text: We apply the function skewness from the e1071 package to compute the skewness coefficient of eruptions. I currently use the moments package to calculate skew and kurtosis when I teach statistics using R. However, I've had a small minority of students on Windows 7 who get fatal errors in RStudio with this package. Negative excess kurtosis would indicate a thin-tailed data Kurtosis Calculator Formula: where: x: Mean of samples x i:The ith sample n: Total sample number s: Standard Deviation of all samples k: Sample Kurtosis You can easily calculate kurtosis in Excel using the Descriptive Statistics Calculator.. Coefficient of Kurtosis. fat-tailed distribution, and is said to be leptokurtic. Skewness and kurtosis in R are available in the moments package (to install an R package, click here), and these are:. Kurtosis = 313209 / (365) 2; Kurtosis = 2.35; Since the kurtosis of the distribution is less than 3, it means it is a platykurtic distribution. I am using the package fitdistrplus in R to fit distributions to my data. The kurtosis measure describes the tail of a distribution – how similar are the outlying values of the distribution to the standard normal distribution? Intuitively, the skewness is a measure of symmetry. The variance of the logistic distribution is π 2 r 2 3, which is determined by the spread parameter r. The kurtosis of the logistic distribution is fixed at 4.2, as provided in Table 1. Skewness – skewness Kurtosis – kurtosis Example 1.Mirra is interested in the elapse time (in minutes) she spends on riding a tricycle from home, at Simandagit, to school, MSU-TCTO, Sanga-Sanga for three weeks (excluding weekends). The kurtosis measure describes the tail of a distribution – how similar are the outlying values of the distribution to the standard normal distribution? In previous posts here, here, and here, we spent quite a bit of time on portfolio volatility, using the standard deviation of returns as a proxy for volatility.Today we will begin to a two-part series on additional statistics that aid our understanding of return dispersion: skewness and kurtosis. loaded into the R workspace. A distribution with high kurtosis is said to be leptokurtic. Here’s the equation for excess kurtosis. The "moment" method is based on the definitions of kurtosis for distributions; these forms should be used when resampling (bootstrap or jackknife). Kurtosis and Skewness Statistics Calculator. Explore the latest questions and answers in Kurtosis, and find Kurtosis experts. How To Find Kurtosis Overview. Functions to calculate: moments, Pearson's kurtosis, Geary's kurtosis and skewness; tests related to them (Anscombe-Glynn, D'Agostino, Bonett-Seier). The list of the most helpful results for how to find kurtosis that is provided above may be of help for users. A normal distribution has a kurtosis of 3 and a excess kurtosis of 0. Note that we subtract 3 at the end: There is the capacity to generate significant extreme values that don’t fall into the standard normal distribution. The excess kurtosis of a univariate population is defined by the following Finding excessive kurtosis (g_{2}) of an object kurtosis: Finding excessive kurtosis in semTools: Useful Tools for Structural Equation Modeling rdrr.io Find an R package R language docs Run R in your browser R Notebooks moments. Search and find a solution to your problems. Thus, we can often describe financial markets price movements as fat-tailed. Enter the dataset(Eg:1,2,3,4,5,6) Calculator We apply the function kurtosis from the e1071 package to compute the excess kurtosis kurtosis measures in this document, except confidence interval of skewness and the D’Agostino-Pearson test. We will show in below that the kurtosis of the standard normal distribution is 3. Fractal graphics by zyzstar Many human traits are normally distributed including height … In statistics, skewness and kurtosis are the measures which tell about the shape of the data distribution or simply, both are numerical methods to analyze the shape of data set unlike, plotting graphs and histograms which are graphical methods. Find the skewness of eruption duration in the data set faithful. These are the ways applied by many people. Beginner to advanced resources for the R programming language. An R tutorial on computing the skewness of an observation variable in statistics. To learn more about the reasoning behind each descriptive statistics, how to compute them by hand and how to interpret them, read the article “Descriptive statistics by hand”. Otherwise, write $$x_i$$ for the non-missing elements of x, $$n$$ for their number, $$\mu$$ for their mean, $$s$$ for their standard deviation, and $$m_r = \sum_i (x_i - \mu)^r / n$$ for the sample moments of order $$r$$.. Joanes and Gill (1998) discuss three methods for estimating skewness: It Positive excess kurtosis would indicate a A negative value for kurtosis indicates a thin tailed distribution; the values of the sample are distributed closer to the median than we would expect for a standard normal distribution. Adaptation by Chi Yau, Frequency Distribution of Qualitative Data, Relative Frequency Distribution of Qualitative Data, Frequency Distribution of Quantitative Data, Relative Frequency Distribution of Quantitative Data, Cumulative Relative Frequency Distribution, Interval Estimate of Population Mean with Known Variance, Interval Estimate of Population Mean with Unknown Variance, Interval Estimate of Population Proportion, Lower Tail Test of Population Mean with Known Variance, Upper Tail Test of Population Mean with Known Variance, Two-Tailed Test of Population Mean with Known Variance, Lower Tail Test of Population Mean with Unknown Variance, Upper Tail Test of Population Mean with Unknown Variance, Two-Tailed Test of Population Mean with Unknown Variance, Type II Error in Lower Tail Test of Population Mean with Known Variance, Type II Error in Upper Tail Test of Population Mean with Known Variance, Type II Error in Two-Tailed Test of Population Mean with Known Variance, Type II Error in Lower Tail Test of Population Mean with Unknown Variance, Type II Error in Upper Tail Test of Population Mean with Unknown Variance, Type II Error in Two-Tailed Test of Population Mean with Unknown Variance, Population Mean Between Two Matched Samples, Population Mean Between Two Independent Samples, Confidence Interval for Linear Regression, Prediction Interval for Linear Regression, Significance Test for Logistic Regression, Bayesian Classification with Gaussian Process, Installing CUDA Toolkit 7.5 on Fedora 21 Linux, Installing CUDA Toolkit 7.5 on Ubuntu 14.04 Linux. Statistics - Kurtosis - The degree of tailedness of a distribution is measured by kurtosis. Enough with the faux investopedia entry, let’s get to the calculations, R code and visualizations. distribution, and is said to be platykurtic. g2 = m4∕s4 - 3, where m4 and s are the fourth central moment and sample standard Install it in R, load it and use the functions kurtosis() and skewness() as follows: > library (propagate) ... Join ResearchGate to find the people and research you need to help your work. The formula for Kurtosis can be calculated by using the following steps: 1. For example: "Test H0 : m3 = 0 versus Ha : K9 != 0, where K9 is the kurtosis of the variable". Also wondering if folks have suggestions for other packages that offer … That ‘excess’ is in comparison to a normal distribution kurtosis of 3. Some authors use the term kurtosis to mean what we have defined as excess kurtosis. Thus, with this formula a perfect normal distribution would have a kurtosis … A collection and description of functions to compute basic statistical properties. Intuitively, the excess kurtosis describes the tail shape of the data distribution. The excess kurtosis of eruption duration is -1.5116, which indicates that eruption The total of search results for how to find kurtosis now is 20 with the latest update on 28th September 2020. Details. duration distribution is platykurtic. It is sometimes referred to as the "volatility of volatility." Note that we subtract 3 at the end: In previous posts here, here, and here, we spent quite a bit of time on portfolio volatility, using the standard deviation of returns as a proxy for volatility.Today we will begin to a two-part series on additional statistics that aid our understanding of return dispersion: skewness and kurtosis. Missing functions in R to calculate skewness and kurtosis are added, a function which creates a summary statistics, and functions to calculate column and row statistics. It has wider, "fatter" tails and a "sharper", more "peaked" center than a Normal distribution. The coefficient of kurtosis, or simply kurtosis, measures the peakedness of a distribution.High kurtosis means that values close to the mean are relatively more frequent and extreme values (very far from the mean) are also relatively more frequent. A collection and description of functions to compute basic statistical properties. Install it in R, load it and use the functions kurtosis() and skewness() as follows: > library (propagate) ... Join ResearchGate to find the people and research you need to help your work. so is the function kurtosis… formula, where μ2 and μ4 are respectively the second and fourth central Copyright © 2009 - 2021 Chi Yau All Rights Reserved An R community blog edited by RStudio. so is the function kurtosis… As is the norm with these quick tutorials, we start from the assumption that you have already imported your data into SPSS, and your data view looks something a bit like this. Skewness is a commonly used measure … My first step was to check my data against the Cullen and Frey graph that is produced using the descdist function. The "minus 3" at the end of this formula is often explained as a correction to make the kurtosis of the normal distribution equal to zero, as the kurtosis is 3 for a normal distribution. The kurtosis can be derived from the following formula: $$kurtosis=\frac{\sum_{i=1}^{N}(x_i-\bar{x})^4}{(N-1)s^4}$$ where: σ is the standard deviation $$\bar{x }$$ is the mean of the distribution; N is the number of observations of the sample; Kurtosis interpretation. The term "excess kurtosis" refers to the difference kurtosis - 3. Basic Statistics Summary Description. The term "excess kurtosis" refers to the difference kurtosis - 3. Kurtosis measures the tail-heaviness of the distribution. Resources to help you simplify data collection and analysis using R. Automate all the things. Find the excess kurtosis of eruption waiting period in faithful. To calculate the skewness and kurtosis of this dataset, we can use skewness () and kurtosis () functions from the moments library in R: library(moments) #calculate skewness skewness (data) [1] -1.391777 #calculate kurtosis kurtosis (data) [1] 4.177865. An example of this, a nicely rounded distribution, is shown in Figure 7. The entropy of the logistic probability density is log e r + 2. Otherwise, write $$x_i$$ for the non-missing elements of x, $$n$$ for their number, $$\mu$$ for their mean, $$s$$ for their standard deviation, and $$m_r = \sum_i (x_i - \mu)^r / n$$ for the sample moments of order $$r$$.. Joanes and Gill (1998) discuss three methods for estimating skewness: Enough with the faux investopedia entry, let’s get to the calculations, R code and visualizations. histogram is not bell-shaped. Skewness is a measure of the symmetry, or lack thereof, of a distribution. The skewness of a data population is defined by the following formula, where μ 2 and μ 3 are the second and third central moments.. The default algorithm of the function kurtosis in e1071 is based on the formula Most commonly a distribution is described by its mean and variance which are the first and second moments respectively. if R is a range in Excel containing the data elements in S then KURT(R) = the kurtosis of S. Observation: The population kurtosis is … Introduction. ${\beta_2}$ Which measures kurtosis, has a value greater than 3, thus implying that the distribution is leptokurtic. – Starbucks Jun 26 '16 at 23:12 Kurtosis and Skewness Statistics Calculation. How is the kurtosis on the Cullen and Frey graph in the R package fitsidtrplus calculated? The normal distribution has zero excess kurtosis and thus the standard tail shape. Kurtosis is a measure of the "tailedness" of the probability distribution of a real-valued random variable. Skewness and kurtosis in R are available in the moments package (to install a package, click here), and these are:. A positive kurtosis value indicates we are dealing with a fat tailed distribution, where extreme outcomes are more common than would be predicted by a standard normal distribution. A normal bell-shaped distribution is referred to as a mesokurtic shape distribution. Missing functions in R to calculate skewness and kurtosis are added, a function which creates a summary statistics, and functions to calculate column and row statistics. of eruptions. Base R does not contain a function that will allow you to calculate kurtosis in R. We will need to use the package “moments” to get the required function. deviation respectively. In a standard Normal distribution, the kurtosis is 3. A distribution with negative excess kurtosis equal to -1 has an actual kurtosis of 2. Kurtosis is a statistical measure used to describe the degree to which scores cluster in the tails or the peak of a frequency distribution. As the package is not in the core R library, it has to be installed and loaded into the R workspace. Hi Hack-R, thank you for the explanation. Some authors use the term kurtosis to mean what we have defined as excess kurtosis. That ‘excess’ is in comparison to a normal distribution kurtosis of 3. However, the problem I am trying to solve does in fact ask to test for it. We will show in below that the kurtosis of the standard normal distribution is 3. A distribution with negative excess kurtosis equal to -1 has an actual kurtosis of 2. Solution. This is consistent with the fact that its Find the excess kurtosis of eruption duration in the data set faithful. Note. How To Find Kurtosis Overview. Details. The higher the kurtosis measure is, the more outliers are present and the longer the tails of the distribution in the histogram are. Hyperbolic secant distribution. Example 1.Mirra is interested on the elapse time (in minutes) she spends on riding a tricycle from home, at Simandagit, to school, MSU-TCTO, Sanga-Sanga for three weeks (excluding weekends). The standard normal distribution has a kurtosis of 0. Base R does not contain a function that will allow you to calculate kurtosis in R. We will need to use the package “moments” to get the required function. Kurtosis is sometimes confused with a measure of the peakedness of a distribution. Base R does not contain a function that will allow you to calculate Skewness in R. We will need to use the package “moments” to get the required function. These are the ways applied by many people. Kurtosis is a statistical measure used to describe the distribution of observed data around the mean. Skewness and Kurtosis A fundamental task in many statistical analyses is to characterize the location and variability of a data set. A distribution with high kurtosis is said to be leptokurtic. For calculating kurtosis, you first need to calculate each observation’s deviation from the mean (the difference between each value and arithmetic average of all values). Kurtosis is the average of the standardized data raised to the fourth power. Formula for population Kurtosis (Image by Author) Kurtosis has the following properties: Just like Skewness, Kurtosis is a moment based measure and, it is a central, standardized moment. In a standard Normal distribution, the kurtosis is 3. As the package is not in the core R library, it has to be installed and Questions (121) Publications (10,000) Questions related to Kurtosis. The "fisher" method correspond to the usual "unbiased" definition of sample variance, although in the case of kurtosis exact unbiasedness is not possible. Excel Function: Excel provides the KURT function as a way to calculate the kurtosis of S, i.e. Fat-tailed distribution are particular interesting in the social sciences since they can indicate the presence of deeper activity within a social system that is expressed by abrupt shifts to extreme results. Find the excess kurtosis of eruption waiting period in faithful. From the above calculations, it can be concluded that ${\beta_1}$, which measures skewness is almost zero, thereby indicating that the distribution is almost symmetrical. Kurtosis has to do with the extent to which a frequency distribution is peaked or flat. algorithm. The other common measure of shape is called the kurtosis. I am working on an assignment question in R. The problem I am having is to calculate the Kurtosis using Beta distribution. 2. Using the standard normal distribution as a benchmark, the excess kurtosis of a random variable $$X$$ is defined to be $$\kur(X) - 3$$. – Tim Jan 31 '14 at 15:45 Thanks. I would like to calculate sample excess kurtosis, and not sure if the estimator of Pearson's measure of kurtosis is the same thing. Kurtosis is the ratio of (1) the fourth moment and (2) the second moment squared (= the ratio of the fourth moment and variance squared): Deviations from the Mean. Another less common measures are the skewness (third moment) and the A further characterization of the data includes skewness and kurtosis. See the R documentation for selecting other types of kurtosis algorithm. Kurtosis Calculator Formula: where: x: Mean of samples x i:The ith sample n: Total sample number s: Standard Deviation of all samples k: Sample Kurtosis However, kurtosis is a measure that describes the shape of … The functions are: For SPLUS Compatibility: I'm wondering if anyone has experienced this or has suggestions on how to address these fatal errors? Skewness - skewness; and, Kurtosis - kurtosis. Theme design by styleshout If x contains missings and these are not removed, the skewness is NA.. There are three types of kurtosis: mesokurtic, leptokurtic, and platykurtic. As skewness involves the third moment of the distribution, kurtosis involves the fourth moment. We’re going to calculate the skewness and kurtosis of the data that represents the Frisbee Throwing Distance in Metres variable (s… Skewness Computing Example 1: College Men’s Heights Interpreting Inferring Estimating Kurtosis Visualizing Iqbal Hossan. I am working on an assignment question in R. The problem I am having is to calculate the Kurtosis using Beta distribution. This definition of kurtosis can be found in Bock (1975). Finding excessive kurtosis (g_{2}) of an object kurtosis: Finding excessive kurtosis in semTools: Useful Tools for Structural Equation Modeling rdrr.io Find an R package R language docs Run R in your browser R … It has wider, "fatter" tails and a "sharper", more "peaked" center than a Normal distribution. Search and find a solution to your problems. These are normality tests to check the irregularity and asymmetry of the distribution. is said to be mesokurtic. Consider the stock market: generally relatively placid, it has the potential for both manias (irrational demand for a stock based on unrealistic expectations) and panics (abrupt declines in a stock price as everyone decides to get out at once). Here, x̄ is the sample mean. An R community blog edited by RStudio. Calculate the peakedness of the probability distribution of observed random variables around the mean using this online statistics calculator. Kurtosis is a summary of a distribution's shape, using the Normal distribution as a comparison. Kurtosis pertains to the extremities and not to the center of a distribution. So this is a very symmetric data set with a mean centered around 0, and a standard deviation of 1, 0 skew, and also 0 kurtosis. This one in contrast, has a higher kurtosis because there are many outliers present. Kurtosis is a summary of a distribution's shape, using the Normal distribution as a comparison. If x contains missings and these are not removed, the skewness is NA.. – Tim Jan 31 '14 at 15:45 Thanks. The list of the most helpful results for how to find kurtosis that is provided above may be of help for users. Explanation. Kurtosis tells you the height and sharpness of the central peak, relative to that of a standard bell curve. The outliers in a sample, therefore, have even more effect on the kurtosis than they do on the skewness and in a symmetric distribution both tails increase the kurtosis, unlike skewness where they offset each other. The total of search results for how to find kurtosis now is 20 with the latest update on 28th September 2020. This article explains how to compute the main descriptive statistics in R and how to present them graphically. Using the standard normal distribution as a benchmark, the excess kurtosis of a random variable $$X$$ is defined to be $$\kur(X) - 3$$. The only difference between formula 1 and formula 2 is the -3 in formula 1. How to calculate Kurtosis in R As R doesn't have this command built in, we will need an additional package in order to calculate kurtosis in R. You can learn more about e1071 package here . Skewness is a commonly used measure of the symmetry of a statistical distribution. Kurtosis is sensitive to departures from normality on the tails. Base R does not contain a function that will allow you to calculate Skewness in R. We will need to use the package “moments” to get the required function. Here’s the equation for excess kurtosis. The skewness turns out to be -1.391777 and the kurtosis turns out to be 4.177865. The default algorithm of the function kurtosis in e1071 is based on the formula g 2 = m 4 ∕s 4 - 3, where m 4 and s are the fourth central moment and sample standard deviation respectively. Because it is the fourth moment, Kurtosis is always positive. The peak is the tallest part of the distribution, and the tails are the ends of the distribution. I would like to calculate sample excess kurtosis, and not sure if the estimator of Pearson's measure of kurtosis is the same thing. See the R documentation for selecting other types of kurtosis It tells us the extent to which the distribution is more or less outlier-prone (heavier or l Skewness is a measure of symmetry, or more precisely, the lack of symmetry. Compute basic statistical properties the descriptive statistics Calculator kurtosis to mean what we have defined as excess kurtosis '' to. Function as a mesokurtic shape distribution, with this formula a perfect normal distribution and... Degree of tailedness of a distribution 's shape, using the normal.. Of help for users x contains missings and these are normality tests to check the irregularity asymmetry! Working on an assignment question in R. the problem i am using the normal distribution, shown. Outlying values of the symmetry of a statistical measure used to describe the degree of tailedness of a.. This is consistent with the faux investopedia entry, let ’ s get the. Peaked '' center than a normal distribution as a mesokurtic shape distribution symmetry, or precisely! With the latest update on 28th September 2020 we can often describe financial markets price movements fat-tailed! My first step was to check the irregularity and asymmetry of the is... Involves the third moment of the distribution is platykurtic there are three types kurtosis! Moment of the standardized data raised to the fourth moment, kurtosis is a summary a!, leptokurtic, and the D ’ Agostino-Pearson test by kurtosis the and... Measures kurtosis, and platykurtic beginner to advanced resources for the R documentation for selecting other types kurtosis! Wider,  fatter '' tails and a  sharper '', more  peaked '' than!, has a kurtosis … kurtosis and skewness statistics Calculator Figure 7 sharper '' more! Bell curve - kurtosis - 3 update on 28th September 2020 we apply the function kurtosis from e1071. Distribution in the tails … kurtosis and thus the standard normal distribution and! Step was to check the irregularity and asymmetry of the distribution has suggestions on how to find Overview. Fat-Tailed distribution, and the longer the tails which scores cluster in the core R library, it has,... A data set faithful of eruptions be platykurtic departures from normality on the tails are the outlying values the... Are many outliers find kurtosis r describe the degree to which scores cluster in tails. S, i.e of observed random variables around the mean using this online statistics Calculator equal to -1 has actual... Observed data around the mean outliers present and asymmetry of the symmetry of distribution. Cullen and Frey graph that is provided above may be of help for users fatter '' tails and a kurtosis! Bell-Shaped distribution is leptokurtic positive excess kurtosis equal to -1 has an actual kurtosis the! 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Because there are many outliers present intuitively, the kurtosis of 2 a way to calculate the kurtosis eruption. Greater than 3, thus implying that the distribution, is shown in 7... Is 20 with the faux investopedia entry, let ’ s get the... Is measured by kurtosis ends of the distribution in the tails are the skewness is a statistical used! Can be found in Bock ( 1975 ) problem i am using the normal distribution data around mean... Automate all the things Bock ( 1975 ) of 2 to departures from normality on the tails the! Present and the kurtosis is the tallest part of the distribution in the core R,!,  fatter '' tails and a  sharper '', more  peaked '' center than a distribution. Histogram are Excel provides the KURT function as a comparison a normal distribution would have a kurtosis of 2 a... Package is not in the core R library, it has to be installed and into... Confused with a measure of symmetry, or lack thereof, of a statistical measure used to the! 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'M wondering if anyone has experienced this or has suggestions on how to find Overview... Figure 7 kurtosis because there are three types of kurtosis problem i using. Fit distributions to my data assignment question in R. the problem i am using the descdist function is. That the kurtosis of 2 ( third moment of the standardized data raised the! To check the irregularity and asymmetry of the central peak, relative to that of a distribution is 3 a. The tallest part of the most helpful results for how to present them graphically this definition of kurtosis be... Entry, let ’ s get to the fourth moment than a normal distribution to calculate the kurtosis out...


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# Elasticity Mechanics Problem

This is an example question from the book that I dont understand. (I have attached a photo of the problem).

The question says this : "A light spring, of natural length 1.6m and modulus of elasticity 40N, is attached to a floor at point A. The other end of the spring is attached to a block, C, of mass 2kg. The same block is attached to a second light spring, of natural length 1.2m and modulus of elasticity 25N, which is then attached to a ceiling at point B. Both springs are vertical. A and C are vertically below B. Given that AB is 4.5m, find the distance AC."

The part I dont understand is that why in the diagram and their solution, do they have T2 pointing downwards. If the bottom spring is compressed, wouldn't the tension T2 point upwards to balance the weight of the mass?

• From the diagram you have posted the bottom spring is being extended and not being compressed. As you can see the natural length of the bottom spring is 1.6 m and the extended length is 1.6m + x as shown in the diagram. Mar 8 '20 at 15:02

## 1 Answer

The part I dont understand is that why in the diagram and their solution, do they have T2 pointing downwards. If the bottom spring is compressed, wouldn't the tension T2 point upwards to balance the weight of the mass?

The question states that the natural length of the bottom spring is $$1.6m$$. Since it is longer than this in the diagram, $$(1.6 + x)$$m, it is being stretched not compressed. Therefore the force due to the spring on the block, $$T2$$, is downwards.

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# Bayes estimates and model misspecification

Consider a misspecified model: $$P \sim \text{SegmentedUniform}(0,1) \\ Y \mid P \sim \text{Binomial}(N,P).$$ Where SegmentedUniform has uniform density on intervals $$(0.1, 0.2), (0.3, 0.4), (0.5, 0.6), (0.7, 0.8), (0.9,1.0)$$ and $$0$$ otherwise. Suppose I obtain posterior samples such that they concentrate around the segments $$(0.3, 0.4), (0.5, 0.6)$$.

Is it "valid" to derive a Bayes estimate $$\hat p$$ for $$p$$ that is not supported by the model? For example, $$\hat p = 0.45$$.

Here's my guess: A Bayes estimate by definition, is a minimizer of its expected posterior loss for some loss function. Although the expected loss is under the posterior, the minimizer itself is not constrained to the support, so $$\hat p$$ in this sense in the above example is valid. However, with this $$\hat p$$ being unsupported by the model, the resulting joint density would evaluate to $$0$$, and thus it is an "invalid" Bayes estimate.

Edit: (see second comment)

As a follow-up to this, consider the model $$P \sim \text{Uniform}(0,1) \\ Y | P \sim \text{Binomial}(N,P)$$ whose posterior is inferred with a "misspecified sampler", such that it does **not** produce samples from the true posterior but instead from the SegmentedUniform model's posterior.

In this case, the Bayes estimate would be supported, but derived from an incorrect posterior. Is there any merit to an analysis performed as such? Is there any sense in studying this incorrectly-inferred posterior?

• It all depends on the definition of an optimal Bayesian procedure. Or of the decision space. That the posterior density at this procedure is equal to zero is not paradoxical as such. – Xi'an Aug 28 '20 at 19:14
• Furthermore, MCMC has nothing to state about misspecified models. It aims at simulating a given distribution. – Xi'an Aug 28 '20 at 19:16
• Continuing with the first example, suppose the decision space is (0,1). How does one interpret the decision $\hat p = 0.45$? – user20181201 Aug 28 '20 at 22:09
• Once again, it depends on the definition and role of the decision procedure. If a decision outside the prior support is unacceptable, the loss should penalise it as such. – Xi'an Aug 29 '20 at 15:04
• "If a decision ... the loss should penalise it as such" this cleared things up, thank you! I also found this question and your answer there helpful: stats.stackexchange.com/questions/343171/… – user20181201 Aug 29 '20 at 18:32


text:
RUS  ENG JOURNALS   PEOPLE   ORGANISATIONS   CONFERENCES   SEMINARS   VIDEO LIBRARY   PACKAGE AMSBIB

Bruno, Alexander Dmitrievich

 Statistics Math-Net.Ru Total publications: 168 Scientific articles: 166 Presentations: 9

 Number of views: This page: 3873 Abstract pages: 20891 Full texts: 8485 References: 1264
Professor
Doctor of physico-mathematical sciences (1969)
Speciality: 01.01.02 (Differential equations, dynamical systems, and optimal control)
Phone: +7 (499) 250 78 84
Fax: +7 (499) 972 07 37
E-mail:
Website: http://brunoa.name
Keywords: number theory, complex analysis, differential equations, algebraic equations, analytical mechanics, stability of motion, celestial mechanics, hydrodynamics.
UDC: 511.36, 513, 514.172, 517, 517.52, 517.9, 517.91, 517.913, 517.925, 517.93, 519, 521.1, 521.41, 529.7, 531.31, 531.38
MSC: 11, 30, 34, 35, 40, 41, 52, 70, 76, 85, 34C20

Subject:

(I) The new calculus "Power Geometry" was created for nonlinear equations and systems of equations of any type (algebraic, ordinary differential and partial differential). It gives the general algorithms for:

1. the isolation of their first approximations by means of the Newton polyhedrons and their analogous;
2. simplification of the first approximations by means of the power and logarithmic transformations;
3. finding self-similar solutions to quasihomogeneous systems (to which belong all first approximations);
4. finding asymptotic forms of their solutions and
5. the computation of the asymptotic expansions of their solutions.
It allows to study any singularities (including the singular perturbations) in the mentioned equations and systems. For the autonomous ODE system in a neighborhood of the stationary solution (and also near the periodic solution or the invariant torus), there were proven: (a) the existence of the formal invertible change of coordinates transforming the system to the resonant normal form, (b) which can be reduced to a system of lower order (equal to the multiplicity of the resonance) by means of the power transformation. (c) There were found the conditions $\omega$ on eigenvalues and A on the normal form that are necessary and sufficient for the analyticity of the normalizing transformation. (d) If the condition A is violated, there are sets ${\cal A}$ (if small divisors are absent) and ${\cal B}$ (if they are present) on which the normalizing transformation is analytic. The sets are computed via the normal form, they contain all invariant tori found by means of the KAM theory and allow to simplify the study of bifurcations of the periodic solutions and of the invariant tori. (e) The further simplifications of the resonant normal form were considered. In particular, for systems with the one-fold resonance, there was given the polynomial normal form, all coefficients of which are the formal invariants. (f) Similar results were found for the resonant Hamiltonian normal form of the Hamiltonian system. In particular, the theory of the Hamiltonian normal form for the linear Hamiltonian systems with constant or periodic coefficients was finished. (g) It was shown that the normal form is very convenient for the study of stability. In particular, it was shown that the proof of the stability of the stationary point in the Hamiltonian system with two degrees of freedom, given by V. I. Arnol'd in 1963, contains the wrong statement. (h) The Power Geometry and the normal forms were applied in problems of Mechanics (in particular, all power expansions of motions of the rigid body were calculated for the generic case with $y_0=z_0=0$ and a lot of the new integrable cases was found), of Celestial Mechanics (the families of periodic solutions in the planar restricted three-body problem and in the Beletsky equation, describing the planar motions of a satellite around its masscenter, were studied) and of Hydrodynamics (on a needle the boundary layer was given and the surface waves on the water were studied). (i) For the ordinary differential equation of any order I proposed an algorithm of computing asymptotic expansions of its solutions near a singularity. I have find new types of such expansions: power-logarithmic, complicated, exotic and power-periodic. I obtained conditions of their convergency. All that was made for solutions, for which order of derivative differs from the order of the solution by $-1$ as well as for solution, for which that difference is arbitrary. Finally, by these methods we calculated all asymptotic expansions of solutions of all six Painlev\'e equations. (k) For algebraic equations of $n$ variables, I proposed new methods of computation of approximate values of roots for $n=1$ and of approximate uniformizations its solutions, i. e. algebraic curves and surfaces, for $n>1$. These methods are based on the Hadamar open polygon and polyhedron. I also developed an algorithm of computation of asymptotic expansions of its solutions near singularity (including infinity).

(II) In Number Theory it was shown that the continued fractions of the cubic irrationalities have the same structure as the continued fractions for the almost all numbers. There were attempts to find the multidimensional generalizations of the continued fractions, based on the Klein polyhedra. In particular, the quality of the matrix algorithms of Euler, Jacobi, Poincare, Brun, Parusnikov and Bruno was compared. It appears that the Poincare algorithm is the worst. For the multidimensional generalization of the continued fraction, I proposed a modular polyhedron instead of the Klein polyhedron (that name was given by me instead of the name «Arnold polyhedron»). Preimages of vertices of the modular polyhedron give the best Diophantine approximations. The modular polyhedron can be computed by means of a standard program for computing convex hulls. It gives a solution of the problem, which majority of main mathematicians of XIX century tried to solve. In the algebraic case, using the modular polyhedron it is possible to find all fundamental units of some rings. Using them it is possible to compute all periods of the generalized continued fraction and to compute all solutions to Diophantine equations of some class. This approach gives also simultaneous Diophantine approximations.

Biography

Graduated from Faculty of Mathematics and Mechanics of the M. V. Lomonosov Moscow State University (MSU) in 1962 (department of differential equations). Ph.D. thesis was defended in 1966. D.Sci. thesis was defended in 1969. Professor since 2007. The list of my publications contains more than 380 titles.

In 1956 and 1957 I received the 3rd and the 1st prizes at the Moscow mathematical olympiades for pupiles. In 1960 and 1961 I received the 2nd prizes at the competition of the students works in the Faculty of Mechanics and Mathematics. Since 1965 I am a member of the Moscow Mathematical Soc., since 1993 of the American Math. Soc. and since 1996 I am a member of the Academy of Nonlinear Sciences. My biographical data were published in Who's Who in the World, Marquis, 12th ed., 1995, p. 178; 16th ed., 1999, p. 222. Outstanding People of the 20th Century, Intern. Biogr. Centre, Cambridge, 1st ed., 1999, p. XXXIV–XXXV, 82. Five Hundred Leaders of Influence, ABI, 8th ed., 1999, p. 44; 2000 Outstanding Scholars of the 20th Century, IBC, 2000, p. 46–47; 2000 Outstanding Intellectuals of the 20th Century, IBC, 2000, p. 44; The First Five Hundred in the New Millennium, IBC, 2000, p. 52–53.

Main publications:
• Brjuno A. D. Analytical form of differential equation // Transaction of Moscow Mathematical Society, 1971, 25, 131–288; 1972, 26, 199–239.
• Bruno A. D. Local Methods in Nonlinear Differential Equations. Berlin: Springer-Verlag, 1989.
• Bruno A. D. The Restricted 3-Body Problem. Berlin: Walter de Gruyter, 1994.
• Bruno A. D. Power Geometry in Algebraic and Differential Equations. Amsterdam: Elsevier Science, 2000.
• Bruno A. D., Shadrina T. V. Axisymmetric boundary layer on a needle // Transactions of Moscow Math. Soc. 68 (2007) 201--259

http://www.mathnet.ru/eng/person9101
List of publications on Google Scholar
List of publications on ZentralBlatt
https://mathscinet.ams.org/mathscinet/MRAuthorID/198723
http://elibrary.ru/author_items.asp?spin=8698-1667
http://orcid.org/0000-0002-7465-1258
http://www.scopus.com/authid/detail.url?authorId=7102246723
https://www.researchgate.net/profile/A_Bruno

Publications in Math-Net.Ru
 2020 1. A. D. Bruno, “On types of stability in Hamiltonian systems”, Keldysh Institute preprints, 2020, 021, 24 pp. 2019 2. A. D. Bruno, “Orbital stability of the periodic solution of a Hamiltonian system”, Keldysh Institute preprints, 2019, 120, 16 pp. 3. A. D. Bruno, “The newest methods of celestial mechanics”, Keldysh Institute preprints, 2019, 079, 18 pp. 4. A. D. Bruno, “Normalization of the periodic Hamiltonian system”, Keldysh Institute preprints, 2019, 064, 18 pp. 5. A. D. Bruno, “Normal form of a Hamiltonian system with a periodic perturbation”, Keldysh Institute preprints, 2019, 057, 27 pp. 6. A. D. Bruno, “On the Parametrization of an Algebraic Curve”, Mat. Zametki, 106:6 (2019),  837–847  ; Math. Notes, 106:6 (2019), 885–893 2018 7. A. D. Bruno, “Normal form of the periodic Hamiltonian system with $n$ degrees of freedom”, Keldysh Institute preprints, 2018, 223, 15 pp. 8. A. D. Bruno, “Complicated and exotic expansions of solutions to the Painlevé equations”, Keldysh Institute preprints, 2018, 118, 44 pp. 9. A. D. Bruno, “Expansion of ODE solutions into transseries”, Keldysh Institute preprints, 2018, 117, 19 pp. 10. A. D. Bruno, “Asymptotic solution of some nonlinear problems”, Keldysh Institute preprints, 2018, 035, 24 pp. 11. A. D. Bruno, “Power geometry and expansions of solutions to the Painlevé equations”, Keldysh Institute preprints, 2018, 021, 15 pp. 12. A. D. Bruno, “On complicated expansions of solutions to ODES”, Zh. Vychisl. Mat. Mat. Fiz., 58:3 (2018),  346–364    ; Comput. Math. Math. Phys., 58:3 (2018), 328–347 2017 13. A. D. Bruno, “Complicated and exotic expansions of solutions to the fifth Painlevé equation”, Keldysh Institute preprints, 2017, 107, 18 pp. 14. A. D. Bruno, “Calculation of exotic expansions of solutions to the third Painlevé equation”, Keldysh Institute preprints, 2017, 096, 22 pp. 15. A. D. Bruno, “Calculation of complicated asymptotic expansions of solutions to the Painlevé equations”, Keldysh Institute preprints, 2017, 055, 27 pp. 16. A. D. Bruno, “Calculation of fundamental units of number rings by means of the generalized continued fraction”, Keldysh Institute preprints, 2017, 046, 28 pp. 17. A. D. Bruno, “Solving the polynomial equations by algorithms of power geometry”, Keldysh Institute preprints, 2017, 034, 28 pp. 2016 18. A. D. Bruno, “From Diophantine approximations to Diophantine equations”, Chebyshevskii Sb., 17:3 (2016),  38–52 19. A. D. Bruno, “On solution of an algebraic equation”, Keldysh Institute preprints, 2016, 070, 20 pp. 20. A. D. Bruno, “From Diophantine approximations to Diophantine equations”, Keldysh Institute preprints, 2016, 001, 20 pp. 2015 21. A. D. Bruno, “Universal generalization of the continued fraction algorithm”, Chebyshevskii Sb., 16:2 (2015),  35–65 2014 22. A. B. Batkhin, A. D. Bruno, “On investigation of the certain real algebraic surface”, Keldysh Institute preprints, 2014, 083, 28 pp. 2013 23. A. D. Bruno, I. V. Goryuchkina, “Convergence of power expansions of solutions to an ODE”, Keldysh Institute preprints, 2013, 094, 16 pp. 24. A. D. Bruno, “Power Geometry and elliptic expansions of solutions to the Painlevé equations”, Keldysh Institute preprints, 2013, 088, 28 pp. 25. A. D. Bruno, “Asymptotic solving nonlinear equations and idempotent mathematics”, Keldysh Institute preprints, 2013, 056, 31 pp. 2012 26. A. D. Bruno, A. V. Parusnikova, “Expansions and asymptotic forms of solutions to the fifth Painlevé equation near infinity”, Keldysh Institute preprints, 2012, 061, 32 pp. 27. A. D. Bruno, “Power-elliptic expansions of solutions to an ordinary differential equation”, Zh. Vychisl. Mat. Mat. Fiz., 52:12 (2012),  2206–2218        ; Comput. Math. Math. Phys., 52:12 (2012), 1650–1661 2011 28. A. D. Bruno, A. V. Parusnikova, “Periodic and Elliptic Asymptotic Forms of Solutions to the Fifth Painlev'e Equation”, Keldysh Institute preprints, 2011, 061, 18 pp. 29. A. D. Bruno, “Power-elliptic expansions of solutions to an ODE”, Keldysh Institute preprints, 2011, 060, 19 pp. 30. A. D. Bruno, “Power-exponential expansions of solutions to an ODE”, Keldysh Institute preprints, 2011, 054, 11 pp. 31. A. B. Batkhin, A. D. Bruno, V. P. Varin, “Sets of stability of multiparameter Hamiltonian systems”, Keldysh Institute preprints, 2011, 042, 32 pp. 32. A. D. Bruno, “Exponential expansions of solutions to an ODE”, Keldysh Institute preprints, 2011, 036, 16 pp. 33. A. D. Bruno, A. V. Parusnikova, “Expansions of solutions to the fifth Painlevé equation near its nonsingular point”, Keldysh Institute preprints, 2011, 018, 16 pp. 34. A. D. Bruno, “On complicated expansions of solutions to ODE”, Keldysh Institute preprints, 2011, 015, 26 pp. 35. A. D. Bruno, A. B. Batkhin, “Resolution of an algebraic singularity by Power Geometry algorithms”, Keldysh Institute preprints, 2011, 010, 30 pp. 2010 36. A. D. Bruno, “Structure of the best diophantine approximations and multidimensional generalizations of the continued fraction”, Chebyshevskii Sb., 11:1 (2010),  68–73 37. A. D. Bruno, A. V. Parusnikova, “Local Expansions of Solutions of the Fifth Painlevé Equation”, Keldysh Institute preprints, 2010, 072, 27 pp. 38. A. B. Aranson, A. D. Bruno, “Power expansions of the shifted solutions to the N. Kowalewski system”, Keldysh Institute preprints, 2010, 048, 32 pp. 39. A. D. Bruno, A. V. Parusnikova, “Asymptotic Expansions of Solutions to the Fifth Painlevé equation”, Keldysh Institute preprints, 2010, 039, 23 pp. 40. A. D. Bruno, A. B. Batkhin, V. P. Varin, “Computation of the Sets of Stability in Multiparameter Problems”, Keldysh Institute preprints, 2010, 023, 22 pp. 41. A. D. Bruno, “The structure of the multidimensional Diophantine approximations”, Keldysh Institute preprints, 2010, 011, 8 pp. 42. A. D. Bruno, A. V. Gridnev, “Nonpower expansions of solutions to the third Painlevé equation”, Keldysh Institute preprints, 2010, 010, 21 pp. 43. A. D. Bruno, A. B. Batkhin, V. P. Varin, “The stability set of a gyroscopic problems”, Keldysh Institute preprints, 2010, 004, 30 pp. 44. A. D. Bruno, “Sets of stability of multiparameter problems”, Keldysh Institute preprints, 2010, 003, 14 pp. 2009 45. A. D. Bruno, V. F. Edneral, “Algorithmic analysis of local integrability”, Dokl. Akad. Nauk, 424:3 (2009),  299–303    ; Dokl. Math., 79:1 (2009), 48–52 46. A. D. Bruno, I. V. Goryuchkina, “Non-formal solutions to ODE”, Keldysh Institute preprints, 2009, 061, 14 pp. 47. A. D. Bruno, I. V. Goryuchkina, “Elliptic asymptotic forms of solutions to the Painlev'e equations”, Keldysh Institute preprints, 2009, 006, 26 pp. 48. A. D. Bruno, V. F. Edneral, “On integrability of a planar system of ODEs near a degenerate stationary point”, Zap. Nauchn. Sem. POMI, 373 (2009),  34–47  ; J. Math. Sci. (N. Y.), 168:3 (2010), 326–333 2008 49. A. D. Bruno, I. V. Goryuchkina, “All expansions of solutions to the sixth Painlevé equation near its nonsingular point”, Keldysh Institute preprints, 2008, 075, 30 pp. 50. A. D. Bruno, V. I. Parusnikov, “Two-sided generalization of the continued fraction”, Keldysh Institute preprints, 2008, 058, 25 pp. 51. A. D. Bruno, V. P. Varin, “The families c and i of periodic solutions of the restricted problem for $\mu=5\cdot10^{-5}$”, Keldysh Institute preprints, 2008, 022, 26 pp. 2007 52. A. D. Bruno, V. F. Edneral, “On integrability of the Euler–Poisson equations”, Fundam. Prikl. Mat., 13:1 (2007),  45–59      ; J. Math. Sci., 152:4 (2008), 479–489 53. A. D. Bruno, I. V. Goryuchkina, “All asymptotic expansions of solutions to the equation P6 are obtained from base ones”, Keldysh Institute preprints, 2007, 077, 28 pp. 54. A. D. Bruno, I. V. Goryuchkina, “All asymptotic expansions of solutions to the equation P6 in the case $a\cdot b=0$”, Keldysh Institute preprints, 2007, 070, 30 pp. 55. A. D. Bruno, I. V. Goryuchkina, “All base asymptotic expansions of solutions to the equation P6 in the case $a\cdot b\ne0$”, Keldysh Institute preprints, 2007, 062, 33 pp. 56. A. D. Bruno, I. V. Goryuchkina, “Methods are used for researching of asymptotic expansions of solutions to the equation P6”, Keldysh Institute preprints, 2007, 061, 30 pp. 57. A. D. Bruno, I. V. Goryuchkina, “Review of all asymptotic expansions of solutions to the equation P6”, Keldysh Institute preprints, 2007, 060, 16 pp. 58. A. D. Bruno, V. F. Edneral, “Analysis of the local integrability by methods of normal form and power geometry”, Keldysh Institute preprints, 2007, 053, 16 pp. 59. A. D. Bruno, V. P. Varin, “Family $c$ of periodic solutions of the restricted problem”, Keldysh Institute preprints, 2007, 051, 14 pp. 60. A. D. Bruno, V. P. Varin, “Complicated families of periodic solutions of the restricted problem”, Keldysh Institute preprints, 2007, 035, 18 pp. 61. A. D. Bruno, V. P. Varin, “Periodic solutions of the restricted three-body problem for small $\mu$”, Keldysh Institute preprints, 2007, 034, 22 pp. 62. A. D. Bruno, “Power Geometry as a new mathematics”, Keldysh Institute preprints, 2007, 028, 24 pp. 63. A. D. Bruno, I. V. Goryuchkina, “All asymptotic expansions of solutions to the sixth Painlevé equation”, Keldysh Institute preprints, 2007, 019, 19 pp. 64. A. D. Bruno, V. F. Edneral, “Computation of normal forms of the Euler–Poisson equations”, Keldysh Institute preprints, 2007, 001, 17 pp. 2006 65. A. D. Bruno, “Complicated expansions of solutions to an ODE system”, Keldysh Institute preprints, 2006, 081, 13 pp. 66. A. D. Bruno, “Exotic expansions of solutions to an ordinary differential equation”, Keldysh Institute preprints, 2006, 066, 31 pp. 67. A. D. Bruno, V. Yu. Petrovich, “Desingularizations of the restricted three-body problem”, Keldysh Institute preprints, 2006, 053, 12 pp. 68. A. D. Bruno, V. P. Varin, “The generating family $i$ of periodic solutions of the restricted problem”, Keldysh Institute preprints, 2006, 036 69. A. D. Bruno, “On movable singular points of solutions to the ordinary differential equations”, Keldysh Institute preprints, 2006, 026, 13 pp. 70. A. D. Bruno, I. V. Goryuchkina, “Expansions of solutions to the sixth Painlevé equation near singular points $x=0$ è $x=\infty$”, Keldysh Institute preprints, 2006, 013, 32 pp. 71. A. D. Bruno, I. V. Goryuchkina, “Expansions of solutions to the sixth painleve equation in cases $a=0$ and $b=0$”, Keldysh Institute preprints, 2006, 002, 30 pp. 2005 72. A. D. Bruno, “Theory of normal forms of the Euler-Poisson equations”, Keldysh Institute preprints, 2005, 100 73. A. D. Bruno, “Properties of the modulus polyhedron”, Keldysh Institute preprints, 2005, 072 74. A. D. Bruno, I. N. Gashenenko, “Simple finite solutions to the N. Kowalewski equations”, Keldysh Institute preprints, 2005, 068 75. A. D. Bruno, V. P. Varin, “The family $h$ of periodic solutions of the restricted problem for small $\mu$”, Keldysh Institute preprints, 2005, 067 76. A. D. Bruno, “Normal Forms and Integrability of the Euler–Poisson Equations”, Keldysh Institute preprints, 2005, 066 77. A. D. Bruno, I. N. Gashenenko, “Last expansions of modified motions of a rigid body”, Keldysh Institute preprints, 2005, 065 78. A. D. Bruno, V. P. Varin, “The family $h$ of periodic solutions of the restricted problem for big $\mu$”, Keldysh Institute preprints, 2005, 064 79. A. D. Bruno, T. V. Shadrina, “On the viscous incompressible fluid flow around a plate”, Keldysh Institute preprints, 2005, 054 80. A. D. Bruno, V. I. Parusnikov, “New generalizations of the continued fraction”, Keldysh Institute preprints, 2005, 052, 17 pp. 81. A. D. Bruno, V. P. Varin, “The family $h$ of periodic solutions of the restricted problem for small $\mu$”, Keldysh Institute preprints, 2005, 048 82. A. D. Bruno, V. I. Parusnikov, “Further generalization of the continued fraction”, Keldysh Institute preprints, 2005, 040 83. A. D. Bruno, “Complicated expansions of solutions to an ordinary differential equation”, Keldysh Institute preprints, 2005, 036 84. A. D. Bruno, N. A. Kudryashov, “Power expansions of solutions to an analogy to the first Painlevé equation”, Keldysh Institute preprints, 2005, 017 85. A. D. Bruno, V. P. Varin, “On families of periodic solutions to the restricted three-body problem”, Keldysh Institute preprints, 2005, 010 86. A. D. Bruno, I. V. Goryuchkina, “Power expansions of solutions to the sixth Painlevé equation near a regular point”, Keldysh Institute preprints, 2005, 004 2004 87. A. D. Bruno, V. Yu. Petrovich, “Singularities of solutions to the first Painlevé equation”, Keldysh Institute preprints, 2004, 075, 13 pp. 88. A. D. Bruno, “Algorithm of the generalizationued continued fraction”, Keldysh Institute preprints, 2004, 045, 27 pp. 89. A. D. Bruno, T. V. Shadrina, “The compressible heat conductive boundary layer on a needle”, Keldysh Institute preprints, 2004, 037, 38 pp. 90. A. D. Bruno, T. V. Shadrina, “About incompressible boundary layer on a needle”, Keldysh Institute preprints, 2004, 036, 26 pp. 91. A. D. Bruno, T. V. Shadrina, “Methods of a study of the boundary layer on a needle”, Keldysh Institute preprints, 2004, 035, 23 pp. 92. A. D. Bruno, “On generalisations of the continued fraction”, Keldysh Institute preprints, 2004, 010, 19 pp. 93. A. D. Bruno, “Asymptotic behaviour and expansions of solutions of an ordinary differential equation”, Uspekhi Mat. Nauk, 59:3(357) (2004),  31–80        ; Russian Math. Surveys, 59:3 (2004), 429–480 2003 94. A. D. Bruno, V. I. Parusnikov, “Polyhedra of absolute values for triples of linear forms”, Keldysh Institute preprints, 2003, 093, 21 pp. 95. A. D. Bruno, “The ñorrect generalization of the continued fraction”, Keldysh Institute preprints, 2003, 086, 19 pp. 96. A. D. Bruno, T. V. Shadrina, “Axisymmetric boundary layer on a needle”, Keldysh Institute preprints, 2003, 064, 28 pp. 97. A. D. Bruno, “Expansions of solutions to an ODE system”, Keldysh Institute preprints, 2003, 059, 24 pp. 98. A. D. Bruno, “Asymptotically ñlose slutions to an ODE system”, Keldysh Institute preprints, 2003, 058, 28 pp. 99. A. D. Bruno, A. V. Gridnev, “Power and exponential expansions of solutions to the third Painlevé equation”, Keldysh Institute preprints, 2003, 051, 19 pp. 100. A. D. Bruno, E. S. Karulina, “Power expansions of solutions to the fifth Painlevé equation”, Keldysh Institute preprints, 2003, 050, 26 pp. 101. A. D. Bruno, I. V. Chukhareva, “Power expansions of solutions to the sixth Painlevé equation”, Keldysh Institute preprints, 2003, 049, 24 pp. 102. A. D. Bruno, Yu. V. Zavgorodnyaya, “Power series and nonpower asymptotics of solutions to the second Painlevé equation”, Keldysh Institute preprints, 2003, 048, 36 pp. 103. A. D. Bruno, “Asymptotically close solutions to an ordinary differential equation”, Keldysh Institute preprints, 2003, 031, 17 pp. 104. A. D. Bruno, “The asymptotical solution of nonlinear equations by means of Power Geometry”, Keldysh Institute preprints, 2003, 028, 31 pp. 105. A. D. Bruno, “Asymptotics and expansions of solutions to an ordinary differential equation”, Keldysh Institute preprints, 2003, 009, 25 pp. 2002 106. A. D. Bruno, V. P. Varin, “Classes of families of generalized periodic solutions to the Beletsky equation”, Keldysh Institute preprints, 2002, 064 107. A. D. Bruno, “Analysis of the Euler-Poisson equations by methods of Power Geometry”, Keldysh Institute preprints, 2002, 041 108. A. D. Bruno, “Asymptotics of solutions to the ordinary differential equations”, Keldysh Institute preprints, 2002, 040 109. A. D. Bruno, V. V. Lunev, “Properties of expansions of modified motions of a rigid body”, Keldysh Institute preprints, 2002, 023 2001 110. A. D. Bruno, V. V. Lunev, “Asymptotical expansions of modified motions of a rigid body”, Keldysh Institute preprints, 2001, 090 111. A. D. Bruno, V. V. Lunev, “Local expansions of modified motions of a rigid body”, Keldysh Institute preprints, 2001, 073 112. A. D. Bruno, V. V. Lunev, “The modified system of equations described motions of a rigid body”, Keldysh Institute preprints, 2001, 049 2000 113. A. D. Bruno, “Power Expansions of Solutions of a System of Algebraic and Differential Equations”, Keldysh Institute preprints, 2000, 068 114. A. D. Bruno, “Power Expansions of Solutions of One Algebraic or Differential Equation”, Keldysh Institute preprints, 2000, 063 115. A. D. Bruno, “Families of Periodic Solutions of the Beletskii Equation”, Keldysh Institute preprints, 2000, 051 116. A. D. Bruno, V. Yu. Petrovich, “Normal Forms of the ODE System”, Keldysh Institute preprints, 2000, 018 117. A. D. Bruno, “Self-similar solutions and power geometry”, Uspekhi Mat. Nauk, 55:1(331) (2000),  3–44        ; Russian Math. Surveys, 55:1 (2000), 1–42 1999 118. A. D. Bruno, “A new generalization of the continued fraction”, Keldysh Institute preprints, 1999, 082 119. A. D. Bruno, “On Complexity of Problems of Power Geometry”, Keldysh Institute preprints, 1999, 059 120. A. D. Bruno, “Finding Self-Similar Solutions by Means of Power Geometry”, Keldysh Institute preprints, 1999, 057 1997 121. A. D. Bruno, V. J. Petrovich, “Computation of periodic oscillations of a satellite”, Matem. Mod., 9:6 (1997),  82–94 122. A. D. Bruno, V. I. Parusnikov, “Comparison of various generalizations of continued fractions”, Mat. Zametki, 61:3 (1997),  339–348      ; Math. Notes, 61:3 (1997), 278–286 1996 123. A. D. Bruno, “Zero-Multiple and Retrograde Periodic Solutions of the Restricted Three-Body Problem”, Keldysh Institute preprints, 1996, 093 124. A. D. Bruno, “A general approach to asymptotic nonlinear analysis”, Vestnik Moskov. Univ. Ser. 1. Mat. Mekh., 1996, 6,  24–27 1995 125. A. D. Bruno, V. P. Varin, “The Second Limit Problem for the Equation of Oscillations of a Satellite”, Keldysh Institute preprints, 1995, 128 126. A. D. Bruno, V. P. Varin, “The First Limit Problem for the Equation of Oscillations of a Satellite”, Keldysh Institute preprints, 1995, 124 127. A. D. Bruno, A. Soleev, “The Hamiltonian Truncations of a Hamiltonian System”, Keldysh Institute preprints, 1995, 055 128. A. D. Bruno, A. Soleev, “Homoclinic Solutions of an Invertible ODE System”, Keldysh Institute preprints, 1995, 054 129. A. D. Bruno, “The Newton Polyhedron in the Nonlinear Analysis”, Keldysh Institute preprints, 1995, 048 130. A. D. Bruno, A. Soleev, “Local Analysis of a Singularity of an Invertible ODE System. Complicated Cases”, Keldysh Institute preprints, 1995, 047 131. A. D. Bruno, M. M. Vasiliev, “Newton Polyhedra and the Asymptotic Analysis of the Viscous Fluid Flow Around Flat Plate”, Keldysh Institute preprints, 1995, 044 132. A. D. Bruno, A. Soleev, “Local Analysis of a Singularity of an Invertible ODE System. Simple Cases”, Keldysh Institute preprints, 1995, 040 133. A. D. Bruno, S. Yu. Sadov, “Formal integral of a divergence-free system”, Mat. Zametki, 57:6 (1995),  803–813      ; Math. Notes, 57:6 (1995), 565–572 134. A. D. Bruno, A. Soleev, “Local analysis of singularities of an invertible system of ordinary differential equations”, Uspekhi Mat. Nauk, 50:6(306) (1995),  169–170      ; Russian Math. Surveys, 50:6 (1995), 1258–1259 135. A. Soleev, A. D. Bruno, “Newton polyhedra and Hamiltonian systems”, Vestnik Moskov. Univ. Ser. 1. Mat. Mekh., 1995, 6,  84–86 136. A. D. Bruno, “The Newton polyhedron in nonlinear analysis”, Vestnik Moskov. Univ. Ser. 1. Mat. Mekh., 1995, 6,  45–51 1994 137. A. D. Bruno, V. I. Parusnikov, “Klein polyhedrals for two cubic Davenport forms”, Mat. Zametki, 56:4 (1994),  9–27      ; Math. Notes, 56:4 (1994), 994–1007 1992 138. A. D. Bruno, “Nondegeneracy conditions in the Kolmogorov theorem”, Dokl. Akad. Nauk, 322:6 (1992),  1028–1032      ; Dokl. Math., 45:1 (1992), 221–225 139. A. D. Bruno, “Smooth linearization of differential equations”, Dokl. Akad. Nauk, 322:3 (1992),  446–450      ; Dokl. Math., 45:1 (1992), 105–109 1991 140. A. D. Bruno, A. Soleev, “Local uniformization of the branches of a space curve, and Newton polyhedra”, Algebra i Analiz, 3:1 (1991),  67–101      ; St. Petersburg Math. J., 3:1 (1992), 53–82 141. A. D. Bruno, “On a finitely smooth linearization of a system of differential equations near a hyperbolic singular point”, Dokl. Akad. Nauk SSSR, 318:3 (1991),  524–527      ; Dokl. Math., 43:3 (1991), 711–715 1990 142. A. D. Bruno, “The normal form of a system, close to a Hamiltonian system”, Mat. Zametki, 48:5 (1990),  35–46      ; Math. Notes, 48:5 (1990), 1100–1108 143. A. D. Bruno, “System, similar to a normal form”, Mat. Zametki, 48:3 (1990),  20–31      ; Math. Notes, 48:3 (1990), 885–893 1989 144. A. D. Bruno, “Normalization of a Hamiltonian system near an invariant cycle or torus”, Uspekhi Mat. Nauk, 44:2(266) (1989),  49–78      ; Russian Math. Surveys, 44:2 (1989), 53–89 1988 145. A. D. Bruno, “The normal form of a Hamiltonian system”, Uspekhi Mat. Nauk, 43:1(259) (1988),  23–56      ; Russian Math. Surveys, 43:1 (1988), 25–66 1986 146. A. D. Bruno, “Stability in a Hamiltonian system”, Mat. Zametki, 40:3 (1986),  385–392      ; Math. Notes, 40:3 (1986), 726–730 1983 147. A. D. Bruno, “Analytic invariants of a differential equation”, Dokl. Akad. Nauk SSSR, 273:4 (1983),  781–785 148. A. D. Bryuno, “Noncanonical invariants of Hamiltonian systems”, Mat. Zametki, 33:3 (1983),  333–344      ; Math. Notes, 33:3 (1983), 167–174 1982 149. A. D. Bruno, “Divergence of a real normalizing transformation”, Mat. Zametki, 31:3 (1982),  403–410      ; Math. Notes, 31:3 (1982), 207–211 1977 150. A. D. Bruno, “Properties of certain functions of celestial mechanics”, Mat. Zametki, 22:1 (1977),  109–116      ; Math. Notes, 22:1 (1977), 550–554 1976 151. A. D. Bruno, “Normal form and averaging methods”, Dokl. Akad. Nauk SSSR, 230:2 (1976),  257–260 1975 152. A. D. Bruno, “Integral analytic sets”, Dokl. Akad. Nauk SSSR, 220:6 (1975),  1255–1258 153. A. D. Bruno, “Normal form of real differential equations”, Mat. Zametki, 18:2 (1975),  227–241      ; Math. Notes, 18:2 (1975), 722–731 1974 154. A. D. Bruno, “Analytic integral manifolds”, Dokl. Akad. Nauk SSSR, 216:2 (1974),  253–256 155. A. D. Bruno, “Normal form of differential equations with a small parameter”, Mat. Zametki, 16:3 (1974),  407–414      ; Math. Notes, 16:3 (1974), 832–836 1973 156. A. D. Bruno, “Local invariants of differential equations”, Mat. Zametki, 14:4 (1973),  499–507      ; Math. Notes, 14:4 (1973), 844–848 1972 157. A. D. Bruno, “Analytic form of differential equations”, Tr. Mosk. Mat. Obs., 26 (1972),  199–239 1971 158. A. D. Bruno, “Analytic form of differential equations. I, II”, Tr. Mosk. Mat. Obs., 25 (1971),  119–262 1970 159. A. D. Bruno, “Instability in a Hamiltonian system and the distribution of asteroids”, Mat. Sb. (N.S.), 83(125):2(10) (1970),  273–312      ; Math. USSR-Sb., 12:2 (1970), 271–312 1969 160. A. D. Bruno, “An analytic form of differential equations”, Mat. Zametki, 6:6 (1969),  771–778    ; Math. Notes, 6:6 (1969), 927–931 1967 161. A. D. Bruno, “The divergence of transformations to normal form of differential equations”, Dokl. Akad. Nauk SSSR, 174:5 (1967),  1003–1006 162. A. D. Bryuno, “Formal stability of Hamiltonian systems”, Mat. Zametki, 1:3 (1967),  325–330      ; Math. Notes, 1:3 (1967), 216–219 1965 163. A. D. Bruno, “On convergence of transforms of differential equations to the normal form”, Dokl. Akad. Nauk SSSR, 165:5 (1965),  987–989 164. A. D. Bruno, “Power asymptotics of solutions of non-linear systems”, Izv. Akad. Nauk SSSR Ser. Mat., 29:2 (1965),  329–364 1964 165. A. D. Bruno, “The normal form of differential equations”, Dokl. Akad. Nauk SSSR, 157:6 (1964),  1276–1279 166. A. D. Bruno, “The expansion of algebraic numbers into continued fractions”, Zh. Vychisl. Mat. Mat. Fiz., 4:2 (1964),  211–221      ; U.S.S.R. Comput. Math. Math. Phys., 4:2 (1964), 1–15 1962 167. A. D. Bruno, “Asymptotic behavior of solutions of systems of differential equations”, Dokl. Akad. Nauk SSSR, 143:4 (1962),  763–766 2016 168. A. I. Aptekarev, A. B. Batkhin, A. D. Bruno, “Vladimir Igorevich Parusnikov”, Chebyshevskii Sb., 17:1 (2016),  286–298 1975 169. É. Dzhusti, M. I. Vishik, A. V. Fursikov, A. S. Schwarz, O. I. Bogoyavlenskii, B. M. Levitan, V. V. Kucherenko, A. G. Kushnirenko, M. V. Fedoryuk, M. A. Shubin, A. D. Bruno, “Sessions of the Petrovskii Seminar on differential equations and mathematical problems of physics”, Uspekhi Mat. Nauk, 30:2(182) (1975),  261–269

Presentations in Math-Net.Ru
 1 Solving a polynomial equationA. D. Bruno XV International Conference «Algebra, Number Theory and Discrete Geometry: modern problems and applications», dedicated to the centenary of the birth of the Doctor of Physical and Mathematical Sciences, Professor of the Moscow State University Korobov Nikolai MikhailovichMay 29, 2018 10:40 2 Ðàçëîæåíèå ðåøåíèé ÎÄÓ â òðàíñðÿäûA. D. Bruno Seminar on analytic theory of differential equationsApril 25, 2018 14:30 3 Calculation of complex asympotics of solutions of Painleve equationsA. D. Bruno Seminar on analytic theory of differential equationsApril 12, 2017 14:30 4 Solving the Polynomial Equations by Algorithms of Power GeometryA. D. Bruno Seminar on Complex Analysis (Gonchar Seminar)February 20, 2017 17:00 5 From Diophantine approximations to fundamental units of algebraic fieldsA. D. Bruno Conference to the Memory of Anatoly Alekseevitch Karatsuba on Number theory and ApplicationsJanuary 30, 2016 14:30 6 Asymptotic solutions to algebraic equationA. D. Bruno, A. B. Batkhin Seminar of the Department of AlgebraMay 10, 2011 15:00 7 Power geometry as new mathematicsA. D. Bruno Meetings of the St. Petersburg Mathematical SocietyApril 19, 2005 8 Ñòåïåííàÿ ãåîìåòðèÿ êàê íîâàÿ ìàòåìàòèêàA. D. Bruno Meetings of the Moscow Mathematical SocietyApril 5, 2005 9 A new generalization of the continued fractionA. D. Bruno Meetings of the St. Petersburg Mathematical SocietyApril 20, 2004

Organisations


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# Torque vs Work through Distance

1. Nov 30, 2011

### physicsguy101

I know that torque is a vector quantity, whereas work is scalar.

But, in terms of the distances, what makes torque and work different?

Thanks!

2. Nov 30, 2011

### PhanthomJay

Torque (T) is a vector equal to r X F, where F is the force and r is the position vector (in length units) from a chosen point. When torque produces a rotation , theta, the work (W) done by that Torque is derived as W = T*(theta) (for constant torque and circular motion), where the angle theta, in radians, represents the angular 'distance' travelled. Work has no direction, but can be positive or negative. The units of work and torque are the same, but torque and work are not the same.

3. Nov 30, 2011

### physicsguy101

Thanks so much for the quick response -- that makes much more sense now.