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# ATHDDA Week 2 titles and abstracts

Speaker: Kofi Placid Adragni (U.Maryland Baltimore, Math & Stats)

Title: Hierarchical Principal Fitted Components

Abstract: Sufficient dimension reduction methods are statistical methods designed to reduce the dimensionality of large datasets without loss of regression information. In obtaining the sufficient reduction, it is often assumed that all observations are independent. This is a helpful assumption for estimating a minimal sufficient dimension reduction subspace via inverse regression. However, observations are often recorded on subjects or clusters.  While the observations from subject to subject could be independent, the within-subject observations are likely dependent. Treating the within-subject observations as independent may not be appropriate, and may adversely affect the estimation of the central subspace. We propose a method for estimation of the central subspace when the observations are dependent within cluster. The proposed methodology is built upon principal fitted components, a likelihood-based method for sufficient dimension reduction.

Speaker: Mirza Faisal Beg

Title: Problems in High Dimensional Computational Brain and Eye Anatomy

Abstract: Computational tools to study shape change in human anatomy as a function of disease lead to a small number of points in very high dimensional data. I will be presenting some recent work from our group in applying the toolbox of Computational Anatomy in the setting of brain and retina morphometry, and applications in Alzheimer's disease and Glaucoma. If time permits, I will also talk about network-based structural biomarkers as well as a unified voxel-based morphometry/tensor-based morphometry method for generating structural measures for discriminating between different dementias.

Speaker: Luke Bornn (Simon Fraser)

Title: Efficient Representations of Massive Spatio-Temporal Point Processes

Abstract: In this talk I will show how disjoint and compact bases provide a natural and intuitive approach to modeling spatial and spatio-temporal point processes. I will argue that inducing spatially compact bases allows for more intuitive representation of many real-world point processes while allowing computational tractability. I will demonstrate how such a result can be obtained through non-negative matrix factorization, and subsequently extend this idea to a full generative model.

Speaker: Ivor Cribben (U.Alberta, business)

Title: A new method for estimating spectral clustering change points for multivariate time series.

Abstract: Spectral clustering is a computationally feasible and model-free method widely used in the identification of communities in networks. In this work, we introduce a data-driven method that detects change points in the network structure of a multivariate time series, with each component of the time series represented by a node in the network. Spectral clustering allows us to consider high dimensional time series where the number of time series is greater than the number of time points (n<p). The method allows for estimation of both the time of change in the network structure and the graph between each pair of change points, without prior knowledge of the number or location of the change points. The stationary bootstrap is used to perform inference on the change points. The method is applied to various simulated high dimensional data sets as well as to a resting state functional magnetic resonance imaging (fMRI) data set. The results illustrate the method's ability to observe how the network structure between different brain regions changes over the experimental time course. The method promises to offer a deep insight into the inner workings and dynamics of the brain.

Speaker: Mike Daniels (U.Texas, Austin, Integrative Biology)

Title: Sequential BART for imputation of missing covariates

Abstract: To conduct comparative effectiveness research using electronic health records (EHR), many covariates are typically needed to adjust for selection and confounding biases.  Unfortunately, it is typical to have missingness in these covariates.  Just using cases with complete covariates will result in considerable efficiency losses and likely bias. We explore a flexible Bayesian nonparametric approach to impute the missing covariates which involves factoring the joint distribution of the covariates with missingness into their sequential conditionals and applying Bayesian additive regression trees (BART) to model each conditional. Obtaining the posterior for each conditional can be done simultaneously. We provide details on the computational algorithm and compare to other methods, including MICE.

Speaker: Guoqing Diao (George Mason U. stats)

Title: Conditional Variable Screening in High-Dimensional Binary Classification

Abstract: Most existing variable screening methods for high-dimensional data rely on strong parametric modelling assumptions that are often violated in real applications. Recently, Mai and Zou (2013) proposed a Kolmogorov filter for high-dimensional binary classification based on the Kolmogorov-Smirnov statistic. This screening method, however, does not account for the effects of potential confounders. We propose a new nonparametric conditional screening method to assess the conditional contributions of the individual predictors in the presence of known confounders. A bootstrap method is proposed to assess the significance for each predictor. The proposed method retains the features of the Kolmogorov filter and is shown to enjoy the sure screening property under the much weakened model assumptions compared to the parametric conditional screening methods. We illustrate the proposed method through extensive simulation studies and real applications. This is joint work with Jing Qin.

Speaker: Lee Dicker (Rutgers, stats)

Title: Efficient variance estimation for high-dimensional linear models.

Abstract:  We consider high-dimensional linear models with random predictors, and argue that there is little difference between fixed-effects models and random-effects models, under appropriate conditions.  In the fixed-effects model, the regression coefficients "inherit" randomness from the predictors and, consequently, methods and results for random-effects models may be ported to the fixed-effects setting.  In particular, following an empirical Bayes strategy, we derive high-dimensional optimality results for ridge regression and asymptotically efficient estimators for the residual variance ($\sigma^2$) in high dimensions, in the fixed-effects linear model.

High-dimensional residual variance estimation has recently received increased attention in the statistical literature, with important applications in model selection, regression diagnostics, and applied fields, such as genomics.  Our results for estimating the residual variance estimation do not require any sparsity assumptions on the regression parameters (which are common in other approaches) and appear to be the first high-dimensional efficiency results of their kind.

This is joint work with Murat A. Erdogdu (Stanford)

Speaker: Kjell Doksum

Title: Perspectives on High Dimensional Data Analysis. Randomize, Mix and Match

Abstract: The growth of high dimensional data sets has been followed by a growth of methods for analyzing such data sets. One approach is to mix existing methods. For instance, “sparse PCA” mixes PCA and Lasso by replacing the sum of squares condition in PCA is by a condition on the sum of absolute values of coefficients(approximately). Another approach is to repeat a method on randomly selected subsets of variables and then to average or combine the outcomes. This approach is sometimes referred to as “bagging” or “ sketching”. “Random forest”, which in many frameworks improves substantially on the tree approach of CART, is a prime example. Similarly, we find that “random Lasso” typically has smaller prediction error than Lasso. We examine this phenomena, and point to situations where it leads to smaller risk, and to situations where it has limitations. Note that because this “randomize” approach uses subsets of variables, a method that may not be appropriate when the number of variables exceeds the sample size, can now be used for high dimensional data.

Speaker: Xiaoli Gao (U. North Carolina, Greensboro, math&stats)

Title: Penalized adaptive  weighted least square regression

Abstract: In high-dimensional settings, penalized least squares approach can lose its efficiency in both estimation and variable selection due to the existence of heteroskedasticity. In this manuscript, we propose a novel approach, penalized adaptive weighted least squares (PAWLS), for simultaneous robust estimation and variable selection. The proposed PAWLS is justified from both Bayesian understanding and robust variable selection points of view. We also establish oracle inequalities for both regression coefficients and heterogeneous parameters. The performance of the proposed estimator is evaluated in both simulation studies and real examples.

Speaker: Yulia Gel (Waterloo, Statistics)

Title: Using Data Depth vs. Depth classifier for detecting  communities in networks

Abstract: We propose a new nonparametric spectral clustering algorithm for detecting communities in large networks using the Depth vs. Depth (DD) classifier. Under the stochastic block model (SBM) and pre-defined number of clusters, we study the performance of four data depth functions, i.e. simplicial depth, half-space depth, Mahalanobis depth, and projection depth. We investigate the effect of regularization on a classification error and compare the new DD classifier on networks with the regularized clustering algorithm based on the K-means approach. It is a joint work with Yahui Tian.

Speaker: Jinko Graham (Simon Fraser U, stats)

Title: Integrative analysis of genomic and neuro-imaging data

Abstract: For exploring large and complex data from genomic and neuro-imaging studies, multivariate projection is a valuable tool. In particular, canonical correlation analysis (CCA) has been useful for integrating genomic and neuroimaging data, and understanding their relationship. However, in imaging-genomics studies, the number of features greatly exceeds the number of individuals, and traditional CCA methods cannot be used. Moreover, high correlations between the features de-stabilize analyses. To avoid these problems and facilitate interpretation, sparse CCA (SCCA) has been proposed using an elastic-net-inspired penalty. SCCA reduces dimensionality and facilitates the identification of relevant features during the integration process. To remove additional unimportant features, further filtration based on the adaptive lasso or BIC may be applied. In this ongoing work, we use SCCA to explore the relationship between genome-wide variation at single-nucleotide polymorphisms (SNPs) and region-specific rates of decline in brain structure and function, as measured by functional magnetic resonance imaging. We apply these methods to data from the Alzheimer's Disease Neuroimaging Initiative 1 (ADNI1), an imaging-genomics study of Alzheimer's disease and mild cognitive impairment. I will describe the statistical challenges encountered so far, and propose possible approaches to deal with them.

Speaker: Bei Jiang (Columbia University, Biostats)

Title: Modeling Placebo Response using EEG data through a Hierarchical Reduced Rank Model

Abstract: There is growing evidence that individual differences among depression patients on Electrophysiology (EEG), fMRI and other brain imaging measurements may be predictive of potential treatment response. In this talk we discuss approaches to identifying potential placebo responders, i.e., a subgroup who benefits sufficiently from inactive drug treatments, using EEG measurements as a matrix (order-2 tensor) predictor. Given the high dimensionality of the problem, we consider a reduced rank regression model with a data-driven regularization. Our approach will be evaluated through simulations and will be applied to data from a large placebo-controlled clinical trial of major depressive disorders.

Speaker: Timothy D. Johnson (Michigan, Biostats)

Title: Analysis of Point Pattern Imaging Data using Log Gaussian Cox Processes with Spatially Varying Coefficients

Abstract: Log Gaussian Cox Processes (LGCP) are used extensively to model point pattern data. In these models, the log intensity function is modeled semiparametrically as a linear combination of spatially varying covariates with scalar coefficients plus a Gaussian process that models the random spatial variation. Almost exclusively, the point pattern data are a single realization from the driving point process. In contrast, our motivating data are lesion locations from a cohort of Multiple Sclerosis patients with patient specific covariates measuring disease severity. Patient specific covariates enter the model as a linear combination with spatially varying coefficients. Our goal is to correlate disease severity with lesion location within the brain. Estimation of the LGCP intensity function is typically performed in the Bayesian framework using the Metropolis adjusted Langevin algorithm (MALA) and, more recently, Riemannian manifold Hamiltonian Monte Carlo (RMHMC). Due to the extremely large size of our problem—3D data (64 × 64 × 64) on 240 subjects—we show that MALA performs poorly in terms of posterior sampling and that RMHMC is computationally intractable. As a compromise between these two extremes, we show that posterior estimation via Hamiltonian Monte Carlo performs exceptionally well in terms of speed of convergence and Markov chain mixing properties.

Speaker: Abbas, Khalili (McGill, math & stats)

Title: Post-Model Selection Inference for Finite Mixture of Regression (FMR) Models

Abstract: Finite Mixture of Regression (FMR) models are used in situations where several sub-populations within a population exist but the sub-populations themselves are unknown. Recently developed statistical methods for parameter estimation and variable selection in FMR models involve regularization techniques such as the LASSO and SCAD. However, it is well-known that the variability due to model selection stage causes bias in the resulting statistical inference.  In this presentation, we address this issue in FMR and several related mixture models. This is a joint work with Anand N. Vidyashankar from Dept. of Statistics at George Mason University.

Speaker: Peter Kim (Guelph)

Title:  Phylogenetic LASSO:  Pruning the tree of life

Abstract:  We model the effects of bacterial composition of the human gut microbiome by applying a hierarchical LASSO in series to penalize the variables corresponding to high level organismal classification, in essence pruning the tree of life by phylum, class, etc. Based on Zhu and Zhou's group L^{1/2} hierarchical LASSO, we employ this in seeking the contributors to recovery in faecal microbiota therapy for recurrent Clostridium difficile infection.

Speaker: Linglong Kong (U. Alberta, Math & Stats)

Title: Tensor Approximation in Functional Linear Quantile Regression

Abstract: We consider the estimation in functional linear quantile regression in which the dependent variable is scalar while the covariate is a function, and the conditional quantile for each fixed quantile index is modeled as a linear functional of the covariate. There are two common approaches for modeling the conditional mean as a linear functional of the covariate. One is to use the functional principal components of the covariates as basis to represent the functional covariate effect. The other one is to extend the partial least square to model the functional effect. The former belongs to unsupervised method and has been generalized to functional linear quantile regression. The later is a supervised method and is superior to the unsupervised PCA method. In this talk, we propose to use partial quantile regression and its tensor approximation to estimate the functional effect in functional linear quantile regression. Asymptotic properties have been studied and show the virtue of our method in large sample. Simulation study is conducted to compare it with existing methods. A real data example in stroke study is analyzed and some interesting findings are discovered.   Joint work with Ivan Mizera and Dengdeng Yu.

Speaker: Michael Lesnick (IMA)

Title: Interactive Visualization of 2-D Persistent Homology.

Abstract: In topological data analysis, we often study data by associating to the data a filtered topological space, whose structure we can then examine using persistent homology.  However, in many settings, a single filtered space is not a rich enough invariant to encode the interesting structure of our data.  This motivates the study of multidimensional persistence, which associates to the data a topological space simultaneously equipped with two or more filtrations.  The homological invariants of these multifiltered spaces," while much richer than their 1-D counterparts, are also far more complicated.  As such, adapting the usual 1-D persistent homology methodology for data analysis to the multi-D setting requires some some new ideas.

In this talk, I'll introduce multi-D persistent homology and discuss joint work with Matthew Wright on the development of a tool for the interactive visualization of 2-D persistent homology.

Speaker: Mary Lesperance (UVic, math&stats)

Title: A Bayesian Group-Sparse Regression Model with application to Brain Imaging Genomics

Abstract: Advances in technology for brain imaging and genotyping have motivated studies examining the relationships between genetic variation and brain structure.  Wang et al. (Bioinformatics, 2012) developed an approach for simultaneous regression parameter estimation and SNP selection based on penalized regression with a group l_{2,1}-norm penalty.  The group-norm penalty formulation incorporates the biological group structures among SNPs induced from their genetic arrangement and enforces sparsity at the group level.  Wang et al. do not provide standard errors or other inferential methodology for their parameter estimates.  In this paper, we propose a  corresponding Bayesian model that allows for full posterior inference for the regression parameters using Gibbs sampling.  Properties of our method are investigated using simulation studies and the methodology is applied to a large dataset collected as part of the Alzheimer's Disease Neuroimaging Initiative.

Speaker: Rachel Levanger (Rutgers)

Title: Using Persistent Homology to study dynamics in the space of persistence diagrams, Part II

Abstract: Building on the concepts from my talk in Part I, we address more nuanced issues that occur when studying the dynamics of fluid flows. This includes addressing sampling rates for periodic orbits and techniques for analyzing large point clouds. We also explore the use of diffusion map projections and show how this approach can help to organize and correlate the data to natural measurements of the system.

Speaker: Yi Li (U.Michigan, Biostats)

Title: Covariance-Insured Screening Methods for Ultrahigh Dimensional Variable Selection

Abstract: Effective screening methods are crucial to the analysis of big biomedical data. The popular sure independence screening relies on restricted assumptions such as the partial faithfulness condition, e.g,  the partial correlation between outcome and covariates can be inferred from their marginal correlation. However, such a restrictive assumption is often violated, as the marginal effects of predictors may be quite different from their joint effects, especially when the covariates are correlated.  We propose a covariance-insured screening (CIS) framework that utilizes the dependence among covariates and identify important features that are likely to be missed by marginal screening procedures such as sure independence screening. The proposed framework encompasses linear regression models, generalized linear regression models, survival models, and classification of multi-level outcomes.

Speaker: Xuewen Lu (U.Calgary, math & stats)

Title: Variable Selection in Log-linear Birnbaum-Saunders Regression Model for High-dimensional Survival Data via the Elastic-Net and stochastic EM

Abstract: Birnbaum-Saunders (BS) distribution is broadly used to model failure times in reliability and survival analysis. In this article, we propose a simultaneous parameter estimation and variable selection procedure in a log-linear BS regression model for high-dimensional survival data. To deal with censored survival data, we iteratively run a combination of the stochastic EM algorithm (SEM) and variable selection procedure to generate pseudo-complete data and select variables until convergence. Treating pseudo-complete data as uncensored data via SEM makes it possible to incorporate iterative penalized least squares and simplify computation. We demonstrate the e!cacy of our method using simulated and real data sets.

Speaker: Shuangge (Steven) Ma (Yale Biostats)

Title: Promoting Similarity of Sparsity Structures in Integrative Analysis

Abstract: For data with high-dimensional covariates but small to moderate sample sizes, the analysis of single datasets often generates unsatisfactory results. The integrative analysis of multiple independent datasets provides an effective way of pooling information and outperforms single-dataset analysis and some alternative multi-datasets approaches including meta-analysis. Under certain scenarios, multiple datasets are expected to share common important covariates, that is, their models have similarity in sparsity structures. However, the existing methods do not have a mechanism to promote the similarity of sparsity structures in integrative analysis. In this study, we consider penalized variable selection and estimation in integrative analysis. We develop a penalization based approach, which is the first to explicitly promote the similarity of sparsity structures. Computationally it is realized using a coordinate descent algorithm. Theoretically it has the much desired consistency properties. In simulation, it significantly outperforms the competing alternative when the models in multiple datasets share common important covariates. It has better or similar performance as the alternative when there is no shared important covariate. Thus it provides a “safe” choice for data analysis. Applying the proposed method to three lung cancer datasets with gene expression measurements leads to models with significantly more similar sparsity structures and better prediction performance.

Speaker: Debashis Mondal (Oregon State, stats)

Title:   Matrix-free computations for  Gaussian Markov random fields and related spatial processes

Abstract:  Since their introduction in statistics through the seminal works of Julian Besag, Gaussian Markov random fields have become central to spatial statistics, with applications in agriculture, epidemiology, geology, image analysis and other areas of environmental science.  Specified by a set of conditional distributions, these Markov random fields provide a very rich and flexible class of spatial processes, and  their adaptability to  fast statistical calculations, including  those based on Markov chain Monte Carlo computations, makes them very attractive to statisticians. In recent years,  new perspectives have emerged in connecting Gaussian Markov random fields with geostatistical models, and in advancing vast statistical computations.  In this talk, I will briefly discuss the scaling limit of lattice-based Gaussian Markov random fields, namely,  the de Wijs process that originates in the famous work of George Matheron on gold mines in South Africa. I will then explore how this continuum limit connection holds out further possibilities to fit a wide range of new continuum models by using Gaussian Markov random fields. The main focus of the talk will be on various novel matrix-free computations for these models. In particular, for spatial mixed linear models, I will present novel frequentist residual maximum likelihood inference via matrix-free h-likelihood computations. I will draw applications both from areal-unit and point-referenced spatial data. Part of this talk is based on joint work with PhD student Somak Dutta.

Speaker: George Michailidis (U.Michigan, stats) No talk Mon or Tues.

Title: Estimation of Multi-Granger Network Causal Models

Abstract: Network Granger causality focuses on estimating Granger causal effects from p times series and it can be operationalized through Vector Autoregressive Models (VAR). The latter represent a popular class of time series models that has been widely used in applied econometrics and finance and more recently in biomedical applications. In this work, we discuss joint estimation and model selection issues of multiple Granger causal networks. The modeling framework captures two different settings: in the first, different types of measurements (e.g. returns and volatility) are measured over the same set of entities (e.g. financial assets), while in the second the same type of measurement (e.g. gene expression) is measured across the same set of entities (e.g. genes), but over different but related diseases. We introduce appropriate structural penalties that capture the multi-Granger structure, discuss estimation issues and illustrate the results on synthetic and real financial data.

Speaker: Bin Nan (U.Michigan, Biostats)

Title: Large covariance/correlation matrix estimation for temporal data.

Abstract: We consider the estimation of high-dimensional covariance and correlation matrices under slow-decaying temporal dependence. For generalized thresholding estimators, convergence rates are obtained and properties of sparsistency and sign-consistency are established. The impact of temporal dependence on convergence rates is also investigated. An intuitive cross-validation method is proposed for the thresholding parameter selection, which shows good performance in simulations. Convergence rates are also obtained for banding method if the covaraince or correlation matrix is bandable. The considered temporal dependence has longer memory than those in the current literature and has particular implications in analyzing resting-state fMRI data for brain connectivity studies.

Speaker: Yingli Qin (U.Waterloo, stats)

Title: Testing the order of a population spectral distribution for high-dimensional data

Abstract: Large covariance matrices play a fundamental role in high-dimensional statistics. Investigating the behavior of their eigenvalues can reveal informative structures of large covariance matrices. In this paper, we propose to test the number of distinct population eigenvalues, i.e. the order of the Population Spectral Distribution (PSD). The proposed statistic is based upon a series of bias-reduced estimators of PSD moments. We develop the limiting distributions of our test statistic and the moment estimators. We also prove the (n; p) strong consistency of these estimators, which are clearly demonstrated in our simulation study.

Speaker: Annie Qu (UIUC, stats)

Title: Weak Signal Identification and Inference in Penalized Model Selection

Abstract: Weak signal identification and inference are very important in the area of penalized model selection, yet they are under-developed and not well-studied.  Existing inference procedures for  penalized estimators are mainly focused on strong signals. In this paper, we propose an identification procedure for weak signals in finite samples,  and provide  a transition phase in-between noise and strong signal strengths. We also introduce a  new two-step inferential method  to construct  better confidence intervals  for  the identified weak signals. Both theory and numerical studies indicate that the proposed method  leads to better confidence coverage  for weak signals, compared with those using asymptotic inference. In addition, the proposed  method  outperforms the  perturbation  and bootstrap resampling approaches. We  illustrate our method  for  HIV antiretroviral drug susceptibility data to identify genetic mutations associated with HIV drug resistance. This is joint work with Peibei Shi.

Speaker: L. Leticia Ramírez Ramírez (ITAM and CIMAT, Mexico)

Title: Nonparametric inference on random networks

Abstract: We propose a new nonparametric patchwork'' resampling approach on network inference. This procedure is  based on the adaptation of blocking'' argument, developed for bootstrapping of time series and re-tiling for spatial data, to random networks. We discuss how this procedure can be used to quantify estimation uncertainty for network statistics that are functions of degree distribution. We present the developed computationally efficient and data-driven cross-validation algorithm for selecting an optimal patch'' size. We illustrate its implementation for inference on simulated random networks, flight and Wikipedia networks.

Name: Matias Salibian-Barrera (UBC, stats)

Title: Outlier Detection for Functional Data Using Principal Components

Abstract: Principal components analysis is a widely used technique that provides an optimal lower-dimensional approximation to multivariate observations in mean square error retaining as much information as possible. In the functional case, a new characterization of elliptical distributions on separable Hilbert spaces allows us to obtain an equivalent stochastic optimality property for the principal component subspaces of random elements on separable Hilbert spaces. This property holds even when second moments do not exist. Furthermore, these lower-dimensional approximations can be very useful in identifying potential outliers among high-dimensional or functional observations. In this talk, we discuss the problem of estimating these finite-dimensional approximating linear subspaces robustly. The new class of robust estimators for principal directions is consistent for elliptical random vectors, and Fisher-consistent for elliptically distributed random elements on arbitrary Hilbert spaces. We illustrate our method on two real functional data sets, where the robust estimator is able to discover atypical observations in the data that would have been missed otherwise. Through a simulation study, we also study its performance when used to detect outlying observations.

Speaker: Ali Shojaie (U. Washington, Seattle, Biostats)

Title: A Significance Test for Graph-Constraint Estimation

Abstract: External information on associations between covariates could be helpful in making more accurate statistical estimations. Such information is usually presented using graphs. Those graph-guided estimation methods may also be able to perform automatic variable selection. However, there is no available inference method that calculates p-values for graph-guided estimates. In this paper, we present a testing framework called the Grace test. Grace test could produce coefficient estimates and corresponding p-values with incorporated external graphical information. We show in theory that our method asymptotically controls the type-I error rate regardless of the choice of the graph. However, the power of the test would benefit from using an informative graph. We further propose a more general Grace-ridge test that delivers a higher power than the Grace test when the choice of the graph is not informative. We show via simulations that with a reasonably informative graph, we achieve a substantial gain in power, compared to several existing inference methods that ignore the external graphical information.

Speaker:  Peter X.K. Song (U.Michigan, Biostats)

Title: FLAPO: Fused Lasso with the Adaptation of Parameter Ordering – An approach to parameter fusion in merging multiple studies with repeated measurements

Abstract: Combining multiple studies is frequently undertaken in biomedical research to increase sample sizes for statistical power improvement. We consider the marginal model for the regression analysis of repeated measurements collected in several similar studies with potentially different variances and correlation structures. It is of great importance to examine whether there exist common parameters across study-specific marginal models so that simpler models, sensible interpretations and meaningful efficiency gain can be obtained. Merging multiple studies using the classical means of hypothesis testing involves a large number of simultaneous tests for all possible subsets of common regression parameters, in which it results in unduly large degrees of freedom and low statistical power to assess covariate effects. We develop a new fused lasso method, using estimated parameter ordering, to scrutinize only adjacent-pair parameter differences, leading to a substantial reduction for the number of needed comparisons. Our method enjoys the oracle property as does the full fused lasso that involves all pairwise parameter differences. We show that the proposed procedure has smaller error bounds and better finite sample performance than the full fused lasso. We illustrate our method through extensive simulation studies as well as an analysis of merging HIV surveillance cohorts collected over five geographic regions in China, in which the presence or absence of common covariate effects are reflective to relative effectiveness of regional policies on HIV control and prevention. Work joint with Fei Wang and Lu Wang.

Title: Bayesian and empirical Bayesian forests

Abstract: We derive ensembles of decision trees through a nonparametric Bayesian model, allowing us to view random forests as samples from a posterior distribution. This insight motivates a class of Bayesian forest (BF) algorithms that yield small gains in performance and large gains in interpretability. Based on the BF framework, we are able to show that high-level tree hierarchy is stable in large samples. This leads to an empirical Bayesian forest (EBF) algorithm for building approximate BFs on distributed massive datasets and we show that EBFs outperform sub-sampling based alternatives by a large margin.

Speaker: Anand N.Vidyashankar (George Mason U, Statistics)

Title: Big Data in Healthcare: Privacy vs Statistical Efficiency

Abstract: Over the past few decades, healthcare has experienced a proliferation of databases associated with a massive collection of medical information.  Combinations of such information tend to lead to privacy and security risk. On the other hand, statistical analyses of such combined information is necessary to develop products and services for patient benefits.  Thus a natural question is: is it possible to provide both privacy and statistical guarantees when working with big data. In this presentation, we provide new metrics to measure privacy and security risk and develop data analyses techniques accounting for privacy risk.  In the process, we illustrate a trade-off between statistical efficiency and privacy.  Using this, we develop an algorithm for decision making  within a healthcare organization.  Our results showcase how healthcare organizations can leverage modern statistical techniques to simultaneously address the twin challenges of creating business value and  addressing regulatory guidelines that are emanating from laws like HIPAA/HITECH.

Speaker: Sijian Wang (U.Wisconson, Madison, Biostats)

Title: Integrative analysis of high-dimensional genomic data

Abstract: Two types of integrations of multiple genomic datasets are considered in this talk.  The first type of integration is based on the datasets from multiple studies.  For example, an increasing amount of gene expression datasets for similar biomedical problems is available through public repositories. Integrating data from different but independent studies may facilitate discovery of new biological insights. We propose a meta-lasso method for gene selection with multiple expression data. Through a hierarchical decomposition on regression coefficients, our method not only borrows strength across multiple datasets to boost the power to identify important genes, but also keeps the selection flexibility among datasets to take into account data heterogeneity.  The second type of integration is based on the datasets of multiple types. Recently, many genome-wide datasets capturing somatic mutation patterns, DNA copy number alterations, DNA methylation changes and gene expression are simultaneously obtained in the same biological samples. These samples render an integrated data resolution that may not be available with any single data type. We propose an iCluster+ method for pattern discovery (clustering) by integrating diverse data types. The core idea is motivated by the hypothesis that diverse molecular phenotypes can be predicted by a set of orthogonal latent variables that represent distinct molecular drivers, and thus can reveal tumor subgroups of biological and clinical importance.

Speaker: Matthew Wright (IMA, Minnesota)

Title: Euler Characteristic and Data Analysis

Abstract: Euler characteristic is a simple topological invariant that is useful for data analysis. In particular, Euler characteristic extends to a topological integration theory for real-valued functions. I will give an introduction to this theory of Euler integration and demonstrate intriguing applications to topological enumeration problems. The observation that Euler characteristic is one of the intrinsic volumes (which appear in the classic Hadwiger Theorem) prompts generalizations of this integration theory, which I will mention. I will also describe recent work applying Euler characteristic to data analysis, especially in the areas of sensor networks, image processing, and object recognition.

Speaker: Matthew Wright (IMA, Minnesota)

Title: Computing Multidimensional Persistent Homology

Abstract: Multidimensional persistent homology is highly relevant in various applications in which data is simultaneously filtered by two or more parameters. However, the algebraic complexity of multidimensional persistence modules makes it difficult to extract useful invariants in this setting. In this talk, I will describe recent work with Mike Lesnick to efficiently compute and visualize both the multigraded Betti numbers and the rank invariant of multidimensional persistence modules. In addition, I will demonstrate a new software program for computing and visualizing these invariants. This talk will follow Mike Lesnick's talk from Tuesday, focusing on computational, rather than theoretical, aspects of our work.

Speaker: Ruben H. Zamar  (UBC, Statisitcs)

Title: Robust and sparse hierarchical clustering

Abstract:  We consider the problem of hierarchical clustering when the dataset has a large number of variables and a relatively small number of observations. Non-informative noise variables and outlying observations are likely to occur in this context. The inclusion of non-informative noise variables may impede the finding of the underlying clusters. Sparse hierarchical clustering (SHC), which uses a subset of adaptively chosen informative variables, outperforms classical methods based on all the variables. However, in the presence of outliers, both classical methods and SHC yield poor results. We propose two approaches for robust sparse hierarchical clustering (RSHC) to deal with both noise variables and outliers. The proposed RSHC methods are based on Tau-estimator and Lasso-type penalty. The first approach is a direct robustification of SHC. The second approach introduces a fast robust version that scales better with the number of observations. Experiments with both simulated and real datasets show that the RSHC approaches work well on clean data and outperform the non-robust alternatives when the dataset contains outliers.

Main conference webpage.

Note to speakers: please send me your actual titles when you can.  Above is my current best guess for your title based on our discussions.


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# How do you solve x^2= 8x - 15 by completing the square?

Jul 11, 2017

Rearranging first

#### Explanation:

${x}^{2} - 8 x + 15 = 0$

${\left(x - 4\right)}^{2} - {4}^{2} + 15 = 0$

${\left(x - 4\right)}^{2} - 1 = 0$

${\left(x - 4\right)}^{2} = 1$

$\left(x - 4\right) = \pm 1$

$x = 4 \pm 1$

This gives two values of x, one where you add the 1 and one where you subtract 1

Jul 12, 2017

$x = 3 \mathmr{and} x = 5$

#### Explanation:

${x}^{2} = 8 x - 15$

${x}^{2} - 8 x = - 15 \text{ } \leftarrow$ leave the constant on the right side

Make a perfect square by adding ${\left(\frac{b}{2}\right)}^{2}$ to both sides
$b = - 8 \text{ } \rightarrow {\left(\frac{- 8}{2}\right)}^{2} = 16$

${x}^{2} - 8 x + 16 = - 15 + 16$

${\left(x - 4\right)}^{2} = 1$

$x - 4 = \pm 1$

$x - 4 = 1 \text{ } \rightarrow x = 5$

$x - 4 = - 1 \text{ } \rightarrow x = 3$

format: plain


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## 305 – Homework I

January 23, 2012

This homework set is due Wednesday, February 1st, at the beginning of lecture, but feel free to turn it in earlier if possible.

## David Hilbert

January 23, 2012

Today is the 150th anniversary of Hilbert’s birthday.

Hilbert is one of the mathematicians I admire the most. I believe I first learned about him while in High School, through “Higher Geometry” by N.V. Efimov. My copy of Reid’s “Hilbert” is one of the first books I bought when I arrived in the States (in 1997).

Here is a link to the reprint in the Bulletin of the AMS of Mathematical Problems.

I would love to have a poster sized copy of the picture above. Springer sold them years ago, but now they seem impossible to find.

## 515 – The Dehn-Sydler theorem

January 23, 2012

As mentioned in lecture, Hilbert’s third problem was an attempt to understand whether the Bolyai-Gerwien theorem could generalize to ${\mathbb R}^3:$

## 305 – Solving cubic and quartic equations

January 23, 2012

Ars Magna, “The Great Art”, by Gerolamo Cardano.

## 414/514 – The Schoenberg functions

January 23, 2012

Here is Jeremy Ryder’s project from last term, on the Schoenberg functions. Here we have a space-filling continuous map $f:x\mapsto(\phi_s(x),\psi_s(x))$ whose coordinate functions $\phi_s$ and $\psi_s$ are nowhere differentiable.

The proof that $\phi_s,\psi_s$ are continuous uses the usual strategy, as the functions are given by a series to which Weierstrass $M$-test applies.

The proof that $f$ is space filling is nice and short. The original argument can be downloaded here. A nice graph of the first few stages of the infinite fractal-like process that leads to the graph of $f$ can be seen in page 49 of Thim’s master thesis.


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# How do you find the exponential model y=ae^(bx) that goes through the points (1, 1.2) and (4, 0.0768)?

Jul 19, 2016

$y = 3 {\left(0.4\right)}^{x}$

#### Explanation:

The Exponential curve $y = a {e}^{b x}$ passes thro. pts. $\left(1 , 1.2\right)$ and

$\left(4 , 0.0768\right)$

Accordingly, the respective co-ords. must satisfy the eqn. of the curve.

$\therefore 1.2 = a {e}^{b} , \mathmr{and} , 0.0768 = a {e}^{4 b} \Rightarrow \frac{0.0768}{1.2} = \frac{a {e}^{4 b}}{a {e}^{b}}$

$\Rightarrow 0.064 = {e}^{3 b} = {\left({e}^{b}\right)}^{3} \Rightarrow {0.064}^{\frac{1}{3}} = {\left\{{\left({e}^{b}\right)}^{3}\right\}}^{\frac{1}{3}}$

$\Rightarrow 0.4 = {e}^{b}$. Then, from $1.2 = a {e}^{b}$, we have, $1.2 = 0.4 a$

$\Rightarrow a = 3$

Hence, the reqd. eqn.$: y = a {e}^{b x} = a {\left({e}^{b}\right)}^{x}$,i.e., $y = 3 {\left(0.4\right)}^{x}$


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# This seems strange

Yesterday, after physics, I looked into my history book and found this strange note:

IMPORTANT SECRET MESSAGE
Dtzw Htruzyjw mfx gjjs jshwduyji. Knsi tzy ymj htwwjhy pjd. F qnyyqj mnsy: nrlzw ity htr xqfxm LRVfn

After I solved the riddle, I felt a bit angry because the message was promoting a website but it was a StackExchange website so everything was ok for me

Can you Find oUt What the name oF the stAckexchange website is? (Tip: you might need imgur)

• Wait... why would you look into your history book after physic? That seems suspicious.... – North Læraðr Apr 9 '18 at 18:44
• @North i am a student and they are school books. And hints – MEE Apr 9 '18 at 19:49
• in the first image rot13()Gur gvzr qvivqrq ol gur qbyynef tvirf gur ebg xrl jvgu n yvggyr punatr. – Alan Hoover Apr 9 '18 at 22:59
• @AlanHoover yes, but the only way to get there is to have already solved that part :S – WELZ Apr 10 '18 at 0:59

and

using ROT12

you get :

THECORRECTSOLUTIONISQUANTUMCOMPUTINGSTACKEXCHANGE

THE CORRECT SOLUTION IS QUANTUM COMPUTING STACKEXCHANGE

This is my first answer. Sorry if any wrong format, bad english or other mistakes.

• My intended solution was: 1. find out that there is an imgur with the capital leters from the last sentence 2. go to my profile page, click on the Get it in a blue moon (=impossible) question, divide the qid by six, the modulo 26 is 5, 26 - 5=21. This is the intended rot. 3. decrypt the message and go to the page. Take the time (20:56:22) and concat them, modulo 26 gives 14, the 26-14=12, the rot required to decrypt the final message. – MEE Apr 10 '18 at 17:18
• Sorry, but i didn't thought as far as go to your profile. I've tried all method I know with the time (20:56:22) and ended with 14. With 9781 too. And the missing "d" at address look curious. Thanks, I'll be more careful next time. – nathoenk Apr 11 '18 at 2:08

If you take the:

Capital letters from your last sentence (minus the first one C): FUWFA which can be converted to an imgur link by messing with the case http://i.stack.imgur.com/fUwFa.png
Which brings up:

Which brings up:

HvsQcffsqhGczihwcbWgEiobhiaQcadihwbuGhoqySlqvobus

And if you:

Take the Capital letters from the address you get :HQGWEQGS

• I am sorry but this is not the right direction. Maybe there is some other hint in the image allowing you to decrypt the adress. And you also don't explain correctly why you can decrypt it via rot21. – MEE Apr 9 '18 at 17:41
• Ummm why do we exactly have to answer why? Any decipherer can crack that even without a computer by simply brute-forcing it. Also the fact that F stands by itself makes it obvious that F=A or F=I. Using F=A, you can simply decode it very easily. Seems to be an unnecessary step to me to try and explain Why a rot21 works (maybe because it decodes it?) – North Læraðr Apr 9 '18 at 18:30
• @North but there is an hint inside the question and your solution is not complete if you can't explain why it is correct. Hint: the capitalized characters in the last sentence (except C) give you a new imgur link that will allow you to find out what you need to do (by the way history is a hint too) – MEE Apr 9 '18 at 18:41
• @North you got the right picture. What has something to do with SE and history and impossible... – MEE Apr 9 '18 at 19:51

By decrypting it using rot21 you get:

"Your Computer has been encrypted. Find out the correct key. A little hint: imgur dot com slash GMQai."

This was easily deciphered by finding out that f=a, working backwards from there. Find the opposite of rot6, decrypts it.
Thanks to @Welz, the second imgur picture gives you the rot # it was encrypted to --> "You'll only get sixth... if F=A, that uses a rot6 cipher.

• yes, but it does not contain the hint the last sentence gives and you have not explained why rot21 and what the answer to my question is. – MEE Apr 9 '18 at 17:27
• Fudge I can't go to the imgur website, its blocked for me. – North Læraðr Apr 9 '18 at 17:28
• this still does not explain why this decryption is correct in the context of the riddle (tip: look at the last sentence) – MEE Apr 9 '18 at 17:33
• Updated the answer @MEE – North Læraðr Apr 9 '18 at 20:27
• oh There is an shortcut. The sixth on the picture was originally something different 😀 – MEE Apr 10 '18 at 5:23


text:
# excenter of a triangle

Cite. What's the word for changing your mind and not doing what you said you would? Let be the circumradius and the exradius. Here are the 4 most popular ones: Centroid, Circumcenter, Incenter and Orthocenter. How would I bias my binary classifier to prefer false positive errors over false negatives? And in the last video, we started to explore some of the properties of points that are on angle bisectors. If we think the external angle bisector as a line instead of a ray it can exist till three intersection points. The excenter is the center of the excircle. it.wikipedia.org/wiki/Ex_falso_sequitur_quodlibet. Here $I$ is the excenter which is formed by the intersection of internal angle bisector of $A$ and external angle bisectors of $B$ and $C$. Show that L is the center of a circle through I, I. The point of concurrency of these angle bisectors is known as the triangle’s excenter. An excircle is a circle outside the triangle that is tangent to the three sides of the triangle. Let a be the length of BC, b the length of AC, and c the length of AB. Excenters of a Triangle An excenter of a triangle is a point at which the line bisecting one interior angle meets the bisectors of the two exterior angles on the opposite side. Triangle inscribed in a circle where: a, b, and c are the sides of the triangle r is the radius of the circle 10. Now, the incircle is tangent to AB at some point C′, and so $\angle AC'I$is right. An excenter, denoted , is the center of an excircle of a triangle. @User9523: computing the angles is one way to prove/disprove they are similar. Where all three lines intersect is the center of a triangle's "circumcircle", called the "circumcenter": Try this: drag the points above until you get a right triangle (just by eye is OK). Consider $\triangle ABC$, $AD$ is the angle bisector of $A$, so using angle bisector theorem we get that $P$ divides side $BC$ in the ratio $|AB|:|AC|$, where $|AB|,|AC|$ are lengths of the corresponding sides. Here is the Incenter of a Triangle Formula to calculate the co-ordinates of the incenter of a triangle using the coordinates of the triangle's vertices. Problems Introductory How does pressure travel through the cochlea exactly? Given triangle ABC with side lengths a, b, and c. Let circle O be an excircle and let … It is also the center of the circumscribing circle (circumcircle). If the distance between incenter and one of the excenter of an equilateral triangle is 4 units, then find the inradius of the triangle. Is there any means of transportation available to tourists that goes faster than Mach 3.5? Improve this answer. Let’s observe the same in the applet below. Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. The distance from the "incenter" point to the sides of the triangle are always equal. This gives $$D=\frac{aA+bB-cC}{a+b-c}\tag{2}$$ Share. The center of the incircle The longest edge of a right triangle, which is the edge opposite the right angle, is called the hypotenuse. From Wikimedia Commons, the free media repository. It is possible to find the incenter of a triangle using a compass and straightedge. Did Gaiman and Pratchett troll an interviewer who thought they were religious fanatics? It is also known as an escribed circle. There are three excenters for a given triangle, denoted J_1, J_2, J_3. You find a triangle’s incenter at the intersection of the triangle’s three angle bisectors. There are actually thousands of centers! An excircle is a circle tangent to the extensions of two sides and the third side. Thanks for contributing an answer to Mathematics Stack Exchange! A right triangle is a triangle in which one of the angles is 90°, and is denoted by two line segments forming a square at the vertex constituting the right angle. Protection against an aboleth's enslave ability. All content on this website, including dictionary, thesaurus, literature, geography, and other reference data is for informational purposes only. Hence there are three excentres I1, I2 and I3 opposite to three vertices of a triangle. Adjust the triangle above by dragging any vertex and see that it will never go outside the triangle: Finding the incenter of a triangle. I have triangle ABC here. A, B, C. A B C I L I. A triangle (black) with incircle (blue), incenter (I), excircles (orange), excenters (J A,J B,J C), internal angle bisectors (red) and external angle bisectors (green) In geometry, the incircle or inscribed circle of a triangle is the largest circle contained in the triangle; it touches (is tangent to) the three sides. How can I disable OneNote from starting automatically? The formula first requires you calculate the three side lengths of the triangle. Now using the above facts we get the point $P$ as $P(\frac{|AB|x_3+|AC|x_2}{b+c},\frac{|AB|y_3+|AC|y_2}{b+c})$. The radius of excircle is called the exradius. In a $\Delta ABC$ with incenter $I$, prove that the circumcenter of $\Delta AIB$ lies on $BI$, In a triangle $\Delta ABC$, let $X,Y$ be the foot of perpendiculars drawn from $A$ to the internal angle bisectors of $B$ and $C$, Find the ratio of the lengths of the bisectors of internal angles of $B$ and $C$, What is the angle of $\angle BPC$ in $\triangle BPC$, Need advice or assistance for son who is in prison. Thus the radius C'Iis an altitude of $\triangle IAB$. Disclaimer. File:Triangle excenter proof.svg. These three angle bisectors are always concurrent and always meet in the triangle's interior (unlike the orthocenter which may or may not intersect in the interior). $$I_A = \frac{-aA+bB+cC}{-a+b+c}=\frac{-|BC|(x_1,y_1)+|AC|(x_2,y_2)+|AB|(x_3,y_3)}{-|BC|+|AC|+|AB|}.$$ Follow answered Jan 9 '15 at 11:31. robjohn ♦ robjohn. Triangle, Circles, Circumcircle, Sagitta, Incircle, Excircle, Inradius, Exradius, Metric Relations. These three angle bisectors are always concurrent and always meet in the triangle's interior (unlike the orthocenter which may or may not intersect in the interior). An exradius is a radius of an excircle of a triangle. How to tell if a song is tuned a half-step down? The circumcircle of the extouch triangle XAXBXC is called th… The EXCENTER is the center of a circle that is tangent to the three lines exended along the sides of the triangle. The area of a triangle determined by the bisectors. Given a triangle , the points , , and lie on a line, where is the incenter and is the excenter corresponding to . An excenter, denoted , is the center of an excircle of a triangle. The Gergonne triangle (of ABC) is defined by the 3 touchpoints of the incircle on the 3 sides.The touchpoint opposite A is denoted T A, etc. Furthermore, the circle with as the diameter has as its center, where is the intersection of with the circumcircle of , and passes through and . 2) The -excenter lies on the angle bisector of . An excenter is a point on the outside of a triangle that connects the intersections of the angle bisectors. Asking for help, clarification, or responding to other answers. Incircles and Excircles in a Triangle. Excenter. The incenter I and excenters J_i of a triangle are an orthocentric system. Press the play button to start. Each of these classical centers has the property that it is … An excenter is the center of an excircle.An excircle is one of three circles that touches a triangle - one for each side. How can I motivate the teaching assistants to grade more strictly? Related Formulas. The Nagel triangle of ABC is denoted by the vertices XA, XB and XC that are the three points where the excircles touch the reference triangle ABC and where XA is opposite of A, etc. The centroid is the triangle’s center of gravity, where the triangle balances evenly. There are in all three excentres of a triangle. In the case of the right triangle, circumcenter is at the midpoint of the hypotenuse. When choosing a cat, how to determine temperament and personality and decide on a good fit? On the worksheet below, you can move the pink points A, B, and C, to see how the excenters and excircles change depending on the movement of the points. The incenter and excenters of a triangle are an orthocentric system. The coordinates of the centroid are also two-thirds of the way from each vertex along that segment. File; File history; File usage on Commons; File usage on other wikis; Metadata; Size of this PNG preview of this SVG file: 400 × 350 pixels. This is just angle chasing. AREA OF A TRIANGLE 6. The radii of the incircles and excircles are closely related to the area of the triangle. The coordinates of the incenter are the weighted average of the coordinates of the vertices, where the weights are the lengths of the corresponding sides. This geometry video tutorial explains how to identify the location of the incenter, circumcenter, orthocenter and centroid of a triangle. By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy. Here $I$ is the excenter which is formed by the intersection of internal angle bisector of $A$ and external angle bisectors of $B$ and $C$. Had we drawn the internal angle bisector of B and the external ones for A and C, we would’ve got a different excentre. The touchpoint opposite A is denoted T A, etc. Let’s observe the same in the applet below. ExCenter point at center of the circle exscribed opposite 1st point in the 3 points' triangle constructors: ExCenter (point1, point2, point3 ,EXCENTER ) ExCenter (triangle ,EXCENTER ) Had we drawn the internal angle bisector of B and the external ones for A and C, we would’ve got a different excentre. Let ABC be a triangle with incenter I, A-excenter I. Definition. And let me draw an angle bisector. . The distance from the "incenter" point to the sides of the triangle are always equal. A place for students to explore mathematics. he points of tangency of the incircle of triangle ABC with sides a, b, c, and semiperimeter p = (a + b + c)/2, define the cevians that meet at the Gergonne point of the triangle The EXCENTER is the center of a circle that is tangent to the three lines exended along the sides of the triangle. Therefore $\triangle IAB$ has base length c and height r, and so has ar… No other point has this quality. An excenter is the center of an excircle of a triangle. Use GSP do construct a triangle, its incircle, and its three excircles. There are three excircles and three excenters. Is there a book about the history of linear programming? آبادیس - معنی کلمه excenter of a triangle. You find a triangle’s incenter at the intersection of the triangle’s three angle bisectors. @User9523: what's the issue in computing the angles of $BIP$ and $BIA$? Developer keeps underestimating tasks time. These results are vital to most excenter problems. It follows that in general $BAI_A$ and $BI_A P$ are not similar. It only takes a minute to sign up. where is the circumcenter, are the excenters, and is the circumradius (Johnson 1929, p. 190). An excenter of a triangle is a point at which the line bisecting one interior angle meets the bisectors of the two exterior angles on the opposite side. If we extend two of the sides of the triangle, we can get a similar configuration. (A1,B2,C 3). OI^_^2+OJ_1^_^2+OJ_2^_^2+OJ_3^_^2=12R^2, where O is the circumcenter, J_i are the excenters, and R is the circumradius (Johnson 1929, p. 190). We can have three hyperbolic excenters for a fixed triangle. Abstract. Two angles of $BAI_A$ are $\frac{A}{2},\frac{\pi+B}{2}$. It is also known as an escribed circle. PERIMETER OF A TRIANGLE The Perimeter, P, of a triangle is the sum of the lengths of its three sides P = a + b + c where: a, b and c are the lengths of the sides of the given triangle 5. Every triangle has 3 excircles and excenters. If the circle is tangent to side of the triangle, the radius is , where is the triangle's area, and is the semiperimeter. Proof. Properties of the Excenter. Recall that the incenter of a triangle is the point where the triangle's three angle bisectors intersect. Circles and meet at other than The circumcle of triangle meet line again at other than Prove that lies on the excircle of triangle opposite . The three angle bisectors in a triangle are always concurrent. Let A = (x1, y1), B = (x2, y2) and C = (x3, y3) are the vertices of a triangle ABC, c, a and b are the lengths of the sides AB, BC and AC respectively. If A (x1, y1), B (x2, y2) and C (x3, y3) are the vertices of a triangle ABC, Coordinates of … Does Kasardevi, India, have an enormous geomagnetic field because of the Van Allen Belt? Other resolutions: 274 × 240 pixels | 549 × 480 pixels | 686 × 600 pixels | 878 × 768 pixels | 1,170 × 1,024 pixels. Triangle circumscribing a circle where: r is the radius of the circle and 11. ExCenter point at center of the circle exscribed opposite 1st point in the 3 points' triangle constructors: ExCenter (point1, point2, point3 ,EXCENTER ) ExCenter (triangle ,EXCENTER ) It lies on the angle bisector of the angle opposite to it in the triangle. Search. Where is the center of a triangle? The point of concurrency of these angle bisectors is known as the triangle’s excenter. An excenter is the center of the excircle. A. For example the centroid, circumcenter, incenter and orthocenter were familiar to the ancient Greeks, and can be obtained by simple constructions. Draw a line (called a "perpendicular bisector") at right angles to the midpoint of each side. See the derivation of formula for radius of incircle.. Circumcenter Circumcenter is the point of intersection of perpendicular bisectors of the triangle. By Mary Jane Sterling . Triangle Centers. The three angle bisectors in a triangle are always concurrent. This is readily seen to be a triangle center function and (provided the triangle is scalene) the corresponding triangle center is the excenter opposite to the largest vertex angle. And in the last video, we started to explore some of the properties of points that are on angle bisectors. I have triangle ABC here. This Gergonne triangle T A T B T C is also known as the contact triangle or intouch triangle of ABC.. If you draw lines from each corner (or vertex) of a triangle to the midpoint of the opposite sides, then those three lines meet at a center, or centroid, of the triangle. This location gives the incenter an interesting property: The incenter is equally far away from the triangle’s three sides. Then coordinates of center of ex-circle opposite to vertex A are given as I1(x, y) = (– ax1 + bx2 + cx3 – a + b + c, – ay1 + by2 + cy3 – a + b + c). آبادیس از سال 1385 فعالیت خود را در زمینه فن آوری اطلاعات آغاز کرد. The other two excenters can be picked out by similar functions. It is also the center of the triangle's incircle. To learn more, see our tips on writing great answers. Always inside the triangle: The triangle's incenter is always inside the triangle. The excenters and excircles of a triangle seem to have such a beautiful relationship with the triangle itself. And now, what I want to do in this video is just see what happens when we apply some of those ideas to triangles or the angles in triangles. This Gergonne triangle T A T B T C is also known as the contact triangle or intouch triangle of ABC. To subscribe to this RSS feed, copy and paste this URL into your RSS reader. Denote the midpoints of the original triangle … For each of those, the "center" is where special lines cross, so it all depends on those lines! Can we get rid of all illnesses by a year of Total Extreme Quarantine? Note that these notations cycle for all three ways to extend two sides (A1, B2, C3). rev 2021.1.21.38376, The best answers are voted up and rise to the top, Mathematics Stack Exchange works best with JavaScript enabled, Start here for a quick overview of the site, Detailed answers to any questions you might have, Discuss the workings and policies of this site, Learn more about Stack Overflow the company, Learn more about hiring developers or posting ads with us. Then . Incenter-Excenter Circle. An excircle is a circle tangent to the extensions of two sides and the third side. Let be a triangle with circumcircle Point lies on side such that Let denote the excenter of triangle opposite and let denote the circle with as its diameter. Triangle, Circles, Circumcircle, Sagitta, Incircle, Excircle, Inradius, Exradius, Metric Relations. For every triangle there are 3 excircles and 3 excenters. The incenter is the center of the incircle. ChemDraw: how to change the default aromatic ring style for drawing from SMILES. ALTITUDE OF A TRIANGLE ALTITTUDE of a triangle is a line segment drawn from a vertex perpendicular to the opposite side ORTHOCENTER is the point of intersection of the altitudes … Thanks for your response, but I am not really aware of that 'barycentric' stuff.. Let ABC be a triangle with circumcenter O and let E be the excenter of the excircle opposite A. Many centers of the triangle are solutions to a variety of extremal problems. The trilinear coordinates of the incenter are $[1;1;1]$ and the trilinear coordinates of the $A$-excenter are $[-1;1;1]$, hence the barycentric coordinates of the $A$-excenter $I_A$ are $[-a;b;c]$ and Adjust the triangle above by dragging any vertex and see that it will never go outside the triangle : Finding the incenter of a triangle. Every triangle has three excenters and three excircles. A, and denote by L the midpoint of arc BC. This triangle XAXBXC is also known as the extouch triangle of ABC. In an isosceles triangle, all of centroid, orthocentre, incentre and circumcentre lie on the same line. This is the center of a circle, called an excircle which is tangent to one side of the triangle and the extensions of the other two sides. As you can see in the figure above, circumcenter can be inside or outside the triangle. Let's look at each one: Centroid. site design / logo © 2021 Stack Exchange Inc; user contributions licensed under cc by-sa. I am not trying to compute those angles, I am trying to see whether $\triangle BIP$ and $\triangle BIA$ are similar or not ! Just wanted to know are the triangles.$BIP,BIA$ really similar ? It lies on the angle bisector of the angle opposite to it in the triangle. Circumcenter. What are the odds that the Sun hits another star? Knowing these lengths, which repeat often, we can com-pute … So let's bisect this angle right over here-- angle BAC. The center of the incircle is called the triangle's incenter. Excenter of a triangle - formula A point where the bisector of one interior angle and bisectors of two external angle bisectors of the opposite side of the triangle, intersect is called the excenter of the triangle. The altitude of a triangle is a segment from a vertex of the triangle to the opposite side (or to the extension of the opposite side if necessary) that’s perpendicular to the opposite side; the opposite side is called the base. Given a triangle ABC with a point X on the bisector of angle A, we show that the extremal values of BX CX occur at the incenter and the excenter on the opposite side of A. Related Geometrical Objects. Consider $\triangle ABC$, $AD$ is the angle bisector of $A$, so using angle bisector theorem we get that $P$ divides side $BC$ in the ratio $|AB|:|AC|$, where $|AB|,|AC|$ are lengths of the corresponding sides. Triangles classified based on their internal angles fall into two categories: right or oblique. This circle has radius Stack Exchange network consists of 176 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. In the following applet , the internal bisector of angle B of triangle ABC and bisectors of exterior angles at A and C meet at E. An excenter is the center of an excircle.An excircle is one of three circles that touches a triangle - one for each side. This is not surprising: in your diagram, too, $BPI$ is acute-angled while $ABI$ is not. Jump to navigation Jump to search. Definition. Now, if we know the ratio in which $P$ divides $AI$ we are done, but I can't think of anything that will help me do it. Hyperbolic Excenter The excenter of a triangle is the intersection point of the three external angle bisectors. The triangle's incenter is always inside the triangle. The incenter is one of the triangle's points of concurrency formed by the intersection of the triangle's 3 angle bisectors.. Excentre of a triangle is the point of concurrency of bisectors of two exterior and third interior angle. Since each of the triangles in $(1)$ has the same altitude, which is the radius of the excircle, their areas are proportional to the lengths of their bases, which are the sides of $\triangle ABC$. Calculate the excenter of a triangle at the specified vertex: Calculate all of the excenters: Calculate the foot of an altitude of a triangle at the specified vertex: Calculate the incenter of a triangle: Calculate the midpoint of a side of a triangle: Calculate the nine-point center of a triangle: of the Incenter of a Triangle. Excenter, Excircle of a triangle - Index 2 : Geometry Problem 942. There are in all three excentres of a triangle. See Incircle of a Triangle. The incenter is the center of the incircle. Hypothetically, why can't we wrap copper wires around car axles and turn them into electromagnets to help charge the batteries? It is also the center of the circumscribing circle (circumcircle). Let A = \BAC, B = \CBA, C = \ACB, and note that A, I, L are collinear (as L is on the angle bisector). Excenter of a triangle - formula A point where the bisector of one interior angle and bisectors of two external angle bisectors of the opposite side of the triangle, intersect is called the excenter of the triangle. In any given triangle, . Orthocentre, centroid and circumcentre are always collinear and centroid divides the line joining orthocentre and circumcentre in the ratio 2:1. Use the calculator above to calculate coordinates of the incenter of the triangle ABC.Enter the x,y coordinates of each vertex, in any order. The center of this excircle is called the excenter relative to the vertex A, or the excenter of A. 1:08 1.2k LIKES Two angles of $BI_A P$ are $\frac{\pi-B}{2}$ and $\frac{A+B}{2}=\frac{\pi-C}{2}$. (Source: Problem 13.2 - MOSP 2007) Here is the Incenter of a Triangle Formula to calculate the co-ordinates of the incenter of a triangle using the coordinates of the triangle's vertices. Excenter, Excircle of a triangle - Index 2 : Geometry Problem 942. In geometry, the incircle or inscribed circle of a triangle is the largest circle contained in the triangle; it touches (is tangent to) the three sides. Excenter, Excircle of a triangle - Index 1 : Triangle Centers.. Distances between Triangle Centers Index.. Gergonne Points Index Triangle Center: Geometry Problem 1483. Saw a proof somewhere which says the same, but I am not really sure, could you comment on that ? I am just trying to solve it using similarity/congruence. EQUIANGULAR TRIANGLE – a triangle with three congruent angles c. OBTUSE – a triangle with one obtuse angles and two acute angles 4. In geometry, a triangle center is a point in the plane that is in some sense a center of a triangle akin to the centers of squares and circles, that is, a point that is in the middle of the figure by some measure. Use MathJax to format equations. MathJax reference. In any given triangle, . The Gergonne triangle (of ABC) is defined by the 3 touchpoints of the incircle on the 3 sides. Circumcenter is the point of intersection of perpendicular bisectors of the triangle. where A t = area of the triangle and s = ½ (a + b + c). Illustration: If (0, 1), (1, 1) and (1, 0) are middle points of the sides of a triangle, find its incentre. This is the center of a circle, called an excircle which is tangent to one side of the triangle and the extensions of the other two sides. Suppose $\triangle ABC$ has an incircle with radius r and center I. There are three excircles and three excenters. Finding the incenter. of the Incenter of a Triangle. And now, what I want to do in this video is just see what happens when we apply some of those ideas to triangles or the angles in triangles. Press the play button to start. Use GSP do construct a triangle, its incircle, and its three excircles. Incenters, like centroids, are always inside their triangles.The above figure shows two triangles with their incenters and inscribed circles, or incircles (circles drawn inside the triangles so the circles barely touc… The incenter is the center of the triangle's incircle, the largest circle that will fit inside the triangle and touch all three sides. His interest is scattering theory. Properties of the Excenter. The incenter is one of the triangle's points of concurrency formed by the intersection of the triangle's 3 angle bisectors.. I thought of somehow proving $\triangle BIP$ and $\triangle BIA$ to be similar, to get something, but that isn't the case. Making statements based on opinion; back them up with references or personal experience. Think the external angle bisectors B, c. a B C I L.... … Abstract at some point C′, and other reference data is for informational only. } { a+b-c } \tag { 2 }  Share cookie policy so all! Your RSS reader changing your mind and not doing what you said you would,... Classical centers has the property that it is also the center of an excircle... And denote by L the midpoint of each side I bias my binary classifier to prefer false positive errors false... Temperament and personality and decide on a line ( called a  perpendicular bisector '' ) at angles! By the intersection of perpendicular bisectors of the triangle 's incenter agree to our terms service. Bai_A $and$ BI_A P $are not similar lengths, which is the center the! Gergonne triangle T a T B T C is also the center of this excircle is a point the.: in your diagram, too,$ BPI $is not surprising: in your diagram, too$! That connects the intersections of the excircle opposite a 2021 Stack Exchange is a question and answer site people! Lengths, which repeat often, we started to explore some of excenter of a triangle triangle it follows in... Inradius, Exradius, Metric Relations the figure above, circumcenter, orthocenter and centroid of a triangle one... There are in all three excentres of a triangle intersection of perpendicular bisectors the! You said you would also the center of the incircle is tangent to three! Thought they were religious fanatics circumcircle ) in an isosceles triangle, denoted J_1, J_2, J_3,. Field because of the incircle is tangent to the vertex a, B, and denote by L the of. A circle tangent to the extensions of two sides ( A1, B2, )... $BI_A P$ are not similar angle right over here -- angle BAC with incenter I A-excenter! Compass and straightedge an answer to mathematics Stack Exchange Inc ; user contributions licensed under cc.... Two-Thirds of the circumscribing circle ( circumcircle ) excenter, denoted, is the radius of incircle circumcenter! Field because of the angle bisector of the three side lengths a, and so $\angle AC I.: the triangle ’ s center of a triangle using a compass and straightedge and 11 the are! From each vertex along that segment point on the angle opposite to it in the ratio.. The third side let ’ s three angle bisectors$ $D=\frac { aA+bB-cC } { }. Given triangle, we can com-pute … Definition also known as the contact triangle or intouch of... Of gravity, where the triangle excenter, denoted, is called the triangle are always equal BC,,! Triangles classified based on their internal angles fall into two categories: right or.! By L the midpoint of arc BC and its three excircles ( of ABC BIA. At 11:31. robjohn ♦ robjohn XAXBXC is also known as the contact triangle or intouch of. Orthocentre, incentre and circumcentre in the ratio 2:1 location of the.. Similar configuration circumcentre lie on the angle bisector as a line instead of a triangle are an orthocentric system –. Feed, copy and paste this URL into your RSS reader circle:! Response, but I am not really aware of that 'barycentric ' stuff grade more strictly inside the.. To change the default aromatic ring style for drawing from SMILES follows in! In your diagram, too,$ BPI $is right of linear?. Same line are three excentres I1, I2 and I3 opposite to it in the applet below to find incenter... These classical centers has the property that it is also the center of the three external angle bisectors s the. 'S bisect this angle right over here -- angle BAC relationship with the triangle I2 and I3 opposite to in... I3 opposite to three vertices of a triangle radii of the incenter and orthocenter were to!, where the triangle 's 3 angle bisectors that touches a triangle - Index 2: Geometry Problem 942 of. 'Barycentric ' stuff O be an excircle of a triangle with three congruent c.! The centroid, orthocentre, centroid and circumcentre in the triangle ’ s sides. Transportation available to tourists that goes faster than Mach 3.5 ancient Greeks and! Are not similar the radii of the incircle is tangent to the extensions of sides. Opposite the right angle, is the circumradius ( Johnson 1929, p. 190 ), and! Where special lines cross, so it all depends on those lines issue in computing the angles$! I3 opposite to it in the applet below ways to extend two the! Bai_A $and$ BI_A P $are not similar hyperbolic excenters for a given triangle all... Along the sides of the incenter an interesting property: the triangle are an orthocentric system or personal experience:. Triangle determined by the 3 touchpoints of the triangle ’ s three angle bisectors$... With incenter I and excenters of a triangle statements based on opinion ; back them up with references or experience... Three side lengths of the triangle 's 3 angle bisectors triangle of... At the intersection point of concurrency of these angle bisectors RSS feed, copy and this... Orthocentre, incentre and circumcentre lie on the angle opposite to it in the ratio 2:1 angles two. A cat, excenter of a triangle to tell if a song is tuned a half-step down BAC... S observe the same, but I am just trying to solve it using similarity/congruence similar functions three Circles touches! Math at any level and professionals in related fields XAXBXC is also known as the contact triangle or triangle. Intersection point of concurrency of these classical centers has the property that it is … Incenter-Excenter.. Sides and the third side point to the three side lengths a, B, is... R and center I vertex along that segment two-thirds of the Van Allen Belt where lines... 2 }  Share orthocenter and centroid divides the line joining orthocentre and circumcentre are always equal is! Answered Jan 9 '15 at 11:31. robjohn ♦ robjohn be inside or the... Compass and straightedge triangle are always collinear and centroid of a  center '' is where special lines cross so! 2 }  D=\frac { aA+bB-cC } { a+b-c } \tag 2. The edge opposite the right angle, is the point of intersection of angle. Circumcenter O and let … Abstract you find a triangle ’ s three angle bisectors the. And denote by L the midpoint of arc BC for people studying at... Sides of the triangle: the incenter is equally far away from the  incenter '' point to the of... For your response, excenter of a triangle I am not really aware of that 'barycentric ' stuff circle the! To change the default aromatic ring style for drawing from SMILES C3 ) the longest edge of triangle. Grade more strictly of formula for radius of incircle.. circumcenter excenter of a triangle is at midpoint... What are the 4 most popular ones: centroid, orthocentre excenter of a triangle centroid and circumcentre lie on the line! To our terms of service, privacy policy and cookie policy incircle.. circumcenter circumcenter is the triangle solutions! Abc be a triangle, Circles, circumcircle, Sagitta, incircle, and lie on a fit... Closely related to the three side lengths of the way from each vertex along segment. Third side and other reference data is for informational purposes only three excentres of circle... Obtained by simple constructions that goes faster than Mach 3.5 aA+bB-cC } { a+b-c \tag... And Pratchett troll an interviewer who thought they were religious fanatics and orthocenter lines... Changing your mind and not doing what you said you would circumcentre are always equal by a year of Extreme! Incenter and excenters of a triangle with circumcenter O and let … Abstract { a+b-c } \tag 2! Orthocenter were familiar to the area of the excircle opposite a using a compass and straightedge and. Including dictionary, thesaurus, literature, geography, and excenter of a triangle reference is. L the midpoint of arc BC from each vertex along that segment applet below,... We wrap excenter of a triangle wires around car axles and turn them into electromagnets help... Triangle itself the teaching assistants to grade more strictly be the length of AB AC I... Contributions licensed under cc by-sa has an incircle with radius r and center I instead of a circle to...: r is the center of the incircle on the angle opposite to it the... Centroid, orthocentre, incentre and circumcentre are always collinear and centroid of a where! Instead of a triangle using a compass and straightedge IAB $we get rid of all illnesses by year! All content on this website, including dictionary, thesaurus, literature geography. So$ \angle AC ' I $is acute-angled while$ ABI $is not surprising in... I L I answered Jan 9 '15 at 11:31. robjohn ♦ robjohn touchpoints of the triangle and of! I bias my binary classifier to prefer false positive errors over false negatives balances evenly that in general$ $. Learn more, see our tips on writing great answers radius of incircle circumcenter. Right angle, is called the hypotenuse mathematics Stack Exchange Inc ; user contributions licensed under cc by-sa cc.! The circumcenter, incenter and is the center of the incenter and orthocenter, BIA$ really similar of,..., are the triangles. $BIP, BIA$ and centroid of a triangle opposite a is denoted a!, the incircle on the angle bisectors … Definition BIP, BIA \$ draw a line called...


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# Revision history [back]

### On algebraic number fields in Sage

Assume I have an irreducible monic polynomial $f$ over $\mathbb{Z}$ of degree n and let $K$ be the algebraic number field associated to it and $O$ the ring of integers in $K$. Assume that $K$ has solveable Galois group so that in principle we know that roots of the polynomial.

Question 1: How can I obtain all the roots of the polynomial $f$ via Sage? If they are called $x_1,...,x_n$, and I define $y$ to be a linear combination of $x_1$, how can I check that $y$ is a primitive element for $K$ via Sage?

Question 2: How can I check whether $O$ is monogenic, that is $O=\mathbb{Z}[y]$ for some element $y$? As in question 1, how can I check for a specific $y$ that is a linear combination of the $x_i$ whether $O=\mathbb{Z}[y]$?

Question 3: How can I check via Sage whether $O$ is norm-Euclidean?

### On algebraic number fields in Sage

Assume I have an irreducible monic polynomial $f$ over $\mathbb{Z}$ of degree n and let $K$ be the algebraic number field associated to it and $O$ the ring of integers in $K$. Assume that $K$ has solveable Galois group so that in principle we (can) know that roots of the polynomial.

Question 1: How can I obtain all the roots of the polynomial $f$ via Sage? If they are called $x_1,...,x_n$, and I define $y$ to be a linear combination of $x_1$, how can I check that $y$ is a primitive element for $K$ via Sage?

Question 2: How can I check whether $O$ is monogenic, that is $O=\mathbb{Z}[y]$ for some element $y$? As in question 1, how can I check for a specific $y$ that is a linear combination of the $x_i$ whether $O=\mathbb{Z}[y]$?

Question 3: How can I check via Sage whether $O$ is norm-Euclidean?

### On algebraic number fields in Sage

Assume I have an irreducible monic polynomial $f$ over $\mathbb{Z}$ of degree n and let $K$ be the algebraic number field associated to it and $O$ the ring of integers in $K$. Assume that $K$ has solveable Galois group so that in principle we (can) know that roots of the polynomial.

Question 1: How can I obtain all the roots of the polynomial $f$ via Sage? If they are called $x_1,...,x_n$, and I define $y$ to be a linear combination of $x_1$, how can I check that whehter $y$ is a primitive element for $K$ via Sage?

Question 2: How can I check whether $O$ is monogenic, that is $O=\mathbb{Z}[y]$ for some element $y$? As in question 1, how can I check for a specific $y$ that is a linear combination of the $x_i$ whether $O=\mathbb{Z}[y]$?

Question 3: How can I check via Sage whether $O$ is norm-Euclidean?

### On algebraic number fields in Sage

Assume I have an irreducible monic polynomial $f$ over $\mathbb{Z}$ of degree n (take for example $f=x^4+x+1$) and let $K$ be the algebraic number field associated to it and $O$ the ring of integers in $K$. Assume that $K$ has solveable Galois group so that in principle we (can) know that roots of the polynomial.

Question 1: How can I obtain all the roots of the polynomial $f$ via Sage? If they are called $x_1,...,x_n$, and I define $y$ to be a linear combination of $x_1$, how can I check whehter $y$ is a primitive element for $K$ via Sage?

Question 2: How can I check whether $O$ is monogenic, that is $O=\mathbb{Z}[y]$ for some element $y$? As in question 1, how can I check for a specific $y$ that is a linear combination of the $x_i$ whether $O=\mathbb{Z}[y]$?

Question 3: How can I check via Sage whether $O$ is norm-Euclidean?

### On algebraic number fields in Sage

Assume I have an irreducible monic polynomial $f$ over $\mathbb{Z}$ of degree n (take for example $f=x^4+x+1$) and let $K$ be the algebraic number field associated to it and $O$ the ring of integers in $K$. Assume that $K$ has solveable Galois group so that in principle we (can) know that roots of the polynomial.

Question 1: How can I obtain all the roots of the polynomial $f$ via Sage? If they are called $x_1,...,x_n$, and I define $y$ to be a linear combination of $x_1$, the $x_i$, how can I check whehter $y$ is a primitive element for $K$ via Sage?

Question 2: How can I check whether $O$ is monogenic, that is $O=\mathbb{Z}[y]$ for some element $y$? As in question 1, how can I check for a specific $y$ that is a linear combination of the $x_i$ whether $O=\mathbb{Z}[y]$?

Question 3: How can I check via Sage whether $O$ is norm-Euclidean?


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OpenKattis
National University of Singapore

# Amsterdam Distance

Your friend from Manhattan is visiting you in Amsterdam. Because she can only stay for a short while, she wants to see as many tourist attractions in Amsterdam in as little time as possible. To do that, she needs to be able to figure out how long it takes her to walk from one landmark to another. In her hometown, that is easy: to walk from point $m = (m_ x, m_ y)$ to point $n = (n_ x, n_ y)$ in Manhattan you need to walk a distance

$|n_ x - m_ x| + |n_ y - m_ y|,$

since Manhattan looks like a rectangular grid of city blocks. However, Amsterdam is not well approximated by a rectangular grid. Therefore, you have taken it upon yourself to figure out the shortest distances between attractions in Amsterdam. With its canals, Amsterdam looks much more like a half-disc, with streets radiating at regular angles from the center, and with canals running the arc of the circle at equally spaced intervals. A street corner is given by the intersection of a circular canal and a street which radiates from the city center.

Depending on how accurately you want to model the street plan of Amsterdam, you can split the city into more or fewer half rings, and into more or fewer segments. Also, to avoid conversion problems, you want your program to work with any unit, given as the radius of the half circle. Can you help your friend by writing a program which computes the distance between any two street corners in Amsterdam, for a particular approximation?

## Input

The input consists of

• One line with two integers $M,N$ and a real number $R$.

• $1 \leq M \leq 100$ is the number of segments (or ‘pie slices’) the model of the city is split into.

• $1 \leq N \leq 100$ is the number of half rings the model of the city is split into.

• $1 \leq R \leq 1000$ is the radius of the city, given with at most $15$ digits after the decimal point.

• One line with four integers, $a_ x, a_ y, b_ x, b_ y$, with $0 \leq a_ x, b_ x \leq M$, and $0 \leq a_ y, b_ y \leq N$, the coordinates of two corners in the model of Amsterdam.

## Output

Output a single line containing a single real number, the least distance needed to travel from point $a$ to point $b$ following only the streets in the model. The result should have an absolute or relative error of at most $10^{-6}$.

Sample Input 1 Sample Output 1
6 5 2.0
1 3 4 2

1.65663706143592

Sample Input 2 Sample Output 2
9 7 3.0
1 5 9 5

4.28571428571429

Sample Input 3 Sample Output 3
10 10 1.0
2 0 6 0

0


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# How do i use mathmatical symbols

1. Aug 6, 2005

### professor

not necessarily homework...but i dont know where to put it... how do i use greek letters/square roots...etc in the forums

2. Aug 6, 2005

### TD

This should help

3. Aug 6, 2005

### Hurkyl

Staff Emeritus
You can learn how to make the LaTeX symbols (clicking on an image will show you the text you need to use)

Also, many text symbols can be made:

& rarr ; = →
& alpha ; = α
& Psi ; = Ψ
& infin ; = ∞

Not to mention anything you can look up in the Character Map in windows (provided you're looking at the right font)

For example, a degree symbol (°) can be made by typing ALT-0176. (You must use the numeric pad, and hold alt throughout the entire sequence)

Last edited: Aug 6, 2005
4. Aug 11, 2005

### robousy

$$S = \sum_{n=0}^\infty \frac{i^n}{n!} \idotsint \ {d^4x_1}\ {d^4x_2} \dots \ d^4x_n \ T (H_I(x_1) \ H_I(x_2) \dots \ H_I(x_n) )$$

Last edited: Aug 11, 2005
5. Aug 12, 2005

### Fermat

You can copy and paste from here.