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# Birdsalls Covariance Of Two Random Variables Example

## Covariance Encyclopedia of Mathematics

### Covariance & Correlation вЂ“ mayank aggarwal вЂ“ Medium

Covariance {cov(X Y)} Calculator Formula & Example. Chapter 5: JOINT PROBABILITY DISTRIBUTIONS Part 2: Covariance When two random variables are being consid- Example: Book problem 5-43 p, I am trying to find the covariance of two random variables but I am not having any lucky. Just for simplicity lets say that my random variables are : X = value rolled.

### Covariance Correlation Variance of a sum Correlation

Covariance Correlation Variance of a sum Correlation. In general terms, correlation and covariance measure whether two random variables have a linear relationship. Statistical independence is about whether the variables, All of the above results can be proven directly from the definition of covariance. For example, correlation coefficient of two random two random variables.

WORKED EXAMPLES 3 COVARIANCE CALCULATIONS EXAMPLE 1 Let Xand Y be discrete random variables with joint mass Hence the two variables have covariance and In probability theory and statistics, covariance is a measure of how much two random variables change together. the covariance is positive. For example,

WORKED EXAMPLES 3 COVARIANCE CALCULATIONS EXAMPLE 1 Let Xand Y be discrete random variables with joint mass Hence the two variables have covariance and Correlation is a scaled version of covariance; note that the two parameters always In statistical terms, the variables form a random sample from the common

As an example, the variation in a collection of random points in two since it is the variance of a real-valued random variable. A covariance matrix is . one needs to model relationships between two or more random variables вЂ“ for example. your CPI at IITB The covariance of two random variables X and

Often times, we want to know the relationship between two variables. For example, The covariance of a random variable with itself (own covariance) For a pair of random variables X and Y It turns out that the covariance always lies between two limits For example, if the variables are X and Y and

Calculating Covariance. The equation above reveals that the correlation between two variables is the covariance between then the two stocks move in random Covariance and Correlation - What is a sample covariance? Find out how Numpy helps calculate these two stats with random variable as input.

Covariance & Correlation The covariance between two variables is defined Combinations of Variables Suppose we have two random variable X and Y For example, at I have two random matrices (matrix-valued random variables) $X$ and $Y$, both of dimension $n \times n$. Is there a notion of covariance between the two random

As an example, the variation in a collection of random points in two since it is the variance of a real-valued random variable. A covariance matrix is Note that while calculating a sample two random variables in a data set will change together. A positive covariance means that the two variables at

Chapter 5: JOINT PROBABILITY DISTRIBUTIONS Part 2: Covariance When two random variables are being consid- Example: Book problem 5-43 p ... Covariance Let X and Y be two RVвЂ™s Example Positive Covariance: on the inп¬Ѓnite dimensional vector space of random variables with п¬Ѓnite variance.

Correlation is a scaled version of covariance; note that the two parameters always In statistical terms, the variables form a random sample from the common Chapter 3: Expectation and Variance example, X might be the Covariance is a measure of the association or dependence between two random variables X and Y

A Gentle Introduction to Expected Value, Variance, to Expected Value, Variance, and Covariance with of the joint probability for two random variables. For example, cov(A,'omitrows') The covariance matrix of two random variables is the matrix of pairwise covariance calculations between each variable, C =

Calculating Covariance. The equation above reveals that the correlation between two variables is the covariance between then the two stocks move in random . one needs to model relationships between two or more random variables вЂ“ for example. your CPI at IITB The covariance of two random variables X and

... to quantify the dependence between two random variables X and Y by investigating what is called the covariance between the two random variables. Example In the example below the covariance of two random variables is calculated, yielding the result: -1.64 . These two variables are also displayed in the

It will calculate covariance in Excel of two given it will calculate the joint variability of two random variables, a few examples: Example 1 вЂ“ Covariance Chapter 3: Expectation and Variance example, X might be the Covariance is a measure of the association or dependence between two random variables X and Y

Covariance of two jointly continuous random variables. Covariance of Two Dependent Variables. 0. What is an example of a proof by minimal counterexample? This covariance calculator can help you determine the covariance factor which is a measure of how much two random variables (x,y) change together and find as well

4 Covariance The covariance between two random variables X and Y is deп¬Ѓned as follows: Cov(X,Y) = E[(X в€’ E(X))(Y в€’ E(Y))] Simple example: (1) ... Covariance Let X and Y be two RVвЂ™s Example Positive Covariance: on the inп¬Ѓnite dimensional vector space of random variables with п¬Ѓnite variance.

Covariance and Correlation Correlation is a scaled version of covariance; note that the two parameters always have the the variables form a random sample from the 1.10. TWO-DIMENSIONAL RANDOM VARIABLES 41 1.10.5 Covariance and Correlation Covariance and correlation are two measures of the strength of a relationship be-

... Covariance Let X and Y be two RVвЂ™s Example Positive Covariance: on the inп¬Ѓnite dimensional vector space of random variables with п¬Ѓnite variance. Calculating Covariance. The equation above reveals that the correlation between two variables is the covariance between then the two stocks move in random

1.10. TWO-DIMENSIONAL RANDOM VARIABLES 41 1.10.5 Covariance and Correlation Covariance and correlation are two measures of the strength of a relationship be- For a pair of random variables X and Y It turns out that the covariance always lies between two limits For example, if the variables are X and Y and

### Random_Variables.pdf Covariance Probability Distribution

GrГјss-Type Bounds for the Covariance of Transformed Random. ... to quantify the dependence between two random variables X and Y by investigating what is called the covariance between the two random variables. Example, Covariance definition: a measure of the association between two random variables , equal to the expected value... Meaning, pronunciation, translations and examples.

probability Covariance of two random variables. Chapter 5: JOINT PROBABILITY DISTRIBUTIONS Part 2: Covariance When two random variables are being consid- Example: Book problem 5-43 p, In general terms, correlation and covariance measure whether two random variables have a linear relationship. Statistical independence is about whether the variables.

### Functions of Two Continuous Random Variables LOTUS

Random_Variables.pdf Covariance Probability Distribution. 5.2.4 Functions of Two Continuous Random Variables. Example Let $X$ and $Y$ be two jointly continuous random variables with joint PDF That is why covariance is normalized into correlation coefficient to measure the strength of linear dependence between two random variables. sample covariance.

• Covariance of X and Y STAT 414 / 415 Statistics
• What's the difference between covariance and mutual
• Covariance & Correlation UNIGE

• Random Variables and Probability Distributions in Business Statistics. The sample covariance between two variables, X and Y, is. An explanation of Variance, Covariance and Correlation in rigorous yet clear Covariance is a great tool for describing the variance between two Random Variables.

Topic 5: Functions of multivariate random variables вЂ  Functions of several random variables вЂ  Random vectors { Mean and covariance matrix { Cross-covariance Calculating Covariance. The equation above reveals that the correlation between two variables is the covariance between then the two stocks move in random

In the example below the covariance of two random variables is calculated, yielding the result: -1.64 . These two variables are also displayed in the Covariance and Correlation Correlation is a scaled version of covariance; note that the two parameters always have the the variables form a random sample from the

s y = sample standard deviation of the random variable y. you must know the covariance for the two variables and the standard deviations of each variable. For example, in introductory The covariance of two random variables is deп¬Ѓned to be Cov(X;Y)=E which shows that a random walk is not covariance stationary

For example, in introductory The covariance of two random variables is deп¬Ѓned to be Cov(X;Y)=E which shows that a random walk is not covariance stationary Two random variables X and Y are said to be independent if <4.2> Example. Suppose a random variable X has a discrete Chapter 4 Variances and covariances

I have two random matrices (matrix-valued random variables) $X$ and $Y$, both of dimension $n \times n$. Is there a notion of covariance between the two random Covariance & Correlation The covariance between two variables is defined Combinations of Variables Suppose we have two random variable X and Y For example, at

That is why covariance is normalized into correlation coefficient to measure the strength of linear dependence between two random variables. sample covariance 5.2.4 Functions of Two Continuous Random Variables. Example Let $X$ and $Y$ be two jointly continuous random variables with joint PDF

All of the above results can be proven directly from the definition of covariance. For example, correlation coefficient of two random two random variables A Gentle Introduction to Expected Value, Variance, to Expected Value, Variance, and Covariance with of the joint probability for two random variables.

Covariance formula. The covariance between two random variables and can be computed using the definition of covariance: where the capital letter This covariance calculator can help you determine the covariance factor which is a measure of how much two random variables (x,y) change together and find as well

... Covariance Let X and Y be two RVвЂ™s Example Positive Covariance: on the inп¬Ѓnite dimensional vector space of random variables with п¬Ѓnite variance. Covariance and Correlation Correlation is a scaled version of covariance; note that the two parameters always have the the variables form a random sample from the

## Random_Variables.pdf Covariance Probability Distribution

Calculating Covariance For Stocks Investopedia. ... to quantify the dependence between two random variables X and Y by investigating what is called the covariance between the two random variables. Example, Note that while calculating a sample two random variables in a data set will change together. A positive covariance means that the two variables at.

### Random Variables Independence and Covariance

Random_Variables.pdf Covariance Probability Distribution. Covariance of two jointly continuous random variables. Covariance of Two Dependent Variables. 0. What is an example of a proof by minimal counterexample?, Estimation Error in the Correlation of Two Random Variables: A Spreadsheet-Based Exposition 1 Introduction The statistical term correlation is well-known to students.

An explanation of Variance, Covariance and Correlation in rigorous yet clear Covariance is a great tool for describing the variance between two Random Variables. Covariance & Correlation The covariance between two variables is defined Combinations of Variables Suppose we have two random variable X and Y For example, at

MASSACHUSETTS INSTITUTE OF TECHNOLOGY MORE ON DISCRETE RANDOM VARIABLES AND THEIR Expected values of some common random variables 3. Covariance and Chapter 4 Variances and covariances random variables. Example <4.5> Comparison of spread in the de nitions of variance and covariance, and their

I have two random matrices (matrix-valued random variables) $X$ and $Y$, both of dimension $n \times n$. Is there a notion of covariance between the two random In probability theory and statistics, two real-valued random variables, X,Y, are said to be uncorrelated if their covariance, E(XY) в€’ E(X)E(Y), is zero.

covaraince {cov(X, Y)} calculator, formula & example to estimate the nature of association between two random variables X & Y in probability & statistics experiments. covariance of normal distribution. (x,y)\$ fully to find the covariance. For example, Two dependent random variables with standard normal distribution and zero

Random Variables, Independence and Covariance Jack Xin Examples (discrete r.v): (1) two point r.v, formly distributed random variables by rand(1; Calculating Covariance. The equation above reveals that the correlation between two variables is the covariance between then the two stocks move in random

What is covariance? Definition and examples. If you are finding the covariance of just two random variables, just divide by n. Back to top. In general terms, correlation and covariance measure whether two random variables have a linear relationship. Statistical independence is about whether the variables

Chapter 5: JOINT PROBABILITY DISTRIBUTIONS Part 2: Covariance When two random variables are being consid- Example: Book problem 5-43 p What's the difference between covariance and mutual What are examples in science of variables which The covariance of two random variables measures

As an example, the variation in a collection of random points in two since it is the variance of a real-valued random variable. A covariance matrix is A Gentle Introduction to Expected Value, Variance, to Expected Value, Variance, and Covariance with of the joint probability for two random variables.

covaraince {cov(X, Y)} calculator, formula & example to estimate the nature of association between two random variables X & Y in probability & statistics experiments. A Gentle Introduction to Expected Value, Variance, to Expected Value, Variance, and Covariance with of the joint probability for two random variables.

Estimation Error in the Correlation of Two Random Variables: A Spreadsheet-Based Exposition 1 Introduction The statistical term correlation is well-known to students ... the Joint Density and Covariance between Two Random covariance between two random variables with random variables is normal. For examples of

Random Variables, Independence and Covariance Jack Xin Examples (discrete r.v): (1) two point r.v, formly distributed random variables by rand(1; For example, cov(A,'omitrows') The covariance matrix of two random variables is the matrix of pairwise covariance calculations between each variable, C =

Covariance definition: a measure of the association between two random variables , equal to the expected value... Meaning, pronunciation, translations and examples Correlation is a scaled version of covariance; note that the two parameters always In statistical terms, the variables form a random sample from the common

... random variables, which can, for example, the Covariance of Transformed Random Variables. for the covariance of two transformed random variables A Gentle Introduction to Expected Value, Variance, to Expected Value, Variance, and Covariance with of the joint probability for two random variables.

In general terms, correlation and covariance measure whether two random variables have a linear relationship. Statistical independence is about whether the variables In general terms, correlation and covariance measure whether two random variables have a linear relationship. Statistical independence is about whether the variables

Random Variables and Probability Distributions in Business Statistics. The sample covariance between two variables, X and Y, is. . one needs to model relationships between two or more random variables вЂ“ for example. your CPI at IITB The covariance of two random variables X and

Correlation is a scaled version of covariance; note that the two parameters always In statistical terms, the variables form a random sample from the common 1.10. TWO-DIMENSIONAL RANDOM VARIABLES 41 1.10.5 Covariance and Correlation Covariance and correlation are two measures of the strength of a relationship be-

Calculating Covariance. The equation above reveals that the correlation between two variables is the covariance between then the two stocks move in random What is covariance? Definition and examples. If you are finding the covariance of just two random variables, just divide by n. Back to top.

MASSACHUSETTS INSTITUTE OF TECHNOLOGY MORE ON DISCRETE RANDOM VARIABLES AND THEIR Expected values of some common random variables 3. Covariance and A numerical characteristic of the joint distribution of two random variables, In order to statistically estimate the covariance one uses the sample covariance

### Covariance Moments - Statistics in C C++ - codecogs.com

Covariance Moments - Statistics in C C++ - codecogs.com. Two random variables X and Y are said to be independent if <4.2> Example. Suppose a random variable X has a discrete Chapter 4 Variances and covariances, As an example, the variation in a collection of random points in two since it is the variance of a real-valued random variable. A covariance matrix is.

Uncorrelated random variables Wikipedia. WORKED EXAMPLES 3 COVARIANCE CALCULATIONS EXAMPLE 1 Let Xand Y be discrete random variables with joint mass Hence the two variables have covariance and, ... Covariance Let X and Y be two RVвЂ™s Example Positive Covariance: on the inп¬Ѓnite dimensional vector space of random variables with п¬Ѓnite variance..

### probability Covariance of two random variables

Covariance of Returns Definition Formula Example. WORKED EXAMPLES 3 COVARIANCE CALCULATIONS EXAMPLE 1 Let Xand Y be discrete random variables with joint mass Hence the two variables have covariance and Covariance and Correlation - What is a sample covariance? Find out how Numpy helps calculate these two stats with random variable as input..

A Gentle Introduction to Expected Value, Variance, to Expected Value, Variance, and Covariance with of the joint probability for two random variables. WORKED EXAMPLES 3 COVARIANCE CALCULATIONS EXAMPLE 1 Let Xand Y be discrete random variables with joint mass Hence the two variables have covariance and

For example, in introductory The covariance of two random variables is deп¬Ѓned to be Cov(X;Y)=E which shows that a random walk is not covariance stationary ... random variables, which can, for example, the Covariance of Transformed Random Variables. for the covariance of two transformed random variables

s y = sample standard deviation of the random variable y. you must know the covariance for the two variables and the standard deviations of each variable. It will calculate covariance in Excel of two given it will calculate the joint variability of two random variables, a few examples: Example 1 вЂ“ Covariance

Covariance and Correlation - What is a sample covariance? Find out how Numpy helps calculate these two stats with random variable as input. Binomial random variable examples page 5 Here are a Covariance for discrete random variables page 19 probability of exactly two successes is P

covaraince {cov(X, Y)} calculator, formula & example to estimate the nature of association between two random variables X & Y in probability & statistics experiments. Random Variables and Probability Distributions in Business Statistics. The sample covariance between two variables, For example, suppose you take a

In probability theory and statistics, covariance is a measure of how much two random variables change together. the covariance is positive. For example, Covariance definition: a measure of the association between two random variables , equal to the expected value... Meaning, pronunciation, translations and examples

Calculating Covariance. The equation above reveals that the correlation between two variables is the covariance between then the two stocks move in random In probability theory and statistics, covariance is a measure of how much two random variables change together. the covariance is positive. For example,

Correlation in Random Variables Suppose that an experiment produces two random vari- The covariance of a random variable with itself is equal to its vari- 2.8 Expected values and variance For example, a uniformly-distributed random variable X The covariance between two random variables X and Y is deп¬Ѓned as

Covariance and Correlation and correlation of two random variables. 2 Covariance Covariance is a measure of how much two random variables vary together. For example, covaraince {cov(X, Y)} calculator, formula & example to estimate the nature of association between two random variables X & Y in probability & statistics experiments.

... to quantify the dependence between two random variables X and Y by investigating what is called the covariance between the two random variables. Example ... the Joint Density and Covariance between Two Random covariance between two random variables with random variables is normal. For examples of

Guidelines for Describing Volunteer Activities you can apply your volunteer experience to your professional life. Volunteer work enhances a resume; Where do you put volunteer work on a resume example Wilcox Corners Volunteer Work and Professional Affiliations section of your job resume.

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Scientific Visualization and Computer Graphics

Scientific Visualization and Computer Graphics > BI > FSE > RUG

# Skeletonization Benchmark

## Introduction

In the past decade (or more), tens of different algorithms have been published for the extraction of 2D skeletons or medial axes, 3D surface skeletons, and 3D curve skeletons. However, to date, few if any benchmarks exist for comparing the quality of such methods. As such, deciding whether a (new) skeletonization method fits its purpose is, in practice, quite hard.

## Objectives

To assist such an assessment, we propose a skeletonization benchmark. The benchmark consists of the following ingredients:

• for 2D skeletonization: a set of 2D binary images
• for 3D skeletonization: a set of 3D shapes, both in polygonal and volumetric formats

## Shape selection

The shapes included in this benchmark have been selected to match several criteria:

• known: shapes frequently used in skeletonization literature
• public: shapes present in well-known repositories, such as Aim@Shape, McGill's shape benchmark, and Stanford scan repository (among others)
• complex: shapes have challenging structures, such as
• highly variable amounts of noise
• high surface detail
• zero and higher genus
• natural (smooth) and synthetic (faceted) objects
• variate: shapes from various sources (3D laser scanning, 3D contouring of volume scanning, CAD/CAM modeling)
• relevant: shapes suitable for skeletonization, i.e. described by surfaces which are
• closed
• orientable
• having no duplicate vertices (for mesh representations)
• having tubular symmetry (for curve skeleton) and planar symmetry (for surface skeletons)

## Repository

For simplicity, we store all shapes in easy-to-read, accessible, file formats. Three representations are used:

• image: 2D shapes are represented in PGM binary image format, with white for foreground, and black for background.
• surface: 3D shapes are represented in PLY format. Only triangle primitives are used. All properties except vertex coordinates are ignored.
• volume: 3D shapes are represented in the binvox volume format.

By providing three representation types, we aim to cover 2D skeletonization methods (typically, these assume image inputs), and 3D mesh-based and voxel-based skeletonization methods.

## 2D Image shapes

The first set of test images provided below serves as a benchmark for 2D skeletonization methods. The set includes both natural and synthetic shapes, with and without (multiple) holes, varying thickness, and smooth and noisy boundaries. Image resolutions range between 2002 and 10002 pixels. All images are in PGM grayscale format, with black for the shape foreground and white for the shape background.

## 3D Mesh shapes

The second set of test shapes provided below serves as a benchmark for 3D surface and curve skeletonization methods. The set includes both natural and synthetic shapes, with and without (multiple) hopes, varying thickness, and smooth and noisy boundaries. Shapes are provided in PLY triangle-mesh format. All meshes have been cleaned to ensure a consistent orientation, closeness, no duplicate or T vertices, and no degenerate faces. Each file describes a single connected shape. Mesh resolutions range between a few thousand vertices and over a million vertices. Some meshes are uniformly (densely) sampled, while others are highly non-uniform. All meshes can be visually examined using MeshLab.

• After downloading all files, rename files 3dmeshes_file1.zip up to and including 3dmeshes_file9.zip to 3dmeshes_file.z01 .. 3dmeshes_file.z09 respectively. Leave the master file 3dmeshes_file.zip as it is. Next, execute

zip -F 3dmeshes_file.zip --out 3dmeshes.zip
unzip 3dmeshes.zip
The above command create (and unzip) a single large 290MB ZIP archive containing around 125 3D mesh models from the ten ZIP parts.

## Software

Besides shapes, we also provide (pointers to) a number of 2D and 3D skeletonization algorithm implementations. These methods can be readily used on the corresponding image, mesh, and volumetric shapes provided above. By providing these implementations, we aim to give everybody a quick and easy way for comparing existing (and new) skeletonization methods with earlier such methods.

Note: The list of algorithms provided here is, of course, not exhaustive. It was determined based on methods that are

• available: in either source code or executable format, publicly redistributable
• simple: easy to install and run by a typical end-user
• complete: are self-contained, i.e., provide all code required to run them
• scalable: handle images of at least 5002 pixels or 3D shapes of at least 10K polygons or 3003 voxels in (tens of) seconds on a typical PC
• robust: were successfully tested by us by running them on a wide variety of third-party input shapes

• thinvox: Curve skeletonization based on thinning (Windows/Linux source)
• Reniers et al.: Geodesic-based curve and surface skeleton extraction for 3D voxel models (Windows binary and source)
• IMA Integer medial axis for 3D surface skeletons (Linux source code; original from authors; our version).
• AFMM: 2D skeleton extraction from binary images (Windows/Linux binary and source)
• CUDASkel: CUDA-accelerated version of AFMM (Linux binary and source)
• Siddiqi et al. Hamilton-Jacobi 3D curve and surface skeletons (Linux source)
• Gorgon: Iterative thinning process (Windows binary)
• Liu et al.: Robust thinning (Windows binary)
• Au et al.: Curve skeleton extraction for 3D mesh models (Windows binary executable)
• Cao et al.: Curve skeleton extraction for 3D mesh models (Linux source code).
• MCF: Mean curvature curve-skeletons (Windows/Linux/Mac source code).
• MCF-CGAL: Mean curvature curve-skeletons with CGAL library (Windows/Linux/Mac (source code).

## Evolution

We are continuously investing effort to refine the proposed benchmark by adding new shapes, refining the proposed shapes, and adding (pointers to) new 2D and 3D skeletonization software. Be sure to visit this page frequently to get the latest version of our benchmark.

format: plain


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Search…
Vaults
The daily farming yields in vaults automatically get bonded. Farmers earn more yields in ALD through the bonding process and can have this income auto compounded in the Staking Pool as well. This will significantly improve farmers’ APY and capital efficiency. The vaults are all quality strategies selected by the Boule Council.
The number of ALD bond obtained by farmers every day is calculated as follows:
Harvest once per epoch (6400 blocks), and automatically bond the farming yields minerProfit into ALD according to the ratio to be bonded expressed as
$BondPercent_{vault}$
. The ratio is 100% for now, the number of ALD that farmer i can get in each epoch is:
$minerBondald_i= aldSupply((1+\cfrac{minerProfit_i*BondPercent_{vault}}{Reserve_{bond}})^{LR}-1)*Discount_{asset}$
$minerProfit_i$
is farmer i’s total farming yield in the epoch. For example, farmer i gets 100 CRV and 50 LDO (CRV = $4 & LDO =$5) from staking stETH in the Convex vault for the epoch, then total farming yield is = $4100 +$550= \$650.
At the end of the epoch, the amount of ALD that farmer will receive is
$minerBondald_i$
, and is auto staked into the staking pool for compounding interests. When
$BondPercent_{vault}$
<100%,
$minerProfit_i*(1-BondPercent_{vault})$
is no longer entering the Bond Pool, but pays back to farmers.
Bonding profit will be transferred to the Treasury and POL Pool in proportion. For example,
$profitPOLPercent_{cry}=50\%$
, which means 50% of bonding profit will be stored in POL, which can be used to purchase ALD from the secondary market.


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# Resolve the 250 N force into components

Resolve the 250 N force into components acting along the u and v axes and determine the magnitudes of these components.

Image from: Hibbeler, R. C., S. C. Fan, Kai Beng. Yap, and Peter Schiavone. Statics: Mechanics for Engineers. Singapore: Pearson, 2013.

#### Solution:

We will first draw the components along the u and v axes like so:

The angles were found using  alternate interior angles theorem, corresponding angles theorem and consecutive angles theorem.

We can now draw the components tail to tail as follows:

We can now use sine law to figure out the forces.

$\dfrac{250}{\sin120^0}\,=\,\dfrac{F_u}{\sin40^0}$

(solve for $F_u$)

$F_u\,=\,185.5$ N

$\dfrac{250}{\sin120^0}\,=\,\dfrac{F_v}{\sin20^0}$

(solve for $F_v$)

$F_v\,=\,98.7$ N

$F_u\,=\,185.5$ N
$F_v\,=\,98.7$ N


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# SciREX: A Challenge Dataset for Document-Level Information Extraction

Extracting information from full documents is an important problem in many domains, but most previous work focus on identifying relationships within a sentence or a paragraph. It is challenging to create a large-scale information extraction (IE) dataset at the document level since it requires an understanding of the whole document to annotate entities and their document-level relationships that usually span beyond sentences or even sections. In this paper, we introduce SciREX, a document level IE dataset that encompasses multiple IE tasks, including salient entity identification and document level $N$-ary relation identification from scientific articles. We annotate our dataset by integrating automatic and human annotations, leveraging existing scientific knowledge resources. We develop a neural model as a strong baseline that extends previous state-of-the-art IE models to document-level IE. Analyzing the model performance shows a significant gap between human performance and current baselines, inviting the community to use our dataset as a challenge to develop document-level IE models. Our data and code are publicly available at https://github.com/allenai/SciREX

PDF Abstract ACL 2020 PDF ACL 2020 Abstract


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# Heisenberg uncertainty principle fluctuations and dimensions less than 3+1

I've been looking at the 1+1 dimensional Dirac equation recently and am wondering if the Heisenberg uncertainty principle would allow application of said model to spin-1/2 fermions in 3+1 dimensions if the system were required to be localized within 3+1 D spacetime?

My reasoning for this doubt is as follows: If the uncertainty principle applies to 3 spatial dimensions then setting, say, the $$p^1$$ and $$p^2$$ momentum components equal to zero would reduce their corresponding uncertainty to 0. This then seems to require that the positions $$x^1$$ and $$x^2$$ be completely unknowable meaning that one could not confine the 1+1 D system within any finite region inside 3+1 D spacetime... This appears to contradict the localizability assumption.

Yet, there are clear situations where 2+1 D or 1+1 D models apply and make successful predictions... So I am wondering if I have made a mistake somewhere and, if so, where?


text: The essays in this thesis are concerned with two main themes in port logistics. The first theme is the coordination of transport arrivals with the distribution processes and the use of storage facilities. We study this for both containerized and bulk chemical transport. The second theme is the uncertainty associated with the arrival time of ships with bulk chemicals and the impact on port logistics. Each essay describes a case study where quantitative methods, especially simulation, are used. The operation of container terminals and in particular the way in which containers are stacked in a yard is influenced by information about the departure of a container. We find that even inaccurate information is valuable and helps to reduce unproductive moves. Next, we present the floating stocks'' distribution concept which uses intermodal transport to deploy inventories in a supply chain in advance of retailer demand. We demonstrate that a main drawback of intermodal transport, a longer transit time, can be mitigated using this concept. This concept also influences the choice of a port: we provide a quantitative interpretation of routing flexibility in port selection.


text:
# Problem: A flask contains a mixture of He and Ne at a total pressure of 2.6 atm. There are 2.0 mol of He and 5.0 mol of Ne in the flask The partial pressure of He is ____ atm. A. 6.5 B. 1.04 C) 0.74 D) 9.1E) 1.86

###### FREE Expert Solution

$\overline{){{\mathbf{P}}}_{{\mathbf{He}}}{\mathbf{=}}{{\mathbf{\chi }}}_{{\mathbf{He}}}{{\mathbf{P}}}_{{\mathbf{total}}}}$

###### Problem Details

A flask contains a mixture of He and Ne at a total pressure of 2.6 atm. There are 2.0 mol of He and 5.0 mol of Ne in the flask The partial pressure of He is ____ atm.

A. 6.5

B. 1.04

C) 0.74

D) 9.1

E) 1.86


text:
# Friday math movie: Fractals without a computer

By Murray Bourne, 12 Nov 2010

This is neat - a simple way to produce fractals without even turning on a computer.

A key concept of fractals is that many of them are "self-similar". What this means is if you zoom in far enough, you'll see the same patterns you saw when you started. And likewise, when you zoom out far enough, you will once again see the same shapes. (You can see some of this idea in this 3-D fractal journey).

In the video below, they point a video camera at 3 screens, and project the image form the camera onto the screens. It's a bit like being in a room with mirrors on opposite walls - you see infinite versions of yourself going off into space.

Since there is a slight delay as the image signal reaches the 3 screens, it provides opportunities to create Sierpinski triangles (which are examples of "exact self-similarity"), Julia sets and various other fractals, all without a computer. (Normally you need to generate them using complex numbers, various random generators or even chaotic differential equations.)

This 3-screen fractal idea could be a good activity for a math event (or a science fair) in schools or a math club.

### 2 Comments on “Friday math movie: Fractals without a computer”

1. Dalcde says:

This is beautiful.

2. Melvin Goldstein says:

A fractal is the image of chaos. Chaos Theory is one of “Physics Foibles”.

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text:
Category:Beryl

English: The mineral beryl is a beryllium aluminium cyclosilicate with the chemical formula Be3Al2(SiO3)6. The hexagonal crystals of beryl may be very small or range to several meters in size. Terminated crystals are relatively rare. Pure beryl is colorless, but it is frequently tinted by impurities; possible colors are green, blue, yellow, red, and white.

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text:
# Game theory: Mixed Strategies and Nash Equilibrium

So I've recently become interested in game theory, and I've visited this site to help me understand what exactly game theory is and the applications of it.

In the lesson, they use an example of tennis and one of the players is at the net and the other is at the baseline. The player at the baseline is determining where to hit her shot: right or left, and where the volleyer will move left, or right. They called the probability that the volleyer will volley to the left is q, and to the right is 1- q.

They then state that in order to determine the nash equilibrium, the "expected payoffs" must be equal? Could someone please explain to me why? This pdf may also be of help to you. This pdf may help as well.

I also have another question. I'm thinking about using game theory in my science fair project to determine where tennis players should hit their balls (right, left, middle, short, or deep) based on the data from the two players facing one another in order for one of them to win the point. If the data was already given, would it be possible for you to conduct this experiment using game theory?

My try was to use the knowledge that I had learned up above but I didn't know how to handle multiple cases, in the example provided you only had to deal with 2 cases.

Thanks in advance to anyone who can help.

• Why the downvote? – Varun Iyer Jul 16 '14 at 18:28
• Some people give downvote without mentioning their reason(s)! :| It would be better if they tell why? – Mohammad Khosravi Jul 16 '14 at 18:30
• @MohammadKhosravi I understand. Could you help me :). – Varun Iyer Jul 16 '14 at 18:38
• Nice piece of mathematical research!!! – Bourbaki is ALIVE Sep 9 '14 at 17:22

I think you already understand why the players need to take expectations. Now why do the expectations need to be equal? The idea is, if there was one strategy which gave you strictly higher expected payoff, you would just stick to playing that strategy, instead of randomizing between 2 or more strategies, right? So, the only reason that might prompt you to play a mixed strategy is when all strategies give equal expected payoff. $\\$ Also, you can obviously extend this to randomizing over 3 or more strategies. Do you know about the previous applications of game theory in sports? I would encourage you to do some research into that before embarking into the project. For example, https://econ.duke.edu/uploads/assets/dje/2006_Symp/Wiles.pdf


text:
The Total Derivative of a Function From Rn To Rm As A Linear

# The Total Derivative of a Function From Rn To Rm as a Linear Combination of Its Partial Derivatives

Recall from the Differentiable Functions from Rn to Rm and Their Total Derivatives page that if $S \subseteq \mathbb{R}^n$, $\mathbf{c} \in \mathbb{R}^n$, and $\mathbf{f} : S \to \mathbb{R}^m$ then if $\mathbf{f}$ is differentiable at $\mathbf{c}$ then all of the directional derivatives of $\mathbf{f}$ at $\mathbf{c}$ exist and for all $\mathbf{u} \in \mathbb{R}^n$ we have that:

(1)
\begin{align} \quad \mathbf{T}_{\mathbf{c}} (\mathbf{u}) = \mathbf{f}'(\mathbf{c}, \mathbf{u}) \end{align}

In particular, if $\mathbf{e}_1, \mathbf{e}_2, ..., \mathbf{e}_n$ are the standard basis vectors for $\mathbb{R}^n$ then we know that the partial derivatives of $\mathbf{f}$ at $\mathbf{c}$ exist. The following theorem will tell us that for any $\mathbf{v}$, the total derivative evaluated at $\mathbf{v}$ is a linear combination of the partial derivatives of $\mathbf{f}$.

 Theorem 1: Let $S \subseteq \mathbb{R}^n$ be open, $\mathbf{c} \in \mathbb{R}^n$, and $\mathbf{f} : S \to \mathbb{R}^m$. If $\mathbf{f}$ is differentiable at $\mathbf{c}$ and if $\mathbf{v} = v_1\mathbf{e}_1 + v_2\mathbf{e}_2 + ... + v_n\mathbf{e}_n$ where $\mathbf{e}_1, \mathbf{e}_2, ..., \mathbf{e}_n$ are the standard basis vectors for $\mathbb{R}^n$ then $\mathbf{T}_{\mathbf{c}}(\mathbf{v}) = v_1\mathbf{f}'(\mathbf{c}, \mathbf{e}_1) + v_2\mathbf{f}'(\mathbf{c}, \mathbf{e}_2) + ... + v_n\mathbf{f}'(\mathbf{c}, \mathbf{e}_n)$.

Recall the notation $\mathbf{T}_{\mathbf{c}}(\mathbf{v}) = \mathbf{f}'(\mathbf{c})(\mathbf{v})$ and $\mathbf{f}'(\mathbf{c}, \mathbf{e}_k) = D_k \mathbf{f}(\mathbf{c})$. Thus the conclusion of the Theorem above can be rewritten as:

(2)
\begin{align} \quad \mathbf{f}'(\mathbf{c})(\mathbf{v}) = v_1 D_1 \mathbf{f}(\mathbf{c}) + v_2 D_2 \mathbf{f}(\mathbf{c}) + ... + v_n D_n \mathbf{f}(\mathbf{c}) \end{align}
• Proof: Let $\mathbf{f}$ be differentiable at $\mathbf{c}$. For each $\mathbf{v} \in S$ let $\mathbf{v} = v_1\mathbf{e}_1 + v_2\mathbf{e}_2 + ... + v_n\mathbf{e}_n$. Then since the total derivative $\mathbf{T}_{\mathbf{c}}$ is a linear function we have that:
(3)
\begin{align} \quad \mathbf{T}_{\mathbf{c}} (\mathbf{v}) &= \mathbf{T}_{\mathbf{c}} (v_1\mathbf{e}_1 + v_2\mathbf{e}_2 + ... + v_n\mathbf{e}_n ) \\ &= v_1 \mathbf{T}_{\mathbf{c}}(\mathbf{e}_1) + v_2 \mathbf{T}_{\mathbf{c}}(\mathbf{e}_2) + ... + v_n \mathbf{T}_{\mathbf{c}}(\mathbf{e}_n) \end{align}
• But by the Theorem referenced above, for each $k \in \{ 1, 2, ..., n \}$ we have that $\mathbf{T}_{\mathbf{c}} (\mathbf{e}_k) = \mathbf{f}'(\mathbf{c}, \mathbf{u})$ and so:
(4)
\begin{align} \quad \mathbf{T}_{\mathbf{c}}(\mathbf{v}) = v_1\mathbf{f}'(\mathbf{c}, \mathbf{e}_1) + v_2\mathbf{f}'(\mathbf{c}, \mathbf{e}_2) + ... + v_n\mathbf{f}'(\mathbf{c}, \mathbf{e}_n) \quad \blacksquare \end{align}
 Corollary 1: If $S \subseteq \mathbb{R}^n$ is open, $\mathbf{c} \in \mathbb{R}^n$, and $f : S \to \mathbb{R}$ is an $n$-variable real-valued function then $\mathbf{T}_{\mathbf{c}}(\mathbf{v}) = \nabla f(\mathbf{c}) \cdot \mathbf{v}$.


text:
• # question_answer The sum of the digits of a two digit number is 8 and the difference between the number and that formed by reversing the digits is 18. Find the number.

 Let unit digit $=x$ Tens digit $=y$ So, original number $=\text{unit digit}+10\times \text{tens digit}$ $1=x+10y$ According to question, Sum of digits $=8$ so,        $x+y=8$                                                         ...(i) On reversing the digits, unit digit $=y$, Tens digit $=x$ So,   New number $=10x+y$ According to question, Difference $=18$ $\Rightarrow \,x+10y-(10x+y)=18$ $\Rightarrow \,x+10y-10x-y=18$ $\Rightarrow \,9y-9x=18$ $\Rightarrow \,y-x=2$                                                    ...(ii) By adding eq. (i) and (ii) $2y=10$ $y=\frac{10}{2}\Rightarrow y=5$ Put the value of y in eq. (i) $x+y=8$ $\Rightarrow x+5=8$ $\Rightarrow x=8-5$ $\Rightarrow x=3$ $\therefore$ Original number $=10y+x$ $=10\times 5+3$ $=50+3$ $=53$


text:
UK Primary (3-6)
Multiples of unit fractions using number lines and symbols (A x 1/b) (2,3,4,5,6,8,10,12)

## Interactive practice questions

We are going to work out how to represent $5\times\frac{1}{3}$5×13 on the number line.

a

Mark $\frac{1}{3}$13 on the number line.

b

Now mark $5\times\frac{1}{3}$5×13 on this number line.

Easy
Less than a minute
We are going to work out how to represent $7\times\frac{1}{12}$7×112 on the number line.
What fraction has been multiplied by $4$4 to get the value plotted on the number line below?


text:
# Question

A production process manufactures electronic components with timing signals whose duration follows a normal distribution. A random sample of six components was taken, and the durations of their timing signals were measured.
a. The probability is 0.05 that the sample variance is greater than what percentage of the population variance?
b. The probability is 0.10 that the sample variance is less than what percentage of the population variance?

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text:
# 婆羅摩笈多

668年

## 數學

《婆羅摩曆算書》中有四章半講嘅係純數學，第12章講嘅係演算系列同少少幾何學。第18章係關於代數，婆羅摩笈多响呢度引入咗一個解二次丟番圖方程例如nx² + 1 = y²嘅方法。

### 代數

18.44：色同二次項同4相乘嘅積加一次項嘅二次方嘅數，將呢個數開方後減一次項，再將全個數除一次項嘅2倍，就係方程嘅解。[注 1]
18.45：色同二次項嘅積加一次項一半嘅二次方嘅數，將呢個數開方之後減一次項嘅一半，再約全個數除一次項就係方程嘅解。[注 2][5]

${\displaystyle x={\frac {{\sqrt {4ac+b^{2}}}-b}{2a}}}$

${\displaystyle x={\frac {{\sqrt {ac+{\tfrac {b^{2}}{4}}}}-{\tfrac {b}{2}}}{a}}}$

### 計數

#### 零

18.30：正數加正數係正數，負數加負數係負數。正數加負數係佢哋之間嘅差，如果佢哋一樣，結果就係零。負數加係負數，正數加零係正數，零加零係零[注 4]
18.32：負數減零係負數，正數減零係正數，零減零係零，正數減負數係佢哋之間嘅和。[注 5][5]

18.33：正乘負係負，負乘負係負，正乘正係正，正數乘零、負數乘零同零乘零都係零。[注 6][5]

18.34：正數除正數或者負數除負數係正數，正數除負數或者負數除正數係負數，零除零係零[注 7][5]
18.35：正數或者負數除零會有零做嗰個數嘅除數，零除正數或者負數會有正數或者負數做嗰個數嘅除數。正數或者負數嘅平方係正數，零嘅平方係零。[注 8][5]

### 幾何

#### 婆羅摩笈多公式

12.21：一個四邊形或者三角形嘅大約面積係邊同對邊嘅和嘅一半。四邊形嘅準確面積係每一條邊分別咁俾另外三條邊減嘅和嘅一半嘅開方。[注 9][6]

#### 圓周率

12.40：直徑同半徑嘅二次方每個乘3分別係圓形大約嘅周界同面積。而準確值就係直徑同半徑嘅二次方乘開方10。[注 10][6]

## 原文引注

1. 英文原文係：“18.44. Diminish by the middle [number] the square-root of the rupas multiplied by four times the square and increased by the square of the middle [number]; divide the remainder by twice the square. [The result is] the middle [number].”
2. 英文原文係：“18.45. Whatever is the square-root of the rupas multiplied by the square [and] increased by the square of half the unknown, diminish that by half the unknown [and] divide [the remainder] by its square. [The result is] the unknown.”
3. 英文原文係：“12.20. The sum of the squares is that [sum] multiplied by twice the [number of] step[s] increased by one [and] divided by three. The sum of the cubes is the square of that [sum] Piles of these with identical balls [can also be computed]”
4. 英文原文係：“18.30. [The sum] of two positives is positives, of two negatives negative; of a positive and a negative [the sum] is their difference; if they are equal it is zero. The sum of a negative and zero is negative, [that] of a positive and zero positive, [and that] of two zeros zero. [...]”
5. 英文原文係：“18.32. A negative minus zero is negative, a positive [minus zero] positive; zero [minus zero] is zero. When a positive is to be subtracted from a negative or a negative from a positive, then it is to be added [...]”
6. 英文原文係：“18.33. The product of a negative and a positive is negative, of two negatives positive, and of positives positive; the product of zero and a negative, of zero and a positive, or of two zeros is zero.”
7. 英文原文係：“18.34. A positive divided by a positive or a negative divided by a negative is positive; a zero divided by a zero is zero; a positive divided by a negative is negative; a negative divided by a positive is [also] negative.”
8. 英文原文係：“18.35. A negative or a positive divided by zero has that [zero] as its divisor, or zero divided by a negative or a positive [has that negative or positive as its divisor]. The square of a negative or of a positive is positive; [the square] of zero is zero. That of which [the square] is the square is [its] square-root.”
9. 英文原文係：“12.21. The approximate area is the product of the halves of the sums of the sides and opposite sides of a triangle and a quadrilateral. The accurate [area] is the square root from the product of the halves of the sums of the sides diminished by [each] side of the quadrilateral.”
10. 英文原文係：“12.40. The diameter and the square of the radius [each] multiplied by 3 are [respectively] the practical circumference and the area [of a circle]. The accurate [values] are the square-roots from the squares of those two multiplied by ten.”

## 參考

1. Seturo Ikeyama (2003). Brāhmasphuṭasiddhānta (CH. 21) of Brahmagupta with Commentary of Pṛthūdhaka, critically edited with English translation and notes. INSA.
2. "Brahmagupta biography". School of Mathematics and Statistics University of St Andrews, Scotland. 喺2013-07-15搵到.
3. David Pingree. Census of the Exact Sciences in Sanskrit (CESS). American Philosophical Society. 第 p254頁.
4. Boyer (1991). "The Arabic Hegemony". 第 226頁. By 766 we learn that an astronomical-mathematical work, known to the Arabs as the Sindhind, was brought to Baghdad from India. It is generally thought that this was the Brahmasphuta Siddhanta, although it may have been the Surya Siddhanata. A few years later, perhaps about 775, this Siddhanata was translated into Arabic, and it was not long afterwards (ca. 780) that Ptolemy's astrological Tetrabiblos was translated into Arabic from the Greek. 缺少或空白|title= (詳情)
5. Template:Harv
6. Template:Harv
7. Boyer (1991). "China and India". 第 220頁. However, here again Brahmagupta spoiled matters somewhat by asserting that ${\displaystyle 0\div 0=0}$, and on the touchy matter of ${\displaystyle a\div 0}$, he did not commit himself: 缺少或空白|title= (詳情)
8. Template:Harv Brahmagupta does not explicitly state that he is discussing only figures inscribed in circles, but it is implied by these rules for computing their circumradius.
9. "Brahmagupta, and the influence on Arabia". School of Mathematical and Computational Sciences University of St Andrews. 2002-05. 喺2013-07-15搵到. 檢查日期值喺: |date= (詳情)


text:
# Find the remainder of $f(x)$ when divided by two binomials

I got stuck on the following two questions:

1. Find the remainder if the polynomial $f(x)$ is divided by $ax+b.$
2. Find the remainder if the polynomial $f(x)$ is divided by $(x-a)(x-b).$

I got as far as rewriting question 1's divisor as $a(x+\frac{b}{a}).$ From there, I didn't know where to proceed.

Whether it's a hint or full solution, I would love some directions on how to proceed.

By the way, the answers are $f(-\frac{b}{a})$ and $(\frac{f(a)-f(b)}{a-b})x+\frac{af(b)-bf(a)}{a-b}.$

• Note that $f(x)=(ax+b)g(x)+r$ [r is constant as the divisor is one-degree]. Put $x=-\frac{b}{a}$ to get $r=f(-\frac{b}{a})$. Try solving the second problem similarly. – JDF Sep 13 '16 at 14:31

In the 1st problem, let $f(x)=(ax+b)g(x)+R$ for some polynomial $g$ and constant $R$.

$=a(x+\frac{b}{a})g(x)+R$

Then the remainder is given by $f(-b/a)=R$

For the 2nd problem,

let $f(x)=(x-a)(x-b)q(x)+Ax+B\quad...(*)$ for some polynomial $q$ and arbitrary constants $A,B$.

We substitute $x=a$ and $x=b$ successively in the identity $(*)$.

Then we have $f(a)-f(b)=Aa-Ab\Rightarrow A=\frac{f(a)-f(b)}{a-b}$

Putting this value in $f(a)$ we get, $f(a)=Aa+B=\frac{af(a)-af(b)}{a-b}+B$

$\Rightarrow B=\frac{af(b)-bf(a)}{a-b}$

So the remainder is given by $Ax+B$

Both problems are elementary applications of the division algorithm and the remainder theorem. Note that the degree of the remainder in both cases can be predetermined - when the divisor is linear, the remainder is constant and when the divisor is quadratic, the remainder is linear.


text:
# On which the function is a diffeomorphism to its image?

1. Jan 7, 2007

### cristo

Staff Emeritus
Are the following functions $\mathbb{R}^2\rightarrow\mathbb{R}^2$ diffeomorphisms. If not is there an open set containing the origin on which the function is a diffeomorphism to its image?

1. $$(x,y)\mapsto(x+y^3, y)$$
2. $$(x,y)\mapsto(x+x^3,x)$$

I have the definition of a diffeomorphism: $f:X\rightarrow Y$ is a diffeomorphism if f-1 exists, and f and f-1 are $C^{\infty}$ maps.

Where f is a $C^{\infty}$ map iff coordinates of a point $y \in Y$ are infinitely differentiable functions of $f^{-1}(y)\in X$.

So, for 1. I have tried the following:

$$f:\mathbb{R}^2\rightarrow \mathbb{R}^2: (x,y)\mapsto(x+y^3, y) =(u,v) \\ \Rightarrow x+y^3=u , y=v \Rightarrow x=u-v^3, y=v$$

So $f^{-1}: (u,v)\mapsto (u-v^3,v)=(x,y)$, and thus f-1 exists and both f and f-1 are $C^{\infty}$, therefore f is a diffeomorphism.

Is this right? I've tried a similar method for Q2, but to no avail!

Any help would be much appreciated!

2. Jan 7, 2007

### andytoh

Solution 1 looks good, but you should mention the bijectivity of f (isn't that part of the definition of diffeomorphism that you were given?) Because f inverse is defined on all of R^2, then f is surjective.

Are you sure you typed out Question 2 correctly? The image of R^2 under the map in question 2 is a curve, which is a 1-dimensional manifold. Diffeomorphisms preserve dimension of manifolds, so if you typed the question correctly, the answer would have to be no.

Note: You don't have to work out explicitly the inverse. Just calculate the Jacobian matrix and determine it has rank 2. For the second question, the Jacobian is

(1+3x^2 0)
(1 0)

which has rank 1 for all (x,y) in R^2, so the mapping cannot be a diffeomorphism.

For the first question, the Jacobian is

(1 3y^2)
(0 1)

which has rank 2 for all (x,y) in R^2. This means that the map is everywhere a local diffeomorphism. You went further and found a global inverse.

Last edited: Jan 7, 2007
3. Jan 7, 2007

### cristo

Staff Emeritus
Ok, thanks.

The question is definitely typed out correctly, so the answer must be no.

Thanks for this; it's definitely a better method than struggling to find the inverse!

However, how do I deduce the rank of the Jacobian matrix? I've not studied linear algebra for a while, and am quite rusty! Would I be right in thinking that the rank of a matrix is the number of linearly independent columns in the matrix- or am I completely off the mark here?

(Also, should the first entry in your first matrix above not be 1+3x^2)?

4. Jan 7, 2007

### andytoh

I've editted my first response. It is always good to find the explicit inverse if you can. Resort to the Jacobian if you cannot find an explicit inverse or want to prove that it is not a diffeomorphism. Remember to consider all possible values for x and y when determining the rank of the Jacobian. Sometimes two rows may look linearly independent but then they are linearly dependent for certain values of x and y. Fortunately those traps don't occur in your questions.

5. Jan 7, 2007

### cristo

Staff Emeritus
Ok, thanks for your help. I have one more question along the same lines; here's my attempt: $$(x,y)\mapsto(x^2+yx,y)=(u,v) \\ \Rightarrow x^2+yx=u, y=v \Rightarrow y=v, x(x+v)=u$$

So, the inverse function cannot be calculated. The Jacobi matrix is $$\left( \begin{array}{cc}2x+y&x\\0&1\end{array}\right)$$

This is not a diffeomorphism, since for the origiin and the points (1/2,-1) and (-1/2,1) the rows are linearly dependent (and hence the rank of the matrix is 1 for these points)

Is this correct? Thanks again!

Last edited: Jan 7, 2007
6. Jan 7, 2007

### andytoh

The Jacobian matrix has rank 1 only when 2x+y=0. So the map is a local diffeomorphism on all of R^2 except for the line {(x,y):2x+y=0}.

Now back to the original question, since the line 2x+y=0 goes through the origin, then there is NO local diffeomorphism at the origin. However it is a local diffeomorphism everywhere except on that line.

Last edited: Jan 7, 2007
7. Jan 7, 2007

### cristo

Staff Emeritus

8. Jan 7, 2007

### andytoh

HOWEVER, the map is not one-to-one (a diffeomorphism has to be one-to-one and onto). For example, (-1,0) and (1,0) both map to (1,0), so the map is NOT a diffeomorphism on R^2 - {(x,y):2x+y=0}. It is only LOCALLY a diffeomorphism on R^2 - {(x,y):2x+y=0}. This is the trouble that results when you cannot find an explicit inverse.

Last edited: Jan 7, 2007
9. Jan 7, 2007

### cristo

Staff Emeritus
Ok, but since the question asks for an "open set including the origin for on which the function is a diffeomorphism to its image". The set R^2-{(x,y):2x+y=0} includes the origin, and so does not satisfy this criterion.

10. Jan 7, 2007

### andytoh

This whole issue of local vs global confuses me too. For example, let's look at
f(x,y,z)=(x^2, y^2, z^2, yz, xz, xy), with domain {R^3: x,y,z>0}. Now f is one-to-one and clearly differentiable. The question is whether it is a diffeomorphism (not just a local diffeomorphism) from
{R^3: x,y,z>0} to f({R^3: x,y,z>0}). The Jacobian is

2x 0 0
0 2y 0
0 0 2z
0 x y
z 0 x
0 x y

which is of rank 3 since x,y,z>0. So f is locally a diffeomorphism at every point of {R^3: x,y,z>0}? So is f a diffeomorphism on all of {R^3: x,y,z>0} to its image? Can anyone answer?

Last edited: Jan 7, 2007
11. Jan 7, 2007

### andytoh

I said from {R^3: x,y,z>0} to f({R^3: x,y,z>0}), so the image is a 3-dimensional submanifold of R^6, so dimension is preserved and f is bijective, and so the question is valid.

Last edited: Jan 7, 2007


text: AbstractQuestion Answering can be improved by focusing on three areas like the ontology enhanced processing and augmentation, content manipulation approaches, the query and the answer. Ontology enhanced processing could enhance answers to include identified objects satisfying a query. Natural language user questions and information sources with a common ontology are required for ontology-based QA systems. A Question Answering System returns answer to a user question in succinct form. In order to provide a precise answer, the system must know what exactly a user wants. The prior knowledge of the expected answer type helps the Question Answering System to extract correct and precise. Question Answering is one of the major issues in e-learning research on how to provide more interactive activities around the learners and instructors. Every answer to the questions must be relevant to the users query in that context. The input is given to the tree-tagger parser to identify the syntactical information. This syntactical information gives us the lexical constraints like Noun {NN}, Verb {VV} and other terms. The noun and verb keywords are analyzed with the semantic meaning using WordNet and semantic similarity measures. This paper proposes a method for QA system by providing different patterns for the same questions.


text:
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# Calling All Consortium 2013 Applicants

Author Message
Manager
Joined: 19 Aug 2012
Posts: 122
Schools: Booth '15 (M)
GMAT 1: 700 Q48 V38
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Kudos [?]: 17 [0], given: 10

Re: Calling All Consortium 2013 Applicants [#permalink]

### Show Tags

07 Feb 2013, 11:06
haha well id hope there are more interview invites coming for us because it doesnt seem like many of us have received too many invites i figure over the next week and a half well see a good chunk more action
Intern
Joined: 29 Jun 2012
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Concentration: Entrepreneurship
WE: Corporate Finance (Real Estate)
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Re: Calling All Consortium 2013 Applicants [#permalink]

### Show Tags

07 Feb 2013, 11:33
RossMBA2014 wrote:
method wrote:
Also, just a heads up for Ross Consortium applicants: I've heard there have been delays in when some Consortium applications reached Ross. My guess is that there is still a Consortium interview invite wave coming.

I have heard that as well that the consortium has been delayed in sending apps to Ross (and other schools). Best of luck guys

I just got a confirm email yesterday from Ross...so that makes sense.
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Re: Calling All Consortium 2013 Applicants [#permalink]

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07 Feb 2013, 12:05
Hi! New to the conversation, but have been lurking here (and on last year's thread) for a while.

Anybody know anything about Yale interview invites? I'm wondering if not having heard anything yet is a bad sign...Yale is my top choice, and I really want to get to the interview stage at least!

I haven't heard from Cornell yet, either, but was invited to interview with Ross on the 31st (Went to spam! Be on the look out!).

Thanks!
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Re: Calling All Consortium 2013 Applicants [#permalink]

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07 Feb 2013, 12:36
CT2 wrote:
Hi! New to the conversation, but have been lurking here (and on last year's thread) for a while.

Anybody know anything about Yale interview invites? I'm wondering if not having heard anything yet is a bad sign...Yale is my top choice, and I really want to get to the interview stage at least!

I haven't heard from Cornell yet, either, but was invited to interview with Ross on the 31st (Went to spam! Be on the look out!).

Thanks!

I haven't heard from Yale either...I'm leaning towards thinking I'm a ding.
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Re: Calling All Consortium 2013 Applicants [#permalink]

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07 Feb 2013, 12:43
Has anyone from Consortium received a Yale interview invite?
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Re: Calling All Consortium 2013 Applicants [#permalink]

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07 Feb 2013, 13:50
minkathebest wrote:
Has anyone from Consortium received a Yale interview invite?

I received my interview invite on January 7th. The e-mail did ask Consortium applicants to schedule their interview on or before February 14th. I'm not sure how likely they'd be to send out more invites a week prior to the deadline. It might be worth looking into it to make sure all of your Consortium material transferred over correctly?

Last edited by andreak on 07 Feb 2013, 14:03, edited 1 time in total.
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Re: Calling All Consortium 2013 Applicants [#permalink]

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07 Feb 2013, 13:52
MBAConfused wrote:
andreak wrote:
Do any of you guys know when in March Fellowship decisions will be made?

Foster was kind enough to extend my deposit deadline until March 20th, and I'm hoping that this will give me enough time to find out about whether I made it into Yale and Ross, and how much financial aid I can expect from them. Fuqua isn't budging on their deposit deadline, which is February 27th, so even though it's a great school, I'm thinking of letting it go and falling back on Foster if UM and Yale don't work out.

From last year's thread Schools were making calls to consortium fellows around 3/14

Thanks!
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Re: Calling All Consortium 2013 Applicants [#permalink]

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07 Feb 2013, 14:20
andreak wrote:
minkathebest wrote:
Has anyone from Consortium received a Yale interview invite?

I received my interview invite on January 7th. The e-mail did ask Consortium applicants to schedule their interview on or before February 14th. I'm not sure how likely they'd be to send out more invites a week prior to the deadline. It might be worth looking into it to make sure all of your Consortium material transferred over correctly?

Hmmm...it's showing up on Yale page that my GMAT score is missing....
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Re: Calling All Consortium 2013 Applicants [#permalink]

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07 Feb 2013, 14:23
bftw wrote:
andreak wrote:
minkathebest wrote:
Has anyone from Consortium received a Yale interview invite?

I received my interview invite on January 7th. The e-mail did ask Consortium applicants to schedule their interview on or before February 14th. I'm not sure how likely they'd be to send out more invites a week prior to the deadline. It might be worth looking into it to make sure all of your Consortium material transferred over correctly?

Hmmm...it's showing up on Yale page that my GMAT score is missing....

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Re: Calling All Consortium 2013 Applicants [#permalink]

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07 Feb 2013, 15:15
andreak wrote:
minkathebest wrote:
Has anyone from Consortium received a Yale interview invite?

I received my interview invite on January 7th. The e-mail did ask Consortium applicants to schedule their interview on or before February 14th. I'm not sure how likely they'd be to send out more invites a week prior to the deadline. It might be worth looking into it to make sure all of your Consortium material transferred over correctly?
andreak wrote:
minkathebest wrote:
Has anyone from Consortium received a Yale interview invite?

Thanks for your input, that makes sense. My application didn't actually get sent to Yale from the Consortium until later much later than Jan 7th--around the 24th or so--due to a delay in ETS score reporting (GRE taker here). The Consortium assured me that it would not reflect badly on me since I got all of my materials in on time, and that basically all of the people who reported GRE scores were in the same boat. I'm going to go ahead and assume that that delay is the reason for this delay (gotta keep hope alive until the ding comes, right?).

I remember when I visited the school the Director of Admissions said that they send out interview invites right up to their deadlines, so not receiving one right away doesn't necessarily mean you won't be asked. I hope that is true!
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Re: Calling All Consortium 2013 Applicants [#permalink]

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07 Feb 2013, 15:43
Finally got an invite to Tepper today! Any ladies traveling to the Women at Tepper event on Febraury 18th?
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Re: Calling All Consortium 2013 Applicants [#permalink]

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07 Feb 2013, 17:46
Anyone else scheduled for Consortium interview days at Johnson tomorrow and terrified about your travel plans/ getting stranded it Ithaca in a blizzard?
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Re: Calling All Consortium 2013 Applicants [#permalink]

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07 Feb 2013, 17:50
1
KUDOS
Expert's post
sonjeroo wrote:
Anyone else scheduled for Consortium interview days at Johnson tomorrow and terrified about your travel plans/ getting stranded it Ithaca in a blizzard?

Welcome to life in Ithaca
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Re: Calling All Consortium 2013 Applicants [#permalink]

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08 Feb 2013, 06:50
helpmehelpme wrote:
then one could argue that your investment in the consultants may have led to better essays and application which may have been responsible for your full ride to vandy. if you ranked darden as your #1 then maybe its worth the $1k to travel there and eat that costs to save the 55k a year in tuition from a potential full ride. its an investment. no free lunches but consortium does a damn well good job at offering great opportunities, network, and a high fellowship chance compared to traditional applications. definitely not a bait and switch So wrong kid.... Vandy was my final application also without a consultant's help... nice try... Manager Status: 2013 Joined: 26 Jun 2010 Posts: 62 Followers: 0 Kudos [?]: 17 [2] , given: 1 Re: Calling All Consortium 2013 Applicants [#permalink] ### Show Tags 08 Feb 2013, 07:48 2 This post received KUDOS ...what a way to end the week! Accepted to Tuck...and Cornell... #excited!! Manager Joined: 19 Aug 2012 Posts: 122 Concentration: Finance, International Business Schools: Booth '15 (M) GMAT 1: 700 Q48 V38 Followers: 2 Kudos [?]: 17 [1] , given: 10 Re: Calling All Consortium 2013 Applicants [#permalink] ### Show Tags 08 Feb 2013, 08:29 1 This post received KUDOS rrossjd wrote: helpmehelpme wrote: then one could argue that your investment in the consultants may have led to better essays and application which may have been responsible for your full ride to vandy. if you ranked darden as your #1 then maybe its worth the$1k to travel there and eat that costs to save the 55k a year in tuition from a potential full ride. its an investment. no free lunches but consortium does a damn well good job at offering great opportunities, network, and a high fellowship chance compared to traditional applications. definitely not a bait and switch

So wrong kid.... Vandy was my final application also without a consultant's help... nice try...

what am i nice trying? youre the one who made the comment about using the two consultants and seemed like your comment about that was a bit snarky to machichi but even if its without a consultant youre still gonna gain some insight from your consultants assistance in other apps for your very last application..

tldr the whole process is expensive and costs money for everyone involved no matter what and we all spend money on alternate transcripts, gmat scores, gmat prep, travel, time, etc everything else. ultimately it is up to us to pick the schools and go with the methods that give us the best return on investment for our own desired goals. if something doesnt work for you then you have to make that decision
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Re: Calling All Consortium 2013 Applicants [#permalink]

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08 Feb 2013, 08:30
jay4den wrote:
...what a way to end the week! Accepted to Tuck...and Cornell... #excited!!

congrats jay
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Re: Calling All Consortium 2013 Applicants [#permalink]

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08 Feb 2013, 09:39
CT2 wrote:
andreak wrote:
minkathebest wrote:
Has anyone from Consortium received a Yale interview invite?

I received my interview invite on January 7th. The e-mail did ask Consortium applicants to schedule their interview on or before February 14th. I'm not sure how likely they'd be to send out more invites a week prior to the deadline. It might be worth looking into it to make sure all of your Consortium material transferred over correctly?
andreak wrote:
minkathebest wrote:
Has anyone from Consortium received a Yale interview invite?

Thanks for your input, that makes sense. My application didn't actually get sent to Yale from the Consortium until later much later than Jan 7th--around the 24th or so--due to a delay in ETS score reporting (GRE taker here). The Consortium assured me that it would not reflect badly on me since I got all of my materials in on time, and that basically all of the people who reported GRE scores were in the same boat. I'm going to go ahead and assume that that delay is the reason for this delay (gotta keep hope alive until the ding comes, right?).

I remember when I visited the school the Director of Admissions said that they send out interview invites right up to their deadlines, so not receiving one right away doesn't necessarily mean you won't be asked. I hope that is true!

also that is probably for first rounders not 2nd round seeing as i dont think consortium event sent round 2s out by then?
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Re: Calling All Consortium 2013 Applicants [#permalink]

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08 Feb 2013, 11:27
Applied R2 to Darden, email confirmation Feb 1, just got interview invite on Feb 8.
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Re: Calling All Consortium 2013 Applicants [#permalink]

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08 Feb 2013, 13:37
So excited! Have my Marshall Interview in less than two hours, and also got an interview invite from Anderson earlier this afternoon! Now I'm just holding my breath for Haas.
Re: Calling All Consortium 2013 Applicants   [#permalink] 08 Feb 2013, 13:37

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# Calling All Consortium 2013 Applicants

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text:
# Posts published in “Hashtable”

pangram is a sentence where every letter of the English alphabet appears at least once.

Given a string sentence containing only lowercase English letters, returntrue if sentence is a pangram, or false otherwise.

Example 1:

Input: sentence = "thequickbrownfoxjumpsoverthelazydog"
Output: true
Explanation: sentence contains at least one of every letter of the English alphabet.

Example 2:

Input: sentence = "leetcode"
Output: false

Constraints:

• 1 <= sentence.length <= 1000
• sentence consists of lowercase English letters.

## Solution: Hashset

Time complexity: O(n)
Space complexity: O(26)

## Python3

You are given the logs for users’ actions on LeetCode, and an integer k. The logs are represented by a 2D integer array logs where each logs[i] = [IDi, timei] indicates that the user with IDi performed an action at the minute timei.

Multiple users can perform actions simultaneously, and a single user can perform multiple actions in the same minute.

The user active minutes (UAM) for a given user is defined as the number of unique minutes in which the user performed an action on LeetCode. A minute can only be counted once, even if multiple actions occur during it.

You are to calculate a 1-indexed array answer of size k such that, for each j (1 <= j <= k), answer[j] is the number of users whose UAM equals j.

Return the array answer as described above.

Example 1:

Input: logs = [[0,5],[1,2],[0,2],[0,5],[1,3]], k = 5
Output: [0,2,0,0,0]
Explanation:
The user with ID=0 performed actions at minutes 5, 2, and 5 again. Hence, they have a UAM of 2 (minute 5 is only counted once).
The user with ID=1 performed actions at minutes 2 and 3. Hence, they have a UAM of 2.
Since both users have a UAM of 2, answer[2] is 2, and the remaining answer[j] values are 0.

Example 2:

Input: logs = [[1,1],[2,2],[2,3]], k = 4
Output: [1,1,0,0]
Explanation:
The user with ID=1 performed a single action at minute 1. Hence, they have a UAM of 1.
The user with ID=2 performed actions at minutes 2 and 3. Hence, they have a UAM of 2.
There is one user with a UAM of 1 and one with a UAM of 2.
Hence, answer[1] = 1, answer[2] = 1, and the remaining values are 0.

Constraints:

• 1 <= logs.length <= 104
• 0 <= IDi <= 109
• 1 <= timei <= 105
• k is in the range [The maximum UAM for a user, 105].

## Solution: Hashsets in a Hashtable

key: user_id, value: set{time}

Time complexity: O(n + k)
Space complexity: O(n + k)

## C++

You are given an array nums that consists of non-negative integers. Let us define rev(x) as the reverse of the non-negative integer x. For example, rev(123) = 321, and rev(120) = 21. A pair of indices (i, j) is nice if it satisfies all of the following conditions:

• 0 <= i < j < nums.length
• nums[i] + rev(nums[j]) == nums[j] + rev(nums[i])

Return the number of nice pairs of indices. Since that number can be too large, return it modulo 109 + 7.

Example 1:

Input: nums = [42,11,1,97]
Output: 2
Explanation: The two pairs are:
- (0,3) : 42 + rev(97) = 42 + 79 = 121, 97 + rev(42) = 97 + 24 = 121.
- (1,2) : 11 + rev(1) = 11 + 1 = 12, 1 + rev(11) = 1 + 11 = 12.

Example 2:

Input: nums = [13,10,35,24,76]
Output: 4

Constraints:

• 1 <= nums.length <= 105
• 0 <= nums[i] <= 109

## Solution: Two Sum

Key = x – rev(x)

Time complexity: O(n)
Space complexity: O(n)

## C++

You are given a string s that contains some bracket pairs, with each pair containing a non-empty key.

• For example, in the string "(name)is(age)yearsold", there are two bracket pairs that contain the keys "name" and "age".

You know the values of a wide range of keys. This is represented by a 2D string array knowledge where each knowledge[i] = [keyi, valuei] indicates that key keyi has a value of valuei.

You are tasked to evaluate all of the bracket pairs. When you evaluate a bracket pair that contains some key keyi, you will:

• Replace keyi and the bracket pair with the key’s corresponding valuei.
• If you do not know the value of the key, you will replace keyi and the bracket pair with a question mark "?" (without the quotation marks).

Each key will appear at most once in your knowledge. There will not be any nested brackets in s.

Return the resulting string after evaluating all of the bracket pairs.

Example 1:

Input: s = "(name)is(age)yearsold", knowledge = [["name","bob"],["age","two"]]
Output: "bobistwoyearsold"
Explanation:
The key "name" has a value of "bob", so replace "(name)" with "bob".
The key "age" has a value of "two", so replace "(age)" with "two".

Example 2:

Input: s = "hi(name)", knowledge = [["a","b"]]
Output: "hi?"
Explanation: As you do not know the value of the key "name", replace "(name)" with "?".

Example 3:

Input: s = "(a)(a)(a)aaa", knowledge = [["a","yes"]]
Output: "yesyesyesaaa"
Explanation: The same key can appear multiple times.
The key "a" has a value of "yes", so replace all occurrences of "(a)" with "yes".
Notice that the "a"s not in a bracket pair are not evaluated.

Example 4:

Input: s = "(a)(b)", knowledge = [["a","b"],["b","a"]]
Output: "ba"

Constraints:

• 1 <= s.length <= 105
• 0 <= knowledge.length <= 105
• knowledge[i].length == 2
• 1 <= keyi.length, valuei.length <= 10
• s consists of lowercase English letters and round brackets '(' and ')'.
• Every open bracket '(' in s will have a corresponding close bracket ')'.
• The key in each bracket pair of s will be non-empty.
• There will not be any nested bracket pairs in s.
• keyi and valuei consist of lowercase English letters.
• Each keyi in knowledge is unique.

## Solution: Hashtable + Simulation

Time complexity: O(n+k)
Space complexity: O(n+k)

## C++

You are given a string word that consists of digits and lowercase English letters.

You will replace every non-digit character with a space. For example, "a123bc34d8ef34" will become " 123  34 8  34". Notice that you are left with some integers that are separated by at least one space: "123""34""8", and "34".

Return the number of different integers after performing the replacement operations on word.

Two integers are considered different if their decimal representations without any leading zeros are different.

Example 1:

Input: word = "a123bc34d8ef34"
Output: 3
Explanation: The three different integers are "123", "34", and "8". Notice that "34" is only counted once.

Example 2:

Input: word = "leet1234code234"
Output: 2

Example 3:

Input: word = "a1b01c001"
Output: 1
Explanation: The three integers "1", "01", and "001" all represent the same integer because
the leading zeros are ignored when comparing their decimal values.

Constraints:

• 1 <= word.length <= 1000
• word consists of digits and lowercase English letters.


text:
21 svar
112 visningar
Nox_M är nöjd med hjälpen
Nox_M 62

# Linjär Algebra

(a)När jag försöker lösa 'det(A-λI)=0' så får jag '-x³+7x²+ax-15x+10-2a=0'. Hur kan jag lösa den när jag har fler okända variabler?

Finns det något annat sätt att svara på (a) än att först hitta eigenvärde och eigenvektorer?

(Ps: I am not sure if my Swedish is correct so if my question is confusing, please ask me!

To the mods: is it allowed to write in English instead of Swedish?)

Dracaena Online 4937 – Moderator

Although you could attempt to calculate the eigenvalues/vectors and check the condition A=PDP^(-1), it is not at all necessary.

A matrix is diagonalizable if and only if for each eigenvalue the dimension of the eigenspace is equal to the multiplicity of the eigenvalue.

That is, suppose we have a 2x2 square matrix. It should be obvious that if n denotes rows, then n=2. If we can find two (n distinct) distinct eigenvalues for this given 2x2 matrix then we can conclude that it is indeed diagonalizable without having to show that A=PDP^(-1).

As far as I know, writing your thread in English is not per se breaking the rules but keep in mind you'll be far less likely to recieve help.

Nox_M 62

My main issue is how to find the eigenvalues. If I use det(A-λI)=0 then I get an equation that has too many unknowns. Is there an alternative method I can use to find the eigenvalues?

oneplusone2 551

There is a lucky root to your characteristic polynomial, x=2. You can then factor the polynomial using polynomial division.

Once you have done that you will be able to see that there are 3 distinct eigenvalues in all cases but one. With 3 distinct eigenvalues, diagonalizability follows from gm <= am with a minimum of 1 for gm. In the special case you can simply substitute that value for a into your original matrix and manually evaluate the situation.

I have ignored complex solutions here.

Nox_M 62

So I factored the polynomial and got:

(λ-2)(-λ²+5λ-5+a)=0

I don't know where to go from here. How do I factor (-λ²+5λ-5+a)?

oneplusone2 551

complete the square

Nox_M 62

I did that and I got

λ = a + 3,75

So does that mean that A is diagonalizable for all real number values of a? is -3,75 included?

oneplusone2 551

write out the polynomial with only linear factors

Nox_M 62

I completed the square but I made a wrong assumption along the way

-λ² + 5λ - 5 + a=0

λ² - 5λ = a - 5

λ² - 5λ + 6,25 = a - 5 + 6,25

(λ - 2,5)(λ - 2,5) = a + 1,25

(λ - 2,5)² = a + 1,25

λ - 2,5 = ±√(a + 1,25)

λ  = ±√(a + 1,25) + 2,5

Have I done it correctly? and if so can I assume that A is diagonalizable for all numbers when a ≥1,25?

oneplusone2 551

it looks correct,

now write the polynomial like this -(x-a1)(x-a2)(x-a3)

Nox_M 62

(λ  + √(a + 1,25) - 2,5)(λ - √(a + 1,25) - 2,5)(λ - 2)= 0

So, I see that for any choice of a≥1,25 there are 3 distinct eigenvalues except in the case when a=1,25, there are two eigenvalues, λ=2 and λ=2,5 (multiplicity 2)

Am I on the right track?

oneplusone2 551

yes now u must evaluate the case a=1.25 manually by inserting it into your original matrix

Nox_M 62

Why do I have the evaluate the special case of a=1,25 manually?

Can I not assume that since a=1,25 gives me two eigenvalues one of which has multiplicity 2, it automatically satisfies the condition that there are 3 eigenvalues for the 3 x 3 matrix?

oneplusone2 551

u dont know what the geometirc multiplicity of the a=1.25 is. you need to determine which vectors are associated with the a=1.25 case

Nox_M 62

I will put everything in writing and post it and if you have time later you can give me feedback.

Thank you very much for all the help so far! : )

Nox_M 62

I skipped the calculation steps (polynomial division, completing the square, row reduction) so that the information on the page is readable. Is my answer correct?

oneplusone2 551

I get the impression that you need to have good look in your textbook reguarding diagonalization and geometric and algebraic multiplicity.

For the case of a=-1.25 we have established that there are 2 eigenvalues, x=2 and 2.5. You still have to investigate what vectors belong to the x=2 eigenvalue.

Nox_M 62

Yeah, my knowledge of linear algebra is very superficial. I need to do a whole lot of revision.

There is one eigenvector that belongs to λ=2

I automatically assumed that since λ=2 has multiplicity 1, it could only give me one eigenvector. Was my assumption wrong?

Now that I found that there are only 2 eigenvectors when a=-1.25, does that change my answer?

Thank you for being so patient with me : )

oneplusone2 551

However my idea was to give you the follow up question: Can you diagonalize A using the eigenvectors you have discovered for a=-1.25 ?

Nox_M 62

I need 3 linearly independent eigenvectors to diagonalize a 3 x 3 matrix. In this case, I only have 2 eigenvectors so I would assume they are not enough to diagonalize A. Is my reasoning correct?

oneplusone2 551


text:
## Chemistry (4th Edition)

$Ka = 4.801 \times 10^{-9}$
1. Drawing the equilibrium (ICE) table we get these concentrations at equilibrium: -$[H_3O^+] = [Conj. Base] = x$ -$[Acid] = [Acid]_{initial} - x$ 2. Convert the pH into hydronium concentration: $[H_3O^+] = 10^{-pH}$ $[H_3O^+] = 10^{- 4.52}$ $[H_3O^+] = 3.01 \times 10^{- 5}M$ 3.. Write the Ka equation, and find its value: $Ka = \frac{[H_3O^+][Conj. Base]}{ [Acid]}$ $Ka = \frac{x^2}{[Initial Acid] - x}$ $Ka = \frac{( 3.02\times 10^{- 5})^2}{ 0.19- 3.02\times 10^{- 5}}$ $Ka = \frac{ 9.12\times 10^{- 10}}{ 0.19}$ $Ka = 4.801\times 10^{- 9}$


text:
# 3.1. Overall

A model has two parts, a descriptor that maps atomic configuration to a set of symmetry invariant features, and a fitting net that takes descriptor as input and predicts the atomic contribution to the target physical property. It’s defined in the model section of the input.json, for example,

    "model": {
"type_map":	["O", "H"],
"descriptor" :{
"...": "..."
},
"fitting_net" : {
"...": "..."
}
}

The two subsections, descriptor and fitting_net, define the descriptor and the fitting net, respectively.

The type_map is optional, which provides the element names (but not necessarily same with the actual name of the element) of the corresponding atom types. A model for water, as in this example, has two kinds of atoms. The atom types are internally recorded as integers, e.g., 0 for oxygen and 1 for hydrogen here. A mapping from the atom type to their names is provided by type_map.

DeePMD-kit implements the following descriptors:

1. se_e2_a: DeepPot-SE constructed from all information (both angular and radial) of atomic configurations. The embedding takes the distance between atoms as input.

2. se_e2_r: DeepPot-SE constructed from radial information of atomic configurations. The embedding takes the distance between atoms as input.

3. se_e3: DeepPot-SE constructed from all information (both angular and radial) of atomic configurations. The embedding takes angles between two neighboring atoms as input.

4. loc_frame: Defines a local frame at each atom, and the compute the descriptor as local coordinates under this frame.

5. hybrid: Concate a list of descriptors to form a new descriptor.

The fitting of the following physical properties are supported

1. ener: Fit the energy of the system. The force (derivative with atom positions) and the virial (derivative with the box tensor) can also be trained.

2. dipole: The dipole moment.

3. polar: The polarizability.