K.V. Emtsev, T. Seyller, K. Horn, S. D. Kevan, and E. Rotenberg, Phys. Rev. Lett. 103, 056404 (2009).
17. J. Baringhaus, M. Ruan, F. Edler, A. Tejeda, M. Sicot, A. Taleb-Ibrahimi, A.-P. Li, Z. Jiang, E. H. Conrad, C. Berger, C. Tegenkamp, and W. A. de Heer, Nature 506, 349 (2014).
18. J. H. Bardarson, J. Tworzydlo, P. W. Brouwer, and C. W. J. Beenakker, Phys. Rev. Lett. 99, 106801 (2007).
19. K. Nomura, M. Koshino, and S. Ryu, Phys. Rev. Lett. 99, 146806 (2007).
20. S.-J. Xiong and Y. Xiong, Phys. Rev. B 76, 214204 (2007).
21. T. Senthil and M. Fisher, Phys. Rev. B 61, 9690 (1999).
22. N. H. Shon and T. Ando, J. Phys. Soc. Jpn. 67, 2421 (1998).
23. E. Fradkin, Phys. Rev. B 33, 3257 (1986); ibid 3263 (1986).
24. K. Ziegler, Phys. Rev. Lett. 102, 126802 (2009); Phys. Rev. B 79, 195424 (2009).
25. L. Brey and H. A. Fertig, Phys. Rev. B 73, 235411 (2006).
26. A. R. Akhmerov and C. W. J. Beenakker, Phys. Rev. Lett. 98, 157003 (2007).
27. K. Wakabayashi, K.-i. Sasaki, T. Nakanishi, and T. Enoki, Sci. Technol. Adv. Mater. 11, 054504 (2010).
28. A. Sinner and K. Ziegler, Phys. Rev. B 86, 155450 (2012).
29. E. Pallecchi, A. C. Betz, J. Chaste, G. Fève, B. Huard, T. Kontos, J.-M. Berroir, and B. Plaçais, Phys. Rev. B 83, 125408 (2011).
30. A. H. Castro Neto, F. Guinea, N. M. R. Peres, K. S. Novoselov, and A. K. Geim, Rev. Mod. Phys. 81, 109 (2009).
31. E. Y. Andrei, G. Li, and X. Du, Rep. Prog. Phys. 75, 056501 (2012).
32. M. V. Medvedyeva, J. Tworzydło, and C. W. J. Beenakker, Phys. Rev. B 81, 214203 (2010).
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# Tag Info

8

In computational electromagnetism, there is a famous (or infamous because of the difficulties in some) set of test problems: Testing Electromagnetic Analysis Methods (T.E.A.M.). Some of them really need seriously state-of-art numerical techniques to get the correct simulation results aligned with the experimental data. For example, the conductor-coil ...

7

There are many good open-source implementations of root-finding methods out there already. One example is boost, whose implementation of Newton-Raphson and related methods you can find here. If you read its source code, you will be able to see what issues the authors wanted to address, and how they dealt with issues like convergence, user-specified tolerance ...

5

You can use Plot Digitizer and extract the data points in your image graph as a xml file and then you can parse it by using Python. It's pretty straightforward. You need to import the image of your graph into the software. Then calibrate the X and Y axes by specifying the $x_{min}$, $x_{max}$, $y_{min}$, and $y_{max}$ which are the min and max of X and Y ...

3

The most trivial pointsets are regular grid of points. They look innocent, but are the most difficult to handle in Voronoi/Laguerre codes because the Delaunay triangulation is non-unique, so they can be used to detect bugs in your code. They can be handled by adapted geometric predicates (see my publication [6] on geometric predicates and the references ...

3

One thing you can do for smoke testing is to take a known solution $x$ (you could generate it randomly or pick something convenient) and a known matrix $A$, and then set $b$ equal to $Ax$, and solve the equation $Ax = b$ to see if you recover your known solution. This method is known as the method of manufactured solutions. To avoid the influence of ...

3

There is a collection of reference PDE-constrained optimization problems maintained by Roland Herzog at TU-Chemnitz here.

2

As I mentioned before, you can try with the Visible Human Project. You can also go with 3D Slicer sample data, e.g., the Knee Atlas or the Brain Atlas. Their terms are quite permissive: By downloading these data, you agree to acknowledge our contribution in any of your publications that result form the use of this atlas. There is also OpenfMRi, but you ...

2

If you're looking for large graphs or network data to test on. The Stanford Network Analysis Project (SNAP) has many large graph datasets typically in the form of an anonymized adjacency list. Some of their options include: Data Social networks: Facebook, Gplus, Twitter, bitcoin, etc. Citation networks: nodes represent papers, edges are citations. Web ...

1

You might want to give a try to DataHarvester. This tool claims to support ROOT, HBOOK, HDF, XML, SQLITE and input/output between them. See the following publication: W. Waltenberger, G. Richter, and H. V. Riedel, "DataHarvester—an abstraction layer for accessing scientific data from various sources," in Int. Conf. Computing High Energy Nuclear Phys., ...

1

The problem you proposes must be treated with care. The function $f$ must have zero mean in the domain $\Omega$,i.e. $$\int_\Omega{f\, dV}=0$$ Otherwise the problem is not well posed. Physically the terms of your equation mean: $\partial_tu$: the energy change of your system. $-\Delta u$:the diffusion of the temperature through the domain. $f$: Heat ...

1

The dataset is huge. So I would like to be more memory-efficient. The main point to consider is the following: […] most of the cells will be “unactivated”, i.e most of the elements of the matrix will store an empty list (no signal). This sounds as if an array of lists (or variably sized arrays) is just fine. In this case, the crucial factor for memory ...

1

Useful inclusion measures depend on whether you want to check pairwise similiarity or containment. For similiarity the Jaccard index is intuitive. For containment I would adapt it to what ratio of columns of my 'gold standard file' are in the file to check which is a common feature used in natural language problems. For the order of columns in a file, an ...

1

A practical complication of such travel time estimates is the time spent transferring from one line/train to another. If all you aim for is the minimum travel time, then there are simple relatively fast algorithms to find one or all of the "optimal" routes. See Dijkstra's shortest path algorithm for a single start and destination, or Floyd-Warshall ...

1

Have you looked at the Matrix Market? http://math.nist.gov/MatrixMarket/

1

Without knowing more about your data, it is not possible to do better than the exhaustive search. This would be memory efficient, but linear in time. If your data set does not contain local minima, you could simply follow the steepest descent. This still has a worst case linear time however. A possibly better solution would be the downhill-simplex ...

1

To convert from Rodrigues vector to a rotation matrix (and back) please check the MATLAB code here: http://www.cs.ucla.edu/~soatto/vision/courses/268/rodrigues.m So this gives you the rotation matrix. You use the translation parameters directly. Additionally, you might want to form the camera matrix. It would be a good idea to get a fully calibrated K (...

1

As pointed out in the previous answer, the concept you are looking for is sensitivity. You take the formula that computes the parameters of your regression line and compute the derivatives with respect to your point positions. These should be large for the three isolated points, indicating that moving them will have big influence.

1

(I hope there is no objection to my tacking onto the end of this discussion. I'm new here, so please let me know if I have transgressed!) Test functions for evolutionary algorithms are now much more complicated than they were even 2 or 3 years ago, as can be seen by the suites used in competitions at conferences like the (very recent) 2015 Congress on ...

Only top voted, non community-wiki answers of a minimum length are eligible


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# Szemerédi–Trotter theorem

The Szemerédi–Trotter theorem is a mathematical result in the field of combinatorial geometry. It asserts that given n points and m lines in the plane, the number of incidences (i.e. the number of point-line pairs, such that the point lies on the line) is

$O \left ( n^{\frac{2}{3}} m^{\frac{2}{3}} + n + m \right ),$

this bound cannot be improved, except in terms of the implicit constants.

An equivalent formulation of the theorem is the following. Given n points and an integer k > 2, the number of lines which pass through at least k of the points is

$O \left( \frac{n^2}{k^3} + \frac{n}{k} \right ).$

The original proof of Szemerédi and Trotter[1] was somewhat complicated, using a combinatorial technique known as cell decomposition. Later, Székely discovered a much simpler proof using the crossing number inequality for graphs.[2] (See below.)

The Szemerédi–Trotter theorem has a number of consequences, including Beck's theorem in incidence geometry.

## Proof of the first formulation

We may discard the lines which contain two or fewer of the points, as they can contribute at most 2m incidences to the total number. Thus we may assume that every line contains at least three of the points.

If a line contains k points, then it will contain k − 1 line segments which connect two of the n points. In particular it will contain at least k/2 such line segments, since we have assumed k ≥ 3. Adding this up over all of the m lines, we see that the number of line segments obtained in this manner is at least half of the total number of incidences. Thus if we let e be the number of such line segments, it will suffice to show that

$e = O \left ( n^{\frac{2}{3}} m^{\frac{2}{3}} + n + m \right).$

Now consider the graph formed by using the n points as vertices, and the e line segments as edges. Since all of the line segments lie on one of m lines, and any two lines intersect in at most one point, the crossing number of this graph is at most m2. Applying the crossing number inequality we thus conclude that either e ≤ 7.5n, or that m2e3 / 33.75n2. In either case e ≤ 3.24(nm)2/3 + 7.5n and we obtain the desired bound

$e = O \left ( n^{\frac{2}{3}} m^{\frac{2}{3}} + n + m \right ).$

## Proof of the second formulation

Since every pair of points can be connected by at most one line, there can be at most n(n − 1)/2 lines which can connect at k or more points, since k ≥ 2. This bound will prove the theorem when k is small (e.g. if kC for some absolute constant C). Thus, we need only consider the case when k is large, say kC.

Suppose that there are m lines that each contain at least k points. These lines generate at least mk incidences, and so by the first formulation of the Szemerédi–Trotter theorem, we have

$mk = O \left ( n^{\frac{2}{3}} m^{\frac{2}{3}} + n + m \right ),$

and so at least one of the statements $mk = O( n^{2/3} m^{2/3} ), mk = O(n)$, or $mk = O(m)$ is true. The third possibility is ruled out since k was assumed to be large, so we are left with the first two. But in either of these two cases, some elementary algebra will give the bound $m = O( n^2 / k^3 + n/k )$ as desired.

## Optimality

Except for its constant, the Szemerédi–Trotter incidence bound cannot be improved. To see this, consider for any positive integer NZ+ a set of points on the integer lattice

$P = \left \{ (a, b) \in \mathbf{Z}^2 \ : \ 1 \leq a \leq N; 1 \leq b \leq 2N^2 \right \},$

and a set of lines

$L = \left \{ (x, mx + b) \ : \ m, b \in \mathbf{Z}; 1 \leq m \leq N; 1 \leq b \leq N^2 \right \}.$

Clearly, $|P| = 2N^3$ and $|L| = N^3$. Since each line is incident to N points (i.e., once for each $x \in \{1, \cdots, N\}$), the number of incidences is $N^4$ which matches the upper bound.[3]

## Generalization to Rd

One generalization of this result to arbitrary dimension, Rd, was found by Agarwal and Aronov.[4] Given a set of n points, S, and the set of m hyperplanes, H, which are each spanned by S, the number of incidences between S and H is bounded above by

$O \left (m^{\frac{2}{3}}n^{\frac{d}{3}}+n^{d-1} \right ).$

Equivalently, the number of hyperplanes in H containing k or more points is bounded above by

$O\left( \frac{n^d}{k^3} + \frac{n^{d-1}}{k} \right ).$

A construction due to Edelsbrunner shows this bound to be asymptotically optimal.[5]

Solymosi and Tao obtained near sharp upper bounds for the number of incidences between points and algebraic varieties in higher dimensions. Their proof uses the Polynomial Ham Sandwich Theorem.[6]

## References

1. ^ Szemerédi, Endre; Trotter, William T. (1983). "Extremal problems in discrete geometry". Combinatorica 3 (3–4): 381–392. doi:10.1007/BF02579194. MR 0729791.
2. ^ Székely, László A. (1997). "Crossing numbers and hard Erdős problems in discrete geometry". Combinatorics, Probability and Computing 6 (3): 353–358. doi:10.1017/S0963548397002976. MR 1464571.
3. ^ Terence Tao (March 17, 2011). "An incidence theorem in higher dimensions". Retrieved August 26, 2012.
4. ^ Agarwal, Pankaj; Aronov, Boris (1992). "Counting facets and incidences". Discrete and Computational Geometry (Springer) 7 (1): 359–369. doi:10.1007/BF02187848.
5. ^ Edelsbrunner, Herbert (1987). "6.5 Lower bounds for many cells". Algorithms in Combinatorial Geometry. Springer-Verlag. ISBN 3-540-13722-X.
6. ^ Solymosi, J.; Tao, T. (September 2012). "An incidence theorem in higher dimensions". Discrete and Computational Geometry 48 (2). doi:10.1007/s00454-012-9420-x.


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## Overview

This page is about one of the most prominent errors of pascal: unit not found or can't find unit. This error has a lot of reasons and this page tries to gather most of the reasons, how to find out, which problem it is, and how to solve it.

There is a tool to test for many common misconfigurations: Project / Project Options ... / Test. Run it and see the warnings it produces. Normally it should only warn about some duplicate ppu files in the fpc sources (the files depends on the fpc version and distribution).

## Quick list

For experts here is a list of possible causes for the error. This list is not sorted for probability.

• FPC installation
• You named one of your units like one of the FPC units.
• Wrong fpc.cfg. $HOME/.fpc.cfg instead of /etc/fpc.cfg. Find out with -vut. • Missing or wrong search path in fpc.cfg. Find out with -vut. • Added a search path to fpc.cfg that contains sources (.pas,.pp) • Cross compiling: Forgot to install the cross compiled FPC ppus. Find out with -vut. • Broken FPC installation. Remove FPC directory manually and install again. • Compiler symlink points to wrong compiler. • FPCDIR environment variable is set • You named one of your include files like one of the FPC include files and your FPC installation was not compiled with -Ur. • Lazarus • Forgot to add a package dependency. Check project inspector or package editor. • A ppu exists twice. Find out with -vut. Remove the wrong ppu. Check if a package usage unit paths is only$(PkgOutDir) and the usage include path is empty. Check if two packages or a package and the project contain the same unit search path in their compiler options.
• Typo or missing search paths. Check Project / Project Options / Compiler Options, pages Paths and Inherited and check source editor / popup menu / file settings / unit info / unit paths.
• Wrong case. If the unit filename uses upper and lower case letters all uses sections must use the correct case. Before 0.9.29 the IDE did not search 'in' filenames in lower and upper case. Check that the 'in' filenames in uses sections.
• Wrong unit: If there are two units with the same name in the search path the compiler may use the wrong one without warning. The IDE normally warns unless disabled in Tools / Options / Environment / Naming / ambiguous file action.
• Forgot to install the FPC sources and/or forgot to set the path in Tools / Options / Environment / Files / FPC source directory.
• Using the wrong version of FPC sources
• Using the wrong compiler. Check Tools / Options / Environment / Files / compiler path.
• Using the wrong lazarus directory. Check Tools / Options / Environment / Files / Lazarus directory.
• Cross compiling: Forgot to cross compile lcl, synedit for the new platform. These are not compiled automatically (They are since 0.9.31).
• Wrong package: you have a package (.lpk) twice on your disk and the IDE is still using the old

## Notations, terms

• .pas file: This is a place holder for any pascal unit source. For example unit1.pas or unit1.pp. Both extensions are allowed and there is no difference. There are other compilers, which ignore .pp files, but this page is about the Free Pascal compiler.
• .pp file: Same as .pas. Just another extension for the same thing.
• .p file: Same as .pas. Just another extension for the same thing. Commonly utilized on Macintosh.
• .ppu file: This is place holder for any compiled pascal unit, created by the Free Pascal compiler.

## Step 1: Compiler or IDE or both

The IDE and the compiler do not use the exact same way to locate units.

### Compiler

If the compiler can not find it, the compilation fails (Build, Ctrl+F9) with the error 'unit not found'.

### IDE

If the IDE can not find, the codetools will fail. For example Find Declaration (Alt+Up or Ctrl+Left mouse button) or Identifier completion (Ctrl+Space) will fail with the error 'unit not found'. If the error message is about units you don't use directly, then simply add the unit name to a uses section and do a find declaration. After that you can remove it.

### Search strategies

#### What both search

• Both search units in the search path.
• Both search first for pascal sources (.pas, .pp) files. Under MacPAS mode: search for .p files too.
• Both search for several cases: normal case, lower case, upper case. For example: if the uses section contains 'Unit1', then it searches:
 1. Unit1.pp
2. unit1.pp
3. UNIT1.PP
4. Unit1.pas
5. unit1.pas
6. UNIT1.PAS

They are searched in every directory of the search path.

#### Differences between IDE and FPC unit search

• FPC uses the fpc.cfg for additional search paths. The IDE uses them only to find files of the FPC sources. So adding a search path to the fpc.cfg is ignored by Lazarus.
• FPC always starts in one directory and every relative filename is relative to this directory. For the Lazarus IDE each project and each package has its own base directory. So the search path of a package is relative to the package directory. The package directory is the directory of the .lpk file. The base directory of a project is the projects main source file (e.g. the .lpr file). If there is no main source file, the directory of the .lpi file is taken. Normally both are the same and the IDE has no graphical frontend to split. But by editing the .lpi file manually you can split them. At the moment there is only one prominent example for this case: the ppxxx.lpi files of the Free Pascal compiler.
• FPC and Lazarus have different macros.
• You can define additional source search paths for the IDE, which are not given to the compiler. (Compiler options -> Other source files)

## Case 1: Both can not find a unit

• Check if the unit exists: If not, then see chapter Searching packages, projects of a unit (ToDo: write me).
• Check if the unit is readable: (You can open it in the IDE). A unit source does not need to be writable. If not, then fix the permissions or get a readable copy.
• Check if the unit has a valid pascal name: (identifier plus extension). Valid examples: unit1.pas, Bogus2.pp, _1.pas, LAZ.PAS. Wrong examples: 1.pas (starts with number), a%.pas (contains special character), unit.PAS (extension uppercase, but filename lowercase).
• Check if directory name contains special characters
  1. slashes / and backslashes \ are used as directory delimiter and should never be used for directory names
2. Semicolons are used to separate paths in the IDE.
3. dollar \$ is used for macros in many scripting languages

• Check for right case: If your unit name is all lowercase or all uppercase you don't need to worry. If not, then you must make sure, that every uses section in every source code uses exactly this case. For example: if your unit is named Unit1.pas, then every uses section must use Unit1. This is not so important under windows, but under any other platform (Linux, MacOSX). The IDE gives warnings when saving files with mixed case.
• Check if the unit is in the search path. If the compilation stopped while compiling a package, look in the compiler options of that package (open the package, there is a button 'compiler options'). If the compilation stopped while compiling the project, look in the project / compiler options.

## Case 2: Compiler finds a unit, but IDE does not

The next sub cases assumes, that Run -> Build (Ctrl+F9) in the Lazarus IDE works, but Find Declaration fails with the error: unit not found. If unsure, check Case 1 as well.

• Check if the unit source exists: The IDE searches for pascal source (.pas, .pp) and ignores the .ppu files. The compiler can use a .ppu, but the Lazarus IDE not. This feature is planned, but there is no time frame.
• Check if unit directory was added to your fpc.cfg: The compiler uses all paths defined in the fpc.cfg file. The IDE does not read this file. Add the search path to the compiler options of your projects and/or packages.
• Check for macros and special compiler flags: The compiler understands the asterisk * character in search paths. The IDE treats this character as normal character. You can add the -Fu, -FU, -FE flags manually to the custom compiler flags in the lazarus IDE, but they are not parsed. Always use the right fields for options. The custom options are only for flags like the -D flag or uncommon compiler options, like new features of the unstable developer compiler.
• Rescan the FPC source directory: sometimes after a new/fresh install the IDE has problems finding the FPC sources. Selecting this option from the main menu (Environment/Rescan the FPC directory) might fix the problem.

## Case 3: IDE finds a unit, but compiler does not

This is the case, when Find Declaration finds the unit, but building (Ctrl+F9) fails.

There are two cases: Either the compiler didn't find anything, or it does not like, what it found. In both cases it returns the same error: unit xyz not found.

You can see where the compiler sarches and what it finds by adding the -vu -vt options (short: -vut). Beware: This gives easily thousands of lines of output. The -vut must be applied to the step, that creates the error. If it happens when compiling the project, enable the -vt in the compiler options of the project. If it happens when building a package, use the compiler options of the package. If it happens when building the IDE, use the Tools / Configure Build Lazarus / Advanced Build Options / Options. Then compile again. It will slow down the compilation. Then right click on on messages window and click on 'Copy all and hidden messages to clipboard'. Then paste it i the source editor. Now search for all places where the unit name is mentioned.

### Case 3.1: The compiler finds the .ppu file, but does not like it

This can happen, when the found .ppu file is older than a required unit (.pas, .pp or .ppu). For example if unit1 uses unit2 and unit2 was changed, then unit1 needs to be recompiled as well.

It can also happen, if you installed newer files without deleting the old ones. The compiler stops at the first found.

You can see, what the compiler finds and what is missing by giving the -vt command line switch. But this gives a lot of output and it is hard to read for newbies.

The easiest way is often to first recompile the packages and project clean (Build all). In case of the Lazarus sources itself: The configure build lazarus dialog has a button to setup all options for a clean build.

### Case 3.2: The compiler cannot find source (.pas, .pp) or .ppu file

In this case you have to extend the search path. The question is: add the .ppu directory or the source directory?

Warning: If you add the wrong search path, it can happen, that the compiler will create multiple copies of files and you will later get the 'unit not found' error of Case 3.1.

Many projects and packages have distinct source directories (.pas, .pp) and output directories (.ppu). If a package has an output directory, then only this directory should be added to the search path of any depending project and/or package. The IDE will do this automatically. For example: Adding the LCL as dependency to your project adds two output directories to your unit search path. You can see this in Compiler Options -> Inherited. Never add any other LCL directory to your search paths or you will get Case 3.1 errors.

• Check if the unit is in the compiler search path: The search path given to the compiler are the -Fu options of the command line parameters. You can see the options given to the compiler in Compiler Options -> Show Options.

### Case 3.3: The wrong compiler is being used

Perhaps when cross compiling the correct cross .ppu is found, but the wrong compiler is used by lazarus.

Lazarus uses the compiler specificied in the environment options. It is easy to forget to change the compiler, if you change the target CPU. Therefore it is recommended to set the compiler in the environment options to the fpc executable, which is a front end to the ppcXXX compiler backends. The fpc executable chooses the right ppcXXX compiler based on the -P parameter or the native pccXXX compiler if no such parameter is given. The drawback of using fpc instead of for example ppc386 directly, is that an extra process needs to be started (one for fpc and one for ppcXXX) which slows down compilation slightly, but this is hardly measurable.

Run lazarus from a text console (with startlazarus) so you an see the debugging output and look at the output generated when you compile.

It may look like this:

TPkgManager.GetWritablePkgOutputDirectory APackage=DBFLaz 0.1.1 AnOutDirectory="/home/sam/.lazarus/lib/DBFLaz/powerpc-linux"
TLazPackageGraph.CheckIfPackageNeedsCompilation  No state file for JPEGForLazarus 1.0.1
[TExternalToolList.Run] CmdLine="/usr/bin/ppc386  -S2cgi -OG1 -Ppowerpc -gl -vewnhi -l -Fu../../lcl/units/powerpc-linux/ -Fu../../lcl/units/powerpc-linux/gtk2/ -Fu../../packager/units/powerpc-linux/ -Fu. -FU/home/sam/.lazarus/lib/JPEGForLazarus/powerpc-linux/ -dLCL -dLCLgtk2 jpegforlazarus.pas"
WorkDir="/usr/share/lazarus/components/jpeg/"
Hint: Start of reading config file /etc/fpc.cfg
Hint: End of reading config file /etc/fpc.cfg
Warning: You are using the obsolete switch -OG
Free Pascal Compiler version 2.3.1 [2007/12/10] for i386
Copyright (c) 1993-2007 by Florian Klaempfl
Target OS: Linux for i386
Compiling jpegforlazarus.pas
Compiling lazjpeg.pas
PPU is compiled for another processor
Fatal: Can't find unit IntfGraphics used by LazJPEG
TExternalToolList.Run Exception: /usr/share/lazarus/components/jpeg/lazjpeg.pas(1,1) Fatal: Can't find unit IntfGraphics used by LazJPEG

The clues are these lines:

[TExternalToolList.Run] CmdLine="/usr/bin/ppc386  -S2cgi -OG1 -Ppowerpc -gl -vewnhi -l -Fu../../lcl/units/powerpc-linux/ -Fu../../lcl/units/powerpc-linux/gtk2/ -Fu../../packager/units/powerpc-linux/ -Fu. -FU/home/sam/.lazarus/lib/JPEGForLazarus/powerpc-linux/ -dLCL -dLCLgtk2 jpegforlazarus.pas"
...
Target OS: Linux for i386
...