So last week I was looking at some things around Reliability and I realized how big the Reliability Engineering field is. Then I thought why not share it with others on what I have learned so far and at the same time with a hope that experts in this field may read this and correct me if there are any mistakes in my understanding.
In this blog, we will start with some basic concepts, touch on Reliability Mathematics and follow up with some examples in part-II. A basic background in Probability and basic calculus will come handy, but if that’s not the case, then ignoring is always an option.
A general caution about Metrics
Metrics are a quantitative measure of a system or network behavior which allows us to characterize the system behavior so that we can make decisions based on that. But before we start using it, it’s important to understand how they were determined and the assumptions made during calculating those metrics. It’s possible that the assumptions made during calculation may not apply to your situation and thus making those metrics useless. This will allow you to understand how much bullshite those metrics are.
Intro to Reliability Engineering
Design Life: Design life is the intended period of use of a component/system which is expected to be failure free.
Failure: We consider a component/system to be failed, If it is not able to perform its required function as specified. For instance, a light bulb is no longer glowing or a router stops routing.
A failure rate, F, defines failure frequency in terms of failures per unit time, like percentage of failures per 1000 hours. We can estimate Failure rate by looking at the operating hours of all the units, including the failed ones and the numbers of units failed.
Example 1: Assume that there are 50 units which operated for 1000 hours on test and 2 of them failed. Then the Failure Rate (ƛ) will be
Categories of Items:
- Repairable items: Repairable items are the ones which can be repaired once they fail and once fixed they resume their required function.
- Non-Repairable items: Non-Repairable items are the ones which cannot be fixed once they fail and are generally replaced. This is more common in the semiconductor/Telecommunication industry. For instance, let’s say a router line-card gone bad, we typically replace them (hardware failures).
Patterns of Failures with Time
There are three basic ways in which the pattern of failures can change with time. Hardware failures are typically characterized by a “bath tub curve”.
- Infant Mortality Stage: Decreasing failure rates is observed in items which becomes less likely to fail as their survival time increases. This is common in electronics equipment’s and often caused due to manufacturing defects. These kinds of defects are usually caught during the “Burn-In” period of the item.
- Wear out Stage: Wear out failure modes follow an increasing failure rate, This is basically when the component is getting closer to his life time end.
- Constant Failure Rate (Random Failures): A constant failure rate is a characteristic of failures where they can happen randomly. This is the useful life span of the equipment which will be the focus.
As you may have noticed that how Failure is a function of time i.e. Things tend to fail over a period of time. This is represented by Failure Rate ƛ(t). Also, we are making an assumption that during the useful life of an item, the failure-rate is constant to model the reliability of electronic systems as they typically do not experience wear out type failures. There are studies and papers out there that talk about how it may not be true, but for now we will conveniently ignore that.
Reliability is defined as the probability that an item will perform a required function without failure for a stated period of time. Another way to state is that It’s a measure of how long it takes for a network (or a system) to fail.
So why Reliability is defined in terms of probability?. The reason is because in the universe of Reliability, we are dealing with uncertainty. For instance, let’s say the data shows that a certain type of Line Card fails at a constant average rate of one per 1,000,000 units. Now If we build 1000 units, and operate for 100 hours, we cannot say with certainty whether any will fail during that time. However, we can make a statement about the likelihood (probability) of a failure.
Now we will be briefly look at the Reliability Math’s to see how the equations are derived before we start using it. I will try to introduce necessary concepts of probability before start using them, but please keep in mind that there is no way I can do justice in explaining every fundamental if the reader has no background in Probability/Calculus.
Also, I am going to take some luxuries while explaining to keep things simple, so it’s possible that some statements may not be mathematically accurate.
Quick 25,000 foot view of Probability basics:
- As you already know that in the universe of probability, an event occurrence is expressed as a number between 0 and 1. The probability of an event A happening is represented by P (A).
- The sum of probabilities of all possible outcomes is equal to One. For instance, if an experiments can have three possible outcomes A, B and C, then P(A)+P(B)+P(C)= 1
- The Probability that an event A will occur is equal 1 minus the probability that event A will not occur. Assuming that P(A
) is the probability that event A will not occur. Then P(A) = 1 – P(A). Intuitively, Probability of Event A happening and Probability of Event A not happening should be equal=1. We are 100% sure that either it will happen or it will not happen.
A random variable, usually written as X, is a variable whose possible values are numerical outcomes of a random experiment.There are two types of random variables, discrete and continuous.(https://www.mathsisfun.com/data/random-variables.html)
Discrete Random Variables:
A discrete random variable is one which may take on only a countable number of distinct values such as 0,1,2,3,4,… If a random variable can take only a finite number of distinct values, then it is Discrete Random Variable. Examples of discrete random variables include the number of children in a family, the number of defective light bulbs in a box of ten.
Continuous Random Variables:
A continuous random variable is one which takes an infinite number of possible values. Continuous random variables are usually measurements. Examples include height, weight, time. A continuous random variable is not defined as specific values. Instead, it is defined over an interval of values, and is represented by the area under a curve (calculated with the help of integral). The probability of observing any single value is equal to 0, since the number of values which may be assumed by the random variable is infinite.
Probability Distribution Function
In simple words, A Probability Distribution function tells us the probability of an outcome happening. Let’s say we conduct an experiment and the possible outcomes are 1,2,3,4, below are the probabilities of each of the outcome happening. As you can see the outcomes are discrete numbers, we are looking at Discrete random variables.
This is an example of a Probability Distribution function and in the case of a Discrete Random Variable, it is called as a Probability Mass Function (P.M.F.).It allows us to answer questions like:
- What’s the probability of 2 happening P (X=2), we can say its 0.25 or 25%
- What’s the Probability of 1 or 2 happening i.e. P(X=2 or X=1) = P(X=1)+P(X=2) = 0.20+0.25 = 0.45
- What’s the Probability of 1 and 2 happening i.e. P(X=2 and X=1) = P(X=1) * P(X=2) = 0.20 * 0.25 = 0.05
You can also see the sum of all probabilities (0.20+0.25+0.30+0.25) = 1, meaning that we are covering all the potential outcomes of this experiment. In general, P.M.F involves summation to calculate all the probabilities of an experiment.
When it comes to the continuous random variables, the same concept applies as Discrete random variables. The difference is in how we calculate the probabilities.
A probability distribution function for continuous random variables describes the probabilities of possible values of a continuous random variable. Probabilities of continuous random variables (X) are given as the area under the curve of its P.D.F. And if you recall, the way we can find an area under a curve is by using Integrals. Probability distribution function for continuous random variables are known as Probability Density function.
Here is an example of Probability distribution of weight of adult males. We can calculate the probability that a man weighs between 160 and 170 pounds by calculating the area of the shaded range, which in this comes out to be 0.135905 or 13.59 %
The area under the curve is always equal to 1, since it describes the total probability of all possible values of x. Therefore
Cumulative Distribution function (CDF)
The cumulative distribution function (cdf) is the probability that a random variable takes a value less than or equal to x. It helps us in answering questions like what is the probability that an outcome will happen with value less than or equal to X. Like what is the probability that an adult male weighs less than or equal to 170 pounds. CDF’s can be derived from PDF’s.
The cumulative distribution function (cdf), F(x), gives the probability that a measured value will fall between −∞ and x:
Exponential Probability Distribution
The exponential distribution is often concerned with the amount of time until some specific event occurs. For example, the amount of time (beginning now) until an earthquake occurs has an exponential distribution.
One of the key properties of Exponential distribution function is memorylessness, meaning the probability of occurrence of an event happening in the future doesn’t depend on how much time has elapsed. Another way to say is that the probability of an event remains the same and doesn’t change with the time elapsed. Typically questions like: How long do we need to wait before a customer enters our shop? Or how much time will elapse before an earthquake occurs in a given region can be answered in probabilistic terms using exponential distribution.
In Reliability engineering, we can use this distribution as we assume that failure rate is constant, i.e. The probability of failure happening is constant during its “useful lifetime”. In reliability, since we deal with failure times, and times are non-negative values, the lower bound of our functions starts with 0 rather than -∞.
So the Probability Distribution function of Exponential Distribution is reliability universe is given by
Worth a read: An Intuitive Guide To Exponential Functions & e:
C.D.F of Exponential Function
C.D.F. Of exponential function can be given below where y is the dummy variable as we are already using “t” variable.
Back to Reliability
Now before I throw a formula at you for calculating reliability, let’s take a look at an example first which is going to build the intuition
Let’s say if there are 50 components operated for 1000 hrs in a test and two of them failed, then we will say the probability of failure of the component in 1000 hrs is:
And the probability of success for the component in 1000hrs is:
Thus, the Reliability is the probability of no failure within a given operating period. If we want to generalize what we just did in the above example
Where P(s) is the Reliability. Now we know the c.d.f F (t) = 1-e^(-ƛt) gives us the probability of a failure by time “t”. If we subtract that from 1, it will give use the probability of success of a component by time “t” which is Reliability.
In simple words we are saying is that we know the probability of a unit failing by the time “t” from F (t), so the remaining probability (1-F (t)) gives us the probability of it’s not failing by the time “t”.
Another widely used Probability Distribution in Reliability is the Weibull distribution, but we will not cover that.
Mean Time Between Failure(MTBF)
MTBF tells us the mean or average life of a system based on the frequency of system outages or failures. It is the average time of failure for all the failure times in the population. MTBF can be given by the Expected Mean (average) of the exponential probability function. Expected Mean of a continuous random variable is given by following equation where f(x) = ƛe^(-ƛt)
MTBF is commonly used for items which are repairable where MTTF is commonly used for Non-Repairable items in the hardware world. MTTF is the amount of time from the placement of a system or component in service until it permanently fails. It’s typically used for components which aren’t repairable and we just throw it away once it fails. Passive components (like network cables) are often not included in MTTF estimation as they tend to have longer lifetimes compared to Active components.
Many times MTTF and MTBF are used interchangeably and many books or text use them in an interchangeable way.
Mean time to Recovery or Repair (MTTR) is the time required to restore the operations of a repairable component of the system. Most common ways to estimate is to simply obtain the sum total of all observed restoration times and divide by the number of reported outages.
Availability is the probability that a network(or component) is available to users at a any given point of time. It is measured as the average fraction of time over an interval for which the system is up.
The unavailability (U) of a unit is simply given by U =1 – A.
Reliability is the likelihood that a system will continue to provide service without failure. Availability is the likelihood that a system will provide service over the course of its lifetime.
Difference between Reliability and Availability
Let’s say a Car may break down and require maintenance 5% of the time. It’s therefore 95% reliable. Now if the same care is equally shared between two Family members, then the Car is available only 47.5% (95%/2) on average. If the Car was 100% reliable, it’s still 50% available and in order to increase the availability, an additional car is required.
Another example is: consider a system that fails every minute on average, but comes back after only 500ms. Such a system has a mean time between failures (MTBF) of 1 min, and therefore the reliability of the system (= e − t/ MTBF) for 60s is R(60) = e − 60/ 60 = 0.368 (≈ 36.8%), which is low, whereas the availability = MTBF/(MTBF+MTTR)= 59/59+0.500 = 0.991 (≈ 99.1%) is relatively high.
So far we have covered some basics and math’s involved behind some basic Reliability related metrics. In the next post we will try to do something useful with that.


California Course Preview
Each California national standards Mathematics course runs in a comprehensive online learning environment that uses state-of-the-art technology–including animations, simulations, exploratory activities, and videos–to engage students.
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|16180||California national standards Mathematics Grade 6||
|16190||California national standards Mathematics Grade 7||
|16200||California national standards Mathematics Grade 8||
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Click each course name to view the course alignments


MA071 & MA072 UNITS 1 & 2: GENERAL MATHEMATICS
Rationale: This course has been designed for students preparing for work in a service industry, banking or an administrative or clerical job. The content provides sufficient grounding for students wishing to undertake Further Mathematics Units 3 & 4 as preparation for further studies in such areas as Business Studies, Economics, Social Science or Biological Science.
Areas of Study will be chosen from
- Statistics – Univariate + Bivariate
- Linear Equations
- Linear Graphs and Modelling
- Shape and Measurement
- Geometry and Trigonometry
- Financial Arithmetic
- Number Patterns and Recursion
Outcomes: Students should be able to:
- Define and explain key concepts in relation to the topics from the selected areas of study and apply a range of mathematical routines and procedures.
- Apply mathematical processes in non-routine contexts and analyse and discuss these applications of mathematics in at least three areas of study.
- Use technology to produce results and carry out analysis in situations requiring problem-solving, modelling or investigative techniques or approaches in at least three of the areas of study.
Assessment: S or N based on the demonstrated achievement of the outcomes specified above. In addition, a level of achievement A+ to E or N will be reported based on results obtained in assessment tasks which include assignments and homework, tests and an exam for each unit.


Pierre Audétat conceived and constructed empirically the first Diatonic Bell in 2006. This system provides a map of the 462 heptationic modes. Having verified mathematically the correctness of the procedure in 2008, Julien Junod applied it to the pentatonic modes. The procedure, being further generalisable to other temperaments, yields this atlas of modes.
J. Junod et al., A Generalisation of Diatonicism and the Discrete Fourier Transform as a Mean for Classifying and Characterising Musical Scales in Proceedings of the Second International Conference in Mathematics, Computation and Music, New Haven, 2009.
Festival Node, Lausanne, 2016.
J. Junod, Les Cloches diatoniques : un atlas visuel et sonore des modes musicaux, Journées d’informatique musicale, Strasbourg, 2020.
We would like to thank Moreno Andreatta and Carlos Agon for their support and advice. The sounds on the keyboards were adapted to the tunings after a convincing demonstration performed by Thomas Noll and Andrew Milne.


Fourth and fifth graders in D.C. will face off for math challenge sponsored by Harmony DC PCS. It is an extracurricular math event to be held at 62 T St NE, Washington, DC 20002 on Saturday, March 16th, 2019 at 10:00 a.m. The event will provide a challenging, engaging mathematical experience which is both competitive and educational. All interested 4th and 5th graders in Washington, D.C. (District of Columbia) are welcome to participate in the competition. There is no registration fee.Click the link for registration.
Note: This event is only for D.C. Schools, NOT open to Maryland or Virginia Schools.


BayesDA: Functions and Datasets for the book "Bayesian Data Analysis"
Functions for Bayesian Data Analysis, with datasets from
the book "Bayesian data Analysis (second edition)" by Gelman,
Carlin, Stern and Rubin. Not all datasets yet, hopefully
||stats, R (≥ 2.2.0)|
||Compiled by Kjetil Halvorsen|
||Kjetil Halvorsen <kjetil1001 at gmail.com>|
||GPL-2 | GPL-3 [expanded from: GPL (≥ 2)]|


To continue our series of fall-themed activities, I set up an easy Candy Corn Counting Game for the boys today. After all, doesn’t every child love to count using sweet treats?
When I saw a similar idea on another website, I knew that it would be a hit in our house. So last night, while I sat in front of the television watching DVR episodes of the classy reality show The Vineyard, I used green and yellow construction paper (plus markers) to create these stalks of corn. And yes, I just admitted that I watch The Vineyard. Anyone else?! Keith’s on board, too!
While my boys know how to count, learning to read and recognize written numbers is an entirely different skill. To keep it simple, I made each child a set of corn from 1 to 5. If you have an older or more advanced audience, just use larger numbers, or mix up the numbers (for instance, make one corn stalk 23 and the other 12).
especially my candy-loving little one!
Looking for other fun fall-themed activities and recipes? Stay tuned all week, and check out these previous posts:


Are you searching for a Pre-Calculus 12 curriculum that will help your students master its topics so that if and when they hit Calculus they can not only survive but even thrive? And all the while still enjoying the course itself? Your search is over.
StudyForge proudly presents: Pre-Calculus 12.
Pre-Calculus 12 is the culmination of a students’ high school mathematical journey, as they learn the fundamentals of advanced functions, logarithms, trigonometric identities and more. The course thoroughly prepares students for the oncoming rigor of Calculus, yet does so in a manner that is compelling and rewarding, and fosters an appreciation for the real-life applications of each topic explored, as well as for math itself.
With auditory, visual, and hands-on components throughout, this is truly a course that suits all learning styles. The lessons are engaging, interactive and student-friendly, with interactive videos that allow students to pause and rewind at any point and go at their own speed, as well as fun, interactive applets and dynamic graphs scattered throughout that enhance student understanding.
Each video is accompanied by a student-friendly note package that allows students to take notes to whatever level of detail they like. There are also an abundance of practice questions for students to internalize the material, with full, detailed solutions for each, as well as a comprehensive review assignment and practice test. Furthermore, with randomized question banks, students can redo every quiz and test in the course to help them master the material and perform their very best.
This course is a prerequisite for students who might take Calculus in the future, whether in high school (Calculus 12 or AP Calculus) or in post-secondary.
Table of Contents
Lesson 2: Vertical and Horizontal Translations
Lesson 3: Reflections
Lesson 4: Vertical and Horizontal Compressions and Stretches
Lesson 5: Combinations of Transformations
Lesson 6: Transformation of Radical Functions and More
Lesson 2: Composition of Functions
Lesson 3: Inverse Functions
Lesson 4: Inverse Reflections
Lesson 5: Parabolas and Circles
Lesson 6: Ellipses and Hyperbolas
Lesson 2: Transforming Exponential Equations
Lesson 3: Introduction to Logarithms
Lesson 4: Solving Equations Using Logarithms
Lesson 5: Product and Quotient Laws
Lesson 6: Graphing Exponential Functions
Lesson 7: Graphing Logarithmic Functions
Lesson 2: Sum of a Finite Geometric Series
Lesson 3: Infinite Geometric Series
Lesson 4: Sigma Notation
Lesson 2: The Remainder Theorem
Lesson 3: Factor Theorem
Lesson 4: Solve Polynomials by Factoring
Lesson 5: Sketching Polynomials by Finding Zeros
Lesson 6: Graphing Rational Functions
Lesson 2: Angular Measure – Radians
Lesson 3: Trigonometric Ratios of Any Angle
Lesson 4: Discovering Parameters of Sine and Cosine
Lesson 5: Graphing Sine and Cosine
Lesson 6: Graphing Tangent
Lesson 7: Graphing Inverse Trigonometric Functions
Lesson 8: Applications of Trigonometric Functions
Lesson 2: Solving Trigonometric Equations Algebraically
Lesson 3: Using Technology to Solve More Complex Equations
Lesson 4: Trigonometric Identities – An Introduction
Lesson 5: Sum, Difference and Double Angle Identities
Lesson 2: Radical Operations
Experience a lesson as your students would
Don’t Take Our Word for It!
See What Others Have to Say:
Online Teacher at EBUS Academy
Blended Teacher at Brampton Christian School
- Interactive applets which help foster a conceptual and intuitive understanding of fundamental topics.
- Digital DESMOS Graphing Calculator within the course
- Inquiry-Based Projects such as:
- Comparing how acidic (or basic) household products are using logarithms
- Analyzing financial investments using sequences and series.
- Exploring Conic Sections
- Modeling real-life data using trigonometric functions


Compound interest is the interest that accumulates on the principal amount of money plus any interest that has been earned during the course of a loan, deposit or debt. Unlike simple interest, which only accrues on the principal, compound interest accrues on both the principal and interest combined. When interest is compounded, the principal amount grows faster than it would under simple interest. In most cases, interest is calculated on a yearly basis, but the terms may vary among financial institutions. The compounded interest grows higher when interest is added to the principal on a frequent basis. Compound interest is beneficial because it allows for a higher return on the initial monetary investment.
Compound interest is calculated by figuring out the amount of interest for the present value of the investment and then adding that amount to the principal. The new dollar amount can be multiplied against the projected number of years of the investment. This figure will be the total amount earned after the time period is up.
The base formula for calculating interest for one year is:
A = P(1+r)
A is the future value, P is the starting principal and r is the interest rate as a decimal.
The formula for calculating annually compounded interest for multiple years is:
A = P(1+r)Y
Where Y is the number of years to compound over.
If we put $1000 in a term deposit for 5 years at 5.5% we get the equasion:
A = 1000(1+0.055)5
At the end of 5 years we would have $1,306.96.
If you would like to compound more often than annually, you need to modify the equation like so:
A = P(1+r/n)Yn
Where n is the number of times you want to compound each year.
If our $1000 deposit above was compounded monthly instead of annually, we would have:
A = 1000(1+0.055/12)60
At the end of 5 years we would now have $1,315.70.
See also our compound interest calculator.
Simple Interest Formula
Simple interest is the interest that is earned on the principal amount of money over a certain amount of time. In this case, interest only accrues on the original amount of money that is loaned, borrowed or deposited. The simple interest on a loan is calculated by multiplying the principal amount by the rate of interest and the amount of time on the loan.
The formula for calculating simple interest is:
I = Prn
I is the interest earned, P is the principal amount, r is the interest rate as a decimal, and n is the number of years remaining on the loan.
For example, if a person lends $10,000 for five years at the rate of 5 percent, we get:
I = 10000 X 0.05 X 5
In other words, this person will earn $2,500 in interest during the course of the loan.
See also our simple interest calculator.


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Proportion definition, comparative relation between things or magnitudes as to size, quantity, number, etc.; ratio. See more.
On the other hand, a proportion is two ratios which have been set equal to each other; a proportion is an equation that can be solved. When I say that a proportion is two ratios that are equal to each other, I mean this in the sense of two fractions being equal to each other. For instance,
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The following matrix is the two-dimensional Givens Rotation from the axis to the axis:
This matrix rotates two-dimensional column vectors about the origin of the – plane.
For instance, if we rotate the unit column vector using this Givens Rotation, then we have:
Therefore, if we rotate the column vector , then we have:
and to rotate that column vector back to the axis so that the value is zero, we would have:
So, we need to use an angle such that the value goes to zero. Let’s concentrate on that lower equation:
atan2( y, x ) in C/C++ and
ArcTan[ x, y ] in Mathematica.
Therefore, when we rotate the column vector back to the axis so that the value is zero, we have:
Since the radius from the origin to the column vector is (also known as the norm), then we also have:
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I lost my old configuration of Manim first in my Mac then in my PC 6 months after. Both happened after upgrading my Python release to the latest version. Fortunately, the Manim Community made the ManinCE more user friendly to beginners like me. The community website provided the step-by-step instruction on how to install Manim in PC, Mac, and Linux.
The code below will update the function V = 3(x - 6)2 as x increases in value. Notice that the equation is a parabolic curve with independent variable x and dependent variable V. The whole graph actually cannot be fitted on the screen. To improve visibility, the equation was multiplied by a reduction factor whose value is embedded-hard into the equation.
Below is the Python code, calling the Manim library, for creating a circle using parametric equations. The circle is parallel to the xy-plane and has a radius of 4 units . It is located 2 units above the xy-plane.
A cylindrical surface of radius radius 4 and axis on the xz-plane and at a distance of 1.5 units from the xy-plane. The thickness of the cylinder is 4 units and the curvilinear length is half the circle.


In the first post in this series, we outlined the motivation and theory behind principal component analysis (PCA), which takes points in a high dimensional space to points in a lower dimensional space while preserving as much of the original variance as possible.
In this follow-up post, we apply principal components regression (PCR), an algorithm which includes PCA as a subroutine, to a small dataset to demonstrate the ideas in practice.
To understand this post, you will need to be familiar with the following concepts:
Ordinary least squares
In ordinary least squares (OLS), we want to find a line of best fit between the points and the labels .
Denoting by the matrix whose rows are the points and the vector whose entries are the labels, the intercept and slope (a.k.a. gradient) are obtained by minimizing . Some matrix calculus reveals that the minimum is obtained at the values of and for which
where is the vector of all ones.
Principal components regression
The idea behind PCR is simple: instead of doing OLS on the high dimensional space, we first map the points to a lower dimensional space obtained by PCA and then do OLS. In more detail, we
- pick a positive integer ,
- construct the matrix whose columns are the first principal components of ,
- compute , a matrix whose rows are the original points transformed to a lower dimensional “PCA space”, and
- perform OLS to find a line of best fit between the transformed points and .
By the previous section, we know that the minimum is obtained at the values of the intercept and gradient for which
Once we have solved these equations for and , we can predict the label corresponding to a new sample as .
Due to the result below, the linear system involving and is a (permuted) arrowhead matrix. As such, the system can be solved efficiently.
Lemma. where is the diagonal matrix whose entries are the first principal components of in descending order.
Proof. Let denote the -th column of . Since is a principal component of , it is also an eigenvector of with eigenvalue , the square of the -th singular value. Therefore, the -th entry of is
Boston house prices dataset
The Boston house prices dataset from  has 506 samples and 13 predictors. For each , we fit using PCR on the first 405 samples (the training set) and report the root mean squared error (RMSE) on both the training set and the set of remaining 101 samples (the test set).
|Rank (k)||Training set RMSE (in $1000s)||Test set RMSE (in $1000s)|
Both training and test set RMSEs are (roughly) decreasing functions of the rank. This suggests that using all 13 predictors does not cause overfitting.
Code used to generate the table above is given in the appendix.
One way to reduce the test set RMSE is to introduce more predictors into the model. Consider, as a toy example, a dataset where each sample has only three predictors: . We can replace each sample by a new sample . In particular, we have added all possible quadratic monomials in . These new entries are referred to as “derived” predictors. Note that derived predictors need not be quadratic, or even monomials; any function of the original predictors is referred to as a derived predictor.
Returning to the Boston house prices dataset, of all possible derived cubic monomial predictors, we randomly choose roughly 100 to add to our dataset. Since we have approximately 400 training samples, it is reasonable to expect that unlike OLS applied to , OLS applied to the derived matrix will almost certainly overfit. We plot the results of PCR below, observing the effects of overfitting after approximately a rank of greater than 80.
 Harrison Jr, D., & Rubinfeld, D. L. (1978). Hedonic housing prices and the demand for clean air. Journal of environmental economics and management, 5(1), 81-102.
import numpy as np from sklearn.datasets import load_boston from tabulate import tabulate TRAIN_TEST_SPLIT_FRACTION = 0.2 X, y = load_boston(return_X_y=True) N, p = X.shape # Train test split. np.random.seed(123) perm = np.random.permutation(N) X, y = X[perm], y[perm] N_test = int(TRAIN_TEST_SPLIT_FRACTION * N) N_train = N - N_test X_test, X_train = np.split(X, [N_test]) y_test, y_train = np.split(y, [N_test]) # Normalize data. X_mean = np.mean(X_train, axis=0) X_std = np.std(X_train, axis=0) X_test, X_train = [(X_sub - X_mean) / X_std for X_sub in [X_test, X_train]] _, _, V_T = np.linalg.svd(X_train) V = V_T.T rows =  for k in range(1, p + 1): V_k = V[:, :k] Z_k = X_train @ V_k # Solve for alpha_k and beta_k by adding a bias column to Z_k. # This is not efficient (see "Computational considerations" above). Z_k_bias = np.concatenate([np.ones([N_train, 1]), Z_k], axis=1) solution = np.linalg.solve(Z_k_bias.T @ Z_k_bias, Z_k_bias.T @ y_train) alpha_k = solution beta_k = solution[1:] V_k_beta_k = V_k @ beta_k row = [k] for X_sub, y_sub in [(X_train, y_train), (X_test, y_test)]: y_hat = alpha_k + X_sub @ V_k_beta_k error = y_hat - y_sub rmse = np.sqrt(np.mean(error ** 2)) row.append(rmse) rows.append(row) table = tabulate(rows, headers=['Rank (k)', 'Training set RMSE (in $1000s)', 'Test set RMSE (in $1000s)'], tablefmt='github') print(table)


The investigation of random matrices was initiated by the early works of Wishart  in 1928, when he introduced the probability distribution of random matrices now bearing his name. The Wishart distribution generalizes the sum of squares of normal variables (a.k.a. -distribution) and can be used for the maximum-likelihood estimation of the covariance matrix of i.i.d. vectors drawn from a multivariate normal distribution. The Wishart ensemble has degrees of freedom and is characterized by a fixed real positive definite matrix . It consists of real symmetric, positive definite matrices of size with the probability density function
where denotes the gamma function . In the special case , the density (1) takes the form that of the -distribution.
The first surprising employment of random matrices was in nuclear physics. It was Wigner in his 1951 paper , who proposed “a new kind of statistical physics” based on random matrices to model certain properties of excited states of heavy nuclei. His intent was to gain some meaningful information about the high-energy states of complex systems by considering ensembles of different Hamiltonians, which are related by their ‘symmetry’. The theoretical footing of Wigner’s idea is Bohr‘s Compound-nucleus model from 1936 , which was designed to explain nuclear reactions, such as neutrons bombarding a nucleus of not too small atomic weight, as two-stage events:
- Union of the colliding bodies into a single unit, the so-called compound nucleus. This takes about seconds.
- Disintegration of the unstable compound into an ejected small particle and a product nucleus.
Between the two events there is a relatively long period of time, typically to seconds. To demonstrate his model Bohr made a toy model (see Figure 1 below). A number of billiard balls in a shallow basin represent the target nucleus. The struck ball enters the basin hitting other particles and quickly distributing its energy among them. If the basin and the particles are regarded as perfectly smooth and elastic, there will be a point when some particles close to the basin’s edge receive so much energy that they exit the compound.
At a first glance, Wigner’s proposal seems baffling for (at least) two reasons. First, the energy levels and eigenstates of a quantum mechanical system is completely determined by its Hamiltonian, therefore one could ask: “How could a statistical approach be of any use?” One reason is that energy levels of highly excited states of heavy atoms become so dense that precise information about individual levels cannot be obtained using exact methods. Second, the usage of an ensemble of Hamiltonians to describe a single system. This is in opposition with standard statistical mechanics, where one considers copies of identical physical systems, all governed by the same Hamiltonian but differing in initial conditions, and calculates thermodynamic functions by averaging over this ensemble. As Dyson explains in :
“We picture a complex nucleus as a ‘black box’ in which a large number of particles are interacting according to unknown laws. The problem then is to define in a mathematically precise way an ensemble of systems in which all possible laws of interaction are equally probable.”
Wigner considered real symmetric matrices with the algebraically independent entries , being also statistically independent normal variables with the probability distribution
where is a normalization constant. This is called the Gaussian Orthogonal Ensemble (GOE). The word ‘orthogonal’ is meant to indicate that the distribution (2) is invariant under conjugation with orthogonal matrices, that is with . Various spectral statistics of this ensemble can be calculated, such as the joint eigenvalue distribution, nearest neighbour distribution, correlation functions, etc. When compared with experimental data, the effectiveness of GOE is striking (see Figure 2).
In 1962, Dyson showed  that there are three invariant ensembles of random matrices: orthogonal, unitary, and symplectic. Again, orthogonal, unitary, and symplectic refers to the symmetry groups. Those with Gaussian components are called GOE, GUE, and GSE. The joint distribution of their eigenvalues can be written as
with for orthogonal, for unitary, and for symplectic. For more on the topic of random matrices in physics see, for example Mehta’s book  or the 7th John von Neumann Lecture given by Wigner .
An astounding connection was discovered between analytic number theory and random matrices in 1972, when Montgomery gave a talk in Princeton at the Institute for Advanced Study about his results on the non-trivial zeros of the Riemann zeta function
He explained how the zeros on the critical line, viz. such that , tend to repel each other and that he conjectured a formula for the distribution of the gaps between them. After the talk during teatime Montgomery spoke to Dyson, who didn’t attend the talk. Here is Montgomery’s recollection of the conversation:
“Freeman Dyson was standing across the room. I had spent the previous year at the Institute and I knew him perfectly well by sight, but I had never spoken to him. Chowla said: ‘Have you met Dyson?’ I said no, I hadn’t. He said: ‘I’ll introduce you.’ I said no, I didn’t feel I had to meet Dyson. Chowla insisted, and so I was dragged reluctantly across the room to meet Dyson. He was very polite, and asked me what I was working on. I told him I was working on the differences between the non-trivial zeros of Riemann’s zeta function, and that I had developed a conjecture that the distribution function for those differences had integrand . He got very excited. He said: ‘That’s the form factor for the pair correlation of eigenvalues of random Hermitian matrices!’ “
For a more detailed description, see .
Dynamical Systems and Quantum Chaos
A significant boost to random matrix theory was the discovery its connection with classical and quantum chaos. In the late 70’s and early 80’s the quantum spectra of conservative systems, which have chaotic classical analogues, was explored (e.g. Sinai billiard). Sequences of energy levels of the same symmetry class were generated numerically. In 1984, Bohigas, Giannoni, and Schmit  produced a statistically significant set of data consisting of more than 700 eigenvalues of the Sinai billiard. They applied methods of statistical analysis (invented by Dyson and Mehta for problems in nuclear physics) and found that nearest neighbour spacings of levels were distributed like that of the GOE.
They formulated the conjecture that quantum systems with time-reversal symmetry, whose classical counterparts are chaotic, have spectral statistics of the GOE. In the absence of time-reversal symmetry GOE is replaced by GUE. In contrast, it was conjectured by Berry and Tabor in 1977  that if the corresponding classical dynamics is completely integrable, then the nearest neighbour spacing is equal to the waiting time between consecutive events of a Poisson process. Both of these conjectures are still open, but their statement is strongly supported by a vast number of examples.
Consult  for an extensive review about random matrix theories and their applications in quantum physics.
- J. Wishart, The generalised product moment distribution in samples from a normal multivariate population, Biometrika 20A (1-2): 32-52, 1928.
- E.P. Wigner, On the statistical distribution of the widths and spacings of nuclear resonance levels, Mathematical Proceedings of the Cambridge Philosophical Society 47, 790-798, 1951.
- N. Bohr, Neutron capture and nuclear constitution, Nature 137, 344-348, 1936.
- F.J. Dyson, Statistical Theory of the Energy Levels of Complex Systems, I, II & III, J. Math. Phys. 3, 140-175, 1962.
- M.L. Mehta, Random Matrices, Vol. 142, Academic press, 2004.
- E.P. Wigner, Random Matrices in Physics, SIAM Review 9(1) 1-23, 1967.
- P. Bourgade, Tea time in Princeton, Harvard College Math. Review 4, Spring 2012.
- O. Bohigas, M.J. Giannoni, and C. Schmit, Characterization of Chaotic Quantum Spectra and Universality of Level Fluctuation Laws, Phys. Rev. Lett. 52(1), 1984.
- M.V. Berry and M. Tabor, Level clustering in the regular spectrum, Proc. Roy. Soc. A 356, 375-394, 1977.
- T. Guhr, A. Müeller-Groeling, and H.A. Weidenmüeller, Random Matrix Theories in Quantum Physics: Common Concepts, Phys. Rept. 299, 189-425, 1998; arXiv:cond-mat/9707301


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About the ACCUPLACER tests
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Cs61a Hw 8 Solutions
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The graphs have the same shape, but are stretched in the x direction. java: Program2: Write a program called CarRental that computes the cost of renting a car for a day based on the size of the car: economy, medium, or full-size. Currently all checks are sent to. MAT 602 HOMEWORK 8 SOLUTIONS # 2. Homework Reminders Homework reminders, test dates, etc. · 9/13/05 - HW 03 Ch 02. UC Berkeley CS61A, Fall 2014. Need 6 noninferior solutions evenly spaced along the Z1 axis Max is 14; Min is 4. Due Wednesday, October 30th, 2019 by 11:59PM, 50 points. Homework 8 Solutions 1. The max function can take an arbitrary. We are nationally recognized experts in the field of household employment taxes, regularly consulted by media such as the New York Times and Wall Street Journal. Handouts Here are the handouts that I gave out on the first day of class. Bivariate Fit of StockRate By Handicap. Suppose x 2Z. Homework 25%. Algebra -> Customizable Word Problem Solvers -> Misc-> SOLUTION: A student did 3/5 of his homework and 8 more problems. Let = xdx+ ydy+ zdzand = ydx+ zdy+ xdz. Homework 8. The television series Sesame Street is concerned mainly with teaching preschool skills to children age 3-5, with special emphasis on reaching economically disadvantaged children. Check for changes often as it is subject to change at any time. The ALU is exactly the one used from HW 6; see the homework 6 solution for the ALU's details. analysis time _t: time. Unit 9 Quiz #2 Solutions. (a) In the reaction e+H+ ←→H we have νe = 1,νH+ = 1,νH = −1 and so the law of mass action gives us ne nH + nH nQe nQH nQH 1 ZH(int) (1) where ZH(int) = eβ∆E is the internal partition function of the hydrogen atom (we just. My assignment was due at 10. We have a strong 31 year history of providing our clients with quality candidates and service. So when we receive the data bits, the first one is the one to output on the LED. Homework 9: Due 4/8: Work the problems on the Worksheet Read pp. When one scans a number of. From table 4 with df=4, we ﬁnd the critical values t. Do not wait until the night before they're due to undertake the homework problems for the previous week. Homework 5 - Solutions. Math 8 Homework 8 Solutions 1. Homework must be turned in to the Chemical Engineering office (MEB 3290) by 4:00 PM on the day it is due. Note that im(F) is a subspace of V. On this contour plot, that means. Solution Homework #8 • 4. 6k Downloads; Part of the Texts in Computer Science book series (TCS) This is a preview of subscription content, log in. 111 A mounting bracket for electronic components is formed from sheet metal of uniform thickness. Look at the solution for the homework version of this problem - it is attached to the back of the exam 2. Interpretation: If the partial. I will use notation of the text. A completely analogous. No credit will be given for any homework submitted after the solutions are posted. Math 250A Solutions to Homework 8 (III. The magnetic eld is B~= ( 2:50. Unit 9 Test Solutions. Video Supported Review Packet for Exam. Since p 2 = 1 2 "p 2 2 p 3 2 p 2 + p 3 + (p 2. FINC 330 FINC330 Homework 8 Solution. A Morning Filled with 2 to the 32 Suns 2nd Place: Still some opportunities for extra credit, to be announced. CS 61A Weekly Schedule summer 2014 summer 2014. If an individual chooses to be a thief and is not caught by the police, he receives a payoff of W. Homework 8 ; Notes. Homework 1 Solutions. Solutions to current and old homeworks. Due in class next Friday. Chapter 3: Problem Solutions Fourier Analysis of Discrete Time Signals Problems on the DTFT: Definitions and Basic Properties àProblem 3. Office Hours: Tu 3 - 4, and by appointment. If you have any questions about using OK, please refer to this guide. 10 and x(CF4) = 0. Solutions. Power Flow Coding Project. ASK NOW About Slader. -MASTAN versions for pc or. CMSC 203 – Homework 8 Solutions Exercise 6. T = 3? 6 3 0 3 6. 3-3) Show how Quicksort can be made to run in O(nlogn)timeintheworst case. ECE 5463 Homework 8 -Solution Due November 4, 2014 1. ECE 410 Homework 8 -Solutions Spring 2008 Problem 1 a) Complete the following truth table for a 4-bit shift left/right circuit that specifies the number of bits to shift by the 2-bit word SH and the direction of shift by the 1-bit R/L signal (R/L=1 denotes shift right, R/L=0 denotes shift left). Solution 2. Physics 123 Homework Solutions Week #6 Unit T Thermodynamics T9B. You can use them as reference but I would strongly suggest you to attempt these problems without relying on them in your initial attempts. The data are given in the following table. Rochester Institute of Technology. solutions. STATISTICS 371 HOMEWORK 8 Solution 7. Students will have at least two or three weeks to try every single question. Boyd EE263 homework 8 solutions 13. A variable star changes in brightness by a factor of 2. 11/10: Homework 8: Reading and Exercises. Previous finals with my answers & solutions (not available elsewhere!) Homework Solutions. This is a tentative outline and will be updated at least 24 hours in advance before each class. Solutions for Homework 8 Please see below for the key to the homework and some remarks. MAT 602 HOMEWORK 8 SOLUTIONS # 2. The beam is a channel section with dimensions as shown in the figure. CS61A HOMEWORK 8 - OK autogrades programming assignments, facilitates submission, composition feedback, and analytics for your class. Guest additional topics lecture 5-6pm Wed 10/16 in 3106 Etcheverry. Homework 7 - Due April 23, 11:59pm. Pre-Algebra Solutions Homework 8 1) a) a4b2a/a5 = a4+1−5b2 = a0b2 = b2 b) (−6y3) = (−6)2y3·2 = 36y6 c) (3x2)3 (9x3y)2 = 33x6 32x6y2 = 3 y2 d) (4xy)3(z2)−3. The first expression in each pair is a predicate: Each link below leads to a directory with a readme file containing the homework or project, a readme. Construct CFG C to accept all strings. cs61a homework 2. String Theory 230A Homework # 8 Solutions Richard Eager Department of Physics University of California; Santa Barbara, CA 93106 Problem 1 Given the BRST transformation. If fIﬁg is a totally ordered. CS61A HOMEWORK 8 SOLUTIONS - To implement the system, we will use the following data abstraction. The team of highly-qualified, educated specialists in both writing and science are eager to do what you want right now. 16) Find the dimensions of the rectangle of largest area that has its base on the x-axies and its other two vertices above the x-axis and lying on the parabola y = 8−x2. Homework Reminders Homework reminders, test dates, etc. The Python implementation of this procedure is provided for your reference. Solution: Proof. Solutions to homework 1; Solutions to homework 2; Solutions to homework 3; Solutions to homework 4; Solutions to homework 5; Solutions to homework 6; Solutions to homework 7; Solutions to homework 8. can be sent directly to your phone or to an email address. Let M 1 be the Turing machine whose description is given in Example 3. We have provided a starter file for the questions below. Previous Slide  Next Slide  Experience the freedom and flexibility of WileyPLUS Schedule a Demo Study Anytime, Anywhere Learn how WileyPLUS fits your mobile lifestyle. CS 420, Spring 2019 Homework 8 Solutions 1. Homework 8 - Solutions 1st December 2004 Problem 1 (a) The diﬀerence now is that the summation goes over (nx,ny,nz) and for Zs we get Zs = eǫ0/τ (1− e−~ω/τ)3 ≈ τ3eǫ0/τ ~2ω3 where the last equality holds for τ ≫ ~ω. Solutions available here. pdf Homework 9 Posted: 1/15/15 Due. ) Homework #2 Homework #3 Homework #4 Homework #5 Homework #6 Homework #7 Homework #8 Homework #9. These are called regular numbers. CS61A HOMEWORK 8 SOLUTIONS - You may check the solutions if you are stuck, but we highly recommend you work through the problem on your own for practice. ECE 329 - HOMEWORK AND EXAMS Homework 1 - Solutions. Solution: An argument against a small time quantum: Efficiency. (a)Use implicit di erentiation to nd dy dx. Try Chegg Study today!. 2-1 under the condition that nodes 5 and 6 are linked by a 2-mile cable (the corresponding network is shown in Figure 1). lim x!0 x2 1 cosx First notice that this limit is of the form 0 0 so we apply l’hopital’s Rule twice. Since a = 1a and 1 is a unit, a ∼ a. Homework assignments are made available at the end of class and are due the following week by the end of the day. The example we used was factorial shown in both Python and Scheme:. Note that m= nyields the same solution as m= 0; in general, if two choices of mdiffer by nthen they yield the same solution, and thus it sufﬁces to look at th e nspeciﬁed values of m. Formally, De nition 19. Higher-Order Functions Video 1pp 1pps 8pp We'll ask you to explain one part of a lab or homework that you completed. Solutions to Homework 8 Statistics 302 Professor Larget For each R problem, turn in answers to questions with the written portion of the homework. If he chooses not to commit theft, he receives a zero. We have provided a hw3. Resources Aops Wiki AMC 8 Problems and Solutions Page. 21 We are given a system with open loop transfer function G(s) = K s(s/2 +1)(s/6 +1) (1) and unity negative feedback. For an even energy eigenstate ϕ, ϕ(x) = ϕ(x) and ϕ′(x) = ϕ′(x). Each resident has to decide whether to engage in the crime of theft. Homework 9: Due 4/8: Work the problems on the Worksheet Read pp. 1 #2: converges to 0 11. Project Management Homework 8 - Solution manual Project Management. that given f ∈ Aut(G) we have φˆ(f) ∈ Aut(H). View Demo Student. We have a strong 31 year history of providing our clients with quality candidates and service. 111 A mounting bracket for electronic components is formed from sheet metal of uniform thickness. py and vitamin04. If a ≡ b then a = ub for some unit u ∈ D. These links will point to a published google doc once the homework solution homeworo released at least 72 hours after the deadline. In this case borrowing money increases your wealth and lending money. Extra material related to CS61A, Fridays 4: CS 61A Data Science.


Instructor: Dr. William DeMeo
Time/Place: MW 3:55–5:10, LeConte Rm 310
Textbook: Linear Algebra and Its Applications, 2nd Edition, Peter Lax.
- Finite Dimensional Vector Spaces, Paul Halmos.
- Linear Algebra Problem Book, Paul Halmos.
- Advanced Linear Algebra, Steven Roman.
Brief Summary: (from the catalog) Vector spaces, linear transformations, dual spaces, decompositions of spaces, and canonical forms.
Details: This is a beginning graduate level treatment of linear algebra in which we focus on finite dimensional vector spaces from a more general point of view than you might have seen in an undergraduate class. Below is a detailed list of some of the topics we will cover (essentially the first 172 pages of Lax’s book, plus additional topics depending on students’ interests). In addition to these topics, we will cover vector spaces over finite fields, the Jordan canonical form and Smith normal form. Depending on student interest, we might also spend some time demonstrating and applying the theory using computer algebra software (like Sage) and numerical software (like Matlab or Octave/Sage).
- Review of Fundamentals: Linear Space, Isomorphism; Subspace; Linear Dependence; Basis, Dimension; Quotient Space.
- Duality: Linear Functions; Dual of a Linear Space; Annihilator; Codimension; Quadrature Formula.
- Linear Mappings: Domain and Target Space; Nullspace and Range; Fundamental Theorem; Underdetermined Linear Systems; Interpolation; Difference Equations; Algebra of Linear Mappings; Dimension of Nullspace and Range; Transposition; Similarity; Projections.
- Matrices: Rows and Columns, Matrix Multiplication, Transposition, Rank, Gaussian Elimination
- Determinant and Trace: Ordered Simplices; Signed Volume, Determinant; Permutation Group; Formula for Determinant; Multiplicative Property; Laplace Expansion; Cramer’s Rule; Trace
- Spectral Theory: Iteration of Linear Maps; Eigenvalues, Eigenvectors; Fibonacci Sequence; Characteristic Polynomial; Trace and Determinant Revisited; Spectral Mapping Theorem; Cayley-Hamilton Theorem; Generalized Eigenvectors; Spectral Theorem; Minimal Polynomial; When Are Two Matrices Similar; Commuting Maps.
- Euclidean Structure: Scalar Product, Distance; Schwarz Inequality; Orthonormal Basis; Gram-Schmidt; Orthogonal Complement; Orthogonal Projection; Adjoint; Overdetermined Systems; Isometry; The Orthogonal Group; Norm of a Linear Map; Completeness Local Compactness; Complex Euclidean Structure; Spectral Radius; Hilbert-Schmidt Norm; Cross Product.
- Spectral Theory of Self-Adjoint Mappings: Quadratic Forms; Law of Inertia; Spectral Resolution; Commuting Maps; Anti-Self-Adjoint Maps; Normal Maps; Rayleigh Quotient; Minmax Principle; Norm and Eigenvalues.
- Calculus of Vector- and Matrix-Valued Functions: Convergence in Norm; Rules of Differentiation, Derivative of det A(t), Matrix Exponential; Simple Eigenvalues; Multiple Eigenvalues; Rellich’s Theorem; Avoidance of Crossing.
- Matrix Inequalities: Positive Self-Adjoint Matrices; Monotone Matrix Functions; Gram Matrices; Schur’s Theorem; The Determinant of Positive Matrices; Integral Formula for Determinants; Eigenvalues; Separation of Eigenvalues; Wielandt-Hoffman Theorem; Smallest and Largest Eigenvalue; Matrices with Positive Self-Adjoint Part; Polar Decomposition; Singular Values; Singular Value Decomposition;
And a selection of topic from the following, as time permits and depending on students’ interests:
- Convexity: Convex Sets; Gauge Function; Hahn-Banach Theorem; Support Function; Caratheodory’s Theorem; Konig-Birkhoff Theorem; Helly’s Theorem.
- The Duality Theorem: Farkas-Minkowski Theorem; Duality Theorem; Economics Interpretation; Minmax Theorem.
- Normed Linear Spaces: Norm; lp Norms; Equivalence of Norms; Completeness; Local Compactness; Theorem of F. Riesz; Dual Norm; Distance from Subspace; Normed Quotient Space; Complex Normed Spaces; Complex Hahn-Banach Theorem; Characterization of Euclidean Spaces.
- Linear Mappings Between Normed Linear Spaces: Norm of a Mapping; Norm of Transpose; Normed Algebra of Maps; Invertible Maps; Spectral Radius;
- Positive Matrices: Perron’s Theorem; Stochastic Matrices; Frobenius’ Theorem.
- How to Calculate the Eigenvalues of Self-Adjoint Matrices: QR Factorization; Using the QR Factorization to Solve Systems of Equations; The QR Algorithm for Finding Eigenvalues; Householder Reflection for OR Factorization; Tridiagonal Form; Analogy of QR Algorithm and Toda Flow; Moser’s Theorem.


Fingerspelet is a sensor based counting game for children. The player answers mathematical questions with their own fingers. This project was a part of a embodied learning course at Malmö University.
In the research phase of this project we found that counting on fingers helps you learn basic mathematics. I remember my teacher not encouraging me to count on my fingers and I have always been doing it under the table on the sly. So we thought we wanted to create a game that encourages this type of learning.
To create a game where the player is able to answer questions with their fingers we had to use some kind of sensor. We ended up using the Microsoft Kinect sensor in this project. A finger counting library was found but it worked really poorly so we had to customize it. A low pass filter was implemented and the sensor was programmed to only read at a specific depth (see picture)
The questions was made with representations instead of numbers. To use just numbers would not be that funny or appealing for a child. The questions was more of counting cars and apples type of questions.
In this project I had the technical/graphical role. I programmed some of the functionality for the game and the sensor while aslso designing some of the levels


Main Page -> Working Seminars -> Children 3-6 years old
Abstract of the Working Seminar (3-6 years old)
NRICH Primary Coordinator
Centre for Mathematical Sciences
University of Cambridge, UK
In these working seminars, we will consider the features of an inquiry based kindergarten setting in the context of mathematics. By discussing and reflecting on the conference content, and bringing our own experiences to the table, we will aim to come to a shared understanding of how young children can be supported on their mathematical journey.
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Study With Us
The University of Edinburgh is one of only 15 Universities in the UK chosen by Q-Step to develop and deliver new forms of quantitative skills training to contribute to equip future graduates with advanced data analysis skills.
The focus of the Edinburgh centre is the development of undergraduate degree programmes and to embed quantitative methods training into existing courses. This is supported by a range of related activity, including summer schools for secondary school students, internships with key local employers, and a research seminar series. We also offer a number of exciting short courses to support teachers and promote the acquisition of quantitative skills in secondary schools.
The centre also supports the development of advanced quantitative methods courses for MSc and PhD students interested in applying quantitative methods to their social & political science research projects. There are also tutoring opportunities for PhD students.
Read more about our different study opportunities:


The Stevens STEM (Science, Technology, Engineering and Mathematics) Ambassadors Program was developed to increase awareness and excitement about career opportunities in science, technology, engineering and mathematics (STEM) women and minorities in grades 5-10. The program is a collaborative effort between the Charles V. Schaefer School of Engineering and Science, the Office of Undergraduate Admissions, and Women's Programs. Led by Stevens’ undergraduate men and women trained by Stevens faculty and staff, interactive presentations and workshops are given to schools and organizations in the New York-New Jersey Metropolitan area as well as hosted on-campus. STEM Ambassadors represent a broad group of science and engineering disciplines, with a particular emphasis in areas where women and minorities are most underrepresented: computer science, electrical engineering and mechanical engineering. In addition to encouraging young women and minorities to pursue study in STEM fields, this program plays a key role in developing the leadership capacity of the Stevens STEM Ambassadors.
Introduce a Girl to Engineering Day - February 20, 2014
Stevens STEM Ambassadors with the Upper Saddle River Girl Scouts for a day of science and engineering! November 16, 2013
For more information about our STEM Ambassadors, please email


By the secant-tangent theoremthe square of this tangent length equals the power of the point P in the circle C.
Circlesparabolashyperbolas and ellipses do not have any inflection point, but more complicated curves do have, like the graph of a cubic functionwhich has exactly one inflection point, or a sinusoid, which has two inflection points per each period of the sine.
Just like the geometrical approach, visualize that you are traveling from point A to point B.
No tangent line can be drawn through a point within a circle, since any such line must be a secant line. Here is a general rule for taking the derivative of all terms of a polynomial where c is a constant: For comparison, secant lines intersect a circle at two points, whereas another line may not intersect a circle at all.
We need to find a normal vector. This power equals the product of distances from P to any two intersection points of the circle with a secant line passing through P. The third term is eliminated because it does not have an x, which means it is a constant.
It has been dismissed and the modern definitions are equivalent to those of Leibniz who defined the tangent line as the line through a pair of infinitely close points on the curve.
The derivative of a function must have the same limit from left to right for it to be differentiable at that point. The line segments OT1 and OT2 are radii of the circle C; since both are inscribed in a semicircle, they are perpendicular to the line segments PT1 and PT2, respectively.
The reason for this is because the number 3 can be written as 3x0, and when the 0 comes down the whole term becomes 0.
If none of them is vertical thus none of them is horizontalas shown in Fig. Tangent circles Two pairs of tangent circles. This property of tangent lines is preserved under many geometrical transformationssuch as scalingsrotationtranslationsinversionsand map projections.
A line a is drawn from O, the center of the circle, through the radial point T; The line t is the perpendicular line to a. This is a tedious process when you want to find the slope for many points on the graph. The following problems range in difficulty from average to challenging.
Its slope is the derivative ; green marks positive derivative, red marks negative derivative and black marks zero derivative. Note that horizontal tangent lines are classified as non-vertical tangent lines.
More specifically, two curves are said to be tangent at a point if they have the same tangent at a point, and orthogonal if their tangent lines are orthogonal.
GO Differentiation - Taking the Derivative Differentiation is the algebraic method of finding the derivative for a function at any point. Tangent plane to a sphere In geometrythe tangent line or simply tangent to a plane curve at a given point is the straight line that "just touches" the curve at that point.
When you display the graph of f in the default viewing rectangle you see that f has one relative maximum point near -1,4 and one relative minimum point near 2, Click HERE to see a detailed solution to problem Integration is basically a type of summing up each value between 2 limits.
Using the default view, the lowest point found while tracing near the minimum point is 1. The red line is tangential to the curve at the point marked by a red dot. The tangent at A is the limit when point B approximates or tends to A.
A graph of g is sketched in Fig.Nov 05, · Find an equation of the tangent line to the graph of y = ln(x2) at the point (5, ln(25)).?
More questions Find equation of the tangent line to the curve y = ln e^(6x+1) at the point x =cheri197.com: Resolved. (b) Write an equation fcr the tangent line to the graph of f at x = 2.
(c) Write an equation of the graph that is the reflection acros the Y-axis of the graph of f. f(x) —. Calculus Project study guide by zipapezooda includes 30 questions covering vocabulary, terms and more. Use implicit differentiation to find an equation of the tangent line to the graph of the equation at the given point.
arctan(x+y)=y²+π/4, (1,0) y=-1(x-1) Find an equation of the tangent line to the graph of the function at the given. To find the equation of the line perpendicular to the given line, x + 7y = 8, you need two important pieces of information, i.e., 1.) you need to know the coordinates (x, y) of a point that the perpendicular line passes through, and 2.) the slope of the perpendicular line.
Parametric curves in the plane 1. The idea of parametric equations. Up to now, we’ve been used to describing curves in the xy-plane by specifying a single equation that relates xand y, such as y= x2 to de ne a parabola or x2 + y2 = 2 to de ne the circle of radius p.
Using the Point-Slope Form of a Line Another way to express the equation of a straight line.
Point-slope refers to a method for graphing a linear equation on an x-y cheri197.com graphing a linear equation, the whole idea is to take pairs of x's and y's and plot them on the graph.Download


2Nd Grade Clock Face Worksheet. Free pdf worksheets from k5. Our time telling worksheets for your second grade children use an analog clock face in funny and engaging ways.
You may select 1, 2, 4, 6, or 9 clock faces to be printed on these time worksheets. *click on open button to open and print to worksheet. Worksheets are clock face, clock, all faces were found for on the clock, acute and obtuse angles on clock faces, telling time work, blank digital and analogue clock work, telling time work, telling time.
Welcome To The 2Nd Grade Math Salamanders Time Worksheets Hub Page.
You can also make customized worksheets. Matching the time on an analog clock to a digital clock. Telling the time to the full hour.
Grade 2 Interactive Clock Worksheets.
Telling the time lesson plans for kindergarten, 1st grade & 2nd grade; Clock face with hands time worksheets. Draw the time on the clock face (quarter hours), large clocks
Second Grade Time Worksheets Worksheet #3;
Telling time worksheets for 2nd grade.
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Number MagicUnknown - 2014
The real trick in this book is that it will improve your young magicians' arithmetic skills while teaching them fun feats of magic. The simple, step-by-step instructions are fully illustrated and are sure to captivate your reluctant readers. And the mathematical tricks themselves will delight and astound.
Publisher: New York : Windmill Books, 2014.
Edition: First Edition.
Branch Call Number: 793.8 CAN J
Characteristics: 32 pages : color illustrations ; 28 cm.


Summer Scholars Camp
Registration for this camp is now closed.
For students entering grades 2 - 5
Summer Scholars is an instructional school-based program targeting literacy and math intervention for students entering grades 2-5, with identified needs in literacy and math. Students will be engaged in comprehensive and meaningful literacy and math instruction using evidence-based practices to address their individual needs. The goal of Summer Scholars is to accelerate student growth in literacy and math.
Comprehensive Literacy Instruction: phonemic awareness, phonics, comprehension, fluency, vocabulary, and writing.
Literacy Workshop: literacy workstations, whole group, small group, and one-on-one tailored instruction in reading and writing.
Area of Instruction: grade level concepts with a focus on number sense and mathematical fluency.
Math Framework: Activities, games, choice time, and math mindset activities.
Social Emotional Learning (SEL)
literacy mindset and math mindset activities


I retired in 2004 and since then have used some of my spare time reading up on ‘classical’ discoveries in maths, principally Number Theory.
I found that I invariably ended up writing my own ‘take’ on particular topics. I have consolidated my notes on completed topics as pdfs and these can be accessed through the web pages shown in the sidebar. The most substantial of them are those on binary quadratic diophantine equations, and on the prime number theorem, which can be found under ‘Notes on Numbers’.
The blog represents odds and ends and work in progress. The most substantial of the posts are on Weierstrass’ proof of the Lindemann-Weierstrass theorem (February 2010) and Roger Apéry’s proof that zeta(3) is irrational (October 2011), although I must say that the short post on the coin problem/ Frobenius number (January 2012) is also a favourite of mine.
As well there are pages with links to resources I have found useful.
The banner at the top of the blog shows the portion of the manuscript of Riemann’s 1859 memoir ‘Ueber die Anzahl … ‘ (complete with ink smudge) in which he comments on his famous hypothesis: ‘One would of course like to have a rigorous proof of this but I have put aside the search for such a proof after some fleeting vain attempts because it is not necessary for the immediate objective of my endeavours’ (translation from H M Edwards’ book on Riemann’s Zeta Function).
I suspect that if he had decided differently we wouldn’t still be wondering 150 years later.
Anyone wishing to contact me can do so by making a comment below.


Incluye referencias bibliográficas e índices
Introduction -- I. Regression analysis with cross-sectional data. The simple regression model -- Multiple regression analysis: estimation -- Multiple regression analysis: inference -- Multiple regression analysis: OLS asymptotics -- Multiple regression analysis: further issues -- Multiple regression analysis with qualitative regressors -- Heteroscedasticity -- More on specification and data issues -- II. Regression analysis with time series data. Basic regression analysis with time series data -- Further issues in using OLS with time series data -- Serial correlation and heteroscedasticity in time series regressions -- III. Advanced topics. Pooling cross-sections across time: simple panel data methods -- Advanced panel data methods -- Instrumental variables estimation and two stage least squares -- Simultaneous equations models -- Limited dependent variable models and sample selection corrections -- Advanced time series topics -- Carrying out an empirical project -- IV. Appendices. R scripts.
"This book does not attempt to provide a self-contained discussion of econometric models and methods. It also does not give an independent general introduction to R. Instead, it builds on the excellent and popular textbook 'Introductory Econometrics' by Wooldridge (2016). It is compatible in terms of topics, organization, terminology, and notation, and is designed for a seamless transition from theory to practice."--
Texto en inglés


Fibonacci ratio courses
Fibonacci retracements and extensions have become
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Fibonacci ratio basics
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It takes key retracement numbers such as
23.6%, 38.2% and 61.8% to support
entry and exit points when placing a trade.


SAT Section 1 – Mathematics
Mathematics, which is also known as the Quantitative Section or the Calculation Section consists of 3 sections, two 25-minute tests and one 20-minute test. The 20-minute test consists of a total of 16 multiple choice questions. The questions include advanced math concepts such as Basic Algebra, Intermediate Algebra and Advanced Algebra. Questions will form around favorite triangles, fractions, decimals and percentages.
The first 25-minute section consist of 20 multiple-choice questions. The second 25-minute test has a total of 8 multiple-choice questions and 10 Grid-in questions. The 10 Grid-in questions allow you to have a more natural math response than the usual multiple choice questions. With this scoring system, you are supposed to fill in the grid with 4 boxes and a column of ovals below every box.
Calculators are allowed for the SAT math test but only the 4-function scientific and graphing calculators. Calculators with a ‘QWERTY’ keyboard and calculators on laptops, organizers and mobile devices are not allowed.
The average score for the SAT Mathematics section is 515 points.


Once you understand a few important basics, you'll be well on your way to determining not only if a piece of gold jewelry is real, but equally important tidbits like gold content (purity) and thus, value.
The first term we'll look at is "karat", which is a standard unit used to measure the purity of gold. It works on a sliding scale up to 24k (pure gold), although you typically won't see jewelry marked less than 10k (older pieces do go lower). A higher number of karats means the item contains more gold, and lower numbers means less gold. So if you have a 10k gold pendant that is the identical SIZE of another pendant made of 18k gold, the 10k pendant will not only contain LESS GOLD, it will WEIGH LESS.
In order to find the gold content in a piece of jewelry, you first must know the karat purity. You can check out the section on Identifying Real Gold Jewelry Markings if you'd like the become familiar with how to spot real gold and tell exactly how much gold is in an item.
Once you know the karat purity of a piece of gold, simply divide that number by 24 and the result equals the percentage of pure gold contained in the item... represented in decimal form.
Say you have a large men's wedding band, marked "14KT". 14 divided by 24 is 0.58333... Move the decimal two places to the right and the number is now a percentage. In this case, the 14 karat gold wedding band is 58.333% PURE GOLD.
The same can be done no matter the karat marking...
- 10k / 24k = 0.41667 = 41.667% PURE GOLD
- 12k / 24k = 0.500 = 50.00% PURE GOLD
- 14k / 24k = 0.58333 = 58.333% PURE GOLD
- 18k / 24k = 0.7500 = 75.00% PURE GOLD
- 22k / 24k = 0.91667 = 91.667% PURE GOLD
So, once you know for sure that the piece in question IS SOLID GOLD, and the PURITY IS KNOWN, the GOLD CONTENT is quickly found per the table above. Since gold is soft and very malleable, jewelry is rarely made with more than 75% pure gold (18 karat jewelry) but 22k pieces do exist and are out there. (Keep an eye out for those, they offer one heck of a concentration of wealth in a small package!)
The next step is to figure out the value of the gold in your piece of jewelry. This is covered in more depth in another section and is the next recommended step to becoming a Gold Jewelry Guru!