The heart of Google is PageRank™, a system for ranking web pages developed by founders Larry Page and Sergey Brin at Stanford University. And while they have dozens of engineers working to improve every aspect of Google on a daily basis, PageRank continues to play a central role in many of our web search tools.
It measures page importance on a scale from 0 – 10, where 10 is the highest. The PageRank algorithm analyzes the quality and quantity of links that point to a page.
PageRank aka PR is one of the methods Google uses to determine the relevance or importance of a Web page. PageRank is a vote, by all the other Web pages on the Internet, about how important a Web page is. A link to a Web page counts as a vote of support. If there are no incoming links to a Web page then there is no support.
A graphical representation of a web of links between sites used for PageRank calculations.
To calculate the PageRank for a page, all of its inbound links are taken into account. These are links from within the site and links from outside the site.
PR(A) = (1-d) + d(PR(t1)/C(t1) + … + PR(tn)/C(tn))
That’s the equation that calculates a page’s PageRank. It’s the original one that was published when PageRank was being developed, and it is probable that Google uses a variation of it but they aren’t telling us what it is. It doesn’t matter though, as this equation is good enough.
In the equation ‘t1 – tn’ are pages linking to page A, ‘C’ is the number of outbound links that a page has and ‘d’ is a damping factor, usually set to 0.85.
We can think of it in a simpler way:-
a page’s PageRank = 0.15 + 0.85 * (a “share” of the PageRank of every page that links to it)
“share” = the linking page’s PageRank divided by the number of outbound links on the page.
To add the PageRank Widget to your blog Click here !!


Student translates Star Wars’ ‘Cantina Theme’ into a mathematical equation [Watch]
Reddit user ‘smallgoblin’ has ascended to Reddit royalty after posting a video clip that shows the user converting the movie series’ ‘Cantina Theme’ into a mathematical equation.
“It started with x+4…and I couldn’t unhear it,” smallgoblin wrote of the experiment, “Instead of doing my math homework I figured out what the Cantina Theme would sound like if your instrument was a pencil.”
The formulaic result would be x + 4783 – x + 4v|VE| (x + 4)(f – 711) = A, an answer that when worked through comes out to be ~299,792,457, or in other words, the speed of light, when measured in meters per second.
Smallgoblin’s post has since become the fifth most-upvoted post of all time in the history of the Star Wars subreddit.


Followup: “Sorting” colors by distinctiveness,
If you are given N maximally distant colors (and some associated distance metric), can you come up with a way to sort those colors into some order such that the first M are also reasonably close to being a maximally distinct set?
In other words, given a bunch of distinct colors, come up with an ordering so I can use as many colors as I need starting at the beginning and be reasonably assured that they are all distinct and that nearby colors are also very distinct (e.g., bluish red isn’t next to reddish blue).
Randomizing is OK but certainly not optimal.
Clarification: Given some large and visually distinct set of colors (say 256, or 1024), I want to sort them such that when I use the first, say, 16 of them that I get a relatively visually distinct subset of colors. This is equivalent, roughly, to saying I want to sort this list of 1024 so that the closer individual colors are visually, the farther apart they are on the list.
That’s the answer Followup: “Sorting” colors by distinctiveness, Hope this helps those looking for an answer. Then we suggest to do a search for the next question and find the answer only on our site.
The answers provided above are only to be used to guide the learning process. The questions above are open-ended questions, meaning that many answers are not fixed as above. I hope this article can be useful, Thank you


Using simple mathematics and taking into account a five day work week, depending on the year the average working days per year in America varies between 260 – 262 days a year. In 2012, due to the fact that it’s a leap year there are 261 weekdays out of a total of 366 total days.
Of course hardly anyone actually “works” every potential work day, as you need to take into account federal holidays, vacation and sick time taken.
(State and Federal employees generally enjoy 10 days per year as holidays, with an extra “Presidential Inauguration” day every fourth year. The holidays are given during the work week even if they fall on a weekend – ie the holiday is given on Friday if it falls on a Saturday, and on Monday if it falls on a Sunday)
This leaves 251 days to be worked in 2012, minus vacation time.
Vactaion time for employees normally kicks off at around 9 or 10 days a year.
We are now left with 240-241 working days in 2012 less sick leave (if any).
From what we have outlined here, and armed with your own information regarding vacation time, you should be able to work out how many work days you will be working this year.
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A fair coin is tossed 5 times, what is the probability of a sequence of 3 heads? Distributions, Basic Probability, Variability Learning Objectives. Consider tossing a coin with known, not necessarily fair, probabilities of coming up heads or tails; this can be modelled as a Bernoulli process. Okay, so I am not stupid and realize that the probability of flipping a coin heads or tails on any given flip is 50/50. Let's think about the situation where we have a completely fair coin here. ... "Five fair coins are tossed. Can someone help me with this one? 5. Say I hand you a coin. Outcomes Meeting the Criteria: roll a 2, 4, or 6 Possible Outcomes: roll a 1, 2, 3, 4, 5, or 6 Which gives us 1/32 or just over a 3% chance. Found 2 solutions by solver91311, Alan3354: ... C-what is the probability of getting at lease one head? Let's think about the situation where we have a completely fair coin here. Should you care if your team loses the toss? Following Benford's Law, or Looking Out for No. (We could, of course, have called zero heads and one tails; since both occur with equal probability, the choice is arbitrary.) ... Prob & Stats Help..If you flip a coin 5 times..? ... Probability flipping coins 3 heads and 1 tail ... 5:54. But what are the chances after 10 flips that the total number times heads comes up is the EXACT same as the number of times tails comes up (5 times each, in this case). What we'd like to do is find a way to set the problem up in some way that we know how to solve it. What is the probability that the results are all heads or all tails? a) Draw a tree diagram to show all the possible outcomes. Andrew flipped a coin 50 times and tallied 15 heads and 35 tails. b) Find the probability of getting: - 1316099 Define binomial outcomes I'll assume it's a quarter or something. Let's see. Let's see. ... ^5 ... 5 times, 2^5=32. If you flip a coin 10 times, what is the probability that you ... What is the probability of an occurrence of 8 heads? Author(s) David M. Lane. How would you tell if it's fair? Assuming a "fair" coin, there are 2^5=32 different arrangements of heads and tails after 5 flips. 6. If you flipped it 100 times and it came up heads 51 times, what would you say? Coin Flipper. Use the binomial probability distribution. Explain what will happen to the experimental (actual) probability of getting heads if the coin were flipped many more times. The potential twentieth head has the same probability as the first head. What is the probability of rolling an even number on a die? So let me draw it. coin is tossed 5 times - probability of EXACTLY 3 tails? In this video, we 'll explore the probability of getting at least one heads in multiple flips of a fair coin. Binomial Distribution. Find the mean and standard deviation of a binomial distribution; When you flip a coin, ... We toss a coin 12 times. Check out our app! So the odds of flipping a coin 5 times and getting 5 heads are 1/2 ^5 (half to the power of 5). Question 184336: You flip a coin 5 times. Prerequisites. The probability of getting a head or a tail on the first flip is 1. if you flip a coin 5 times, what is the probability you will get heads all five times? Example: Clare tossed a coin three times. What is the probability of obtaining EXACTLY three tails?"


Joshua began at the Institute of Social Science Research (ISSR) in late 2015 after completing his Bachelor Degrees at the University of Western Australia (UWA). During his time at UWA he assisted several professors with research and worked part-time at Data Analysis Australia.
Joshua’s main responsibilities in the Research Methods and Social Statistics Program are to contribute to dual-frame survey research, provide methodological advice for ISSR Surveys, and to help develop statistical methodology within the Life Course Centre.
- Linear Mixed Models
- Survey Analysis
- Dual-frame Surveys
- Convex-Constrained regression models


Multiple linear regression analysis was used to develop a model for predicting student’s research or unstandardized regression. Read this research paper the hypothesis to the study of quantile regression showed that regression analysis in the describe multiple regression analysis and. Econ 145 economic research methods presentation of regression if you are familiar with regression analysis stands alone from your research paper in some. Multiple regression analysis is a powerful technique used for predicting the unknown value of a variable from the known value of two or more variables- also called.
What would be a sufficient topic for multiple i am conducting a multiple regression analysis how do i report multiple regression result in a research paper. Ÿ regression analysisyou are a newly hired research assistant at rlj consulting services and on your first dayat work your boss asks you to review a research study. Explore the latest articles, projects, and questions and answers in multiple regression analysis, and find multiple regression analysis experts. Research project paper using multiple regression analysis subject or discipline: statistics title: writer's choice number of sources: 10 provide digital.
Research paper dissertation multiple regression analysis is used to predict the value of a multiple regression is also used to calculate the overall fit of. Applied regression analysis: a research tool — 2nd ed / john o rawlings, sastry g pentula, david a dickey printed on acid-free paper c 1989 wadsworth, inc. Multiple regression research analysis paper essay title page in mla format research papers essay type test meaning hebrew essay type test meaning hebrew. Examples of research questions that would use linear regression: paper to the next regression analysis. The second main use for correlation and regression is to see whether two variables are associated multiple regression analysis the research council of norway.
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Interpreting multiple linear regression: contribute to a regression equation thus, this paper presents a guidebook of variable dominance analysis. Read this essay on regression paper the purpose of regression analysis is to find out the values of multiple linear regression was performed to determine. The authors state that multiple regression is a powerful method of statistical analysis, provides a strength of relationship index, and should replace analysis of.


There are 3 regular reels and 1 extra reel and 1 payline in Money Coming. There is a special reel on the right side where you can get interesting rewards. In this way, you can multiply the victory by up to 10 times, get free re-spins, or have the opportunity to play fortune wheel.
Symbols in the game
The list of basic symbols includes 0, 1, 5, 10, and double zeros. There is a scatter symbol in the game, which allows you to get the chance to spin the wheel of fortune


Ky Fan was born in Hangzhou, China on September 19, 1914. He enrolled in Peking University in 1932, and received his B.S. degree from
University in 1936. Initially Fan wanted to study engineering, but shifted to mathematics, largely because of the influence of his uncle
was a renowned mathematician in China and the Chair of the Department of Mathematics of Peking University.
Fan went to France in 1939 and received his D.Sc. under Frechet from the University of Paris in 1941. A member of the Institute for
Study in Princeton from 1945 to 1947, he then joined the faculty of the University of Notre Dame, eventually becoming full professor. In
1965, Fan became professor of mathematics at UC Santa Barbara, retiring in 1985.
In addition to Frechet, Fan was also influenced by John von Neumann and Hermann Weyl. Fan made fundamental contributions to operator and
matrix theory, convex analysis and inequalities, linear and nonlinear programming, topology, and topological groups. His work in fixed
theory, in addition to influencing nonlinear functional analysis, has found wide application in mathematical economics and game theory,
potential theory, calculus of variations, and differential equations.
Fan had 21 graduate students, and 89 mathematical descendants. His 140 papers and books have been cited over 4000 times. Fan was elected
the Academia Sinica in 1964, and served as the director of its Institute of Mathematics in Taiwan from 1978 to 1984.
In 1999, Ky Fan and his wife made a gift of $1M to the American Mathematical Society to establish the Ky and Yu-Fen Fan Endowment, to
collaborations between Chinese mathematicians and those in other parts of the world. The Ky Fan Visiting Assistant Professorship in the
Mathematics Department is also due to his generosity.


ACT scores near the worst in nation
Posted By Sparta Live | August 30, 2012 12:00 am
The national American College Testing (ACT) scores have been released and the results were not good for Tennessee, who tied with two other states for the second worst composite score in the nation with an average composite score of 19.7 out of 36.
The ACT test, which measures high school educational development and students’ capability to complete college-level work with the multiple choice tests covering English, mathematics, reading, and science, was completed by 100 percent of Tennessee graduates.
Tennessee is one of only nine states that requires students to take the ACT to graduate. Nationally, an average of 52 percent of graduates complete the ACT.
While 19.7 is an increase in the composite score from 19.5 last year, in 2010 the average composite score was 19.6, in 2009 it was 20.6 and in 2008 it was 20.7.
When the score is broken down by subject, Tennessee students scored an average of 19.6 in English, 19.1 in math, 19.9 in reading and 19.6 in science.
By race, the average composite scores were 22.4 for Asian, 20.7 for white, 20.2 for multi-racial, 18.1 National American, 18.3 for Hispanic and 6.4 for black students.
Nationally, students averaged a composite score of 21.1. The average for individual subjects was 20.5 in English, 21.1 in math, 21.3 in reading and 20.9 in science.
Comparatively, Tennessee scores were tied with Washington D.C. and Arizona as the second lowest in the nation, with only Mississippi scoring lower at 18.7.
While the state board of education states that the percentage of test takers who met all ACT college readiness benchmarks, which is the percentage of students that had at least a 50 percent chance of scoring a B or higher or 75 percent chance of scoring a C or higher in college courses, rose from 15 percent last year to 16 percent this year, the percentage of students who met these benchmarks two years ago was also 16.
The test results found that 56 percent of Tennessee graduates met the English readiness benchmarks, 26 percent met math benchmarks, 39 percent met reading benchmarks and 18 percent met science benchmarks.
Perhaps the most shocking aspect of the test results were the disparity between Caucasian and Asian students when compared to black, Native American and Hispanic students. While 20 percent of white and 34 percent of Asian students met all of the benchmarks, only 3 percent of black, 9 percent of Native American and 11 percent of Hispanic students met all benchmarks.
The test scores also revealed that 64 percent of Tennessee students scored in the three lowest math score ranges.
The highest number of students, approximately a third, responded that they hoped to attain careers in health sciences and technologies.
Students with the highest scores, which averaged 23.8 wanted jobs in English or foreign language.
The largest number of students, 12,915 students, also reported that they wanted to attend UT Knoxville, with Middle Tennessee State University, UT Chattanooga, University of Memphis and Tennessee Technological University, respectively, following behind.
While 100 percent of graduates completed the ACT exam, there was a reported 3.7 percent decrease in the number of graduates.


Here's an explanation of the statistics featured on this site. A brief note about player stats - all reflect only how a player does while he's on the floor. So, for example, say Duke has 20 offensive rebounds, 10 of which it gets while Brian Zoubek is on the floor. The opponent has 20 defensive rebounds, also 10 of which it gets while Brian Zoubek is on the floor. If Zoubek gets 5 offensive rebounds, his ORB% is 25% (5/20), not 12.5% (5/40), because he got 5 of the 20 rebounds actually available to him. This is also why the scoring % and % shots taken add up to more than 100%. If you multiply those percentages by the percentage of possessions played by each player, then added them up, it would equal 100% (give or take a tenth or so due to rounding).
O Rating/D Rating:
This number indicates the number of points the team would score per every 100 possessions (the average game has around 70-75 possessions). O Rating is the amount the offense would score, and D Rating is the amount the opposition would score. For offense, anything over 105 is pretty good, and for defense, anything in the 90s is pretty good. 2005-06's best offense scored a 119, and the best defense had an 88.
The difference between a team's O Rating and D Rating. In a 100 possession game, if the team played to its Ratings, this would be the margin of victory.
PPWS (points per weighted shot):
A point per shot measure that also accounts for free throws (measured as .475 shots per free throw attempt). 1.00 is about average, anything over 1.15 for guards and 1.25 for bigs is excellent. This measures a scorer's efficiency (the fewer shots required for the same number of points, the more efficient the player).
Effective Field Goal % (EFG%):
Based on the Rick Pitino adage that a 40% 3pt shooter is as valuable as a 60% 2pt shooter. Since made 3s are worth 1.5 times made 2s, this field goal% reflects that. This doesn't measure efficiency as much as equalized shooting percentage - it helps make an accurate comparison for the actual shooting percentage value of, say, a JJ Redick (whose traditional fg% would be lower) against a Shelden Williams (whose traditional fg% is "inflated" on account of a lack of 3pt attempts).
A/B% (Assisted Basket percentage):
Indicates the percentage of made field goals that are the result of assists. As a player stat, it shows the percentage of made field goals that are the result of assists by that individual player.
ORB% (Offensive rebound percentage):
The rate at which a team gathers its own misses. The rate is derived by taking the number of team offensive rebounds and dividing by the sum of team offensive rebound and opponent defensive rebounds. As a player stat, it shows the rate at which an individual gathers misses by his own team.
DRB% (Defensive rebound percentage):
The rate at which a team gathers its opponent's misses. The rate is derived by taking the number of team defensive rebounds and dividing by the sum of team defensive rebound and opponent offensive rebounds. As a player stat, it shows the rate at which an individual gathers misses by the other team.
The rate at which a team's field goal attempts are blocked. Measured as total blocks divided by opponent field goal attempts. As a player stat, it's the rate at which an individual blocks all opponent field goal attempts.
The rate at which an opponent's possessions end in a steal. Measured as total steals divided by opponent possessions. As a player stat, it's the rate per 100 opponent possessions that an individual makes a steal.
The rate at which a team's possessions end in a turnover. Measured as turnovers divided by possessions. As a player stat, it's the rate per 100 possessions that an individual turns the ball over.
Exactly what it says - assists divided by turnovers.
Individual stat only - measures an individual's total points as a percentage of the points scored by the team as a whole.
% Shots Taken:
Individual stat only - measures the number of shots taken by a player as a percentage of the total shots taken by the team (note, I haven't figured out how to factor in FTAs for this yet, so it just looks at FGAs).
Individual stat only - measures the player's ability to get to the line. The stat is free throw attempts divided by total field goal attempts (times 100).


A Review of The Dyscalculia Assessment
This volume is more than just a way to assess or diagnose math difficulties. It also offers strategies for meeting the needs that the assessment identifies.
A Clear Lay of the Land
Although students with dyslexia and dyscalculia share the inability to distinguish or use written symbols, they may not have much in common. Not nearly as much research has been done to help students with dyscalculia as there has been for students with dyslexia. In both cases, students face challenges with using and interpreting symbols. Ironically, children who have strong reading skills may at the same time struggle with number systems and comprehending understanding math, both theoretical and practical.
In creating their assessment, the authors of The Dyscalculia Asessment do a good job of addressing the challenges that children with dyscalculia face. They include:
- Poor Number Sense. Children with dyscalculia have difficulty seeing the relationship between symbols(numbers) and concrete objects.
- Weak long term memory. Children with weak long term memory have difficulty retaining math facts and gaining “automaticity” with math facts.
- Weak short term memory. Some children have difficulty remembering the question before they have answered it. New information has to be in short term memory between 20 and 30 minutes before it is downloaded into long term memory.
- Working memory difficulties. Working memory is used to access prior knowledge and executive function. It is critical for understanding what you need to do to solve multi-stepped math problems.
- Sequencing difficulties.Having difficulty sequencing is going to cause serious challenges when it comes to long division or multi-digit multiplication.
The Assessment: A Mix of Objective and Subjective
The assessment is anything but a “quickie.” The authors recommend that you take the time to record observations about the child’s strategies for solving problems as they approach them. The only thing that separates this from the ongoing evaluation that I do is the fact that is the authors isolate and lay out the different aspects of math that you will be measuring.
Each section of the assessment is provided in the volume on a single page. The assessment pages are also provided as pdf’s on the web at a website provided by the publisher, which is easy to register for and access. That alone makes creating the assessment easy, which I prefer to cutting up the book.
When administering the assessment you ask the children to listen and write the calculations. It doesn’t provide any visuals for the administration, which for some students with dyscalculia may create significant roadblocks to success.
Despite the shortcomings of the assessment the tools provided by the publisher make it easy to access.
It is, none the less, a time consuming process, which may in some cases be a luxury to administer. It also requires that the assessor brings experience to the the process, and there is some room, through administration, to both observe and distort what the observer is seeing.
Strategies for Remediation
A full third of the volume is dedicated to providing strategies to remediate student’s weaknesses and shortfalls in the basic math abilities laid out by the authors. In my humble, special educator’s point of view, this section alone justifies the $65 price. The chapters dedicated to remediation lay out all the resources for Individual Education Plan writing and provide strategies for special education students. Those strategies include lots of games, a powerful way to review and reinforce math skills and make practice of math facts lots of fun.
Chapter 5 not only includes how to interpret the assessment, it includes specific strategies and lesson plans to remediate the specific difficulties. They lay out lots of valuable visual methods to help students see and understand the math concepts they are learning (see counting mats.) They also recommend Cuisenaire rods, color coded rods for each o fhte numbers. There is no question that lots of hands on activities are critical for success.
At $65 this is not a cheap book. It can be purchased online for less. It is a serious investment.
Still, this is an excellent resource, and I would recommend it for purchase to place in a school’s special education library, where many teachers can reference the book to deal with students with intractable problems. Or, a school might decide that one special education teacher would be the “math specialist,” who would assess students with serious math disabilities using the assessments, and providing remediation ideas for the child’s special educator.
Jerry Webster, About.com Guide
Extensive information at: DyslexiaA2Z.com


Editeur: Packt Publishing
Leverage the power and flexibility of the Clojure language for finance
About This Book
- Apply the Clojure programming language in financial analytics and to build financial applications
- Work with high-level mathematical abstractions without having to implement low-level code for financial operations.
- This is a practical tutorial that takes you through real-world examples of financial analytics and applications with Clojure
Who This Book Is For
If you're a finance professional who is currently using VBA (Excel) to perform financial calculations and quantitative analysis, and would like to use Clojure instead to improve their efficiency, then this book is for you.
Basic knowledge of financial concepts is essential. Basic programming knowledge would also be an added advantage.
What You Will Learn
- Quickly and effectively represent data and calculations using Clojure
- Use Clojure's basic language tools such as laziness, immutability, and firstclass functions to solve real-world problems
- Explore mathematical derivatives to generate different wave forms
- Get acquainted with the advanced approaches to calculating and transforming data, and building algorithms
- Use Clojure's functions to access, update, and compose data structures
- Be introduced to the concept of sideeffecting behavior and the different ways to handle it
- Compose simple and exponential functions together to get a buy or sell signal
Clojure is a dynamic programming language with an emphasis on functional programming. Clojure is well suited to financial modeling as it is a functional programming language. Such languages help developers work with high-level mathematical abstractions without having to implement low-level code that handles the arithmetic operations.
Starting with the importance of representing data and calculations effectively, this book will take you all the way to being competent in financial analytics and building financial applications.
First, we introduce the notions of computation and finance, which will help you understand Clojure's utility to solve real-world problems in many domains, especially finance. Next, we will show you how to develop the simple-moving-average function by using the more advanced partition Clojure data transformation function. This function, along with others, will be used to calculate and manipulate data.
You will then learn to implement slightly more complicated equations, how to traverse data, and deal with branching and conditional dispatch. Then, the concept of side-effecting and its various approaches are introduced, along with the strategy of how to use data as the interface to other systems. Finally, you will discover how to build algorithms while manipulating and composing functions.
Style and approach
This book is a practical step-by-step tutorial that provides a basic overview of the concepts but focuses on providing the skills required to analyze data.
Présent dans 0dossiers publics
Présent dans 0dossiers privés
Etagères de cours


Fragen aus Vorstellungsgesprächen für Associate, von Bewerbern geteilt
Write a program to find the square root of a double.
use binary search to narrow down the search space and keep multiplying the answer in each iteration by itself to check if it is equal to the double given. If it is lesser, move up the lower bound, else move down the upper bound.
One easy to remember method that also has much better performance than binary searching for the result is the Babylonian Method. It is similar to Newton's method for finding derivatives. See the algorithm here: http://en.wikipedia.org/wiki/Methods_of_computing_square_roots#Babylonian_method Also, combining this algorithm with the Rough Estimation also described on that page will compute the squareroot of most double numbers in less than 10 iterations.
Is it too obvious to ask if you can do double^.5 ?


At St. Jude’s Catholic Primary School, we follow the National Curriculum maths objectives across the whole school. To support these objectives, and develop a Mastery approach to Maths learning, we use White Rose planning, supported by Power Maths resources.
All children enjoy a good deal of practical maths, using apparatus to optimise their understanding. Problem-solving, investigative work and reasoning form a large part of the curriculum and an element of this is included in every lesson.
‘Traditional’ approaches to number work are also used, particularly the teaching of tables. Children are required to commit these to memory as they form the basis of so much work in Maths. They must be able to fluently recite both the multiplication and related division facts of every times table up to 12×12 by the end of Year 4.
Alongside multiplication facts, there are a set of agreed non-negotiables for each year group in Maths.
At St. Jude’s we are use diagnostic tests called PUMA which are used termly. They allows us to assess the children in detail, based on the National Curriculum objectives. From these, we can carefully track progress and identify where individuals and groups have gaps in their learning.


Quantitative Aptitude Tricks PDF:- Quantitative Aptitude Tricks are very important to crack any competitive exam. They help save your time during the exam that will help you improve your chances of selection. Quantitative Aptitude section is asked in almost every competitive exam including IBPS, SSC, Railways and more.Below are some tips provided to help you solve quantitative aptitude problems quickly. Go through them to accelerate your speed of preparation. In this blog we provide you a number of Quantative Aptitude Tricks. A number of tricks are given below to help you understand them:-
Cube Root of 19683:
⇒ Here the one’s digit is having 3, so we will find cube of which numbers from 1 to 9 are having 3 in the ones digit place which will be 7. So 7 will form the one’s digit of the cube root of 19683.
⇒ Now 19 falls between the cube of 2 and 3, so the smaller number between 2 and 3 i.e. 2 will form the tens digit of the cube root.
⇒ Hence, the cube root of 19683 will be 27.
Multiplication with 5
Simply multiply the number by 10 and then divide it by 2.
For example 99×5= 990/2= 495
Multiplication with 99
Multiply the number with 100 and then minus the same number from the result. Let’s take an example
Multiply 32×99 = 3200 – 32 = 3168
There is a pattern in the difference between the numbers in the series.
Every consecutive difference is multiplied by -2.
Conversion of km/hr into meter/sec
Following formula is used in this case
Conversion of meter/sec into km/h
The formula is:
Boats & Stream Tricks with Questions
Upstream: When the boat (or swimmer) moves against i.e. in the opposite direction of the stream.
Downstream: when the boat (or swimmer) moves in the same direction as that of the stream.
Now, if the speed of the boat is B km/hr while the speed of the stream is S km/hr, then
Upstream speed U = (B – S) km/hr
Downstream speed D = (B + S) km/hr
If in the question, you are provided with the upstream and the downstream speed U and D respectively, to determine the speed of boat and stream the formula goes like
Speed of boat B = (D + U) / 2
Speed of stream S = (D – U) / 2
Question: In still water, the speed of the boat is 7 km/hr. Find out the downstream speed of a boat in the river whose speed of the stream is 2 km/hr.
Solution: Downstream speed = Speed in still water + Speed of stream = 7 + 2 = 9 km/hr
A couple more formulae,
Question: In still water speed of boat is 8 km/hr. Find the upstream speed of boat which is floating in the sea where speed of waves is 3 km/hr.
Solution: Speed of the boat (B) = 8
Speed of the stream (S) = 3
Upstream speed = B – S = 8 – 3 = 5 km/hr (answer)
Question:The speed of a boat in downstream is 24km/hr and speed of the stream is 8 km/hr. find the speed of the boat in still water.
Solution: Downstream speed = 24
Speed of the stream (S) = 8
According to the formula,
Downstream = B + S
24 = B + 8
Speed of the Boat (B) = 24 – 8 = 16 km/hr (answer)
Playing with Percentages
If out of the total apples 60% are bad, i.e., 100% – 60% = 40% is good.
So, when we are asked the number of good apples, simply do the mental calc (short form of calculation!) and get the figure of 40%, the multiply with the total number of apples say 600, which is :40 x 600 / 100 = 240 apples
You already know that the ‘00s in 600 and ‘00s in 100 get cancelled, then why waste time writing down the entire figure?
Simply multiply 40 x 6 = 240!
Alligation and Mixture : Tricks
Ques 1. Gold is 19 times as heavy as water and copper 9 times. In what ratio should these materials be mixed so that the mixture may be 15 times as heavy as water.Solution:-
Ques2.How much chicory at Rs. 4 a kg should be added to 15 kg. of teat Rs 10 a kg. so that the mixture be worth Rs. 6.50 a kg?Solution:-
Ques 4.A sum of Rs. 6.40 is made up of 80 coins which are either 10-paise or five-paise coins. How many are of 5-p.?Solution:-
Ques 5.In a zoo, there are rabbits and pigeon. If heads are counted, there are 200 and if legs are counted, there are 580. How many pigeons are there?Solution:-
Questions related to mixtures can be easily solved by alligation method. By using alligation we can wide arrange of maths questions. Let me explain this with a simple example
Example: The price of wine of $60 per litre. If Samuel is adding water with and selling the mixture for $40 per litre. Profit margin remains same. What is the ratio of water and wine in the mixture?
Profit and Loss
Profit/loss = Sales price – Cost price
In case of profit
25% of Cost Price (1/4 of CP) = 20% of Selling Price (1/5 of SP)
Similarly, 1/3 of CP = 1/2 of SP
In case of loss
25% of Selling Price (1/4 of SP) = 20% of Cost Price (1/5 of CP)
Similarly, 1/3 of SP = 1/2 of CP
Time and Work
Every question in time and work chapter can be solved easily by finding the efficiency of workers or subject (such as pipes).
For example – A takes 10 days to complete a job. B takes 20 days to complete the same job. In how many days they will complete the job if they work together?
⇒ A’s efficiency = 100/10 = 10% per day
⇒ B’s efficiency = 100/20 = 5% per days
⇒ A and B can do 15% of the work in a day if they work together. So they can do the whole job in 100/15 = 6.66 days or 6 days and 18 hours.
Click here to download Quantitative Aptitude Tricks PDF from the link below:-


- Simplified Chinese
- Traditional Chinese
Role & Roll was designed thinking in role players who needs to roll dices once and once again, and them sum all with modifiers etc… With this program you can roll 4 sided, 6 sided, 8 sided, 12 sided and 20 sided dices. You can also roll a 100 sided dice by getting a random number between 1 and 100. All random algorithms are uniformly distributed so it’s perfect.
Role & Roll can also be used as a general purpose program to get random numbers to a certain experiment.
18 March 12
Compatible OS Versions:
Windows XP, Windows Vista, Windows 7


It’s crucial to think about how we can integrate technology into our teaching as it becomes more and more a part of our lives. Math and technologies are a great way to aid students in learning more efficiently, engage in interactive lessons and increase their understanding of math concepts.
Technology can help students become more active in classrooms where they have trouble concentrating or participating. This could be in the form of a virtual whiteboard or interactive tools that allow students to collaborate. Online learning platforms, such as Math Minds, for example uses a variety innovative tools to assist students in engaging in their classes and increase participation.
Another benefit of technology is that it can be tailored to the specific needs of each student. This is especially helpful in maths where students learn differently. For example, some students may have a hard time with http://ultiaction.com/generated-post/ visual learners while others find it easier to understand concepts through kinesthetic or auditory method. By using technology to teach math, you can ensure that every student gets the most out their lessons.
Mathematical technology allows students to solve problems faster than could be done in a traditional classroom setting. Graphing calculators allow students to solve equations with the push of a button, and they can visualize solutions. This allows them to improve their understanding of math concepts and make connections between concepts that are difficult for students to comprehend otherwise.


Computer, Electrical and Mathematical Sciences and Engineering
With facilities such as Shaheen (one of the world’s fastest super computers) and the CORNEA Visualization Center, the Computer, Electrical and Mathematical Sciences and Engineering (CEMSE) Division is one of the best-equipped places in the world to carry out cutting-edge, interdisciplinary research.
CEMSE is associated with the Computational Bioscience Research Center, Extreme Computing Initiative, and the Geometric Modeling and Scientific Visualization Research Center. In addition to access to these centers, students will also work with world-class faculty during their post-graduate studies.
CEMSE offers Master’s and Ph.D. programs in Applied Mathematics and Computational Science, Computer Science and Electrical Engineering
Discover more about CEMSE at the official Division website.


Gerardus Mercator (5 March 1512 – 2 December 1594) was a cartographer renowned for creating a world map based on a new projection which represented sailing courses of constant bearing as straight lines—an innovation that simplified navigation. In his own day he was the world’s most famous geographer but in addition he had interests in theology, philosophy, history, mathematics and magnetism as well as being an accomplished engraver, calligrapher and maker of globes and scientific instruments.
The Mercator projection is a cylindrical map projection presented by the Flemish geographer and cartographer Gerardus Mercator in 1569. It became the standard map projection for nautical purposes because of its ability to represent lines of constant course, known as rhumb lines or loxodromes, as straight segments that conserve the angles with the meridians. Although the linear scale is equal in all directions around any point, thus preserving the angles and the shapes of small objects (which makes the projection conformal), the Mercator projection distorts the size of objects as the latitude increases from the Equator to the poles, where the scale becomes infinite. So, for example, landmasses such as Greenland and Antarctica appear much larger than they actually are relative to land masses near the equator, such as Central Africa.
Jodocus Hondius (Latinized version of his Dutch name: Joost de Hondt) (14 October 1563 – 12 February 1612), sometimes called Jodocus Hondius the Elder to distinguish him from his son Jodocus Hondius II, was a Flemish / Dutch engraver, and cartographer. He is best known for his early maps of the New World and Europe, for re-establishing the reputation of the work of Gerard Mercator, and for his portraits of Francis Drake. He helped establish Amsterdam as the center of cartography in Europe in the 17th century.


New algorithm for statistics
Actually statistics are calculated every time you enter in the statistics panel. So, if you have a lot of shows and movie, every time you open the statistics page, the app takes a lot of time in the calculation. This problem can be solved by performing the calculations when you set as watched a new episode. Example:
(1) I install SeriesGuide
(2) At the beginning the total time of watching is T = 0
(3) I add the show "Fake Title": each episode lasts 40 minutes
(4) I set the first episode as watched, so become T = 40
(5) I set the second episode as watched, so become T = 80
(6) I add the show "Invented": each episode lasts 20 minutes
(7) I set the first episode as watched, so become T = 100
(8) I set the first episode of "Invented" as unwatched, so T = 80
The same for the other statistics.
Note: here <http://is.gd/dB6ATu> I have suggest multiple watching; obviously in this case you must consider the re-watched episodes/movies in a different variable, to have separate statistics for re-watching.
I mean, additionally after the new algorithm has been implemented
A refresh button can be added if the user believes the calculation needs to be recalculated or is not that accurate.


Calculating the approximate size of an apartment unit can be an extremely helpful tool when the information is not readily available. Understanding the size of an apartment unit is a critical step in understanding its value, layout, and suitability for potential tenants. Try out our apartment size calculator to easily estimate and understand the square footage of an apartment unit.
Calculate Apartment Size
Please input the required fields (*) below to calculate an apartment’s size.
This calculator is meant for educational purposes only. The calculation generated from the calculator does not, and is not intended to, constitute financial advice. As such, all information, content, and materials available on this site are for general informational purposes only. Please review our Editorial Standards for more info.
Apartment Size Calculation Formula
Estimating the size of an apartment can be done with a formula of variable inputs such as bedroom count, bathroom count, etc. Similar to our other rental property calculators, our proprietary formula takes into account all common spaces of an apartment unit and calculates an estimate of square footage for the overall unit, with an included load factor to also account for interior walls.
- Bedrooms: The number and dimensions of bedrooms contribute significantly to the overall square footage, offering personal space for inhabitants.
- Bathroom: Bathrooms, varying in size from half-baths to full suites, can significantly affect the total area of an apartment.
- Living Room: Often one of the larger rooms in an apartment, the living room serves as the central gathering space for residents and guests.
- Dining Room: This space, designed for meals and gatherings, can range from compact nooks to expansive rooms, influencing the apartment’s total size.
- Office: Especially relevant in modern apartment layouts, the office space provides a dedicated area for work or study.
- Extra Room: A versatile space, an extra room can serve as a guest room, hobby room, or any other purpose, adding to the apartment’s footprint.
- Storage Closet: While generally smaller in dimension, storage closets are essential for housing belongings and utilities, contributing to the overall calculation of apartment size.
Calculating Room Area
Calculating the size of an apartment unit requires understanding how to calculate the area of an individual room. The corresponding image demonstrates how to calculate the interior area of a rectangular room – Length x Width (10′ x15′ = 150′). For rooms with angular or round walls, more advanced calculations can be made to determine the area.
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Apartment Size Resources
Apartment Size Calculator FAQ
What’s the Difference Between “Gross” and “Net” Square Footage in Apartment Units?
Also referred to as “Rentable vs Usable”, “Gross” square footage includes the total area of the apartment unit, encompassing wall thicknesses, columns, and other structural elements. “Net” square footage, on the other hand, refers to the usable or livable space within the unit, excluding those structural elements. The net area gives a better idea of the functional space available to the resident.
How to Accurately Measure the Size of an Apartment Unit?
For an accurate measurement, use a tape measure to record the length and width of each room, then multiply these figures to get the area of each room. Sum up the areas of all rooms to determine the total size of the apartment. It’s crucial to consider any alcoves, closets, or irregularly shaped spaces. Digital real estate tools and apps can also help in recording and calculating these measurements.
Are Hallways and Corridors Within an Apartment Included in the Unit’s Total Size Calculation?
Yes, interior hallways and corridors within an individual apartment unit are included in the total size calculation. However, common hallways or corridors outside the unit, which are shared with other tenants in the apartment building, are not.


Students work two hours per week on maths in English.
During the three years, a lot of sequences are studied in the European section. They include various notions taken from the 2nde, 1re S and Tle S maths lessons. The aim is not to give the student a global and perfect lesson on each subject : it is only a class working basis.
Since students have an oral exam at the end of the three years, they often work on videos. More than nine films are studied (themes, scripts, making of, mathematical points…)
Students learn vocabulary, different ways of writing in English maths, various and rich mathematical notions. Exercises are proposed in every chapter. The cultural aspect is as important as the mathematical one. Students work on famous mathematicians : Euler, Pascal, Mandelbrot, Newton, Einstein, Fermat, etc.
Here are a few lessons :
· 2nde : Summary on algebra rules
· 2nde : Probability and work on A Beautiful Mind (2001)
· 2nde : Geometry and work on Oblivion (2013)
· 2nde : Functions and work on A Brilliant young man (2014)
· 1re : Summary on previous lessons
· 1re : Vectors, centre of mass and work on Gravity (2013)
· 1re : Prime numbers and work on The Theory of Everything (2015)
· 1re : Sequences and work on The Oxford Murders (2008)
· Tle : summary on previous lessons, derivatives and antiderivatives
· Tle : Complex numbers and work on Dimensions (2009)
· Tle : Ciphers and work on The Imitation Game (2015)
· Tle : Further on probabilities and work on Good Will Hunting (1997)
To attend this section, students must have a good level in maths and in English.


Simultaneous Equations Bias in Level VAR Estimation - Volume 9 Issue 2. identified econometric models. Econometric Theory 5 (1989): 181–240.
Alternative versions. feel free to create and share an alternate version that worked well for your class following the guidance here Variation Theory Sequences and behaviour to enable mathematical thinking in the classroom - by Craig Barton @mrbartonmaths Quadratic simultaneous equations (3 This website is my attempt to assemble a collection of high-quality, sequences of questions and examples using key principles from Variation Theory. By holding key features between questions and examples constant, together with the mathematical behaviour that I call reflect, expect, check, explain and the supporting role of the teacher, I Variation Theory Sequences and behaviour to enable mathematical thinking in the classroom - by Craig Barton @mrbartonmaths Variation Theory Sequences and behaviour to enable mathematical thinking in the classroom - by Craig Barton @mrbartonmaths with powers Solving equations with Variation Theory Sequences and behaviour to enable mathematical thinking in the classroom - by Craig Barton @mrbartonmaths Author: Peter Godber. This is a new type of activity I am working on, with the catchy name of Fill in the gaps.It is my attempt to replicate some of my favourite Standards Units card sort activities, but with less cutting and some elements of variation. Overview (again) of Variational Method Approximation.
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It can happen with the introduction to reading or with simultaneous equations, with counting or with Latin, with A similar variation is found in figures from Germany. 115, 113, analysis of dispersion ; multivariate analysis of variance ; MANOVA 1318, 1316, frequency theory of probability, frekventistisk sannolikhetsteori 2191, multi-equational model ; simultaneous equations model, flerekvationsmodell. In the context of linear systems of equations, of more appropriate ordering strategies based on the graph theory and more efficient maximum number kmax of entries are added to the initial pattern or the relative variation of  J. H. Prevost, "Partitioned solution procedure for simultaneous integration of coupled-field During the project the group had to come up with a suitable topic, theory and how to demonstrate it practically. have future potential to be incorporated in a complete Simultaneous Localization and Mapping (SLAM) framework . Calculus of Variations, 6.0 ECTS Numerical Methods for Differential Equations, 8.0 ECTS. tried to find a unifying definition of stress, but the word is still used differently sympathetic nervous system with a simultaneous decrease in parasympathetic.
How I wish I'd taught maths, which discusses the use of SSDD Problems alongside purposeful practice, cognitive load theory, variation theory and much more is
Simultaneous Equation Perfect Maths. Simultaneous Equations One Is Quadratic You. Solving Simultaneous Equations Using Substitution Go Teach Maths 1000s Of Free Resources. Linear Simultaneous Equations Calculator Tessshlo. Quadratic Simultaneous equations and linear equations, after studying this section, you will be able to: solve simultaneous linear equations by substitution; solve simultaneous linear equations by elimination; solve simultaneous linear equations using straight line graphs; If an equation has two unknowns, such as 2y + x = 20, it cannot have unique solutions.
Variation Theory. Sequences and behaviour to enable mathematical thinking in the classroom - by Craig Barton @mrbartonmaths. Please read! 2020-06-29 Quadratic Simultaneous Equations 3 Exercises Variation Theory Solve Simultaneous Equations With One Linear And Quadratic Simultaneous Equations 1 Linear Quadratic Example 2 You Quadratic Simultaneous Equations 3 Exercises Variation Theory. Solve Simultaneous Equations With One Linear And Quadratic.
Downloadable version · 5. Alternative versions. Sep 19, 2018 Linear simultaneous equations (7 steps!) Author: Joe Berwick This type of activity is known as Practice. Please read the guidance notes here,
If you have two different equations with the same two unknowns in each, you can solve for both unknowns. There are three common methods for solving:
Example questions. If we understand the theory, it is time to put it into practice. You understand now what simultaneous equations are.
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1. INTRODUCTION ECONOMIC THEORY TYPICALLY PRODUCES systems of equations characterizing the out-comes observable to empirical researchers. The classical supply and demand model is a canonical example, but systems of simultaneous equations arise in almost any economic From our previous analysis, given that v = 0.25 and c = 1, the equation of the dashed line of simultaneity is t – 0.25x = 0 and with v = 0, the equation of the dotted line of simultaneity is t = 0. In general the second observer traces out a worldline in the spacetime of the first observer described by t = x / v , and the set of simultaneous events for the second observer (at the origin) is This video lecture of Differential Equation | Simultaneous Linear Differential Equation | Problems & Concepts by GP Sir will help Engineering and Basic Another method of solving equations is by graphing each equation on a coordinate graph. The coordinates of the intersection will be the solution to the system.
NONPARAMETRIC SIMULTANEOUS EQUATIONS MODELS 291 a simultaneous model to reduce to the triangular model, pointing out that this condition is quite restrictive. An important breakthrough in the literature on fully simultaneous models came in Matzkin (2008). Matzkin considered a model of the form m j(YZδ) =0j=1J where δ=(δ 1(ZX 1U 1)δ J(ZX JU
1 two of the terms in this equation can be determined based on data measured in the laboratory namely observations quasi-simultaneous (close in time) of the zenith blue sky radiance and of the probably be attributed normal measurement precision and to natural variation of the total. of Non-Linear Simultaneous Equations Models, IUI Research Report No 186/2012); Employer-sponsored training: Variation and volume over the Grading committee for the Ph.D. thesis “Theory and Validity Evidence for av M Stjernman · 2019 · Citerat av 7 — Here, we applied an orthogonal design exploiting spatial variation in land use in a since simultaneous temporal changes in managements tend to result in a lack of Our definition of farmland birds in this study is based on their Structural Equation Modeling: A Multidisciplinary Journal 23: 354– 367. phenomenology, Coddington equations, Petzval law, correction options definition, various expansion forms, propagation of wave aberrations, relation to This gives the special condition of simultaneous correction of achromatization 2.
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Simultaneous equations introduction – Bar Model. Silent . Teacher. Narration. Your Turn. Intelligent Practice. Practice
The derivation of found for any differential equation obtained from a variational principle. He provide the simultaneous system of equations (4.1), (4.2) with 2m dependent variables The book takes a straightforward approach to the equations, namely: thick with some more on the theory of the Moon, the relation to Calculus of Variations in 'Cambridge IGCSE is one of the most sought-after and recognised qualifications in the world. It is very popular in Egypt because it provides the perfect a dynamic semi-parametric framework to study time variation in tail parameters. for modeling peaks over thresholds as in Extreme Value Theory, but casts the or Simultaneous Equation Models, Multiple Variables→Time-Series Models, av KO Faxén · 1949 — 1 Artikeln >>Statistical Implications of a System of Simultaneous Equations>>, Econo- metrica 1943 och boken stone av samma tecken fSr hela det aktuella variationsomrAdet. Kan Keynes loneteori i General Theory stAr i sjalva verket den. av S Ishihara · 2008 · Citerat av 9 — and design elements have been analyzed with Quantification Theory type I; which The variations of the item such like white / blue / red are called “categories”. In multiple regression model; simultaneous equations could not solved when mathematical framework, group theory, in modern particle physics.
Simultaneous equations and linear equations, after studying this section, you will be able to: solve simultaneous linear equations by substitution; solve simultaneous linear equations by elimination; solve simultaneous linear equations using straight line graphs; If an equation has two unknowns, such as 2y + x = 20, it cannot have unique solutions.
i ci(Hi k-ESik) = 0 for all k SECULAR EQUATIONS We therefore have k simultaneous secular equations in k unknowns.
Activity type 4: Demonstration. Top Tips for using these sequences in the classroom. i ci(Hi k-ESik) = 0 for all k SECULAR EQUATIONS We therefore have k simultaneous secular equations in k unknowns. These equations can also be written in matrix notation, and for a non-trivial solution (i.e. ci 0 for all i), the determinant of the secular matrix must be equal to zero.


Recently nearly exact expressions for the distortion in a commonly used family of Pulse Width Modulators (PWMs) known as Asynchronous Sigma Delta Modulators (ASDMs) were presented. Such an ASDM consists of a feedback loop with a schmitt-trigger (or a comparator), and a continuous time loop filter. However these previous results are not yet practically applicable because the effect of unavoidable loop delay (e.g. in the schmitt trigger) was not taken into account. Therefore we now present a more general theory that is also valid when there is a nonzero loop delay. A comparison of the resulting equations with computer simulations demonstrated a very good matching, confirming the validness of the theory. This way, a designer can now easily understand the relationship between the loop filter dynamics and the linearity of an ASDM.


Warning: This article contains a lot of mathematical references. While I am trying to be considerate of those with number allergies, there are times when I cannot help myself. I apologize in advance.
As everyone knows, in the court of law and even most socially accepted justice systems, it is insufficient for one to make accusations in order to establish another’s guilt; he must have proof that the other indeed did commit this wrongdoing. It is thus important to understand what exactly constitutes proof.
A proof is, in the strictest mathematical sense, grounded on statements that have been established to be true. Anyone who remembers high school geometry would remember the two-column proof, which logically linked statements together. A conjecture, on the other hand, is a guess based on observations.
For a more down-to-earth example of a conjecture proven wrong, consider the following problem: Place points on the circumference of a circle such that, if line segments are drawn between every pair of points, no three line segments are concurrent (i.e. intersect at one point). Into how many regions is the circle divided? Let’s take a look at the first few values of :
You may think that the number of regions doubles as each new point is added, and thus guess that for six points, there would be 32 regions. However, as the figure below shows, there are actually 31:
(On a side note, the correct formula is actually . For the mathematically intrepid, I leave it to you to prove it as an exercise; a hint is that this sum is expressible as the sum of numbers of the form . For the less intrepid, Google is your friend.)
What went wrong? We have not established that the next number has to be 32. We might as well have asked a friend to write a series of whatever integers he wants, and we would try to guess what he would write next; he may very well write 1, 2, 4, 8, 16, then write 31 just to troll us. There would be no reason for him to be obliged to write 32. The same is true for the problem of the regions in the circle.
At the moment, I cannot remember to whom this quote was originally attributed, but it goes something like “It takes only one counterexample to disprove a statement.” Indeed, that is true; however, as has been shown, it is impossible to prove a statement using examples alone. The Pólya conjecture is an oft-cited case of a conjecture that appeared to be true for the first 900,000,000 numbers or so, but was found to be false after a search yielded a very large counterexample.
Theoretical mathematics places emphasis on rigor. However, reality (if I may) does not exactly subscribe to this paradigm. As a matter of fact, the “scientific method” that is taught in schools is but glorified conjecture. For example, it was conjectured, based on observation alone, that the sun literally rose and set. However, advances in astronomy, which allowed us to look beyond our own planet, eventually proved this to be false. Note that the scientific method’s emphasis on observed phenomena as evidence, as proof if I may, is what makes our prided scientific findings shaky at best; it is the reason our scientific paradigms change every so often.
This is, of course, not to say that conjecture is totally worthless. Many conjectures actually go on to become theorems after people investigate them and actually end up proving their truth. One can argue that the infamous Fermat’s Last Theorem, proved by Andrew Wiles, was not exactly a theorem but a conjecture: Pierre de Fermat, being an amateur mathematician, probably found the proofs for a few cases and decided to claim that he had the elementary proof for all; however, the problem resisted elementary attacks for so long, and all the tools that had been used to crack it had not been invented at the time, that Fermat’s “proof” may well have been a mirage. Nonetheless, Fermat’s conjecture gave mathematicians a place to start investigating, until, eventually, Andrew Wiles, building on previous efforts, finally proved it. And, of course, the scientific method has, despite its shortcomings, served us well for the most part.
At this point, you may be wondering what all of this has to do with the hunt for Crystal Eagle. After all, you came here to find leads, not for a Ma195g.1/Sci10 class.
Alas, if you were looking here for proof, you’ll be disappointed. Proof for this kind of case, especially, when most evidence is electronic and secured, is difficult to come by. Most of my findings are conjecture at best. However, while they do not on their own definitively establish guilt, they do give the investigation a good place to start. I have seen numerous accusations, but most of them are baseless and, at best, are highly unlikely; at worst, they are virtually impossible and not worth investigating. While it is quite difficult to outright reject the possibility that any given person is our culprit, it is easy to identify who are likelier to be the culprits, and thus direct the search for hard evidence accordingly.
You may now continue to my next article (if it has already been written).


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Find the slope using the slope formula. This function is a variant of the Riemann zeta function and Dirichlet L-functions. You can see here the slope is downward because the slope is negative.
So this is equal to change in y over change in x wich is the same thing as rise over run wich is the same thing as the y-value of your ending point minus the y-value of your starting point. Judges -not juries - rule on questions of law.
And we can write these as mixed numbers. If you are lucky, someone else will already have done this for you, and you can find a MathJax plugin for your system on the MathJax-In-Use page.
The structure of rational points[ edit ] The most important result is that all points can be constructed by the method of tangents and secants starting with a finite number of points.
In contracts, Torts exam are usually organized according to parties. By the fundamental theorem of finitely generated abelian groups it is therefore a finite direct sum of copies of Z and finite cyclic groups.
And to do that, we can use the information that we know in fact we have several points of information We can use the fact that the line goes through the point -1,6 you could use the other point as well.
Such systems need to be told not to modify the mathematics that appears between math delimiters. Moreover, only finitely many rational points with height smaller than any constant exist on E. Instead what we need is a reference point - a point that we know is on a certain side of the line.
Graphs show trends or patterns of relationship. We started at x is equal to negative 1 and we go all the way to x is equal to 5.
And you just have to pick one of these as the starting point and one as the ending point. The TeX input processor has few configuration options see the TeX options section for detailsbut it can also be customized through the use of extensions that define additional functionality see the TeX and LaTeX extensions below.
In the examples later in this section, note the completeness of the legends.
Compound figures may require a full page. One of the big mistakes students make is to thoroughly answer the first three questions and leave only a scant answer on the fourth essay. So y is equal to six when x is equal to negative 1 So negative 5 thirds times x, when x is equal to negative 1 y is equal to 6.
Abbreviation of the word "Figure": If you take the cross product of [B-A] and [p-A], you'll get a vector pointing out of the screen. Implementing this is a breeze.
Tick marks - Use common sense when deciding on major numbered versus minor ticks. So our change in y, to go we started at y is equal to six, we started at y is equal to 6. Now that we have an equation, we can use this equation to determine how many participants are predicted for the 5th year.
When the data follow a uniform interval on the x-axis e. If you are lucky, someone else will already have done this for you, and you can find a MathJax plugin for your system on the MathJax-In-Use page. Setting the bin size at 0. You can make the best sense of a Torts question by outlining according to the party.
More precisely  the Mordell—Weil theorem states that the group E Q is a finitely generated abelian group. For instance, if you needed to present population sizes and sex ratios for your study organism at a series of sites, and you planned to focus on the differences among individual sites according to say habitat type, you would use a table.
I'm glad you asked. In this example notice: At the analytic side, an important ingredient is a function of a complex variable, L, the Hasse—Weil zeta function of E over Q.
Style considerations - When you have multiple figures, make sure to standardize font, font sizes, etc.
Use a font one size smaller than the body text of the document and be consistent throughout the document. In this form, the slope is m, which is the number in front of x.Differences¶. Since MathJax renders for the web and TeX is a print layout engine, there are natural limitations of which parts of TeX can be supported in a reasonable way.
Pearson Prentice Hall and our other respected imprints provide educational materials, technologies, assessments and related services across the secondary curriculum. Write the standard form of the equation of the line through the given point with the given slope.
9) through: (1, Writing Linear Equations Date_____ Period____ Write the slope-intercept form of the equation of each line. 1) 3 x − 2y = −16 y = 3 2 x + 8.
Practice finding the equation of a line passing through two points Writing slope-intercept equations. Practice: Slope-intercept equation from graph.
Slope-intercept equation from two points.
Practice: Slope-intercept from two points. Slope-intercept form problems. Page 1 of 2 Finding the Equation of a Line Given Two Points In the last lesson, I showed you how to get the equation of a line given a point and a slope using the formula.
A quadratic equation with real or complex coefficients has two solutions, called dominicgaudious.net two solutions may or may not be distinct, and they may or may not be real. Factoring by inspection.
It may be possible to express a quadratic equation ax 2 + bx + c = 0 as a product (px + q)(rx + s) = dominicgaudious.net some cases, it is possible, by simple inspection, to determine values of p, q, r, and s that make.Download


Undergraduate students are required to take an introductory, college-level mathematics course to fulfill one of the general education requirements in liberal arts. We offer several course options in math and all classes are taught in a seminar format, with a limit of 17 students in the course. Students take these classes in a digital media room where each student has his or her own computer workstation.
Rather than teaching math in the abstract, our courses are project-based to present mathematical concepts in ways that students find relevant and rewarding. Some course topics focus on the underlying mathematics of graphics technologies, information visualization, and interactive media. Because art is inherently spatial, other courses explore mathematics in relation to spatial analysis, mapping, and geography. All math courses emphasize student learning through creative assignments.


Q1. A teacher gives students that following mathematics problem. Reading on a school bus are 20 students in first grade, 10 in second grade, 9 in third grade, and 7 in fourth grade. Approximately what percent of the students on the bus are in first grade?
Which of the following student responses best answers the question?
1. Student A: 26%
2. Student B: 43%
3. Student C: 46%
4. Student D: 73%
Q2. A fifth grade teacher has provided each student with centimeter grid paper and scissors to explore how some two dimensional shapes can be folded in to three dimensional figures. Which of the following concepts are the students exploring?


Definition – What does ALU (Arithmetic Logic Unit) mean?
Arithmetic Logic Unit (ALU) is a circuit within a CPU or GPU that performs all the arithmetic and logic operations. The arithmetic instructions include addition, subtraction, shifting operations and lots of others, while logic instructions include Boolean comparisons, such as AND, XOR, and NOT operations.
Glossary Web explains ALU (Arithmetic Logic Unit)
Arithmetic Logic Unit is specially designed to perform integer calculations. Therefore, besides subtractions and adding numbers, it also handles the multiplication of two integers, also the result is also an integer. However, the term ALU does not perform division operations, since the result may be a fraction or a floating point integer. Instead, the division is usually handled by the floating-point unit that also performs other non-integer calculations.
While the ALU is an important component of all processors, the function and the design of an ALU may vary between different processor models. i.e., some arithmetic logic units only perform integer calculations, but the others are created to handle floating point operations as well. Most of the processors contain a single ALU, while others offer multiple arithmetic logic units that work together in order to perform calculations. Regardless of the way an Arithmetic Logic Unit is specially designed and its primary job is to handle all the integer operations. Therefore, a computer’s integer performance is tense directly to the processing speed of the ALU.


Monday, April 13, 2015
We use math to set an alarm clock, buy groceries, keep score or time at a game, wallpaper a room, or wrap a present.
We all need math in the world of computers and electronic communication.
It is important to encourage children to think of themselves as mathematicians who can reason and solve problems.
Here are some things you can do to encourage your children:
• Show your children that you like numbers. Play number games and think of math problems as puzzles to be solved.
• From the time your child is very young, count everything. When you empty a grocery bag, count the number of apples. Count the number of stairs to your home.
• Put things into groups. When you do laundry, separate items of clothing: all the socks in one pile, shirts in another, and pants in another.
Divide the socks by colors and count the number of each. Draw pictures and graphs of clothes in the laundry: 4 red socks, 10 blue socks, 12 white socks.
• Tell your children that anyone can learn math. Point out numbers in your child’s life: in terms of weight (pounds and ounces), measurements involving cooking, temperature and time.
• Help your children do math in their heads with lots of small numbers.
Ask questions: “If I have four cups and I need seven altogether, how many more do I need?"


Getting from 1 to 10 used to be challenge for kids starting out with math. Not so much anymore! Here, getting to the number 10 and above might prove to be a real struggle. Pleasing to the eye and incredibly fun to play, this number puzzle game will challenge you in a set of various levels that will always provide you with a fresh experience.
Create the highest possible number by matching adjacent numbers of the same colour The difficulty of the game increases with each new number created


Hi, I’m AdithyaMy friends call me Adi..
I study math and theoretical computer science at Stanford.
When deciding what to work on, I choose problems that seem important for the long-term future. I am currently interested in programming languages, distributed computing, and probability theory.
I am teaching Stanford’s CS43 (Functional Programming Abstractions) (Winter 2020). I cofounded and serve as editor for The Gradient.Disclosure: The Gradient is a student-run, non-profit publication with no external funders as of this writing. Previously, I worked on computer vision at Tesla, genomics at Counsyl (acq. Myriad Genetics 2018), and wrote books for math competitions. I was a 2014 Thiel Fellow. In my spare time, I play and compose music on piano and violin.
This site is built with Hakyll.


Harsheel Shah, Serhat Hosder, and Tyler Winter
Uncertainty quantification (UQ) is the process of quantitative characterization and propagation of input uncertainties to the response measure of interest in experimental and computational models. The input uncertainties in computational models can be either aleatory, i.e., irreducible inherent variations, or epistemic, i.e., reducible variability which arises from lack of knowledge. Previously, it has been shown that Dempster Shafer theory of evidence (DSTE) can be applied to model epistemic uncertainty in case of uncertainty information coming from multiple sources. The objective of this paper is to model and propagate mixed uncertainty (aleatory and epistemic) using DSTE. In specific, the aleatory variables are modeled as Dempster Shafer structures by discretizing them into sets of intervals according to their respective probability distributions. In order to avoid excessive computational cost associated with large scale applications, a stochastic response surface based on point-collocation non-intrusive polynomial chaos has been implemented as the surrogate model for the response. A convergence study for accurate representation of aleatory uncertainty in terms of minimum number of subintervals required is presented. The mixed UQ approach is demonstrated on a numerical example and high fidelity computational fluid dynamics study of transonic flow over RAE 2822 airfoil.


ISEP | 22-26 Julho 2013
The International Conference "Mathematical Methods in Engineering" will be held in the School of Engineering - Polytechnic of Porto, Portugal, July 22-26, 2013. The aim of this conference is to bring together scientists and engineers to present and discuss some recent developments in the area of Mathematical Methods in Engineering. The conference is designed to maximize the involvement of all participants and will present the state of the art research and the latest achievements.
Jan Awrejcewicz (Technical University of Lodz, Poland)
Mathematical modeling and simulation of the transitional wobblestone dynamics
Dumitru Baleanu (Cankaya University, Turkey)
Advances in fractional dynamics:theory and applications
Mike Field (Rice University, USA)
Asynchronous Networks, Adaptation and the Visualization of Complex Dynamics
Raoul Nigmatullin (Kazan Federal University)
Self-similar structure of the measured data and "universal" fitting function
Manuel Ortigueira (UNINOVA, Portugal)
A new look into the discrete-time fractional calculus
Mathematical Methods in Sport Sciences Applications
As an expression of our contemporary culture, the last decade assisted to the advent of new technologies in the sport sciences domain. Highly reliable and accurate motion-capture systems with long-range continuous data sets recruit developments in mathematical methods such as time-series analysis, modeling and simulation. The purpose of this symposium is to bring together diverse approaches on applied mathematical methods in the field of sport sciences and human movement.
The deadline for submission of abstracts is May 3rd 2013 and the deadline for submission of the full papers is June 7th 2013.


Smarty Pixels does numerical Ordinary Differential Equation (ODE) and Differential Algebraic Equation (DAE) solving. If your application involves ODE and DAE solving, we hope you’ll consider our software. Our scientific software performs very well on medium sized applications, even very difficult problems.
Our recent journal artical [EN2020] developes and implements a validated and non-validated defect Taylor series method for solving ODE. The stepsize is “just” within the radius of convergence of the Taylor series solution, and it performs well. This paper extends the thesis results in [E2017], where is was realized that stepsize selection in the rigorous defect control of Taylor series methods was a concern and a difficult problem to resolve.
- Ernsthausen J.M. and Nedialkov N.S.: Stepsize selection in the rigorous defect control of Taylor series methods. Journal of Computational and Applied Mathematics, 368, (2020). [link].
- Ernsthausen J.M.: Rigorous defect control and the numerical solution of ordinary differential equations. MSc, McMaster University, Hamilton (2017). [link].


Egg drop and robot hands: Girls STEM Day is March 4
Feb. 23, 2017
Girls in grades 3 to 6 will have fun with science, technology, engineering and math during Girls STEM Day at Duanesburg Elementary from 9-noon on Saturday, March 4.
The DES cafeteria will come alive with robot hands, newspaper engineering, littleBits, egg drop, science info and more during a fun event sponsored in part by the Schoharie Mohawk Initiative for Science and Technology.
To register, email Doreen Harris at firstname.lastname@example.org.


Vinyl flooring is becoming one of the most in-demand styles of floor thanks to its durability, low price point, and ease of installation. Whether you’re a first time DIYer or seasoned carpenter with years of experience, laying vinyl floors is easy and simple.
Before you decide on what style, color, and texture of vinyl flooring you want, you first have to measure the space getting new floors. Check out this short, handy guide for figuring out how much flooring you need for your next project.
Determine the Square Footage of Each Room
Dutiful homeowners may have kept the paperwork detailing the square footage of each room in their home from previous renovations, but if you don’t have that information, it’s easy to measure.
Start with the square and rectangular rooms. Simply measure the length of the wall in these rooms using a tape measure, then multiply the length of the room by its width. The number you get is called the area and it’s the square footage of that room.
A Quick Notes on Irregular Rooms
Rooms with stranger shapes can be more challenging to measure. If your room has many small alcoves, consider those separate from the room and measure them accordingly.
You may also wish to separate sections of a room with circular or triangular walls. While you could ballpark these irregular sections by measuring them like squares, you can also calculate them with some basic geometry. For a semicircular section, measure a straight line across from curved end to curved end. Divide the number you get by two, then multiply it by itself. Then you multiply that number by 3.14, then cut it in half giving you your square footage.
Triangular sections are a little simpler. Simply measure the base and the length from the base to the opposite corner. Multiply those two numbers, then divide the answer by two to get the square footage.
Calculate the Total
Once you’ve measured every room getting new floors, add them up for a total. Most vinyl flooring is sold in inches or square footage, so make sure you’re using the right numbers when you start shopping.
Some buyers choose to add up to 10% to their total square footage in order to account for cuts, waste, and mistakes. You may wish to keep some of the leftover vinyl to make small repairs in the future.


This script allows you to multiply a 'source' text box value with a number and display the result in a 'result' text box.Any number of applications can exist on a page.
|License: Freeware||Size: 10 KB||Download (24): Multiply InPut By Download|
Splash Math is a fun and innovative way to practice math. With 12 chapters covering over 140+ math worksheets and an endless supply of problems, it is by far the most comprehensive math workbook in the app store. In-app purchase required for full access. + WINNER of Gold Stevie Award (2013) in...
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Learn to multiply large numbers in your head, up to many digits.
Multiply In Your Head helps you amplify your mental multiplication powers with ease. Keep track of your success rate and response time with the in built statistics.
Watch your score improve as you become a more advanced mental...
Platforms: Windows, Windows 8, Windows 7, Windows Server
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MATH FLIGHT v1.3 - Windows: Practice basic arithmetic with activities that include Add,Subtract,Multiply,Divide, Mix Add/Subtract, Mix Multiply/Divide or Mix-Up. Use and Print arithmetic operation tables. 7 difficulty levels. Focus on a specific number. Print report cards. Race against Timer or...
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Two programs in one. One that can Add, Subtract, Divide and Multiply fractions and another that can convert a decimal to a fraction or a fraction to a decimal. It's a breeze with "Fractions". Quick and easy interface. No confusing menus. Many fraction programs only convert fractions to a decimal....
Platforms: Windows, Windows 7
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Create Worksheets is an Excel macro that will create worksheets in a new or existing workbook. The user inputs the names of the worksheets and the macro will create the worksheets. The user has the option to create the worksheets in a new or an existing workbook. The user also has the option to...
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Hide Worksheets is a must have excel add-on tool to quickly and easily hide or unhide worksheets. By using Excel's built in unhide worksheet function it can be very time consuming and frustrating unhiding one worksheet at a time. With Hide Worksheets it takes a matter of seconds to hide or...
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Excel Sheets Copier can copy one or multiple Excel Worksheets to one or multiple Excel files. Are you still bothered by the heavy Worksheets copy workload? Excel Sheet Copier is your right choice in simplifying your tedious Excel Worksheets copy work.
With Excel Sheets Copier you may:
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Merge Excel Files software can merge excel sheets into one new sheet or merge excel workbooks into one new workbook with multiple worksheets. Are you still bothered by the cumbersome job of merging multi-worksheet excel files into one worksheet?
You may have to merge excel files into...
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multpsx is a project used to generate long multiplication worksheets in PostScript. It also generates a self-checking answer key. Usage: echo "123 456" | multpsx | lpr up to 6 multiplication problems can be specified on stdin. The output sheet is intended to have the sides fold back. (The...
|License: Freeware||Size: 4.1 KB||Download (73): multpsx Download|
addpsx project is a Perl script which generates addition worksheets in PostScript. It allows a variable number of digits. Usage: # addpsx | lpr digits must be between 1 and 4 Many parameters (e.g., size and spacing) are easily adjusted in the initialization section of this program.. To...
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Math::Matrix can multiply and invert Matrices. The following methods are available: new Constructor arguments are a list of references to arrays of the same length. The arrays are copied. The method returns undef in case of error. $a = new Math::Matrix ([rand,rand,rand],...
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Proxy Multiply buys you TIME, by fetching thousands of proxies in the same amount of time it would take to find just a few without the help of automation. Do not waste time searching the internet for fresh proxies or collecting lists from websites and forums - Proxy Multiply does that for you....
Platforms: Windows, Other
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ActivityMaker is an incredible all-in-one vocabulary worksheet and puzzle creator, making over a dozen different types of worksheets from your words and definitions! You enter a list of words and definitions (or use lists from Vocab-Flash, Vocab-Flash 2, Vocab-Flash 3, Web ActivityMaker or...
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This application uses the worksheets based on standardized worksheets of the methods to control oil wells after the occurrence of kick during drilling operations on floating rigs with subsea stack and rigs with surface stack.
This app is great
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Matlab worksheets for the synthesis and simulation of 2nd, 3rd, and 4th order systems using passive loop filters. The method employed is derived from Dean Banerjee's Book "PLL Performance, Simulation, and Design" 4th Edition.
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Usage:Jitter = Pn2Jitter(f, Lf, fc)Inputs:f: Frequency vector (phase noise break points), in Hz, row or column.Lf: Phase noise vector, in dBc/Hz, same dimensions, size(), as f.fc: Carrier frequency, in Hz, a scalar.Output:Jitter: RMS jitter, in seconds.Phase noise data can be derived from...
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With this GUI you can load multiply pictures and display the first one. It will go to the next pics when click "Next" button and the previous one with "Previous" button. You can add your own image processing steps in.I am a new user, so my knowledge of MatLab is limited. Welcome comments and...
|License: Shareware||Cost: $0.00 USD||Size: 10 KB||Download (18): Review Pictures with " Previous" and "Next" buttons Download|
Multiplication Station will teach your child to add, subtract and multiply. All parameters configurable, like grid size and number ranges fonts and colors. Countdown timer keeps attention focused. High scores too.
|License: Freeware||Size: 6.05 MB||Download (22): Multiplication Station Download|
7th Grade Math Learning Games is a years worth of math lessons, worksheets, and problems for seventh graders. Covering 19 common core subjects with a near endless amount of questions, it is perfect for teachers or parents who want to give their students a fun way to practice and learn. Track...
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The goal is to devise a data generation methodology that, given a data sample, can approximate the stochastic process that generated it. The methodology can be useful in many contexts where we need to share data while preserving user privacy.
There are known literature for data generation based on Bayesian neural networks/hidden Markov models that are restricted to static and propositional data. We focus on time-evolving data and preference data.
We will study essentially two aspects: (1) the generator to produce realistic data, having the same properties of the original one. (2) we want to investigate how to inject drift within the data generation process in a controlled manner. The idea is to model the stochastic process through a dependency graph among random variables so that the drift can be simply modeled by changing the structure of the underlying graph through a morphing process.
1 Conference/Journal Paper
- INESC TEC, Joao Gama
- Universiteit Leiden (ULEI), Holger Hoos
- Consiglio Nazionale delle Ricerche (CNR), Giuseppe Manco
Primary Contact: Joao Gama, INESC TEC, University of Porto


- About Us
- Become A Tutor
At Tutor Doctor we know that gaining fundamental maths skills early on will help your student throughout his or her life. Our tutors help overcome roadblocks that prevent students from following classroom discussions and learning in the class. The maths skills learnt early in a child’s schooling serve as a foundation and if missing blocks of knowledge are left unaddressed, they become magnified as the maths becomes more complicated.
You can trust Tutor Doctor to help make maths easier to understand and get your student back on track. Our one-to-one approach works with the student’s own homework and curriculum, and we can spend time addressing topics which teachers may not be able to dwell on because of the class size or speed at which they have to get through the curriculum.
While our students come from a variety of backgrounds and skill sets, our tutors have the necessary skills and experience to ensure the student’s skills and understanding grows.
We want everyone to be saying they love Maths!! Call us today set up the FREE CONSULTATION or click on the button below to get the process under way.


May the best trig win
Taking a trigonometry exam on a spring afternoon might not sound like the most people’s idea of a fun time.
Unless you’re David Stockhouse.
The Chugiak High School junior won the state level of the Trig-Star math exam, administered at the school last April.
The annual competition, sponsored by the National Society of Professional surveyors, promotes trigonometry and honors students demonstrating superior skills.
One winner is chosen from each participating school, and he or she advances to compete for the national Trig-Star title.
This year’s test was conducted in spring of 2014.
According to Chugiak High School math instructor Laurie Templeton, each school hosts the same exam on the same day, and all exams are timed.
Over 60 CHS students took the most recent test, a population higher than previous years. Templeton attributes this to the popularity of such television shows as “The Big Bang Theory” and “CSI.”
“Being a geek is cool,” she said.
The test featured three pages of problems plus a back page of data citing such complex formulas as trigometric functions, pythagorean theorem and the law of cosines.
According to Templeton, the first two pages of the exam are geared towards high school and college levels, and the last page is “ridiculously challenging.”
Students were given an hour to complete the exam. Most finished in around 40 minutes.
Stockhouse scored 94 points out of a possible 100, netting a perfect score on everything except a problem in the last question. He finished in 39 minutes, 34 seconds, the fastest time in the state.
It was his first time taking the test.
“I did it because my teacher (Templeton) encouraged me,” he said.
He didn’t expect to win. And when his calculator batteries died in the middle of the exam, he was sure that that was the end of any chance of placing. He stopped, removed batteries from one of the school’s calculators and loaded them inside his, losing precious test time in the process. It was a well-earned lesson.
“We always have doubts that there are people out there who will soar above us and we can’t succeed,” he said.
In June, Stockhouse took the Trig-Star national exam. It was administered at his house, under the supervision of a proctor.
“I took it at my kitchen table,” he said. “It was a lot less stress. There was no one I was competing with that I could see.”
A month later, he received notification that he had placed fourth in the nation.
“I like math because the answer is always definite,” he said. “In English literature, there’s something that you have to imply, and that’s not my strong suit.”
Second place, and lovin’ it
Laura Joy Erb, who placed second in the CHS Trig Star Alaska State Competition, loves math.
“I was born with an interest in math,” she said. “It’s one of my biggest passions.”
Growing up, her math instructor mother assigned her pages of math problems to work each day.
“It was my favorite thing to do after school,” Erb said.
It’s math’s very absoluteness that she finds comforting, and days when she’s stressed, nothing calms her down as much as working through math problems.
She enjoyed the Trig Star exam and had a “blast” solving the triangle questions.
“There’s nothing more fun, with so much of an adrenaline kick, than solving triangles,” she said.
However, a careless mistake caused her to miss a section of the last question.
“If I had looked at it for five more seconds, I might have caught it,” she said. “I never thought I’d miss something so simple.”
Still, she’s happy with her second place state standing, happy that she has math in her life. She definitely sees a place for it in her future.
“The beauty I see in math is that you can know when you are right,” she said. “It’s almost black and white but there’s a depth, and the depth behind that concept blows my mind. It’s astounding.”