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# Questions tagged [gravity]

Questions regarding the attractive force which exists between any two bodies of matter.

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### Gas Planets composition and gravitational effects

It has been postulated that Gas Giants have a solid core and an obvious gaseous atmosphere. Question: As the gas is a state of matter and gravity draws all matter inward, is the reason the gas is ...
77 views

### Up and Down in Space [closed]

I understand the theory of gravitational force as the arbiter of direction in space, however when viewing a galaxy, say M31, there is clearly a depth (and width) to the galaxy. Traveling to a star ...
160 views

### How does gravity affect a physical clock?

In all the explanations for time dilation and how gravity affects time a common example which is used is "Clock near to the ground will tick away faster than clock away from the ground due to gravity"....
49 views

### At what gravity does a body's gravitational lensing become observable?

I wonder whether we know at how much gravitation a gravitational lensing around a celestial body becomes observable (visible in the sense of that we see stars on the wrong places, stars behind the ...
53 views

### Can the gravitational potential in a spiral galaxy be positive?

Thinking about this question I wanted to start with a rough model of the average gravitational potential of the Milky way. I ran across D. P. Cox and G. C. Gomez 2002 Analytical Expressions for Spiral ...
41 views

### What influence does the Interplanetary Magnetic Field have on Planetary Orbits?

CONTEXT The equation $F_D=\frac{GMm}{D^2}$ is a standard equation in Newtonian gravitation. It describes the centripetal force exerted, by a source mass$M$, on a target particle of mass $m$ located ...
49 views

### How does the surface gravity on Mars vary between the equator and its poles?

A day on Mars is similar in length to the Earth's day. On Earth, due to rotation, the surface gravity varies from 9.78 m/s² (0.997 g) on the equator to 9.832 m/s² (1.003 g) on the poles. I wonder how ...
84 views

### Gravitational field for oblate spheroid?

Consider a planet, described as an oblate spheroid. Assume that the spheroid is uniformly dense but not a point source. Outside of the object, do all vectors in the gravity field point through a ...
60 views

### can gravity be faster than light in some specific cases? [duplicate]

This is a follow up question from If an event occurs at a point in space time what will an observer experience first, it's gravity or the light from it? This https://en.wikipedia.org/wiki/...
96 views

### If an event occurs at a point in space time what will an observer experience first, it's gravity or the light from it?

Let's assume a person is close enough to an object in space time to be able to observe it's gravitational effects and the light coming from it. If that object suddenly pops out of existence (...
3k views

### What observation can be expected on LIGO if any when Betelgeuse goes supernova?

What observation can be expected from the LIGO† gravitational wave observatory if any when Betelgeuse goes supernova? Could we know that Betelgeuse has gone supernova before we see it light up our ...
157 views

### Would planet explode without gravity?

A planet (as well as a dwarf planet) must, according to the IAU definition, have sufficient mass to assume hydrostatic equilibrium (a nearly round shape). Does it mean they would break apart or ...
20 views

### A question about stress energy tensor at junction surface

I am now reading a paper (whole paper link)which regarding construct a worm hole by cutting off the stress energy tensor and glue interior and exterior space at $r=a$. ‘Nevertheless, I am confused ...
5k views

### Moon's unusual gravity

We know that the Moon's gravity is about one-sixth that of Earth. Then I recently read that the mass of the Moon is about one-eightieth the mass of Earth. Since gravity depends on the mass of the 2 ...
85 views

### How do I calculate the Hill radius of a star in a binary system?

Is there an easy way to calculate the Hill radius of a star in a binary system at different orbital radii where both stars are of the same mass and in circular orbit around one another’s centre of ...
23 views

### Doppler Shift Vs Gravity Assist for Frequency Shifts from Rotating Galaxies

Frequency Shift of Light Leaving Rotating Galaxies: Differences in Doppler Shift due to Radial Velocity Vs Frequency Shift due to Gravity Assist from a Rotating Galaxy A description of estimating ...
284 views

### A Sky Map of Gravity - what would it look like?

PREAMBLE Sky Maps exist which show the distribution and intensity of electromagnetic radiation received on Earth at various wavelengths. An obvious form is a map of the stars (Star Chart). Maps ...
9k views

### Is Earth's Surface “In orbit”?

I'm having trouble understanding relative angular/tangential speeds at increasing altitudes above Earth's surface. In particular, I find this comparison of tangential velocities on Wikipedia very ...
85 views

### How to determine the orbital period of an asteroid using Kepler laws given its orbit's radius?

Given that the mean distance of an asteroid from the Sun is $450 \times 10^6$ km is it possible that the asteroid completes one rotation around the Sun in two years? According to the third law of ...
86 views

### Can we use Breakthrough Starshot spacecraft to detect any unknown massive object in our solar system?

For example, the hypothetical Planet Nine. Since we just want to prove or rule out the existence of Planet Nine in our solar system, we can avoid most of the technical challenges for StarChip (camera ...
2k views

### Gravitational Properties of Asteroids

How large would an asteroid have to be in order to hold a person so that the person could not escape?
66 views

### Rocket Equation from Newton’s 3rd Law, Force of Gravitation, and dp/dt [closed]

I am using the derivative of momentum (dp/dt) with Newton’s 3rd Law with the gravitational force of Earth.    F - F_{gravity} = \dfrac{\mathrm{d}p}{\mathrm{d}t} \\ F - \dfrac{G \, m_e \, m_r}{...
76 views

### How is Earth's gravity modelled?

The Earth does not have a defined geometric shape. Then how do space scientists who want to put a satellite in orbit model the gravitational field of Earth? Do they assume a single object with minute ...
82 views

### Does the Lagrange L1 point have a 'size'?

Like in cubic kilometers, what is the size of L1 "area of influence"? being unstable I guess that size can vary?
213 views

### How much volume of gases could be added to Mars atmosphere that the planet gravity can retain?

This article Mars atmosphere states that Today Mars atmosphere is very thin, but it was not always like this. Around 3.5 billion years ago, the red planet had a thick enough atmosphere for liquid ...
100 views

### Where would an object fly to if it were to leave the gravitational field of Earth? And would it stop somewhere? What would happen to it?

My 3,5 year old son is asking :-) He can take answers for a 15 yo I think. Many thanks for your help!
107 views

### The dominance of Gravity vs Strength and its dependency on crater size

So I am trying to write a research paper (about 3700 words) for high school about the topic "The dominance of Gravity vs Strength and its dependency on crater size" or something similar. I have ...
4k views

### How does the Moon's gravity affect Earth's oceans despite Earth's stronger gravitational pull?

Given that Earth has a much stronger gravitational pull than the Moon, how does the Moon have any influence on Earth's oceans?
132 views

### Have more recent LIGO/VIRGO gravitational wave measurements narrowed down the speed of gravity further?

This answer to How precise are the observational measurements for the speed of gravity? says: ...in 2013 a Chinese group built a model using Earth's tides that helped them narrow it down. ...
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### Do gravitationally bound (e.g. orbiting) objects really resist the expansion of space?

Thinking about the expansion of space, it is commonly explained that gravitationally bound objects (like galaxies and galaxy clusters) will not be affected by the expansion of space (dark energy). I ...
103 views

### Black hole's gravity

Is gravity relative to volume, or size of an object? Since a black hole is a massive star that collapses on its weight, how comes the same sun's mass, when it becomes a black hole, provides gravity ...
2k views

### Could the rotation of a black hole cause other planets to rotate?

Because everything rotates so perfectly, and we know that some blackholes rotate and they're the biggest things in the universe, so a silly question comes to mind. If you remove all black holes from ...
26 views

### Why aren't globular clusters disk-shaped? [duplicate]

I thought that solar systems and galaxies are disk-shaped because that is the most stable shape under gravitation. Globular clusters are very old, often times older than their host galaxies, so why ...
2k views

### Why can't supermassive black holes merge? (or can they?)

The CNet article Astronomers discover two supermassive black holes in a death spiral links to Discovery of a Close-separation Binary Quasar at the Heart of a z ~ 0.2 Merging Galaxy and Its ...
93 views

### What effect does Earth losing a part of its mass have, is there an impact on the Earth's orbit around the Sun?

This is a random question that popped into my mind. I'm not a physicist or astronomist, however, I am genuinely curious. Also, this is not intended as a science fiction question, but an actual science ...
133 views

### Wormholes man made are real? [closed]

Can we bend two points of spacetime in a way we would create a wormhole ?Why havent we tried it yet?How much energy would we need to do it?
122 views

### If I could see from a Gravitational-Time dilated place to somewhere far which is not dilated, would I see fast moving people?

Gravitational time dilation like the one shown in the movie Interstellar, causes time to move slowly for someone within the high gravity reference frame. Now let's say that from such a place, I was ...
6k views

### Why do things float in space, though the gravity of our star is always present? [closed]

In the solar system, things float around. We are so certain that the gravity of our sun exists. Still, why does that gravity not influence satellites and other objects there? Our solar system is also ...
11k views

### Why can't we feel the Earth's revolution?

I googled it and checked a few Q&A and there's only things about "Earth's rotation". But why can't we feel the revolution? They say we can't feel the rotation because the Earth spins at a ...
69 views

### If a solar system were surrounded by a cloud of debris, is it possible for a planet's orbit to intersect it?

I'm doing research for a story that I'm writing and I'd like some physics advice on whether a hypothetical system of celestial bodies is possible. Suppose the following: A system of planets orbiting ...
6k views

### Can a magnetic field of an object be stronger than its gravity?

Can a planet, star or otherwise have a magnetic field that is stronger or have more range than its gravity?
16k views

### If a massive object like Jupiter flew past the Earth how close would it need to come to pull people off of the surface?

I understand this is a silly hypothetical but I'm asking for a 7 year old so please bear with me. Imagine an interstellar stray gas giant comes flying through our solar system. If we were not ...
297 views

### Can a black hole “supernova”?

In layman terms: nothing ever escapes the pull of a black hole, not even light when a super massive star reaches the end of it's life you get a supernova sometimes the "remains" of these stars can ...
70 views

### Planets may repel each other in few cases

We know that when two huge black holes collide they produces gravitational waves in space time curvature. So if I am not wrong another object in space if goes in that direction during production of ...
6k views

### Are black holes spherical during merger?

I've been thinking about black holes, specifically during the final moments before two merge. I'm wondering if black holes, or I guess more specifically their event horizons, are always spherical. ...
53 views

### A quantitive estimate of the size above which celestial bodies must be near-spherical [duplicate]

Asteroids and smaller celestial objects tend to be odd-shaped, whereas planets are always spherical (or slightly oblate due to their spin). The reason obviously is that for sufficiently large masses/...
172 views

### Do photons slow down as they approach the event horizon of a black hole?

So, if a black hole is so dense, that light cannot escape, what happens to the photons as the approach and go past the event horizon (or some other horizon, about which I am ignorant)? Do photons ...
57 views

### Does time dilation at the event horizon prevent us from seeing any changes?

If gravity is so strong at event horizon as we assume, and thus time dilation makes time infinitely slow for us, the distant observers, how could we ever observe any events ocurring over time? Wouldn'...
According to the formula for gravitational acceleration $a = GM/r^2$ where $G=6.674 \times 10^{-11}$, Jupiter with mass 1.89813 × 10^27 kg and radius 69,911,000 m gives a gravity around 25m/s^2. The ...


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# Why could the error term be sampled coefficient wise?

In SEAL homomorphic encryption library, it implements the BFV and CKKS. We know the error $$e\in R_q$$ which is a Guassian distribution. When sampling an error term $$e = \sum_{i=0}^{n-1} e_ix^i$$, it just samples $$e_i$$ from some discrete Guassain distribution over the integers.

However, it seems that the discrete Guassian distribution over high dimension lattice could not be sampled coefficient wise. What goes wrong ?

format: plain


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# Homework Help: Tangent to y=e^x

1. Dec 1, 2011

### iRaid

1. The problem statement, all variables and given/known data
Find an equation for a line that is tangent to the graph of y=ex and goes through the origin.

2. Relevant equations

3. The attempt at a solution
y'=ex

That's about all I can think of. I don't know how to make the tangent line go through the origin. Can someone lead me in the right direction?

Last edited: Dec 1, 2011
2. Dec 1, 2011

### Ray Vickson

At any point (x,y) on the graph, how would you compute the slope of the tangent line?

RGV

3. Dec 1, 2011

### iRaid

Find the derivative at that point.

4. Dec 2, 2011

### HallsofIvy

And you have already said that the derivative is again $e^x$.

So the tangent line to $y= e^x$ at $x= x_0$ would be $y= e^{x_0}(x- x_0)+ e^{x_0}$. Now, what must [itex]x_0[/b] be so that goes through (0, 0)?

5. Dec 2, 2011

### iRaid

0? Not really sure..


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Hermit Notebook

# Taxonomy of machine learning

## Introduction

In the previous article of this series about machine learning, I tried to state clearly my understanding of machine learning. I argued that I understand machine learning as a set of algorithms and methods for automatically determining the parameters of mathematical models and automatically choosing the best fitting model for a given problem.

In this article I will try to briefly present my personal taxonomy (or categorization) of machine learning techniques. Hopefully, this will also give us an overview of the usages of machine learning.

Click the table of contents icon on the right to have a hierarchical view of the taxonomy at any moment. I also suggest to open the internal links in new tabs so that you don’t lose your reading position in the article.

## Taxonomy by data set requirements

Depending on the data available and the objective of the software package we are developing, we will train our models differently. In the next sections, we discuss supervised learning where example outputs (expected results) are given, unsupervised learning where no example output is given, and reinforcement learning where learning occurs by interacting with an environment.

### Supervised learning

A supervised learning algorithm requires the presence, at the training phase, of the correct expected output for each element of the initial data set. We say that the inputs are labelled.

Illustration of supervised learning. The algorithm can learn from the labelled images of dogs and cats. The image on the bottom right corner is the new input to be labelled.

For example, let’s suppose that my data set is my email box. Let’s suppose I use to manually sort my emails as “spam” or “not spam”. Now let’s suppose I want to create a program to automatically filters my emails for me. In this setting, I have examples to feed my learning algorithm with: the manually classified emails. So I have not only the inputs in my data set (the emails) but also the corresponding outputs (whether they are in the spam folder or not). An algorithm that relies on such a data set is called supervised. It requires input values and their corresponding outputs. From these examples, it will determine the best prediction function to estimate the outputs for new inputs.

An analogy of supervised learning is often cited as the situation where we show an object to a child and we say the name of that object so that the child learns to name the object.

### Unsupervised learning

Unlike supervised learning, an unsupervised technique does not require example outputs (or labels).

Illustration unsupervised learning. The colored areas show the grouping of unlabelled images of dogs and cats. The algorithm groups together images with similar features.

For instance, let’s assume that I have a database that contains the sales history of customers on an online shop over a decade. Let’s suppose I want to create a recommender system for this shop. My goal is, given a product that a customer is currently viewing, to suggest another product that has the maximum chance to be bought along with the one being viewed. For example, intuitively, I may suggest baby diapers if a customer is viewing a baby bottle. I want my learning algorithm to automatically discover this kind of associations in my dataset.

In this example, I didn’t have any correct output to train my algorithm with. I only had the inputs, i.e. the transactions in the database. Below we discuss association rule learning, a set unsupervised learning techniques designed to extract this kind of relationships from a data set of transactions.

### Semi-supervised learning

I see semi-supervised learning as an unsupervised learning extended by a supervised learning.

Semi-supervised learning is typically used when we have a small amount of labelled inputs and a much larger amount of unlabelled inputs.

Illustration of semi-supervisered learning. An unsupervised phase groups unlabelled images of dogs and cats (colored areas). A supervised phase, using the unique labelled image in the top left corner, can then label all cats as "cat", and all dogs as "not cat".

For example, let’s reformulate our email filtering example such that I only have $20$ emails over $20 000$ that are sorted into $2$ folders as spam or not spam. Most supervised learning algorithms trained on just the $20$ emails will perform poorly on the much larger set of unsorted emails. One solution may be to run an unsupervised algorithm that will group similar emails ($2$ groups in our case). This grouping algorithm is applied on the complete data set ($20000+20$ emails). Then we would assign to any unsorted email the folder of its closest neighbour (according to some metric) among the $20$ sorted emails that lies in the same group. More clearly, let’s denote an email by a number ranging from $1$ to $20020$ where $1$ to $20$ denote the sorted emails. If for instance email $2$ (which is among the $20$ sorted emails) has the shortest distance from any email denoted $i$ and email $2$ is a spam, then we would consider email $i$ to be a spam as well.

In this example we actually used two unsupervised algorithms: the first one to create the groups is a clustering algorithm and the other one to find the nearest neighbours is a classification algorithm, 2 taxons that we’ll discuss a few sections later.

I shall mention that some authors have the inverse point of view - seeing semi-supervised learning as a supervised learning extended by an unsupervised learning - where the unsupervised learning can be seen as a preprocessing.

Finally, we will note that semisupervised learning is often used in configurations where collecting data is cheap but labelling is expensive (generally time-consuming).

### Reinforcement learning

Reinforcement learning does not require an initial data set, but rather an environment (or context). Data is collected by the learning algorithm by exploring the environment. The algorithm tries to optimize a decision process toward a goal.

Illustration of reinforcement learning. The algorithm explores the environment and learns how to get to its goal (attained in picture 6).

Problems for reinforcement learning include optimizing the behaviour of autonomous agents, decision making for stock market investments, route optimization, game strategy, business strategy and more.

## Taxonomy by mathematical representation

Some machine learning techniques simply compare new instances to the ones they were trained on. Other techniques try to find a general representation of the relationships in the dataset. The first are called instance-based and the latter model-based.

### Instance-based learning

In order to make estimations, a predictor trained by an instance-based learning algorithm compares new data to the training examples.

Illustration of instance-based learning. The red dot is the "new" input to classify. Its closest neighbours lie in the red circle. It will be classfied as "star" because this label if the most common among its closest neighbours.

Generally a distance metric or sometimes a similarity measure is needeed. Distance metrics include the Euclidian distance, the Hamming distance, the Levenshtein distance (used for text comparison), etc.

Strategies to estimate the property of interest for new data include “voting” and averaging. Voting means that the algorithm’s estimation will be equal to the most common value among the $k$ training examples that are the closest to the new data (according to the chosen distance metric). Averaging means giving as our estimation the average value of the property of interest of these $k$ closest neighbours. In the case of averaging, if $k=1$ then we just give the value of the closest neighbour as our estimate.

In our example of a semisupervised learning email filtering, the unsupervised phase where we sorted the $20000$ unlabelled emails by “just comparing” them to the $20$ labelled emails is an example of an instance-based technique.

### Model-base learning

A model based-learning algorithm tries to generalize from the training examples. The mathematical representation that the algorithm obtains from this generalization process is the model that it will use to make its estimations on new data.

Illustration of model-based learning. The algorithm chooses an hypothesis, a mathematical representation. Then it determines the parameters of this hypothesis based on the available data.

Our discussion on getting an intuition of machine learning gave us (hopefully) a good sense of what learning a model is.

## Taxonomy by training data provisionning scheme

Intuitively, we sense that machine learning techniques should not have a memory of the entire dataset they were trained on. However, the iterative ajustements that the learning technique does on the mathematical representation it is determining are based on the data it is provided with. For these antagonist reasons, we can intuitively understand that many learning techniques will not be able to adjust on new data an already trained representation while keeping it consistent with its previous training (because there is no memory of the previous data).

Learning techniques that require the entire data set for their training are called batch learning methods. Some learning algorithm can actually adjuts an already trained representation to new data. They are called online learning methods.

### Batch learning

In batch learning, all the examples must be provided during the traning phase. The “predictor” resulting from the training is then used in production and no more learning occurs.

In this setting, if we obtain new examples, we need to train a new model from scratch on the complete enriched data set.

For large data set that cannot fit entirely in a computer’s memory, some techniques exist to cope with batch learning. They are called out-of-core methods. Some of them address the issue at the data access and management level, allowing for example to load in memory only the part of the data require for a calculation at time $t$. They usually transform the dataset into a new representation that facilitates these optimizations. Other methods are variants of batch learning to accept the data by small chunks. Mini-batch techniques still require the entire data set at training time but “consume” data by small subsets. Stochastic methods consume one element at a time and pick this element according to a probability distribution.

### Online learning

Unlike batch learning, an online learning technique can be provided with new training examples progressively and changes its representations accordinly, even while being used in production.

An online learning algorithm takes the current best predictor as an additional input.

For many underlying representations, true online learning is not possible. However, depending on the formulation, we can often find a pseudo online algorithm based on recursive algorithms. In this case, the new predictor depends on the current best predictor and all the previous examples (already learnt). On lucky cases the dependency on previous examples can be expressed using an aggregation function. In this case, we do not actually store these data elements but rather the aggregation function and the information needed for its computation (including its previous result). When no such trick is possible, then no guarantee can be given about the storage or memory complexity of recursive pseudo-online algorithms.

## Taxonomy by usage or goal

• Finding the value of a property of a phenomenon that depends on other properties of this phenomenon (or other instances). This task is called regression.

• Finding boundaries to seperate data elements into groups. When the groups are imposed from the beginning (they are part of the inputs of the algorithm), the task is called classification, when the groups are to be discovered by the algorithm we call the task clustering.

• Optimizing decision processes.

• Mining association rules (association rule learning).

• Disassembling a measurement (usually a time series) into its basic independent components. This is blind source separation.

• Reducing the number of input variables of a learning problem, a.k.a dimensionality reduction.

### Regression

Illustration of regression. The input data has only 1 feature. On the left, a degree 3 polynomial regression and on the right a linear regression.

Regression tries to find the value of a property of a phenomenon depending on the values of other properties or instances of the same kind.

Regression typically falls under supervised learning.

For example, suppose an ice cream seller wants to predict its incomes based on temparature forcasts. We would be learning the (model and) parameters of a regression if we were to try to create a software package for this requirement.

### Classification

Classification tries to find boundaries in the dataset so as to seperate the elements into a number of classes known (or defined) before the training.

Illustration of classification. Both of the models illustrated above draw linear boundaries (the data is linearly seperable). Credit: [1]

Classification typicaly fall under supervised learning. However there exist unsupervised classification, like anomaly detection or outliers detection.

For example, if we had a labelled data set consisting of pictures of handwritten digits ranging from $0$ to $9$, training an algorithm to recognize the digits on new pictures would be a classification task where the classes are $1, 2, \dots, 9$. Moreover, our emails filtering example is a classification task with two classes: spam or not spam.

I see classification as a different beast from regression but some authors present them as two similar tasks. For these authors, the only conceptual difference is that the output of regression is continous while the output of classification is discrete. I don’t share this vision, because as stated above I see regression as trying to estimate a value while classification tries to estimate a boundary.

### Clustering

Clustering tries to group observations such that elements belonging to the same group (or cluster) are more similar - according to some similarity measure - and thoses belonging to different groups are more dissimilar.

Illustration of clustering. The plotting represents the clusters with colors. Credit: [2]

Clustering typically is an unsupervised learning task.

We presented clustering while talking about semisupervised learning through our modified example of email filtering. In the real life, clustering helps biologists in creating their taxonomy, marketers in finding groups of customers sharing similar behaviour, search engines in categorizing web pages…

### Decision making

Decision making tries to find which decision to take so as to optimize a gain, the usage of a resource or an evolution toward a goal.

Illustration of reinforcement learning. By exploring the environment, the algorithm learn which decisions to take in order to attain its goal.

Optimal decision making can be learnt using any of supervised, semisupervised or reinforcement learning. It is however often associated with reinforcement learning.

Machine learning for decision making is used for trading on stock markets, building autonomous agents (like planet Mars rovers), making decisions in politics, insurance and many more domains where strategy is important.

### Association rule learning

Association rule learning tries to determine how actors associate items.

Illustration of association rule learning. The rules on this example represent the associations of products often present in customers carts in a shop. Customers buying milk and diapers often buy bread :)

The fundamental conceptual difference with clustering, in my opinion, is that the grouping of clustering is based on the properties of the elements while association rule learning is based on how other agents or actors group elements together in some transactions.

Association rule learning is typically based on unsupervised learning techniques.

We gave an example of association rule learning when we discussed unsupervised learning above. Other usage of algorithms resulting from association rule learning include ordering books in a library, placing products in a shop, composing service packs (for insurances for example)…

An analogy for association rule learning is how we decipher an ancient language. Knowing the basic common structures of languages, their relationships to one another (ancestry, similarity…), other statistical properties and some historical facts, we learn the association rules used by the ancient users of the language so that the scripts they left make sense to us.

### Blind source separation

Blind source separation tries to decompose a measurement into its basic components. Generally the measurement is a time series, like a sound record.

Illustration of blind source seperation. The signals are correctly recovered (bottom image), but the colors differ. This illustrates the "blind" aspect of the algorithm, which actually knowns nothing of the initial signals (middle image). Credit: [3]

Blind source separation is typically an unsupervised task, but it can be supervised as well.

An often cited example of blind source separation is detaching the speeches of different persons from a recording where they all talk at the same time. Let’s imagine we are at a party and get to the kitchen where there are 5 people talking at the same time. We make a record of this cacophony and we latter want to seperate each person’s speech to a seperated track. This would be a task for blind source separation. Here “blind” means that the algorithm has no information about the sources of the signals neither about the mixing process.

An analogy of blind source separation is how we humans, in a crowd with several people speaking, are able to isolate the voice of someone in particular that we want to listen to.

### Dimensionality reduction

Dimensionality reduction tries to reduce the number of input variables of a learning problem while keeping as much relevant information as possible.

Illustration of dimensionality reduction. Each image represents a hand-written digit. Both feature selection and extraction gave these grayscale pixellate images. The classification algorithm appends to works very well on the reduced dataset. Credit: [4]

The output of dimensionality reduction can be used as input of other learning algorithms or for visualization (if the number of inputs can be reduced to $3$ or less). When used as input of other learning algorithms, it can improve their performances (reduce their generalization error) because less variables often means less complex learning algorithms, less noise and less redundancy in the data.

The input variables of learning algorithms are called features. Usually, dimensionality reduction consists in 3 types of reduction of features:

• Feature selection tries to find a subset of features that are most relevant for a given problem.
• Feature extraction is often used in conjunction with feature selection and tries to infer new features from the existing ones. These new features will hopefully be more informative and less redundant than the initial ones and still accurately describe the initial data set.
• Feature projection reduces the dimension of the inputs by projecting them onto a lower dimensional “geometry”. Feature projection is useful when we suppose that the inputs are described by a lower dimensional “geometry” but folded in a higher dimensional space. For example, in the picture below, the S shaped inputs on the left make a 2D surface “folded” in a 3D space.

Illustration of feature projection. Various projections onto a plane of a 2D surface folded in a 3D space. Credit: [5]

We should note that a big challenge in machine learning is sometimes finding or choosing the representation of our features. For instance, in the example of the email filtering, how could we represent an email numerically such that it can be a relevant input for a learning algorithm ? One method may be to: 1) create a “dictionary” that contains all the words that appear in all emails, 2) represent an email by a vector where each component is the dictionary’s indice of the email’s word. Another option maybe that each component of the vector is the number of occurences in the email of each word of the dictionary. Other representation are possible.

A basic illustration of dimensionality reduction can be seen in our example of digit recognition from the section on classification above. Maybe we didn’t need color informations in order to recognize the digits, so we could drop $2$ color components and reduce the dimensionality of our data from $3$ (R, G, B channels) to $1$ (grayscale value). In the same idea, we can reduce the size of the image and compress it to even more reduce its dimensionality.

Let’s consider another example: a program that says whether a picture shows a face or not. We humans just need a few lines of drawings in order to recognize a face, so we might intuitively think that our algorithm also need only a small amount of the informations from a picture to tell whether it is a face or not. There actually exist neural networks capable of extracting such features from an image and that allow other algorithms to classify the image as face or not face.

## Conclusion

There was a lot of informations in this article, so let’s recap.

There are different axes of taxonomy of machine learning algorithms.

We can differentiate machine learning techniques based on the kind of data they require in order to learn. In this taxonomy we distinguished supervised, semi-supervised, unsupervised and reinforcement learning.

Another axis of taxonomy is the mathematical representation they rely on. From this axis we distinguished model-based and instance-based learning techniques.

A third alternative of differentiating machine learning algorithms is by the way they consume data for training. In this axis we have batch and online learning methods.

Finally, we can base our taxonomy on the type of task the algorithm resulting from the training tries to achieve. These tasks are notably regression, classification, clustering, decision making, association rule learning, blind source separation and dimensionality reduction.

These axes of taxonomy are not exclusive. For example, a supervised learning algorithm can be an instance-based and an online learning technique. Moreover, the taxonomy presented in this article is my personal attempt in linking various qualifications of machine learning techniques, and it is not exhaustive.

Taxonomy is useful when we try to frame a problem and try to determine which learning technique to use for this particular problem. Knowing these taxonomies can guide us in finding a reduced subset of candidate learning algorithms among which a validation technique (for example) can choose.

In the next article, by discussing linear regression, we will make a big step toward actually applying machine learning in practice.


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Revised 1feb15

Absolute Geometry versus Euclidean Geometry

\textit{$\C$  2010, Prof. George K. Francis, Mathematics Department,
University of Illinois}

\begin{document}
\maketitle

\section{Introduction}

The second module of this course addresses the geometry which Euclid
of Alexandria (ca -300) created. In particular, we are concerned with
\textit{Book I} of his \texit{Elements}, which begins with his Postulates,
and ends with his proof of the Pythagorean Theorem (Propositions 47) and
its converse (Proposition 48). For the first 28 propositions, Euclid
carefully avoids using his fifth, the \textit{Parallel Postulate}.
This body of plane geometry is
known as \textit{Absolute Geometry}. Nowadays, it is also called
\textit{Neutral Geometry} because it does not takes sides on whether to
accept or reject the parallel postulate.
David Joyce, of Clark University, summarizes  Euclid's Elements  for your convenience. As you study this lesson, you might want to check these yourself.

It was Euclid's intention of proposing his Postulates, together with
informal definitions and common notions, as given, and to derive the
propositions logically from them. Euclid, however, made a number of
additional assumption without calling any particular attention to them.
Geometers corrected these deficiencies over the subsequent 2300 years.
The final formulation Euclid's axioms was made by David Hilbert (ca 1900).
Robert Campbells praphrase of

In this course we will neither follow Euclid and present his proofs in
the classical progression, nor will we study Hilbert's axiomatic system
in any great detail. The former is purely of historical interest, and
the latter is far too technical and difficult for a first course in
college geometry.
We will discuss the high points of absolute geometry, giving proofs
only as needed to appreciate the Euclid's grand invention. When we
come to the great divide between absolute and Euclidean geometry,
we will study the structure of the theorems and proofs in considerable
detail.

A brief look into the Elements between Propositions 30 and 46 will be
followed by several proofs of Pythagoras' theorem. There are many more
on the WWW. (There is even one by President Garfield.)

The module will conclude with Birkhoff's very economical set of
axioms (only 4 as against Hilbert's 21) . The difference of these two axiomatic
systems is considerable. Birkhoff's axioms are built on top of the
entire edifice of real numbers and their arithmetic. Hilbert's is not.
Indeed, numbers and their arithmetic appear as objects inside Hilbert's
geometry. In other words,  the axiomatic arithmetic of real numbers has a
model inside geometry.
The great achievement of Hilbert and his colleagues was to show that
geometry and arithmetic are equi-consistent. It is not necessary to
choose between them, we accept both arithmetic and geometry as equally
certain.

On the other hand, since Rene Descartes (ca 1600), we have based geometry
on the properties of the real numbers, in the form of analytic geometry,
a.k.a. \textit{Cartesian Geometry}. We will consider Birkhoff's
axioms, because there are fewer of them, and they already assume the
properties of the reals, and show that they have a model inside Cartesian
geometry.

\section{Definition of Absolute Geometry}
We will take the simplest
approach and consider all the definitions, theorems and proofs in Euclid's
Elements up to and including Proposition 28 as constituting \textit{absolute
geometry}. Another, more practical definition would be to say that any
theorem that is true in both Euclidean and Hyperbolic geometry is part
of absolute geometry. But how could we apply this second approach to
deciding whether a geometrical theorem is absolute or not?

For example, that an exterior angle of a triangle is greater than either
opposite interior angle (AXAT) is true Euclidean geometry, and in a
model of Non-Euclidean geometry, and use a geometrical construction package to tdo the relevant experiments. Or we could look into Euclid's Elements and
notice that AXAT is Proposition 16 in Euclid, and $16 \lte 28$.

We will use GGB for this purpose, but Hvidsten's GEX is actually more
suited for this purpope because he provides 3 models of hyperbolic
geometry, a model of spherical geometry, and 3 models of Euclidean geometry
for good measure.

Noet that in Euclidean geometry, there is a stronger version of this
theorem, which says that the exterior angle is actually the sum of the
opposite interior angles (EXAT). But this is false in nonEuclidean geometry,
and therefore EXAT is not absolute, it is strictly Euclidean. Can you
find the number for that proposition in Euclid's Elements? Is is larger than
28?

To repeat a central fact in this course, the \textit{Euclidean Exterior
Angle Theorem} (EXAT) says that the
exterior angle is the sum of the opposite interior angles. This is false
in hyperbolic geometry. Therefore, EXAT cannot be proved inside absolute
geometry. The last lesson of this course will reveal that the amount by
which EXAT fails in hyperbolic geometry, gives the area of the triangle.

So, how can you tell that a theorem is part of absolute geometry, or
requires the Parallel Postulate to be true? One way is to study Euclid's
Elements. Since $48>28$ we know that the Pythagorean Theorem (Prop. 48)
is not a part of absolute geometry.  Another is to use a construcion
pagacke  and check whether or not the theorem holds in both Euclidean
and in non-Euclidean models of geometry, then it must be and absolute theorem.

Question 1: Do you think the theorem that rectangles have four right angles is absolutely true? Experiment with a Euclidean and a hyperbolic model in GEX. Write your results into your Journal. Report a summary here.

\section{Book I of the Elements}

Euclid invites his students to accept a long list of \textit{common notions},
which are not particularly convincing. In fact, Hilbert insisted that the primitives
of an axiomatic system are technically undefined. Of course, once you interpret
them in a model, they have a meaning. But it depends on the model you use.
Until C. F. Gauss (ca 1800) everybody, and I mean everybody was convinced that
there was only one possible geometry. And even if we humans were imperfect in
formulating this ideal geometry, it was as much there as anything we know.

What is much more interesting for us, than arguing about the nature of a point
or a line, is the tremendous care Euclid took in arranging the theorem in a
logical order. This means, none of the conclusions of a theorem was used as
the hypothesis of a prior theorem. And, almost (but not quite perfectly) he
left nothing out between a prior and a posterior proposition.

\subsection{Equilateral Triangle Theorem}

Euclid's first proposition showed how to construct an equilateral triangle
on a given base AB, directly from his postulates.
By Postulate 3 we have two circles, one centered
at A and the other at B, and with the same radius.
Euclid assumed that is was obvious that
the two circles intersected at two points, C and D. In Hilbert's geometry this
was not taken for granted. It had to be proved from prior theorems. But once
we do have a point there, both dashed segments are also radii of their respective circles. Hence the triangle is equilateral.

You should remember from high school geometry, that if we now connect C and D
with a line, this becomes the perpendicular bisector of the segment AB.

Question 2: Do you suppose this is a theorem in absolute geometry, or not?
You'll need to do some constructions in a hyperbolic model of non Euclidean
geometry to decide this experimentally.

\subsection{Side-Angle-Side Congruence}

Euclid's Proposition 4 is an attempt to prove SAS as a theorem.
Unfortunately, this one Euclid really failed to prove in any sense that
would be considered rigorous.
So we'll just accept it as another axiom, as Hilbert and most geometers do
ever since.

Question 3: Where have you seen a use of SAS in this course so far?

\subsection{Pons Asinorum}

You will recall that we really only need SAS to prove that  the base angle of
an isosceles triangle are equal, Proposition 5.
Euclid, as usual, feels obliged to prove the
converse, Proposition 6.
Here, you need to be reminded just what this means. Note that every
proposition can be formulated as an if-then statement.
The proposition $H \implies C$  says that if the
\textit{hypothesis} holds then the \textit{conclusion} follows. Recall from
MA347 or its equivalent that

\begin{eqnarray*}
\mbox{the}  &    &    \\
\mbox{proposition}  & \mbox{in symbols}  &  \mbox{ reads }   \\
& H \implies C  &  \mbox{ if } H \mbox{ then } C   \\
\mbox{its converse}  & C \implies H  &  \mbox{ if } C \mbox{ then } H   \\
\mbox{contrapositive}  & \neg C \implies \neg H  &  \mbox{ if not} C \mbox{ then not } H   \\
\mbox{inverse}  & \neg H \implies \neg C  &  \mbox{ if not} H \mbox{ then not } C\\
H \implies C  & \mbox{ is equivalent to} &  \neg H \or C  \\
\neg( H \implies C)  & \mbox{ is equivalent to}&  H \and \neg C  \\
\end{eqnarray*}

You also need to recall just why a proposition is equivalent to its
contrapositive, and similarly its converse is equivalent to its inverse.
But a proposition is not necessarily equivalent to its converse.

Question 4: State the converse of Pons Asinorum.

Question 5: State the negation of Pons Asinorum.

Can figure out a proof of the converse? A counterexample to its negation?

\subsection{What's next?}

In the next lesson we shall thoroughly study Proposition 28.

\end{document}


text:
# Seasonal and autocorrelated regression residuals: difference raw data or residuals?

I have a multiple linear model for time series data for which the regression residuals are autocorrelated and display seasonal behavior. This seasonal behavior is induced deterministically by a cyclic variable written into the model. In order to calculate corrected standard errors for the regression coefficients, I intend to use generalized least squares with correction for the autocorrelation in the residuals.

But for the seasonality, I am not sure for which time series I should perform the differencing: If for raw original data (observations), or for the regression residuals. For the data, I could superimpose each cycle and take the mean, thus removing season effects, but this would alter the data, and also would generalize bad for other kinds of data. For the regression residuals, I dont know if I have to difference the residuals time series substracting for each period, or to handle it as some kind of seasonal ARMA process.

Any hints?

LAST EDIT: The problem behind this question might have been resolved already, and might have been product of a misconception.

Regression was being done over simulations of the model. These simulations did not contain any stochastic error factor, so the simulations were purely of a deterministic nature. Regression errors were just showing an unperfect fit to the data and the regression residuals were obviously following a deterministic pattern. This pattern was not a result of some neglected explanatory variable in the model, and thus there was no reason to model it (by means of an ARMA model or any kind). When adding white noise to the data -which should be a crucial step on any simulation of real data- regression residuals were mostly dominated by stochasticity, loosing the autocorrelated behavior.

In order to calculate corrected standard errors for the regression coefficients, I intend to use generalized least squares with correction for the autocorrelation in the residuals.

Note that generalized least squares (GLS) would affect not only the standard errors but also the point estimates. Anyhow, you could gain power by estimating regression with an explicitly specified error structure, e.g. regression with ARMA errors as can be done using functions stats::arima or forecast::auto.arima in R. There you use maximum likelihood estimation instead of GLS. See related blog posts by Francis X. Diebold "The HAC Emperor has no Clothes" and "The HAC Emperor has no Clothes: Part 2" where he encourages explicit error specification as a way to get better coefficient estimates and gain predictive power. Although he discusses the case of HAC there, I believe similar conclusions apply here, too.

But for the seasonality, I am not sure for which time series I should perform the differencing: If for raw original data (observations), or for the regression residuals.

Since the problem arises due to a cyclic regressor, you could remove the deterministic component of the cyclic variable before including it in the model, or alternatively you could include some seasonal terms (dummies or Fourier terms) in the model.

For the data, I could superimpose each cycle and take the mean, thus removing season effects, but this would alter the data, and also would generalize bad for other kinds of data.

I am a little confused here, but I will try addressing this nevertheless.
With regards to the regressor, you can adjust using a model, and so altering data is not really a problem because you keep track of how you did it and you can recreate the original variable if you need to.
Regarding generalization, if the cyclic behaviour is unique for this instance, keeping it untreated would not help. If, on the other hand, it is similar across this instance and the ones you want to generalize to, you would not lose by removing the deterministic component before running the regression but then using it to adjust the other cases similarly.

A technical note: If you are doing a regression with ARMA errors, then it is the error that gets differenced. If the errors is some SARIMA process, regular treatment of SARIMA models applies (roughly speaking, you do not have to worry that it is a regression error rather than raw data).

• I had never heard of kernel HAC and I am not sure that is the approach I am taking. Maybe I expressed it wrong in the question, but the approach I would like to take is a full model specification and parameterization of the residual time-series. For the estimation I would use GLS for estimation of the parameters with inclusion of SARIMA residuals (that is what I meant with "correction for autocorrelation"). The last paragraph answers my question. But is there a reason why I shouldnt remove seasonality from the data thus then only treating the residuals as an AR model? – hirschme Jul 29 '16 at 10:33
• After searching for a while I found this en.wikipedia.org/wiki/Newey%E2%80%93West_estimator , which I think is what you meant by kernel-HAC and yeah it is not the approach I am taking. I do want to parameterize the residual time series as an autoregressive model. The main problem is if to adress the seasonality in the raw data and then use an AR model, or if to leave it and then difference the residuals by/or using a SARIMA model (I guess the I = Integrated would come because of the differencing done before, and the Moving Average because I somehow have never seen a "SARI" model.) – hirschme Jul 29 '16 at 12:06
• Thanks for the updates. Let me come back to it later in the day. – Richard Hardy Jul 29 '16 at 12:16
• @RichardHardy Thank you for the anwer. I will update soon with an example. I was intrigued by a couple of points in the answer. First, I have never heard GLS changing the point estimates, just the SE. And the optimization would be max. Likelihood too, where the likelihood function would be adapted for the error model (and thus be other than a gaussian as with OLS). With a proper error specification it could even be transformed into a linear regression with gaussian errors and max.Like = OLS, then transforming the coefficients back (for ex. by Cochrane-orcutt estimation in the case of AR(1) ). – hirschme Jul 30 '16 at 16:09
• @hirschme, GLS not changing the point estimates was actually what was happening in STATA for a long time, not sure if they corrected the mistake. But you can see immediately that the estimates differ: $\beta_{OLS}=(X'X)^{-1}X'y \not \equiv \beta_{GLS}=(X'WX)^{-1}X'Wy$ for some weighting matrix $W$. – Richard Hardy Jul 30 '16 at 16:12


text:
# Use the appropriate Lagrange interpolating polynomials to find the cubic polynomial whose graph passes through the given points. (1,2),(2,1),(3,3),(6,1)(1,2),(2,1),(3,3),(6,1).

Question
Polynomial graphs
Use the appropriate Lagrange interpolating polynomials to find the cubic polynomial whose graph passes through the given points. $$\displaystyle{\left({1},{2}\right)},{\left({2},{1}\right)},{\left({3},{3}\right)},{\left({6},{1}\right)}{\left({1},{2}\right)},{\left({2},{1}\right)},{\left({3},{3}\right)},{\left({6},{1}\right)}$$.

2020-11-10
Step 1
With PSKx_{1}=1, x_{2}=2, x_{3}=3\ and\ x_{4}=6, the Langrange interpolating polynomials give
$$\displaystyle{p}_{{{1}}}{\left({x}\right)}={\frac{{{\left({x}-{x}_{{{2}}}\right)}{\left({x}-{x}_{{{3}}}\right)}{\left({x}-{x}_{{{4}}}\right)}}}{{{\left({x}_{{{1}}}-{x}_{{{2}}}\right)}{\left({x}_{{{1}}}-{x}_{{{3}}}\right)}{\left({x}_{{{1}}}-{x}_{{{4}}}\right)}}}}$$
$$\displaystyle={\frac{{{\left({x}-{2}\right)}{\left({x}-{3}\right)}{\left({x}-{6}\right)}}}{{{\left({1}-{2}\right)}{\left({1}-{3}\right)}{\left({1}-{6}\right)}}}}$$
$$\displaystyle={\frac{{{\left({x}^{{{2}}}-{5}{x}+{6}\right)}{\left({x}-{6}\right)}}}{{{\left(-{1}\right)}{\left(-{2}\right)}{\left(-{5}\right)}}}}$$
$$\displaystyle=-{\frac{{{x}^{{{3}}}-{6}{x}^{{{2}}}-{5}{x}^{{{2}}}+{30}{x}+{6}{x}-{36}}}{{{10}}}}$$

### Relevant Questions

Use the appropriate Lagrange interpolating polynomials to find the cubic polynomial whose graph passes through the given points. $$\displaystyle{\left({2},\ {1}\right)},\ {\left({3},\ {1}\right)},\ {\left({4},\ −{3}\right)},\ {\left({5},\ {0}\right)}$$.
Find the cubic polynomial whose graph passes through the points $$\displaystyle{\left(-{1},-{1}\right)},{\left({0},{1}\right)},{\left({1},{3}\right)},{\left({4},-{1}\right)}.$$

The quadratic function $$\displaystyle{y}={a}{x}^{2}+{b}{x}+{c}$$ whose graph passes through the points (1, 4), (2, 1) and (3, 4).

For the following exercises, use the given information about the polynomial graph to write the equation. Double zero at $$\displaystyle{x}=−{3}$$ and triple zero $$\displaystyle{a}{t}{x}={0}$$. Passes through the point (1, 32).
Consider the curves in the first quadrant that have equationsy=Aexp(7x), where A is a positive constant. Different valuesof A give different curves. The curves form a family,F. Let P=(6,6). Let C be the number of the family Fthat goes through P.
A. Let y=f(x) be the equation of C. Find f(x).
B. Find the slope at P of the tangent to C.
C. A curve D is a perpendicular to C at P. What is the slope of thetangent to D at the point P?
D. Give a formula g(y) for the slope at (x,y) of the member of Fthat goes through (x,y). The formula should not involve A orx.
E. A curve which at each of its points is perpendicular to themember of the family F that goes through that point is called anorthogonal trajectory of F. Each orthogonal trajectory to Fsatisfies the differential equation dy/dx = -1/g(y), where g(y) isthe answer to part D.
Find a function of h(y) such that x=h(y) is the equation of theorthogonal trajectory to F that passes through the point P.
For the equation (-1,2), $$y= \frac{1}{2}x - 3$$, write an equation in slope intercept form for the line that passes through the given point and is parallel to the graph of the given equation.
Graph the polynomial in the given viewing rectangle. Find the coordinates of all local extrema. State each answer rounded to two decimal places. State the domain and range. $$\displaystyle{y}={x}^{{{3}}}-{3}{x}^{{{2}}},{\left[-{2},{5}\right]}{b}{y}{\left[-{10},{10}\right]}$$
Use your knowledge of the graphs of polynomial functions to make a rough sketch of the graph of $$\displaystyle{y}=-{2}{x}^{{{3}}}+{x}^{{{2}}}-{5}{x}+{2}$$


text:
Folder:
115 CFA
File:
115.020.40.08 Reading 9 - 8. The Standard Normal Distribution

# 8. The Standard Normal Distribution

## m. define the standard normal distribution, explain how to standardize a random variable, and calculate and interpret probabilities using the standard normal distribution;

What is the standard normal distribution?
The standard normal distribution is a normal distribution with a mean of 0 and a standard deviation of 1. It is denotes as N(0, 1).

What are the standard deviations and confidence intervals for a standard normal distribution?
This is related to the empirical rule.
Mean = 0
First standard deviation = 1 with 68% confidence
Second standard deviation = 1.96 with 95% confidence
Third standard deviation = 2.58 with 99.7% confidence

What is the formula to transform a normal distribution into a standard normal distribution?
$$z = \frac{X - \mu}{\sigma}$$
- X = score from the original normal distribution
- \mu = mean of the original normal distribution
- \sigma = standard deviation of the original normal distribution

What is a Z-score? How do you calculate it?
- The Z-score is the number of standard deviations away from the mean for a normal distribution. If the mean is 5, the standard deviation is 10, then the z-score for x=20 would be 1.5.
- You could use the formula (X - mean)/std-dev

What is a Z-table and how do you use it?
- A z-table is a table of values that indicate the percentage of area under a normal distribution that lies below the given z-score (# of std dev from the mean).
- The z-table has integers down the left and decimals across the top. Just match your given score down and across to find the percent of area that lies below.

• CFA


text:
## Something interesting concerning parity...

Is there a proof or way of proving that all even numbers (taking into account the definition of an even number as $$n=2k$$) end in 0,2,4,6, or 8?

 PhysOrg.com science news on PhysOrg.com >> 'Whodunnit' of Irish potato famine solved>> The mammoth's lament: Study shows how cosmic impact sparked devastating climate change>> Curiosity Mars rover drills second rock target
 Recognitions: Gold Member Science Advisor Staff Emeritus What do you think? Care to try showing that numbers ending in 1,3,5,7, or 9 can not be even?
 Rewrite n as 10a+b, where 0<=b<10. Then n is even means 2 divides 10a+b, i.e. 2 divides b (let 10a+b=2k and solve for b). Hence b=0,2,4,6,8

## Something interesting concerning parity...

 Quote by TwilightTulip Rewrite n as 10a+b, where 0<=b<10. Then n is even means 2 divides 10a+b, i.e. 2 divides b (let 10a+b=2k and solve for b). Hence b=0,2,4,6,8
An explanation such as this is alright for proving that an integer $$n$$ is even if $$n=2k$$, for some integer $$k.$$ Thus, we have to use our knowledge of even numbers to say that they end in an integer that is divisible by 2. A satisfactory explanation, no doubt, just not what I'm looking for.

Is there a way to show that the fact that all even numbers end in 0,2,4,6, or 8 implies the definition of an even number ($$n=2k$$, where $$k$$ is an integer)?

 Any number that is a multiple of ten is even and divisible by 2. Then any number (base ten) denoted by ...dcba (where a is the ones place, b in the tens, etc.) will be divisible by two if a is divisible by 2 because ...+d*10^3+c*10^2+b*10^1+a represents the number and division is linear. All terms except a are multiples of ten always and therefore divisible by 2, then all that is left is a.


text:
Karnataka Secondary Education Examination BoardKarnataka SSLC

# Ozone Layer Depletion

#### notes

OZONE LAYER:-

Ozone (O3) is a molecule formed by three atoms of oxygen. While O2, which we normally refer to as oxygen, is essential for all aerobic forms of life. Ozone, is a deadly poison. However, at the higher levels of the atmosphere, ozone performs an essential function. It shields the surface of the earth from ultraviolet (UV) radiation from the Sun.

This radiation is highly damaging to organisms, for example, it is known to cause skin cancer in human beings.

Ozone at the higher levels of the atmosphere is a product of UV radiation acting on oxygen (O2) molecule. The higher energy UV radiations split apart some molecular oxygen (O2) into free oxygen (O) atoms. These atoms then combine with the molecular oxygen to form ozone as shown

$\ce{O2 ->[UV] O + O}$

$\ce{O + O2 -> \underset{\text{Ozone}}{O3}}$

The amount of ozone in the atmosphere began to drop sharply in the 1980s. Formation of holes in the ozone layer was reported. This decrease has been linked to synthetic chemicals like chlorofluorocarbons (CFCs) which are used as refrigerants and in fire extinguishers. In 1987, the United Nations Environment Programme (UNEP) succeeded in forging an agreement to freeze CFC production at 1986 levels. It is now mandatory for all the manufacturing companies to make CFC-free refrigerators throughout the world.

#### description

• Depletion of Ozone Layer
• Chemicals responsible for the destruction of the ozone layer
If you would like to contribute notes or other learning material, please submit them using the button below.

### Shaalaa.com

Our Environment part 10 (Ozone layer) [00:15:01]
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text: 6th power needs factors 3 × 2, which we already have. 324. Roots can also include decimal numbers (root 6.4, for example). 144. And 9 is also perfect square, so maybe this is the square of some binomial. How to Find Nth Roots by Hand. MP2. It's good to develop a consistent fingering and learn to play the patterns cleanly. Online surds calculator that allows you to make calculations in exact form with square roots: sum, product, difference, ratio. You can still use either the 3rd or 4th finger on the higher octave of the root. Square root, cubed root, 4th root, and any root are the most common examples of an nth root. Because of this, pentatonic scale shapes (both major and minor) are played using the 1st (index), 3rd (ring), and 4th â¦ Reason abstractly and quantitatively. 169. The nth Root. 256. 361. Free Calculator reduces numbers to nth principal roots and simplest radical form. MP1. If N N N is an integer, then N 2 N^2 N 2 is a perfect square. Just as Latin could turn the neuter (-um) form of the perfect participle into a 2nd declension noun, so could it convert the masculine (-us) form into a regular 4th declension noun.There was originally a contrast between these two, in that the neuter noun was felt to be concrete and the masculine somewhat more abstract; but that contrast is often hard to discern in practice. Improve your math knowledge with free questions in "Square roots of perfect squares" and thousands of other math skills. 9. 8th power needs another factor of 2 (2³ = 8) 9th power needs another factor of 3 (3² = 9). 289. Let us first note that: Digital sum of two-digit cube root Digital sum of A perfect-fifth is one of the Class 1 intervals: perfect-octave, perfect-unison, perfect-fifth, perfect-forth. These are all really extensions of the Binomial Theorem. MP3. Since it's a perfect square, the exponent needs a factor of 2. List of Perfect Cube Numbers 1 to 50 Evaluate square roots of small perfect squares and cube roots of small perfect cubes. Even for perfect root numbers, a root can be difficult to calculate by hand. for 3 squared (or 3 x 3) we write 3 2 Some of the worksheets for this concept are Square roots work, Math 6 notes name, 1 simplifying square roots, Square roots date period, Cubes and cube roots work, Squares and square roots, Square roots teacher notes, Powers and roots. Details on roots and how they are calculated. 6 2. The free calculator will reduce any number to its principal nth root as well as express it in simplest radical form. 225. 25. Hence, 3 âx = y. The nth root calculator below will also provide a brute force rounded approximation of the principal nth root. For example, the fourth root of 81 is 3 as 3 x 3 x 3 x 3 is 81. Syntax : simplify_surd(x), where x represents the square root to calculate. 81. Perfect cube is a number whose cube root is an integer Example : 23, 33, 43, 53, 63, 73 , â¦ are perfect cube i.e. We now consider similar shortcuts for guessing the two-digit cube roots of perfect cubes lying between 1000 and 1000000. 16. Examples : This examples shows how to symplify square roots with the calculator : simplify_surd(sqrt(48)) returns 4*sqrt(3) 7th power needs a factor of 7. e.g. Because of this definition, perfect â¦ Formula â How to calculate a root. to find 6 x 6, enter 6, x, = Writing squares. The perfect fifth is represented almost completely accurately in 12-tone equal temperament, suggesting that its accuracy is fundamental to music. By using this website, you agree to our Cookie Policy. Since the pentatonic doesnât included the 2nd and 6th intervals of the minor scale, you end up with a two-note per string pattern for each position. 4 2. Simplifying Square Roots To simplify a square root, you take out anything that is a "perfect square". Two-digit Cube roots of perfect cubes lying between 1000 and 1000000 We had considered earlier shortcuts to guessing square roots of perfect squares lying between 100 and 10000.with simple techniques. For example, 8 is a perfect cube because 3 â8 = 2. 2 2. e.g. GDEcD. The 6th root of 729 has exactly one real sixth root and has two additional complex 6th roots. Partition your number. Taking the square roots of perfect fractions by taking the square root of the numerator and denominator. If x is a perfect cube of y, then x = y 3.Therefore, if we take out the cube root of a perfect cube, we get a natural number and not a fraction. Perfect for the 4th!! 5 2. 100. And then the middle term is to the second power. Perfect Squares Perfect Cubes Perfect 4th Roots Perfect 5th Roots Perfect 6th Roots 1 1 11² 121 1 1 1 1 1 1 1 1 2 4 12² 144 2 8 2 16 2 32 2 64 3² 9 13² 169 3 27 3 81 3 243 3 729 4² 16 14² 196 4 64 4 256 4 1024 4 4096 5² 25 15² 225 5 125 5 625 5 3125 5 15,625 6² 36 16² 256 6 216 7² 49 17² 289 7 343 8² 64 18² 324 8 512 9² 81 19² 361 9 729 10² 100 20² 400 10 1,000 10 10,000 For example, 100 100 1 0 0 is a perfect square because it is equal to 10 × 10 10\times 10 1 0 × 1 0. So it is the general way of talking about roots (so it could be 2nd, or 9th, or 324th, or whatever) The nth Root Symbol . 8, 27, 64, 125, 216, 343, â¦ are perfect cube We are asked to find the cube root of negative 512. Perfect Squares: 1. MP5. 3 2. is the key of the 5 th or 4 th degrees. But there's something about this that might pop out at you. 1 cup dried cranberries Going apple picking in Georgia is one of those fall traditions you just can't miss! Take a look. Non Perfect Square Roots - Displaying top 8 worksheets found for this concept.. Perfect Squares and their Square Roots Perfect Square: Taking a positive integer and squaring it (multiplying it by itself) equals a perfect square. 4th power needs another factor of 2 (2² = 4) 5th power needs a factor of 5. Construct viable arguments and critique the reasoning of others. The Root Note of the Nearest Key In any given key, the closest tonality The key in which a piece is played, and the relationships of the notes to each other, and most especially to the tonal center. They are described as perfect because their wavelengths perfectly coincide with the wavelength of the fundamental tone. 49. Squares are powers of two. Stars top, S-M-L, \$34 wwwTwistedSouthernRoots.com Taking the square root (principal square root) of that perfect square equals the original positive integer. Adding the digits of 729, we get 7 + 2 + 9 = 18, that mean is a multiple of 9, this tells us that 729 is a multiple of 9. Make sense of problems and persevere in solving them. A perfect square is an integer that can be expressed as the product of two equal integers. So, in our perfect square images, the square roots are the number of squares going across and down. Here is that funny long division-like method for finding square and cube roots generalized to nth roots. 36. Or another way to think about it is if I have some number, and it is equal to the cube root of negative 512, this just means that if I take that number and I raise it to the third power, then I get negative 512. harvest dates of your favorite varieties too. Example: 3 x 3 = 9 Thus: 9 is a perfect square. 196. Know that â2 is irrational. That is, you take out front anything that has two copies of the same factor: Sometimes the argument of a radical is not a perfect square, but it may contain a square amongst its factors. 1 2. You will be playing roots and fifths a lot regardless of style. The sixth root of 729 is a number that multiply by itself 6 times to give us 729. We write squares using the same notation that we use with other powers. 8 2. And the thing that pops out at me at least is that 25 is a perfect square, x to the fourth is a perfect square, so 25x to the fourth is a perfect square. Free roots calculator - find roots of any function step-by-step This website uses cookies to ensure you get the best experience. Perfect Powers . A perfect cube is a number which is equal to the number, multiplied by itself, three times. 400. 121. No headers. When you play these patterns by themselves, use your 1st finger on the root and play the higher 5th with your 3rd finger. 7 2. 4. Perfect Square, Cube, Fourth Power Date: 01/25/2002 at 19:05:54 From: James A. Hopton Subject: Perfect square, cube, fourth power My wife is an 8th Grade Math Teacher, and in the Bonus Section in one of the chapters of the current math book is this problem: Find the least integer greater than 1 that is a perfect square, a perfect cube, and a perfect fourth power. Lesli made her way Instead of talking about the "4th", "16th", etc, if we want to talk generally we say the "nth". what are perfect cubes and how to calculate the cube root of a perfect cube, Evaluate Square Roots of Perfect Squares and Cube Roots of Perfect Cubes, How to simplify radicals with negative radicands and odd indexes, Grade 8 math, with video lessons, examples and step-by-step solutions. Cube, needs a factor of 3. The fourth root of a number is the number that would have to be multiplied by itself 4 times to get the original number. 64. 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That can be expressed as the product of two equal integers is fundamental to music 's a perfect square develop. Nth roots number is the key of the 5 th or 4 th.! Cube roots of perfect squares and cube roots of perfect squares '' and thousands of other math.! Include decimal numbers ( root 6.4, for example ) guessing the two-digit cube roots of perfect cubes found! Because of this definition, perfect â¦ free calculator reduces numbers to nth principal roots and simplest radical form with. Brute force rounded approximation of the fundamental tone 9 Thus: 9 is a number that multiply by 6! Needs another factor of 2 ( 2³ = 8 ) 9th power needs another of. 27, 64, 125, 216, 343, â¦ are perfect cube because â8... As well as express it in simplest radical form, three times numbers to nth roots perfect 4th roots... And 9 is a number which is equal to the second power example: 3 3... Needs a factor of 3 ( 3² = 9 Thus: 9 is perfect! Make calculations in exact form with square roots - Displaying top 8 worksheets found for this... We now consider similar shortcuts for guessing the two-digit cube roots generalized to nth principal roots simplest. Integer, then N 2 is a number which is equal to the number, multiplied by 4. 81 is 3 as 3 x 3 = 9 ) are the most common examples of an nth root free. Write squares using the same notation that we use with other Powers critique the reasoning of.... Solving them to its principal nth root finding square and cube roots of perfect squares '' and thousands of math! Square of some Binomial perfect square, the exponent needs a factor of 2: 3 3! By itself, three times solving them provide a brute force rounded approximation of the Binomial Theorem radical form get! This that might pop out at you perfect root numbers, a root can be difficult calculate... Number which is equal to the number that multiply by itself 4 perfect 4th roots... Cubed root, cubed root, and any root are the most common examples of an nth root almost... Are the most common examples of an nth root as well as express it in radical. Make sense of problems and persevere in solving them that we use with other Powers root numbers a. Of a number that multiply by itself 4 times to get the best experience three times will also a... The exponent needs a factor of 5 we already have is the key the. Common examples of an nth root a brute force rounded approximation of the 5 th or th. When you play these patterns by themselves, use your 1st finger on the root and play the cleanly. In exact form with square roots: sum, product, difference, ratio now consider shortcuts. Be multiplied by itself, three times, multiplied by itself 6 times to get the original positive integer 3²... Reasoning of others arguments and critique the reasoning of others of that perfect square multiply by itself three! Of that perfect square equals the original number: simplify_surd ( x ), where x the! Best experience to get the original number to nth roots arguments and critique the reasoning of others root! Construct viable arguments and critique the perfect 4th roots of others ensure you get the best experience the of! Of 729 is a perfect square roots of perfect fractions by taking the square root to calculate by.. And persevere in solving them free calculator reduces numbers to nth roots are really... Are the most common examples of an nth root calculator below will also a. Where x represents the square root of the numerator and denominator as well as express it in simplest form... Sum, product, difference, ratio key of the fundamental tone form with square roots - Displaying top worksheets... Perfect cube is a number which is equal to the second power that allows to... Â¦ are perfect cube because 3 â8 = 2 with your 3rd finger nth roots suggesting... 343, â¦ are perfect cube because 3 â8 = 2 solving.! X, = Writing squares product, difference, ratio of others ×,... With your 3rd finger with other Powers definition, perfect â¦ free calculator will reduce any number to principal... Make sense of problems and persevere in solving them itself 6 times to give us 729 cube a. 3 as 3 x 3 is 81 4 times to give us 729 the 5 th or th. Reasoning of others this definition, perfect â¦ free calculator reduces numbers to nth.! Roots can also include decimal numbers ( root 6.4, for example, the fourth of! Number to its principal nth root as well as express it in simplest radical.! 'S something about this that might pop out at you Displaying top 8 found! In  square roots of perfect fractions by taking the square root, 4th root, 4th root cubed. Product, difference, ratio to ensure you get the original positive.! 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## Repair Standard Deviation Propagation Of Error Tutorial

Home > Error Propagation > Standard Deviation Propagation Of Error

# Standard Deviation Propagation Of Error

## Contents

Retrieved 2013-01-18. ^ a b Harris, Daniel C. (2003), Quantitative chemical analysis (6th ed.), Macmillan, p.56, ISBN0-7167-4464-3 ^ "Error Propagation tutorial" (PDF). Authority control GND: 4479158-6 Retrieved from "https://en.wikipedia.org/w/index.php?title=Propagation_of_uncertainty&oldid=742325047" Categories: Algebra of random variablesNumerical analysisStatistical approximationsUncertainty of numbersStatistical deviation and dispersionHidden categories: Wikipedia articles needing page number citations from October 2012Wikipedia articles needing Also, an estimate of the statistic is obtained by substituting sample estimates for the corresponding population values on the right hand side of the equation. Approximate formula assumes indpendence ProfessorSerna 7,172 views 7:27 IB Physics- Uncertainty and Error Propagation - Duration: 7:05. http://kldns.net/error-propagation/standard-deviation-vs-error-propagation.html

Peralta, M, 2012: Propagation Of Errors: How To Mathematically Predict Measurement Errors, CreateSpace. PhysicsOnTheBrain 45,578 views 1:36:37 Basic Rules of Multiplication,Division and Exponent of Errors(Part-2), IIT-JEE physics classes - Duration: 8:52. The problem might state that there is a 5% uncertainty when measuring this radius. Pradeep Kshetrapal 33,267 views 1:49:43 Experimental Error Analysis - Duration: 12:26. https://en.wikipedia.org/wiki/Propagation_of_uncertainty

## Error Propagation Calculator

Using Beer's Law, ε = 0.012614 L moles-1 cm-1 Therefore, the $$\sigma_{\epsilon}$$ for this example would be 10.237% of ε, which is 0.001291. Section (4.1.1). Starting with a simple equation: $x = a \times \dfrac{b}{c} \tag{15}$ where $$x$$ is the desired results with a given standard deviation, and $$a$$, $$b$$, and $$c$$ are experimental variables, each Note this is equivalent to the matrix expression for the linear case with J = A {\displaystyle \mathrm {J=A} } .

Uncertainty never decreases with calculations, only with better measurements. Introduction Every measurement has an air of uncertainty about it, and not all uncertainties are equal. For example, the 68% confidence limits for a one-dimensional variable belonging to a normal distribution are ± one standard deviation from the value, that is, there is approximately a 68% probability Error Propagation Excel Management Science. 21 (11): 1338–1341.

John Wiley & Sons. Note that these means and variances are exact, as they do not recur to linearisation of the ratio. Eq.(39)-(40). Given the measured variables with uncertainties, I ± σI and V ± σV, and neglecting their possible correlation, the uncertainty in the computed quantity, σR is σ R ≈ σ V

Sometimes, these terms are omitted from the formula. Error Propagation Average The standard deviation of the reported area is estimated directly from the replicates of area. Le's say the equation relating radius and volume is: V(r) = c(r^2) Where c is a constant, r is the radius and V(r) is the volume. It is important to note that this formula is based on the linear characteristics of the gradient of f {\displaystyle f} and therefore it is a good estimation for the standard

## Error Propagation Physics

Journal of the American Statistical Association. 55 (292): 708–713. Let's say we measure the radius of a very small object. Error Propagation Calculator outreachc21 17,692 views 15:00 IB Physics: Uncertainties and Errors - Duration: 18:37. Error Propagation Chemistry This example will be continued below, after the derivation (see Example Calculation).

Note that these means and variances are exact, as they do not recur to linearisation of the ratio. check over here Generally, reported values of test items from calibration designs have non-zero covariances that must be taken into account if $$Y$$ is a summation such as the mass of two weights, or For example, repeated multiplication, assuming no correlation gives, f = A B C ; ( σ f f ) 2 ≈ ( σ A A ) 2 + ( σ B Published on Nov 13, 2013Educational video: How to propagate the uncertainties on measurements in the physics lab Category Education License Standard YouTube License Show more Show less Loading... Error Propagation Definition

And again please note that for the purpose of error calculation there is no difference between multiplication and division. Propagation of error considerations

Top-down approach consists of estimating the uncertainty from direct repetitions of the measurement result The approach to uncertainty analysis that has been followed up to this doi:10.1007/s00158-008-0234-7. ^ Hayya, Jack; Armstrong, Donald; Gressis, Nicolas (July 1975). "A Note on the Ratio of Two Normally Distributed Variables". his comment is here H.; Chen, W. (2009). "A comparative study of uncertainty propagation methods for black-box-type problems".

Loading... Error Propagation Calculus The derivative of f(x) with respect to x is d f d x = 1 1 + x 2 . {\displaystyle {\frac {df}{dx}}={\frac {1}{1+x^{2}}}.} Therefore, our propagated uncertainty is σ f The derivative of f(x) with respect to x is d f d x = 1 1 + x 2 . {\displaystyle {\frac {df}{dx}}={\frac {1}{1+x^{2}}}.} Therefore, our propagated uncertainty is σ f

## doi:10.1016/j.jsv.2012.12.009. ^ "A Summary of Error Propagation" (PDF).

JSTOR2281592. ^ Ochoa1,Benjamin; Belongie, Serge "Covariance Propagation for Guided Matching" ^ Ku, H. doi:10.2307/2281592. To contrast this with a propagation of error approach, consider the simple example where we estimate the area of a rectangle from replicate measurements of length and width. Error Propagation Square Root Derivation of Arithmetic Example The Exact Formula for Propagation of Error in Equation 9 can be used to derive the arithmetic examples noted in Table 1.

In both cases, the variance is a simple function of the mean.[9] Therefore, the variance has to be considered in a principal value sense if p − μ {\displaystyle p-\mu } University of California. doi:10.1287/mnsc.21.11.1338. http://kldns.net/error-propagation/standard-deviation-using-propagation-error.html For such inverse distributions and for ratio distributions, there can be defined probabilities for intervals, which can be computed either by Monte Carlo simulation or, in some cases, by using the

External links A detailed discussion of measurements and the propagation of uncertainty explaining the benefits of using error propagation formulas and Monte Carlo simulations instead of simple significance arithmetic Uncertainties and Define f ( x ) = arctan  ( x ) , {\displaystyle f(x)=\arctan(x),} where σx is the absolute uncertainty on our measurement of x. doi:10.6028/jres.070c.025. doi:10.1016/j.jsv.2012.12.009. ^ Lecomte, Christophe (May 2013). "Exact statistics of systems with uncertainties: an analytical theory of rank-one stochastic dynamic systems".

In matrix notation, [3] Σ f = J Σ x J ⊤ . {\displaystyle \mathrm {\Sigma } ^{\mathrm {f} }=\mathrm {J} \mathrm {\Sigma } ^{\mathrm {x} }\mathrm {J} ^{\top }.} That JSTOR2629897. ^ a b Lecomte, Christophe (May 2013). "Exact statistics of systems with uncertainties: an analytical theory of rank-one stochastic dynamic systems". Example: Suppose we have measured the starting position as x1 = 9.3+-0.2 m and the finishing position as x2 = 14.4+-0.3 m. This feature is not available right now.

Multivariate error analysis: a handbook of error propagation and calculation in many-parameter systems. ISSN0022-4316. Keith (2002), Data Reduction and Error Analysis for the Physical Sciences (3rd ed.), McGraw-Hill, ISBN0-07-119926-8 Meyer, Stuart L. (1975), Data Analysis for Scientists and Engineers, Wiley, ISBN0-471-59995-6 Taylor, J.


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# listings package-How to highlight something that starts with one @

I would like to highlight any expression made of letters and that starts with one @. The use of moredelim=*[s][\color{blue}]{@}{\ } works only in case like @name is followed by one space. The following code nearly works except that I do not want the parenthesis to be colored in case like @decorator(x=3, y='some text') because that is @decorator( which is caught.

moredelim=[s][\bfseries\color{gray}]{@}{\ },
moredelim=[s][\bfseries\color{gray}]{@}{(},

The use of moredelim=[s][\bfseries\color{gray}]{@}{(} is not a good solution because the search is done one sevral lines... :-(

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In text editors capable of regular expression identification, just search and highlight this string: ^@.*? But how is this a tex question? Do you want LaTeX to do this as it compiles? –  Ariel Apr 11 '12 at 16:38
listings is one LaTeX package. –  projetmbc Apr 11 '12 at 16:49
ah - thanks for the clarification and this is good to know. –  Ariel Apr 11 '12 at 17:19

This is possible. The package provides the command morecomments, which can be used to declare a sequence of arbitrarily delimited symbols to be a comment, and set up its unique formatting. The command

\lstset{morecomment=[s][\color{blue}]{@}{\ }}

means that any sequence of symbols ([s]) delimited by @ and space, will be typeset in blue.

\documentclass{article}
\usepackage{listings,xcolor}
\pagestyle{empty}
\lstset{language=bash}
\lstset{morecomment=[s][\color{blue}]{@}{\ }}
\begin{document}
\begin{lstlisting}
# This is an example
@Blueword # This will be done properly
for x in *~; do
echo $x; done # This too @Blueword x for x in *~; do echo$x;
done
# And this one
@Blueword
for x in *~; do
echo \$x;
done
\end{lstlisting}
\end{document}

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Thanks for {\ }. That's what I've been looking for... –  projetmbc Apr 11 '12 at 17:15
I've updated my question because there are some drawbacks... –  projetmbc Apr 11 '12 at 17:31
Unfortunately listings does not accept multiletter delimiters (yet?), so in your example you need to write @decorator (arg). By the way, I know at least one company, where a space between function name and opening bracket is required by the coding style. –  Boris Apr 11 '12 at 17:40