Alan Turing is most famous to the public for his role in cracking the German Enigma code at Bletchley Park in World War 2. He’s also quite famous for the Turing Test, to compare human with computer intelligence. But computer nerds should also know about his seminal contributions to computers, programming, and computer science, especially Turing Machines, described in his 1936 paper “On Computable Numbers, with an Application to the Entscheidungsproblem” (PDF). It was intended to prove that a general solution to a certain esoteric problem in metamathematics is impossible, and had the side effect of founding the subject of computer science. Turing needed some way to formalise what mathematicians do when they calculate, so he imagined converting human computers into mechanical computers. In 1936, a computer was a person who computed, not the electronic digital devices we use today.
What is notable about this paper? Here is a list of some of the things Turing either invented in this paper, or gave an important new twist to:
- a definition of computability;
- finite state machines (FSMs);
- Turing Machines (TM, the simplest computing model);
- the Universal Turing Machine (an interpreter that can simulate any TM);
- stored-program computers;
- computer programs (after Babbage/Lovelace);
- skeleton tables (= macros/functions/subroutines);
- computer science, esp. the halting problem;
Turing’s paper On Computable Numbers established some of the basic principles of modern digital computers, and showed that there are some things that computers can never compute. Unfortunately for Turing, a proof about the esoteric problem (the Entscheidungsproblem) was published by Alonzo Church just before he finished his paper. So he was scooped. However, his supervisor felt that Turing’s approach to the problem was so original that he should publish it anyway. This seems very reasonable, given that Turing invented computers just to solve his math problem 🙂
This paper is obviously very much concerned with numbers, but you don’t need a Phd in mathematics to appreciate it. Probably the minimum requirement would be to at least understand that there are different kinds of number, such as natural numbers (0, 1, 2, 3, …), integers (natural numbers plus negative numbers), rationals (numbers expressed as ratios of integers), and reals (numbers represented with decimal points). There are many more kinds, and Turing claims that there are also computable and non-computable numbers. He defines computable numbers as the real numbers which can be calculated by finite methods. A number is computable if it can be written down by a machine, and he’s going to demonstrate some automatic machines (we now call them Turing Machines or TMs) that can do this. Each machine is built to solve a specific problem (or algorithm as we now would say), and one of Turing’s amazing insights is that you can design a single machine to do any algorithm, by writing it on a tape to be processed by the machine. We now call this the Universal Turing Machine – or computer…
The “computable” numbers may be described briefly as the real numbers whose expressions as a decimal are calculable by finite means… a number is computable if its decimal can be written down by a machine.
1. Computing machines.
He introduces his automatic machine, which can be in one of a finite number of configurations (or states), and which can scan a tape divided into squares in each of which which there may be a symbol. The machine’s next operation is determined uniquely by the combination of current state, and symbol. It might, for example, write a new symbol on the square, and move along to the next square (left or right), and change to a new state. He claims that this very simple machine can compute any number which can be computed by any other finite means.
Turing defined computable numbers as the real numbers which can be computed to any arbitrary precision by a finite terminating algorithm; i.e. as those whose decimals can be computed in a finite time. This will be done by an imaginary machine that comprises a finite state machine (FSM, not the flying spaghetti monster) connected to an infinite tape divided into squares, via a read-write head that can scan left or right one square at a time, reading and writing as it goes. This was motivated by thinking of a mathematician (or maybe an accountant) doing their math by writing neatly on a sheet of paper. The 2-dimensional nature of the paper can be substituted by a 1-dimensional tape, considering that the computer (who at that time, was a human calculator) would write say, left to right, top to bottom, and might backtrack sometimes, and alter some previous symbols.
We may compare a man in the process of computing a real number to a machine which is only capable of a finite number of conditions q1, q2, …, qR which will be called “m-configurations”. The machine is supplied with a “tape”, (the analogue of paper) running through it, and divided into sections (called “squares”) each capable of bearing a “symbol”. At any moment there is just one square, say the r-th, bearing the symbol S(r) which is “in the machine”. We may call this square the “scanned square”. The symbol on the scanned square may be called the “scanned symbol”. The “scanned symbol” is the only one of which the machine is, so to speak, “directly aware”. However, by altering its m-configuration the machine can effectively remember some of the symbols which it has “seen” (scanned) previously. The possible behaviour of the machine at any moment is determined by the m-configuration qn and the scanned symbol S(r). This pair qn, S(r) will be called the “configuration”: thus the configuration determines the possible behaviour of the machine. In some of the configurations in which the scanned square is blank (i.e. bears no symbol) the machine writes down a new symbol on the scanned square: in other configurations it erases the scanned symbol. The machine may also change the square which is being scanned, but only by shifting it one place to right or left. In addition to any of these operations the m-configuration may be changed. Some of the symbols written down will form the sequence of figures which is the decimal of the real number which is being computed. The others are just rough notes to “assist the memory”. It will only be these rough notes which will be liable to erasure.
The FSM can be specified by a table, each row being indexed by the ‘state’, and each column being indexed by the current symbol. The cell thus indexed contains instructions on which symbol to write back to the current tape position, which way to move the head, and which state to go to next. You may notice that this table sounds very much like a computer program!
The Turing Machine consists of:
- an infinite tape divided into cells along its length. Each cell contains a symbol, such as a letter or a number, taken from a specified set such as ‘0’ and ‘1’.
- a head which can read or write symbols on the tape, and move 1 cell one way or the other.
- a register that stores the current state of the machine. This machine can be in one of a finite number of states, each of which determine what happens next when a given symbol is read.
- a table of instructions that specify what the machine does, such as replace the current symbol with a new one, move along the tape, and what will be the next state. This table is a kind of program indexed by the state and current symbol.
Here’s a simple program to add 2 unary numbers (numbers in base 1, e.g. 5 = 11111).
|1, +1, 1
|1, +1, 0
|0, -1, 2
|1, +1, 1
|0, +1, 2
|0, +1, 9
The machine starts in state 0. If it reads a 1 from the tape, it gets the ‘instruction’ 1, +1, 0. The first element is the symbol to write back to the tape (replacing the current symbol). The 2nd is the direction to move the head: -1 = left, +1 = right. The 3rd is the new state. This is effectively a primitive programming language. You wouldn’t want to use this instead of Python! However, it is, in principle, as powerful as Python, in that it can compute anything that Python can! Here is a modern simulation in Python of this Turing Machine:
If at each stage the motion of a machine (in the sense of § 1) is completely determined by the configuration, we shall call the machine an “automatic machine” (or a-machine).
He distinguishes between automatic machines and choice machines (which we now call interactive).
He explains that his computing machine prints 2 kinds of symbols: figures, which are 0 and 1 (i.e. binary digits), and ‘symbols of the second kind’. The result of the machine’s computation will be sequence of figures (i.e. 0 & 1) printed by the machine, preceded by a decimal point (wrong, Turing – a binary point!).
If an a-machine prints two kinds of symbols, of which the first kind (called figures) consists entirely of 0 and 1 (the others being called symbols of the second kind), then the machine will be called a computing machine. If the machine is supplied with a blank tape and set in motion, starting from the correct initial m-configuration, the subsequence of the symbols printed by it which are of the first kind will be called the sequence computed by the machine. The real number whose expression as a binary decimal is obtained by prefacing this sequence by a decimal point is called the number computed by the machine.
Circular and circle-free machines.
He explains that the machine could get stuck, and not print any more figures (i.e. bits). This is a circular machine. We want circle-free machines.
If a computing machine never writes down more than a finite number of symbols of the first kind, it will be called circular. Otherwise it is said to be circle-free.
Computable sequences and numbers.
A sequence is said to be computable if it can be computed by a circle-free machine. A number is computable if it differs by an integer from the number computed by a circle-free machine.
3. Examples of computing machines.
He shows examples of machines to compute these sequences: 010101…, 001011011101111011111…
4. Abbreviated tables.
There are some processes that turn out to be often useful in constructing these machines, and so he introduces “skeleton tables.”
There are certain types of process used by nearly all machines, and these, in some machines, are used in many connections. These processes include copying down sequences of symbols, comparing sequences, erasing all symbols of a given form, etc. Where such processes are concerned we can abbreviate the tables for the m-configurations considerably by the use of “skeleton tables”.
Some people point out (e.g. Petzold) that this is like a subroutine. I prefer to think of it as a macro, whose text is to be substituted in-place. I don’t see any jump/return mechanism? However, Turing does call them m-functions, but states that they are to be regarded as abbreviations; they aren’t essential. More examples follow. These examples will be useful for the later construction of the UTM.
5. Enumeration of computable sequences.
He introduces a ‘standard description’ (SD) – which I think we can recognise now as a programming language, defined by an agreed standard. This SD can be converted to a ‘description number (DN), and this is reminiscent of Goedel’s trick of encoding logical statements as numbers! So every unique TM corresponds to a unique DN. He now states that:
To each computable sequence there corresponds at least one description number, while to no description number does there correspond more than one computable sequence. The computable sequences and numbers are therefore enumerable.
Turing isn’t interested at this time, in using the machine to do everyday calculations. He wants it as a representation or model of mathematical procedure, and he wants to be able to treat this model as a mathematical item in its own right. This he does by constructing a ‘description number’ from the table of instructions, as follows: take the first row, and append the second row, then the next, and so on, so that the table has become one long row. This row is a unique identifier for the algorithm represented by the original table.
6. The universal computing machine.
This was a huge surprise in 1936 – one machine to rule them all! We are now so familiar with computers that we might find it hard to appreciate just how much of a big deal it was then! Here, Turing outlines how he’s going to construct this machine.
Note that each Turing machine so far can only do the one algorithm it was designed for, but next, Turing constructs the Universal machine. It processes the description number when it’s written on the tape to be read by the Universal machine! This UTM is by no means a simple thing, and in modern terms we might call it a compiler; it compiles description numbers into the ‘assembly language‘ of Turing Machines, namely instructions concerning what to write on the tape, which way to move the head, etc.
It is possible to invent a single machine which can be used to compute any computable sequence. If this machine M is supplied with a tape on the beginning of which is written the S.D of some computing machine M, then U will compute the same sequence as M. In this section I explain in outline the behaviour of the machine. The next section is devoted to giving the complete table for U.
We believe that the UTM can compute anything that can be computed by man or machine. This is the Church-Turing thesis; not formally proven, but strongly believed to be true. No supercomputer can, in principle, compute anything that cannot be computed by the UTM. But keep in mind that this isn’t intended to be a practical device that you would build for serious operation! It’s very much like Einstein’s ‘thought experiments’, a mental model that allows one to derive consequences, much like a set of axioms and rules of inference can be used to derive mathematical theorems.
7. Detailed description of the universal machine.
Here he gives the complete table for the universal machine.
8. Application of the diagonal process.
This section is the heart of the paper. The diagonal process was made famous by Cantor, as a way to show that the real numbers aren’t enumerable. Turing tries to apply this process to the computable numbers and sequences, and fails – thanks to the famous halting problem. So the computable numbers are enumerable. This is critical – you could list them, unlike the reals. There are infinitely more reals than computable numbers, therefore some of them cannot be computed.Turing needs a couple of techniques to get to his main result. One is due to Kurt Gödel, an Austrian-American logician. Gödel published his two incompleteness theorems in 1931. The first incompleteness theorem states that for any self-consistent mathematical system powerful enough to describe the arithmetic of the natural numbers, there are true propositions about the naturals that cannot be proved from the axioms. To prove this theorem, Gödel developed a technique now known as Gödel numbering, which codes formal expressions as natural numbers. Similarly, Turing is going to number his machines (but not the way Gödel did).
He claims that these machines can be enumerated (or listed), and he tries to apply the next technique, due to Georg Cantor to test the claim. Cantor originally used it to to show that the real numbers (i.e. numbers with possibly non-terminating decimals) cannot be enumerated. This is the famous diagonalisation procedure, which works by taking a claimed listing of numbers, and changing the nth decimal of the nth number to something else. This number cannot be in the list, since it differs from every number on it. Add the number to the list, and repeat…
This is a powerful and general technique that has since been used in a wide range of proofs, also known as diagonal arguments by analogy with the argument used in this proof. The most famous examples are perhaps Russell’s paradox, the first of Gödel’s incompleteness theorems, and Turing’s answer to the Entscheidungsproblem.
The heart of Turing’s paper is 8. Application of the diagonal process. He has shown that his machines embody the notion of mathematical computation, and that they can be represented by a table of instructions, which can be ‘Gödelised’ into a ‘description number’ DN. This process can be reversed to recreate the original Turing machine. Now imagine that you have a program that given a DN will tell you if this machine will always halt on all possible data inputs. If this program is used to analyse itself, at some point it must consider itself analysing itself… and so on ad infinitum, and it will never halt.
Alternatively, consider the program H that contains a halt-detector D as a subroutine. After calling D on the supplied input, it goes into an infinite loop if D says the supplied input program always halts, and halts if it doesn’t. If you supply H as the input to H, then if D says that H halts, H goes into the infinite loop – and so cannot halt! But if D says that H doesn’t halt, it does…
The motivation for this is to show that TMs are enumerable, i.e. can be listed, like the integers. In which case, they are infinitely outnumbered by the real numbers and so there are real numbers that cannot be computed. The TMs wouldn’t be enumerable if you could apply Cantor’s diagonalisation process to any purported list of TMs, to generate one not on the list. This cannot be done because the diagonalisation process can’t be guaranteed to terminate.
Here is the actual paragraph from Turing’s paper, explaining the problem. He wants to compute the diagonal digits ala Cantor from a list of TMs. He is considering the machine H which must test the integers i = 1,2,3, …. N to see if they are the D.Ns of circle-free (i.e. halting machines, which are ‘satisfactory’). We may imagine that H is built up from three sub-machines, P, D and U. P parses the supplied integer i to see if it’s a valid TM (most aren’t). D checks for circle-free machines. Valid TM integers that are circle free will be passed to U (a UTM) to be run up to the point where i = N, so that we can extract the nth digit of the nth TM, to add to the sequence for the Cantor diagonal. But the machine H must itself be somewhere in this list; suppose its number is K. R is a tally of the currently known valid circle-free TMs, we need it to keep track of which digit (‘figure’) we’re computing in the diagonal.
Now let K be the D.N of H. What does H do in the K-th. section of its motion ? It must test whether K is satisfactory, giving a verdict “s” or “u”. Since K is the D.N of H – and since H is circle-free, the verdict cannot be “u”. On the other hand the verdict cannot be “s”. For if it were, then in the K-th. section of its motion H would be bound to compute the first R(K—1) + 1 = R(K) figures of the sequence computed by the machine with K as its D.N and to write down the R(K)-th as a figure of the sequence computed by H. The computation of the first R(K) — 1 figures would be carried out all right, but the instructions for calculating the R(K)-th. would amount to “calculate the first R(K) figures computed by H and write down the R(K)-th”. This R(K)-th figure would never be found. I.e., H is circular, contrary both to what we have found in the last paragraph and to the verdict “s” . Thus both verdicts are impossible and we conclude that there can be no machine H.
In computability theory, the halting problem is the problem of determining, from a description of an arbitrary computer program and an input, whether the program will finish running or continue to run forever. Turing proved that a general algorithm to solve the halting problem for all possible program-input pairs cannot exist. Turing never actually called it the halting problem; Jack Copeland (2004) attributes the term to Martin Davis. A consequence of the halting problem’s undecidability is that there cannot be a general algorithm that decides whether a given statement about natural numbers is true or not. The proposition that a certain program will halt given a certain input can be converted into an equivalent statement about natural numbers. If there was an algorithm that could determine the truth of every statement about natural numbers, it could certainly find the truth of this one; but that would determine whether the original program halts, which is impossible, since the halting problem is undecidable.
9. The extent of the computable numbers.
He uses 3 different methods to try to convince us that his computable numbers include all numbers which would naturally be regarded as computable:
- a direct appeal to intuition;
- a proof of the equivalence of two definitions;
- examples of large classes of numbers which are computable.
10. Examples of large classes of numbers which are computable.
11. Application to the Entscheidungsproblem
The Entscheidungsproblem was concerned with whether it is possible, in principle, to find a definite method, or mechanical procedure – now known as an algorithm – to determine if a given mathematical proposition is provable from a given set of axioms and rules. Entscheidungsproblem is German for “decision problem”, that is, the problem of deciding if the proposition can be proven. It originated from David Hilbert‘s desire to show that mathematics is built on solid foundations; for example, there cannot ever be contradictory theorems (one says that P is true, another says that P is false); and all true theorems can be proven (in principle). This desire was shattered in part by Goedel’s Incompleteness Theorems in 1931, and Turing’s 1936 paper (and others).
In 1928 David Hilbert posed the following problem (and several others) to the mathematics community: create an algorithm that scans a statement in first-order logic and answers yes or no to the question of whether it is a true statement. Then, there would not be any such thing as an unprovable theorem.
While the failure of Hilbert’s program was a shattering blow at the time, the context is very arcane and most mathematicians carried on doing math as usual, shaking their heads and muttering “Whodathunkit?”. Math still works, but now we know that there are some theorems that might never be proven. But what turned out to be the real gold nugget of Turing’s paper, was that he effectively founded the field now known as computer science. We might also generously give him the honour of having accidentally invented computers, but as with most claims of priority and credit, the context is a tangled web of other actors and events, such as John von Neumann, Konrad Zuse, World War 2, etc. But most people will agree that Turing’s 1936 paper was seminal to computer science, and contained many of the basic concepts of modern computing, such as stored data, programs and subroutines.
Alonzo Church showed in 1935-6, by means of his lambda calculus, that a general solution to the Entscheidungsproblem was impossible. Slightly later Turing also proved this, by constructing what he called ‘automatic machines’, and we now call Turing Machines. Normally, there wouldn’t be much motivation for publishing the same result twice, but Turing’s approach was so original, intuitive and compelling – Goedel himself favoured it over Church’s method – that it was deemed worthy of publication in the London Mathematical Society‘s proceedings. Turing also demonstrated the equivalence of his and Church’s methods, now known as the Church-Turing thesis.
Who Invented the Computer?
Questions such as who invented the Internet, cars, calculus, television, flying machines often, perhaps even usually, really don’t have a simple answer in terms of one lone superhero. The Internet wasn’t invented by Tim Berners-Lee (he merely placed the world-wide web upon it), cars were originally horse-drawn carriages, Newton & Leibniz both independently invented calculus, and so on. Each invention and discovery usually takes place in a complex web in space and time of several actors and events, and if the commonly acknowledged inventor/discoverer had not put the final piece in place – such as Einstein did for Special Relativity – someone else would have done so later (e.g. Poincare or Lorentz).
In the case of computers, some of the candidates are
- Charles Babbage & Ada Lovelace (1842)
- Alan Turing (1936, 1946)
- Konrad Zuse (1941)
- John von Neumann (1945)
- John Mauchly and J. Presper Eckert (1946)
We could debate endlessly about who “invented computers”, but I think it’s more productive to first ask, what is it about modern computers that makes them so powerful and ubiquitous? Their speed, small size, and low cost are definitely very important, but it’s their flexibility that sets computers apart from all other inventions. They can perform an incredibly wide variety of tasks. They are extremely universal. And it was the Universal Turing Machine that taught us this! The fundamental properties of the digital computer are these:
- an algorithm can be expressed as a program of instructions;
- the instructions can be obeyed in non-sequential order (loops, conditionals);
- programs are stored in the computer as data that can be read by other programs.
The UTM was in effect the first compiler: it scans other TMs and translates them into Turing Machine code which can then be executed. This is the fundamental principle of modern programming, without which your PCs, laptops, smartphones, supercomputers, and embedded computers would be totally dumb. For this reason, we now call programming languages that can truly compute anything, Turing complete.[welcomewikilite wikiurl=”https://en.wikipedia.org/wiki/Universal_Turing_machine” sections=”Short description” settings=””]
- The Alan Turing Internet Scrapbook: Computable Numbers and the Turing Machine, 1936
- The Alan Turing Internet Scrapbook: Who Invented the Computer?
- A geek with a hat Week 1: Turing’s On computable numbers
- The Annotated Turing
- Computable number, From Wikipedia, the free encyclopedia
- Turing’s Landmark Paper of 1936
- The original paper, PDF
- Turing’s Proof (Wikipedia)
- Proof That Computers Can’t Do Everything (The Halting Problem) (YouTube)
- Alan Turing: Is he really the father of computing?
- What is a Turing Machine? By Jack Copeland


SSTable format uses a concept of variant integers borrowed from Protocol Buffers Base 128 Variants. Those encodings aren't exactly the same but they share the idea that an integer does not have to be represented by a fixed number of bytes. In SSTables, a variant integer can be represented by 1-9 bytes and the length is encoded in the first byte. Important characteristic of such representation is that it uses less bytes for small numbers and more bytes for large numbers than a fixed length representation. Both signed and unsigned 64-bit numbers can be translated to bytes using this encoding. Let's investigate unsigned numbers first.
Table below presents the number of bytes used to represent integers from different ranges.
|Range||Length of byte representation|
|from 0 to 2^8 - 1||1|
|from 2^8 to 2^15 - 1||2|
|from 2^15 to 2^22 - 1||3|
|from 2^22 to 2^29 - 1||4|
|from 2^29 to 2^36 - 1||5|
|from 2^36 to 2^43 - 1||6|
|from 2^43 to 2^50 - 1||7|
|from 2^50 to 2^57 - 1||8|
|from 2^57 to 2^64 - 1||9|
There are 3 cases we should look at:
Small numbers, such that are smaller than 2^8, can be represented by a single byte. The representation of such number is it's lowest byte. All the leading bytes are assumed to be 0x00. Such representation can be recognized because it has the most significant bit equal to 0. All multi byte representations have most significant bit of the first byte set to 1.
Big numbers, such that are bigger than 2^57 - 1, are represented using 9 bytes. First byte is 0xFF and the following 8 bytes represent the whole 64-bit number.
It takes 2 to 8 bytes to represent numbers that are bigger than 2^8-1 and smaller than 2^57.
First byte starts with as many leading bits set to one as there are bytes following it in the representation.
Then there is zero separating those leading ones and the actual data.
All remaining bits together with the following bytes form the actual number.
Those bits should be padded with leading zeros.
This diagram presents an example of encoded and decoded version of a number.
Signed numbers are encoded using a ZigZag encoding and then are treated with the procedure described above. Without this additional step, it would always take 9 bytes to represent a negative number. ZigZag encoding transforms a signed number N into unsigned (N << 1) ^ (N >> 63). To recover the initial number from encoded version M (which is unsigned) we need to calculate (M >> 1) ^ -(M & 1).


2-Hr Webinar that took place on November 6, 2018.
This webinar will provide you with information regarding the importance of early math skills and knowledge for young children, and an overview of the major content areas in early math. The development of strong skills associated with this broad concept is indicative of later mathematical achievement as well as later success in school. Much of what we will cover is foundational and is used in other mathematical areas.
*When opening the link, make sure you have downloaded the Adobe Connect Application before viewing the recording.


Here is the finally finished illustration of Autumn Beauty Sunflower... unless I change my mind in the morning:
It's hard to believe all of those flowers come from one variety, but there they are.
So for fun sunflower facts before I go to bed: Apparently there is a mathematical equation for the pattern of florets in the head of a sunflower. I would copy it down here, but it really means nothing to me. It is just kind of neat and crazy that someone, who is named H.Vogel, actually figured that out. I have noticed while drawing sunflowers the interesting pattern in the middle. I always get it down on the page by drawing curves out from the center of the flower like C's all the way around, and then flipping the C and doing curves like that all the way around too, if that makes any sense.
And one last fun fact, International Sunflower Guerrilla Gardening day is on May 1st. "It is a day when guerrilla gardeners around the world plant sunflowers in their neighborhoods, typically in
neglected public places such as tree pits, shabby flower beds and bare
roadside verges. It has taken place since 2007, and was conceived by
guerrilla gardeners in Brussels." I like it :) Perhaps this May 1st, wherever I am I will find myself planting a sunflower.
-info on the math equation and Guerrilla Gardening taken from Wikipedia


I’ve found an interesting article from Robert Walkers which can be found here:
Modern maths has a “Heath Robinson” type approach – at least philosophically – with its many sizes of infinity and logical paradoxes. Would this be the same for ETs? Also, what if they experience time and space differently from us? Perhaps they can only reason using flashes of insight?
Or, perhaps topology is easy, but counting, for them, is an advanced concept few understand? Or perhaps they use quantum logic or some other logic we haven’t thought of yet? Or, might they see everything as fractals?
With no experience of ET mathematicians, we haven’t got much to go on. But, let’s take a look at a few of the ways ET maths could take different approaches from ours, or be hard for us to understand.
INFINITY, SETS AND LOGICAL PARADOXES
This is an area of maths (use of sets…
View original post 6,579 more words


|We've finally derived formulas which incorporate much of the foregoing pages within simple mathematical expressions. When the energy losses in a flute (open end rate, wall losses, boundary layer, etc.) are balanced, a single formula appears. It's unique, in that, it strictly defines the length/bore/hole parameters over a huge range and at the same time contains an inherent equivalent timbre. Use of the formula guarantees your shakuhachi will operate with fundamental acoustical balance--a subtle, yet critical characteristic. Further, it's sound will be of the timbre distinctive to optimal bore geometry--at any length and in any key. For metric measurements only.
General, all purpose metric (mm) equation:
Length = (186500/ Hz) - 5 Bore
You can do these calculations at Google. Just put it into the search box and click search.
Example: Enter (545/17)^(5/6) into the search box, click and you'll get the optimal bore diameter for a 1.8 shakuhachi in millimeters. (545 / 17)^(5 / 6) = 17.9868999
For more Google calculator instructions.
How many octaves should/could your flute play? Aspect Ratio (bore length divided by bore diameter) is the greatest determinate of note range.
AR / 12 = Range (in octaves)
So, a bore with an AR of 24 should play two octaves, an AR of 30 will support 2.5 octaves and so on. Consider this formula an approximation.
Here's another set of formulas--from the Dark Side.
These are approximations which will produce flutes a few percent darker (timbre-wise) than the equations above.
Make all measurements in millimeters.
Comparison of the 'Light' and 'Dark' formulas
See The Synthesis for a final flute design.


Read how to set prize win probability for all the game types!
Prize Win Probability in luck based games
Prize win probability becomes a factor in games which we categorize as luck-based games. The only interaction in the game is to see whether you did or didn’t win something. Our luck-based games are Plinko, Open The Box, Wheel Of Fortune and Scratchcard.
When adding prizes to these campaigns, you have to assign what’s the probability of winning this prize. It can be anywhere from 0.00001% to 100%.
You can also set the default prize, which is won when no other prize is won.
Once the prize probability is assigned then you can assign the prize to the desired outcome in the Game setup. The system will automatically adjust the probability of the prizes in the game.
1. Create a wheel of fortune
2. Add a prize iPhone to the campaign with a probability of 0.001%
3. Connect wheel Sector 1 prize = iPhone
Then every time the wheel is spun there is a 0.001% chance the wheel will stop on Sector 1.
How to choose the correct probabilities?
It’s easier to think about the probabilities in fractions: 0.001% chance to win means 1 in every 1000 spins will land there. If you expect around 10 000 spins then modify it to 0.0001% so 1 in every 10 000 spins will win an iPhone.
How to guarantee the win of the grand prize during the campaign?
You can do so by increasing the probability of the prize during the campaign and if on the last day no prize is given out then eventually have it as 100% so someone will win the prize. But be aware that this has to be communicated through terms and conditions.
How to modify the prize-winning probabilities during the campaign?
You can modify the prize-winning probabilities during the campaign the same way you set them up. After saving all the changes make sure to click “Update live campaign” to apply the changes to the live version.
Although you can do that, be aware that you should communicate this in your campaign terms and conditions under fair play. When you are increasing the probabilities towards the end of the campaign then the players who arrive later would have a higher chance of winning the prize which creates an unfair campaign environment.
How to set the daily limit to a prize?
Daily prize limits assign the total number of given prizes that can be given out during a day. It only applies to lottery (RNG) type campaigns.
You can also manually decide when will the limits be reset.
You want to have a campaign running for 10 days and give out 100 prizes of something during this period.
To make sure that they are distributed equally throughout the campaign then set the Daily prize limit to 10. This means that each day only 10 of the prizes can be assigned.
Learn how to send automated emails to participants from here.
Learn how to set winning probability to instant win prizes.
Prize Win Probability in other game types
Prize winning probability tells you the percentage chance of actually winning the prize.
Winning probability in "score to win"
Decide the score needed and how likely it is to win. If you want someone to get a prize when they reach a certain score, just make sure the chance of winning is set to 100%.
For the second prize, the "Towel," the "score to win" is set at 500 points. This means if your score reaches 500 points, you have a 30% chance of winning the towel and a 70% chance of winning nothing.


Learn Without Direct Instruction
Organisations increasingly use Artificial Intelligence (AI) to solve business problems that prove too challenging or too time-consuming for human beings.
Machine learning (ML) is the process of using mathematical models of data to help a computer learn without direct instruction. It’s considered a subset of artificial intelligence (AI).
Machine learning uses algorithms to identify patterns within data, and those patterns are then used to create a data model that can make predictions.
With increased data and experience, the results of machine learning are more accurate – much like how humans improve with more practice.
Azure Machine Learning
Microsoft’ Azure Machine Learning can equip your data scientists and developers with the capabilities to build, train, and deploy Machine Learning models and foster team collaboration.
We can help organisations use Azure Machine Learning to accelerate their time to market with industry-leading Machine Learning operations or DevOps for Machine Learning.
Organisations can then innovate on a secure, trusted platform designed for responsible Machine Learning.
Features and Benefits
- Predicted patterns of behaviour can inspire a great customer experience and the development of better products
- In today’s competitive world, speed is of the essence, and models that can be created regardless of skill will be most effective
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- CPS and Microsoft can help organisations use machine language to make predictions
- The solution will rapidly build and deploy machine learning models using tools that meet your needs, regardless of skill level
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A stochastic model in some sense is a model which depends on some random process.
A Markov process named after Andrey Markov, is a stochastic process that satisfies the Markov property. Markov property is also refered to as "memorylessness". In probability theory and statistics, memorylessness is a property in which the next state is dependent on present state only.
So as we can see, there are multiple ways in which we can use Markov chains to analyze or simulate real world processes.
Markov chain is a chain and is a type of Markov process with finite or countable state space. The changes of state in a Markov chain are called transition and the probability associated with those transition are known as transition probabilities. We can express a Markov chain with the use of transition matrix with describing the probability in the transition.
Transition Matrix (Stochastic Matrix)
A Stochastic matrix or transition matrix is a matrix used to define the transitions in a Markov Chain. In a Stochastic matrix, each entry is a non negative real number which represents the probability of the state change from a to x, where a is the current state which is given as a row and x is the next possible state reachable from a, this type of stochastic matrix is also known as a right stochastic matrix in which the probability in each row sum's up to 1. Similarly, one can define the transition the other way around i.e. transition where column denotes the current state and the values in each row denotes the transition probability for the jump from current state, This type of transition matrix is coined as a left stochastic matrix. In left stochastic matrix, the probability in each column sums up to 1. These kind of matrix have various names such as probability matrix, transition matrix, stochastic matrix, Markov matrix or substitution matrix.
Code for modeling markov process as described in Fig 2.


We get asked a lot how much weight a particular compactor can hold. To date there are no real official numbers because the weight of material varies so much as well as compaction ratios of equipment. This dilemma is also compounded by the fact that waste with a high liquid content has a very poor compaction ratios since liquid cannot be compressed while material with a high quantity of air compacts into an incredibly small space.
When we draft a savings analysis for a customer the standard calculation is 4:1 as most compactors can compress waste into a space a quarter of is original size. The standard way for calculating how much weight a compactor container can hold has been for a while now (container size X 580 / 2000= Total Weight in Tons). This calculation is based on the fact that most waste is compressed into a compactor container at a rate of between 500 and 600 pounds per cubic yard multiplied by the amount of cubic yards in a container and divided by the amount of pounds in a yard.
The “580” calculation however is of course closer to the high end of the scale and in this industry it is much better to under-promise and over-deliver versus over-promise and under-deliver so if you are looking for a more conservative estimate it may be better to set your calculations between the 500-550 pound per yard range so you aren’t disappointed after having received your new piece of equipment.


This is the second part of the series Linear Regression: Everything From Math to Program.
Part 2: Regression with Two Independent Variable
Earlier in Part-1, we have learned about how to perform linear regression in the case of one independent variable in python and Excel. We have also derived the general expression for calculating slope and intercept of a one independent variable linear regression problem.
Hello All !!!
This is the third and final part of the series Linear Regression: Everything From Math to Program.
Part 3: Regression with Multiple Independent Variables
If you have made it so far then it’s obvious you have understood concepts of Linear regression and how those beautiful libraries on regressions working in the back-end. I don’t want to share anything fancy with you guys at this point and make this blog lengthy and complicated. There are no derivations in this part and we will use…
A few days ago I was trying to solve one of the problems given to me in my school. They asked us to impute some missing values in a given dataset. Those of you who are aware of the term ‘imputation’, you know that there are countless algorithms published in various studies for different kinds of data.
Some of the algorithms for imputing a missing value is:
And many more…
In our problem, it was clearly mentioned that the given data has linear…
Research Student pursuing Master’s In I.E at University Of Windsor, Software Developer by Profession, Bachelors in M.E, Machine Learning Enthusiast


A sportsbook is a gambling establishment where you can place a wager on various events. These bets are based on the probability of each outcome and pay out a sum if the event happens. A sportsbook will also set odds for the underdog and favorite, allowing you to place a bet on the team that you think will win. This type of betting offers higher risk but larger rewards.
The sportsbook industry has grown into a multibillion dollar business. Many states have legalized sports betting and the number of options continues to grow. However, it is important to be aware of the risks involved in this business and to choose a reliable sportsbook. It should have a variety of games and events, offer secure transactions, and be accessible on mobile devices. It should also have a user-friendly design and be easy to navigate.
Sportsbooks use a mathematical model to predict the median margin of victory. They do this by treating the relevant random variable as a distribution that can be fitted with the probability of each event occurring. Then they compare this distribution to the expected probability of winning a given bet. This model allows for an accurate estimate of the magnitude of the sportsbook error needed to permit positive profits to be made by bettors.
In order to estimate the magnitude of this sportsbook error, the sample of matches was stratified by their point spreads. Then the median margin of victory was computed for each stratum. The median value was then compared to the sportsbook’s point spread, and the difference was recorded. The results show that the mean value of the spread is lower than the median, and it requires a much greater sportsbook error to allow for positive returns on a unit bet.
In order to attract more punters, sportsbooks must focus on ensuring that their websites are fully-equipped to provide all the services and features that punters desire. This includes offering a wide range of betting markets with competitive odds, simple navigation, and clear bonus structures. In addition, it is critical to have safe payment methods that enable fast withdrawals and have low transaction charges. Customers also prefer sites that take security seriously, and offer customer service via email or live chat. A sportsbook that is not well-equipped to serve its customers will struggle to stay in business.


There are some powerful math features in Excel that can help you with your date data. First, let’s take a look at how Excel interprets dates.
Although you are seeing a date formatted properly, behind it there is just a number. These are referred to as serial number dates. Each date beginning with January 1, 1900 (date number 1) has its own number. So, for example, if the date is 3/4/2013, the serial number is 41337, or the 41,337th day since 1/1/1900. Knowing that makes it easy to understand that when we are adding or subtracting dates, we are just adding and subtracting numbers, so 41337 + 90 is 41427 or 6/2/2013.
There are actually functions that work with dates, too. Here are a few very useful ones:
- TODAY() Today's date as understood by your computer
- NOW() The date and time, right now.
Let’s have a look at NETWORKDAYS. Let's say you have project start and end dates. The true number of workdays for our example are weekdays Monday through Friday less holidays that fall on those days. If we have our project’s start and end dates in columns A and B, respectively, we can calculate the number of workdays in C with NETWORKDAYS. Before we type in the formula, let’s put our holidays in a list in another worksheet and assign the name Holidays to it using the Name Box.
The formula we’ll type in column C is:
Now, you have the actual number of workdays to determine how resources can be allocated, tasks assigned and man-hours allocated.


When you’re getting into yard inflatables, at some point you’ll probably stop to wonder “How much are these things costing me on my electric bill”? This is not an uncommon question at all, so we did some research and some math to find out how much electricity a yard inflatable uses.
Factors of Inflatable Usage Cost
Without buying your own electricity usage monitor, there’s a few factors we can use to give you a ballpark of how much it’s costing you to run an inflatable.
1. Inflatable Size
Bigger inflatables are going to use require more power to keep them upright than a smaller one. So a 4 foot inflatable will use less energy than a 12 foot inflatable.
2. Animated vs Stationary
If you have a yard inflatable that’s animated, that’s going to suck up more electricity to make the motions happen.
3. Electricity Rates
This is something you’ll need to check your power bill for to find your current rate, it should be listed in xx cents per kilowatt-hour (kWh). The national average in the US is 12 cents, so we will use that for our calculations.
4. Usage Time
How often you keep your inflatable running through the day will obviously play the biggest factor in how much energy it uses.
We will assume your inflatable run 12 hours a day for our estimations, so be sure to modify it accordingly to fit how you plan on using your inflatables.
You may want to keep your best Christmas yard inflatables running 24 hours a day, so in that case, just double the calculations below.
Estimated Costs for Yard Inflatables
The below figures are only estimations based on manufacturer claims, in the real world your costs could be higher or lower. Use an electricity usage monitor if it’s important enough to you that you need exact figures.
4 Foot Inflatable – 52 Watts per hour x 12 hours = .624 kWh per day. If used all 31 days of December for 12 hours a day, this inflatable would add $2.32 to your electric bill.
6 Foot Inflatable – 60 Watts per hour x 12 hours = .72 kWh per day. If used all 31 days of of December for 12 hours a day, this inflatable would cost an additional $2.68 on your electric bill.
8 Foot Inflatable – 76 Watts per hour x 12 hours = .91 kWh per day. If used all 31 days of of December for 12 hours a day, this inflatable would add $3.39 to your electric bill.
12 Foot Inflatable – 85 Watts per hour x 12 hours = 1.02 kWh per day. If used all 31 days of of December for 12 hours a day, this inflatable would add $3.80 to your electric bill.
For animated inflatables, it’s harder to give an accurate estimation due to the variation of complexities of the animated inflatables out there. As a very general rule of thumb, you’d need to double or triple the estimates above to take into consideration the animated power consumption.
So a 6 foot, animated Halloween yard inflatable could use as much as 200 Watts per hour x 12 hours = 2.4 kWh per day. If used all 31 days of December, this would amount to an extra $8.92 on your electric bill.


The scores for both tests are on a 500 point scale, with a designed standard deviation of 50. In the proceeding table, these are are converted to IQ estimates with a mean of 97.5, corresponding to the national average NAEP scores of 281.67 for math and 262.29 for reading, and a standard deviation of 15. The math and reading scores are given equal weighting.
I'm operating under the presumption that the average IQ in the US is 98. The reason the IQ estimates are based on a national mean of 97.5 is to take into account private schools, whose students on average score 15 points higher on the math test and 19 points higher on the reading test, corresponding to an estimated 5.1 IQ advantage over their public school counterparts. Representing a little more than one-tenth of the school age population, this yields a national average of 98. The estimates assume that the private school advantage relative to public schools is equal across states. That is, the public school score, plus 5.1 multiplied by the percentage of a state's students enrolled in private schools, gives the state's overall estimated average IQ.
Again in the spirit of the 2004 IQ hoax, states are colored in accordance to how their populations voted in the 2008 Presidential election. Light red (blue) indicates the margin of victory for McCain (Obama) was less than 10%; dark red (blue) indicates it was 10% or more:
|51.||District of Columbia||91.2|
* "Department of Defense Education Activity", comprised of the children of service members serving outside of the US. Despite a demographic pool (58.1% white, 22.0% black, 9.9% Hispanic, 9.0% Asian/Pacific Islander, 1.0% Native American) that suggests the DoDEA should be found near the bottom of the rankings, if measured as though an independent state, DoDEA schools place 11th, between Maryland and Wisconsin. Military men aren't morons.
An aunt of mine who lives in Alabama has joked that the state's motto is "thank God for Mississippi". The demographic transition the nation is undergoing, however, has already made her remark a bit antiquated. I suspect in a few years' time, both Alabama and Mississippi will be able to thank him for the Southwest.
Although I have substituted reading scores for science scores in converting NAEP scores into IQ estimates four years down the road, the 2005 and 2009 state level results still correlate with one another at a very vigorous .91 (p=0). There is a great deal of consistency in the average state scores among different cohorts of test takers, suggesting that deriving IQ estimates from NAEP testing is sound.
As has historically been the case, the further north one goes, the smarter the population becomes. During the 2008 election cycle, Half Sigma frequently decried the perceived abandonment of the GOP by the cognitive upper crust of the voting population. Indeed, the best-performing red states are only tenuously so. Middling Kansas tops the list of solidly Republican states. I suspect, however, given the staggering growth in government spending that is coming down the pike over the next four years, that 2010 will reveal a reversal in that trend to be occuring.
Parenthetically, follow AE on twitter @AudaciousEpigon.


The Commission on Teacher Credentialing has appointed twenty individuals to serve as panel members to consider the need for, and, as appropriate, propose recommended standards for a preparation program for the Mathematics Specialist Credential. In addition, the Commission has asked the advisory panel to delve into the larger picture related to determining the mathematics needs of students across the K-12 continuum. Considerations then need to be given whether to change authorizations to reflect preparation received or to address teacher preparation to conform to authorizations. The panel begins work on March 30-31, 2009.
CTC Agenda Items Related to Teaching Mathematics
- Authorizations to Teach Mathematics (October 2008) [PDF]
This agenda item presents information on the different documents that authorize an individual to teach mathematics. In addition, the item provides data on the numbers of individuals earning credentials to teach mathematics, the mathematics subject matter requirements, the passing rates of the CSET mathematics examination, pedagogical preparation to teach mathematics, and how students in the public schools are scoring on the standardized mathematics examinations.
- Authorizations for Teaching Mathematics (January 2009) [PDF]
This agenda item continues the discussion begun at the October 2008 Commission meeting related to the teaching of mathematics in California. The focus of this item is the authorization statements for documents that authorize the teaching of mathematics.
- Subject Matter Competence of Teachers of Mathematics (April 2009) [PDF]
This agenda item continues the discussion related to the teaching of mathematics in California. The focus of the item is the process for determining the subject matter competence of K-12 teachers of
- Update on the Work of the Teaching Mathematics Advisory Panel (April 2009) [PDF]
This agenda item presents an update on the work of the Teaching Mathematics Advisory Panel.
- The current Specialist Credential Program Standards[PDF]
- The 2013 K-12 Mathematics Framework
- Mathematics Teacher Preparation in California: Standards of Quality and Effectiveness for Subject Matter Programs (2004) [PDF]
- Commission adopted Multiple and Single Subject Teacher Preparation Program Standards (2009)[PDF]
Other Resources as appropriate such as current credential requirements and standards used by other states or professional organizations for similar credentials
- Mathematical Education of Teachers -- Adding It Up
NCTM's Principles and Standards for School Mathematics (grades 3 - 5 recommendation for math specialists)
- National Math Advisory Panel
- McDaniel College's Math Specialist Program
Dr. Francis (Skip) Fennell
Includes state-by-state overview, list of related research
Current research about effective instructional strategies in the teaching of mathematics
Meetings of the Advisory Panel
Three meetings of the Advisory Panel are scheduled. The agenda for each meeting will be posted prior to the meeting. Other meetings will be scheduled.
- March 30-31, 2009 (CTC) [MS Word]
- May 21-22, 2009 (CTC)
- June 17-18, 2009 (CTC)


Written in EnglishRead online
Includes bibliographical references (p. -63) and index.
|Series||Reading the past ;, v. 2|
|LC Classifications||QA22 .D55 1987|
|The Physical Object|
|Pagination||63 p. :|
|Number of Pages||63|
|LC Control Number||86030803|
Download Mathematics and measurement
"Mathematics and Measurement" is an eclectic collection of information on the origins and applications of mathematics in the ancient world by Oswald A. Dilke, who is a specialist in Latin, Roman land surveying, and ancient cartography. This slim page volume is part of the British Museum's "Reading the Past" series, but it is broader than Cited by: Jul 24, · “This book forced me to use mental muscles I haven't exercised in a long time, but it felt fantastic.
Paul Lockhart is a mathematics evangelist; his passion for his subject is evident on every page, in every line. Looking at the subject of Measurement, he takes the reader on a journey that covers geometry, algebra, trigonometry, and on /5(55). Mathematics & Measurement - Kids: Books. 1 - 20 of results ABCs of Mathematics.
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Aug 20, · Lockhart's Measurement is a work of mathematical beauty. From the very humble beginnings of the mathematics of shape, he develops a mathematical edifice including ideas of Plane Geometry, Solid Geometry, Higher-Dimensional Geometry, Coordinate Geometry, Projective Geometry, Vector Geometry, Elementary Algebra, and Differential and Integral Calculus/5.
Mathematics (from Greek μάθημα máthēma, "knowledge, study, learning") includes the study of such topics as quantity (number theory), structure (), space (), Mathematics and measurement book change (mathematical analysis). It has no generally accepted definition.
Mathematicians seek and use patterns to formulate new conjectures; they resolve the truth or falsity of conjectures by mathematical proof. Measurement Index. Measurement is finding a number that shows the size or amount of something. There are two main "Systems of Measurement": Metric and US Standard.
Metric. Introduction to the Metric System (with Maggie!); The Metric System of Measurement. Apr 26, · Measurement | Paul Lockhart. T his book is a must read for math teachers.
It is about how mathematics should be taught. However, if you have a strong Author: Ali Kayaspor. Elementary Mathematics: Measurement Discover everything you need to successfully teach measurement to elementary students, from references and lesson Mathematics and measurement book to printable worksheets and graphic organizers.
THE MATHEMATICS OF MEASUREMENT. Authors: ROCHE. Buy this book Hardcover 77 Services for this Book. Download Product Flyer. Facebook Twitter LinkedIn Google++. Recommended for you. Bibliographic Information Bibliographic Information. Book Title THE MATHEMATICS OF MEASUREMENT Authors.
ROCHE. Description. Measurement in Mathematics Book 7 aims to further develop the attitudes, understandings and skills students have acquired in working with the measurement strand in previous fencingveterans2013.comic hands-on tasks involve calculations using millimetres, centimetres, metres and kilometres, perimeters, diameters and circumferences of circles; viewing relationships btween the area of rectangles.
Physics: Topic 1 Measurement and Mathematics. STUDY. Flashcards. Learn. Write. Spell. Test. PLAY. Mathematics and measurement book Match.
Gravity. Created by. megan_ma. Chapter 1 of Physics. Terms in this set (30) measurement made during laboratory work which may stand alone or be incorporated into one or more formulas to yield a value for a physical quantity.
Description. Measurement in Mathematics Book 3 is designed to provide young children with simple problem solving situations when students use a number of objects to explore fencingveterans2013.com of hands-on investigations are included in this book.
Specific tasks involve measuring with arbitrary units and rulers, using clocks and calendars to measure time, measuring perimeters and areas.
Note: Citations are based on reference standards. However, formatting rules can vary widely between applications and fields of interest or study. The specific requirements or preferences of your reviewing publisher, classroom teacher, institution or organization should be applied.
To convert between one measurement unit and another, requires a conversion factor. For instance; 1 inch = cm or mm and so 12 inches = 12 x = mm The same conversion factor can also be used to convert mm to inches by using division.
For example; mm =. Picture books with measurement content. Level 1 measurement. Ernest, by Catherine Rayner he and his friend come up with a solution: increase the area of the book. Now he fits perfectly.
Mathematics: Area and size. Guess how much I love you, by Sam McBratney Pipkin the Penguin follows his curiosity to explore the measurement idea of. Nov 24, · 00 This fully illustrated book outlines the ancient systems of mathematics and measurement and describes how they were used in mapping, surveying, telling time, trade and commerce, as well as in leisure pursuits such as games and puzzles, and in the occult.
The eight Standards for Mathematical Practice are an important component of the mathematics standards for each grade and course, K– The Standards for Mathematical Practice describe the varieties of expertise, habits of minds, and productive dispositions that educators seek to develop in all students.
First Steps Mathematics is a series of teacher resource books that is organised around sets of mathematics outcomes for Number, Measurement, Space, and Chance and Data.
The series will help teachers to diagnose, plan, implement and judge the effectiveness of the learning experiences they provide for students. Feb 06, · CBSE Class 5 Mathematics - Learn Measurement Units and their uses. Subscribe to our channel to avail the benefits of Genee India interactive education tutori.
Helping Your Child Learn Mathematics Helping Your Child Learn Mathematics Fore word Contents We know from research that children are more likely to be successful learners of any subject when parents actively support their learning1.
Today, helping children to make the effort to learn, appreciate and master mathematics is more important than ever.
Get this from a library. Mathematics and measurement. [O A W Dilke] -- This fully illustrated book outlines the ancient systems of mathematics and measurement and describes how they were used in mapping, surveying, telling time, trade and commerce, as well as in leisure. Arguing for a better balance between educational and measurement concerns in the development and use of mathematics assessment, this book sets forth three principles--related to content, learning, and equity--that can form the basis for new assessments that support emerging national standards in mathematics education.
Measurement in Mathematics is series of eight books aimed at teaching the topic of ‘measurement’ to pupils from reception through to year Book 6 is aimed at children who are in year 5 and 6. Each book covers the following topics at the appropriate level: Length.
In mathematical analysis, a measure on a set is a systematic way to assign a number to each suitable subset of that set, intuitively interpreted as its size. In this sense, a measure is a generalization of the concepts of length, area, and volume. A particularly important example is the Lebesgue measure on a Euclidean space, which assigns the conventional length, area, and volume of Euclidean.
Mathematics Measurement First Steps in M Improving the mathematics outcomes of students • Understand Units • Direct Measure Book 1 Maths Measure Book 1 fencingveterans2013.com 1 13/5/10 am. A comprehensive and coherent set of mathematics standards for each and every student from prekindergarten through grade 12, Principles and Standards is the first set of rigorous, college and career readiness standards for the 21st century.
Principles and Standards for School Mathematics outlines the essential components of a high-quality school mathematics program.
Search the world's most comprehensive index of full-text books. My library. About the Book. Year Published: Page Count: Prints of this textbook before included a student CD, but any later reprints ( onwards) have moved all of the CD features to Snowflake - our online learning system. Mathematics for Australia 8 has.
About the Book. Fundamentals of Mathematics is a work text that covers the traditional study in a modern prealgebra course, as well as the topics of estimation, elementary analytic geometry, and introductory algebra. It is intended for students who: have had previous courses in prealgebra; wish to meet the prerequisites of higher level courses such as elementary algebra4/4(7).
Measurement is his attempt to change the equation: a conversational book about mathematics as an art that invites the reader to join in the fun. Sounding every bit the teacher whose love for his subject is infectious, he guides us through exercises in geometry and calculus—giving information and hints along the way while always encouraging us.
Measurement offers a permanent solution to math phobia by introducing us to mathematics as an artful way of thinking and living. In conversational prose that conveys his passion for the subject, Lockhart makes mathematics accessible without oversimplifying.
Teaching Children Mathematics (TCM) is an official journal of the National Council of Teachers of Mathematics and is intended as a resource for elementary. Everyday Mathematics Implementation Measurement. Implementation measurement provides information about how teachers are implementing EM materials, what is actually happening in schools, classrooms, and districts “on the ground,” and why.
Learn more. Everyday Mathematics and the Common Core State Standards for Mathematical Practice. Norman Biggs is Professor (Emeritus) of Mathematics at the London School of Economics.
He is the author of 12 books, including a perennial best-selling book Discrete Mathematics (Oxford University Press). He has a special interest in measurement and was Chair of the International Society of Weights and Scales Collectors from Price: $ About the Book.
Year Published: Page Count: Prints of this textbook before included a student CD, but any later reprints ( onwards) have moved all of the CD features to Snowflake - our online learning system.
Mathematics for Australia 10A has. CHAPTER 1 MATHEMATICS AND UNITS OF MEASUREMENT Mathematics is the Engineering Aid’s basic tool. The use of mathematics is found in every rating in the Navy, from the simple arithmetic of counting for inventory purposes to the complicated equations encountered in.
Measurement, Grades 4 to 6is a practical guide that teachers will find useful in helping students to achieve the curriculum expectations outlined for Grades 4 to 6 in the Measurement strand of The Ontario Curriculum, Grades 1–8: Mathematics, This guide provides teachers with. These lesson plans will help teach your students the different forms of measurement for weight, distance, and time.
For access to more resources like these start you free trial today. fencingveterans2013.comtMD.A.2 Use the four operations to solve word problems involving distances, intervals of time, liquid volumes, masses of objects, and money, including problems involving simple fractions or decimals, and problems that require expressing measurements given in a larger unit in terms of a smaller unit.
Represent measurement quantities using diagrams such as number line diagrams. Measurement. Activities. Index. Data. Geometry. Numbers. Puzzles. Money. Algebra. Calculus. Physics. Games. Dictionary. Worksheets. Print out The Times Tables and stick them in your exercise book.
Test Your Tables with an interactive quiz. Have your say. Mathematics is commonly called Math in the US and Maths in the UK. GED "Scoreboost" materials target exactly the skills one needs to pass the General Educational Development (GED) tests.
This book focuses on preparing for the GED mathematics test. It provides 16 strategies for developing the skills needed for measurement and data fencingveterans2013.com: Cathy Hoyt.Mathematics definition is - the science of numbers and their operations, interrelations, combinations, generalizations, and abstractions and of space configurations and their structure, measurement, transformations, and generalizations.
How to use mathematics in a sentence.fencingveterans2013.comtMD.A.1 Convert among different-sized standard measurement units within a given measurement system (e.g., convert 5 cm to m), and use .


Professional and DIY IQ tests are popular because they offer a formula that allows you to compare yourself to other people and see how average (or above average) your intelligence is.
The Cognitive Reflection Test (CRT) is dubbed the shortest IQ test in the world because it only consists of three questions. It assesses your ability to identify that a simple problem may in fact be more difficult than it appears. The faster you do this, the smarter you seem.
Here are the three questions:
1. A bat and ball cost £1.10 in total. The bat costs £1.00 more than the ball. How much does the balloon cost?
2. If it takes five machines five minutes to make five widgets, how long would it take 100 machines to make 100 widgets?
3. In a lake, there is a patch of water lilies. Every day, the patch doubles in volume. If it takes 48 days for the plot to cover the entire lake, how long would it take for the plot to cover half of the lake?
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Here’s what a lot of people think:
1. 10 pence
3. 24 days
These answers would be Wrong.
When you’re ready, scroll down to find the right answers and how to get there:
1. The ball would actually cost 0.05 pence
If the ball costs X and the bat costs £1 more, then it will be:
Bat+ball=X + (X+1) =1.1
2X+1=1.1 and 2X=0.1
2. It would take 5 minutes to create 100 widgets.
Five machines can make five widgets in five minutes; therefore, a machine will also make a widget in five minutes.
So if we have 100 machines all making widgets, they can make 100 widgets in five minutes.
3. It would take 47 days for the patch to cover half the lake
If the patch doubles in size every day, it will shrink back in half. So on day 47 the lake is half full.
In a survey of nearly 3,500 people, 33% got all three wrong, and 83% missed at least one.
While this IQ test had its flaws — its brevity and lack of variation in verbal and nonverbal reasoning — only 48% of sampled MIT students were able to answer all three correctly.
How right were you? Tell us below
Give your opinion on our topical democracy. Click the upvote icon at the top of the page to help push this article up the indy100 rankings.


We used maths factor for our eldest child to increase their confidence in maths. It was extremely useful in cementing times table knowledge and helping with division and multiplication. This was particularly true when it came to year 6 SATs with fractions and percentages.
We will be using maths factor with our younger child to prepare them too. Thank you so much Carol for creating such an invaluable resource for our children
We are very proud of what he has achieved and he never ceases to amaze us, he works very hard and looks forward to doing his session on The Maths Factor.
Thank you for the wonderful programme and for engaging him in such a positive way. He has made so much progress and is committed to his learning.
My daughter really loves using this site! The 30 day challenge has her so motivated! I like that she has been able to go back and redo modules where she wanted to improve her score from last time.
We teach her at home, so I really like receiving emails with her score, each time she has completed a module, as it helps me to keep tabs on her levels of understanding and the areas where we need to give her more help.
I am delighted with my daughter’s progress in maths in the 3 months she has been using The Maths Factor. Her school report this week shows she has improved from 'working towards' in her school report to 'working at' from excellent results in her end of term Maths test. This is purely down to her daily Maths Factor sessions which she has done consistently each day where the small steps have led to large improvements I am totally amazed how quickly this has happened.
I highly recommend this course for anyone whose child is struggling and hates maths as her progress and even enjoyment of maths has improved enormously. Many thanks!
...Carol has put a lot of thought into this program, the videos explain perfectly, Carols tone of voice is lovely and encouraging. The 30 day challenge is fab and my little man was so proud of himself for completing it. Everything about this is just amazing from the emails to update you on your child’s progress to the medals and trophies the children can earn. The free printable certificates and the tick counter is an added bonus!
I wish there was something like this for the other subjects and for higher grade levels.
After a week of using Maths Factor my daughter has grown so much confidence in maths. She came home with a sticker this week being the 2nd fastest to complete her maths school work. Previously she was one of the slowest to complete maths work and being slow to complete work was an issue for her (even though she'd usually do OK with the answers).
The best thing about Maths Factor is it's made her want to complete the maths challenges - the medal helps a lot. She isn't in love with maths yet but she loves the challenge as she's very competitive. Just about to leave for the day and she said "Dad we have to do the maths now in case we're back late!" and happily did her challenge and a few games...
My son absolutely loves doing his Maths Factor - he asks to do it every morning.
Had his school parents evening and if any child is stuck on anything he goes and explains the answer and how to get there. He's 6 years old.
My son scored 97 out of 100 in his maths test, proves to show that the maths factor really does work, 20 months ago his score would of been around 35.
I’m super proud how hard he has worked in the last 20 months. Well done!
Absolutely love Maths Factor! Helped my daughter become a maths whizz.
She learnt her times table by year 2 and has such confidence.
I was very glad when the platform was also available on ipads!


If your child is failing math, you may find yourself feeling helpless and wondering what you can do to help them. If so, use one or more of the following ways you can help when your child is struggling with their math class.
Ask the Teacher for Instructional Materials
One way you can help your child with their math homework is to ask their teacher for instructional materials. These could include explanations of the problems the class is working on and extra assignment worksheets.
Having instructional materials on hand can assist you in two ways. First, you have something to look at so you can review and relearn the material. Depending on the types of problems being taught and how long ago you were taught them, you may need a refresher.
Second, having additional worksheets to give your child at home gives them more practice. As they work through the problems, you can be there with them to explain each step and answer any questions they have.
Discover How Your Child Learns
Another thing you can do to help your child learn is to discover the method with which they learn best. Since everyone retains information differently, you may need to explain the math problems in a different way.
For example, your child may have trouble comprehending verbal explanations of the math problems. However, if you take out some of their toys, explain the math by showing them, and having them perform the same tasks, the combination of visual and hands-on learning may help them.
Hire a Math Tutor
Even if you use one or both methods above, you may still feel that you are not doing all you can or have enough time to help your child bring their math grade up. If so, you could hire a math tutor.
While you can still continue to work with your child at home, having a private tutor who has extensive experience with the math problems can give your child additional learning support. You can schedule sessions once a day or a couple times a week, depending on the amount of help you, the teacher, and the tutor feel your child requires.
Using a combination of the above methods can help you help your child who is failing their math class. If you are interested in hiring a math tutor, you may want to ask your school for a list of services or individuals who are available to regularly meet with and assist your child. For more information, contact a business such as Shandon Tutors.


Electrical test data (e.g. wafer sort data) often consists of dozens of correlated parameters and therefore it is no surprise that the application of multivariate statistics monitoring can significantly improve the effectiveness of the yield control concept.
The YieldWatchDog™ Multivariate Monitoring Module observes the correlation between multiple parameters and summarizes these correlations into a single chart.
What is Multivariate Statistics?
Monitoring of electrical test data for process control purposes is often done using simple univariate statistics. This means that specific or all test parameters are monitored independently one after the other.
Figure 1 is a scatterplot of two parameters and their development over time. Each data point represents the average values of the two parameters on a single wafer. From the diagram we can deduce that each data point is within the control limits for both parameters in the time series charts. Control limits are indicated by the blue and green lines. The multivariate control limit is shown as an ellipse and labelled “control limit T²”.
By performing multivariate statistics, we can see that there is a data point (encircled) in the ellipse that is clearly not behaving like the majority of wafers and therefore stands out. This data point would certainly set off alarm bells. The same data point is encircled in the two time series charts, however these data points do not stand out from the majority of data points in the respective charts and would otherwise go completely unnoticed.
The above example is a bi-variate statistics, meaning exactly two parameters are being considered. Multivariate statistics groups multiple parameters into one T² control limit. Of course, this cannot be illustrated in a two-dimensional plane, but only calculated mathematically in a multi-dimensional space. A standard control chart is known as a Hotelling T² chart (named after the famous statistician Harold Hotelling). It basically plots the distance of data points from the center of their correlation ellipse (in the two-dimensional case) or the multi-dimensional ellipsoid, also known as hyperellipsoid, in the multi-variate case. A major advantage of the T² chart is that monitoring a single chart gives an instant overview over a large number of parameters.
Principal Component Analysis (PCA) is a dimension reduction technique that is implemented to reduce the data into its basic components, stripping away any unnecessary parts. It reduces the data set immensely and saves computation time.
Why Multivariate Monitoring?
The Multivariate Monitoring Module allows you to get the maximum out of your data.
Although univariate statistics has the advantage of being easy to monitor, the analysis of the data is often misleading because the correlation between the parameters is ignored.
To explore beyond what is visible when looking at single parameters, we need multivariate statistical calculations to allow the correlation of multiple parameters thereby detecting behavior that is only visible when looking at a group of parameters. By deploying mechanisms such as PCA, extremely fast processing and calculation times can be assured. Only one chart needs to be monitored and there is a substantial reduction in the number of false alarms.
The diagram below shows an actual case how multivariate monitoring can save a lot of time when it comes to detecting problems.
About DR YIELD software & solutions GmbH
DR YIELD provides the leading-edge advanced analytics software YieldWatchDog for analysis and control of semiconductor manufacturing and test data. This enables semiconductor manufacturers as well as Fabless companies to improve important manufacturing dimensions such as equipment availability, throughput, operating costs and yield. Once installed YieldWatchDog allows you to get actionable insights into your data. For detailed information click here


Canadian Mathematical Bulletin
Published Quarterly | E-ISSN 1496-4287 | ISSN 0008-4395
Appearing quarterly, the Canadian Mathematical Bulletin (CMB) is a highly respected and strictly refereed journal published by the Canadian Mathematical Society (CMS) to disseminate the timely and significant mathematical research vital to the worldwide scientific community. This internationally renowned journal maintains a vigorous publication schedule and is the companion publication to the Canadian Journal of Mathematics, the other flagship publication of the CMS. Founded in 1958, the CMB is an independent peer-reviewed journal devoted to publishing original works of high standard.
Published quarterly by the Canadian Mathematical Society.
Canadian Mathematical Society Publications can be found online at camel.math.ca/CMS/Publications/.
Institutions interested in purchasing combined subscriptions to the Canadian Journal of Mathematics and Canadian Mathematical Bulletin at the following prices, please contact email@example.com:
Institutional Online (Canada) - $1085.00
Institutional Online (US & Foreign) - $1117.00
Institutional Combined - Print and Electronic (Canada) - $1278.00
Institutional Combined - Print and Electronic (US & Foreign) - $1318.00


MS Math Department Contact Information & Vocabulary Words
Sublette Middle School
Lyle Befort, 7th Grade - Math Teacher
Trevor Leverett, 8th Grade - Math Teacher
PO Box 460
Sublette, KS 67877
Click on the link of your choice below and then you can print out the document if you choose:
Download Adobe Reader (click on icon below) if you do not have it on your computer. You will need Adobe Reader to read the documents.
It is a free download.


BMI(Body Mass Index) is measurement of the body fat which is based on the persons height & weight that will applies to both men & women in between the ages of 18 - 65 years. BMI will be used to indicate if a person is overweight, obese, underweight or having normal weight.
A healthy BMI score will be in between 20 - 25. A score below 20 will indicates that person may be in underweight, a value above 25 will indicates that person may be suffering with overweight. You can calculate your body BMI by using BMI Calculator below, or by using the
BMI = ( Weight in Pounds / ( Height in inches x Height in inches ) ) x 703
Metric BMI Formula
BMI = ( Weight in Kilograms / ( Height in Meters x Height in Meters ) )


Breaking the norm of traditional Math workbooks for 1st Graders
SAP Learning Maths is a brilliant interactive workbook that makes math easy to understand for kids who are having learning difficulties in math.
It provides both teachers/parents/guardians and students a summary of learning objectives, formulae sheets for checking/recalling math formulae, work exercises for becoming familiar with various questions, non-routine questions for developing analytical skills and carefully worked out step-by-step solutions.
What's amazing about this courseware is the gradual presentation of lessons and the numerous practice tests it included that will facilitate better absorption of the mathematical lessons for young minds. The questions posted will also ensure that multiple concepts are understood.
Give your child the gift of easy learning for their math subject with Learning Maths now!
Powered by PageWerkz and works on iOS, Android, Windows and Mac with a single purchase.
This is a printable quiz that may be distributed to first grade students to educate them on CCSS.MATH.CONTENT.1.OA.C.5, a First Grade Common Core Standard on Mathematics that teaches about adding and subtracting within 20 by relating counting to addition and subtraction.
Students will be asked to fill the missing boxes with numbers that make up the equation. First grade students will need to count on, count back and subtract all or part of the illustrated objects of the equation to get the correct answers.


It’s amazing how a well laid out book can appeal to your aesthetics or turn you off completely. It’s always best to not judge a book by its cover, but you know what? We do it anyway.
Now these days almost everyone is either stuck on a computer or glued to an e-reader, but there’s just something very fulfilling about holding a proper book in your hands. A well designed book with an appealing page layout will make a great impression the first time that will carry over to the content. So this got me thinking… Can I create a page template that anyone can use for their own work.
Before I went about doing this, I thought I’d start with what’s been around already. The canons of page construction rely heavily on not just visual aesthetic, but also a mathematical aesthetic. The proportions are often inspired by or are directly based on the Golden Ratio for many old texts and there is a very strong use of geometric shapes to get predictable results on even different sizes of parchment or paper.
I started with this image of the canon by Van de Graaf also called the “secret canon”.
What struck me the most is at how simple lines from point to point build up to create a complex layout. I took apart the design to figure out in which order the lines were drawn and afterwards, I numbered them for my own reference. Now unlike the medieval writers, I have access to Photoshop, so finding the perfect center of a layout is fairly simple. The original page designers would have used a 90 degree angle ruler to draw perfectly vertical lines from point to point using the edge of the paper as reference.
Now we can translate this to an actual page we can use. Because 8.5 x 11 inches is the standard letter size in the U.S. although I have a feeling that legal size proportions (8.5 x 14 inches) would look even better if we’re doing a single page in portrait, but I went with letter size for now in landscape and translated these lines onto a blank image with a 300 dpi resolution.
Because I have the ability to find exact center later on thanks to technology, I saved the middle line for last and instead created a guide at the exact middle which was used as a reference for lines 3 and 4. By pressing the Shift key while drawing, you can create straight vertical or horizontal lines in Photoshop. I used this feature to create lines 5 and 7.
Drawing the corner to corner lines first make sense since the corners provide the perfect anchor. The center can be found in Photoshop if you’re using guides since they snap to the exact center of the image. Lines 3 and 4 can be drawn from the botton corners to the guide in the center. The “X” created by the crossing of lines 2 with 4 create a point on which to start line 5. By using the Shift key trick, we can draw a perfectly straight vertical line to the edge of the image. The point created by the intersection of the top edge and line 5 create an anchor to start line 6 which can then be drawn to the intersection of lines 1 and 3.
I also added extra lines on the bottom using the same points as reference because I want to include page numbers and a footer. I didn’t just want to copy the Van de Graaf analysis, but instead wanted to create something appealing for a book that would still look good. I started by extending the vertical lines of the text block on each side until they hit lines 1 and 2 so I can rule off of those. I can then use those to draw more lines horizontally that hit the additional guide lines as well as the other end of the text block vertical lines.
Now the content rectangle appears to have been elongated from the original, but I have plans for the extra space. First, I need to set the actual text region.
And when we momentarily hide the lines, we can see how well it will look.
Now that we have our text blocks, we can add some text to see how it will look. I’m going to use a triangle created at the bottom of the page as a reference point for the page numbers, while using the elongated rectangle from the main content as reference for a page footer/notes section. I used Century Old Style Standard as the text font and Calibri for the page numbers.
And let’s see what the final product looks like without the lines.
The great thing about using this method to find the perfect proportions is that it should be possible to do so with any paper size. You will still end up with a very nice looking layout for a book or single sheet print.


The NFL's rating system is a little bit more complicated that the one used by the NCAA.
In the NCAA system: assuming someone completes every one of his passes and each pass is a 99 yard touchdown pass 1261.6 is the maximum rate that a passer could have.
The NFL's system has arbitrary limits placed upon the values used to calculate their rate. In 1973 the NFL looked at the all of the available records of passing in the NFL and came up with an average in four categories: Completion %, Yards-per-Attempt, Touchdown %, and Interception %. After coming up with averages in each category they set limits on them that where roughly twice their value.
For completion percentage 77.5 % is the maximum allowed value in this equation. For yards-per-attempt 12.5 is the maximum allowed. For touchdown percentage 11.875 % is the maximum value allowed.
Unlike the NCAA rating system, the NFL's does not allow any of the four categories to fall below 0. So 0 is the maximum allowed value for interception %. Conversely 9.5 % is the minimum allowed value for that category as well.
To figure the NFL rating a number is calculated for each category with 2.375 the maximum allowed value for each.
To figure the value for completion percentage, if the value of completions divided by attempts is greater than 77.5 % then 2.375 is the value. If the percentage is not, then the value is calculated by taking the completion percentage - 30 and then dividing by 20.
To figure the value for yards-per-attempt, if the value of yards divided by attempts is greater than 12.5 then 2.375 is the value. If the average is not, then the value is calculated by taking the average - 3 and then dividing by 4.
To figure the value for touchdown percentage, if the value of touchdowns divided by attempts is greater than 11.875 % then 2.375 is the value. If the percentage is not, then the value is calculated by taking the touchdown percentage and then dividing by 5.
To figure the value for interception percentage (keeping in mind that interceptions are BAD and have to be SUBTRACTED), if the value of interceptions divided by attempts is greater than 9.5 % then 2.375 is the value. If the percentage is not, then the value is calculated by taking the interception percentage and then dividing by 4.
Once each of these values have been calculated you add them together, and then multiply 16.667 by that number.
In a nutshell (keeping in mind the maximum and minimum values) the formula is as follows:
(A=Attempts, C=Completions, Y=Yards, T=Touchdowns, I=Interceptions)
[((((C/A)*100)-30)/20) + (((Y/A)-3)/4) + (((T/A)*100)/5) + ((9.5-((I/A)*100))/4)] * 16.667
In other words if Billy Bob, the star quaterback of the hometown Thrashers, throws 20 passes and completes 12 of them (12/20 = .6 * 100 = 60 - 30 = 30 / 20 = 1.5) for 160 yards (160/20 = 8 - 3 = 5 / 4 = 1.25) with 2 touchdowns (2/20 = .1 * 100 = 10 / 5 = 2.0) and 1 interception (-1/20 = -.05 * 100 = -5 + 9.5 = 4.5 / 4 = 1.125) 97.92 (1.5 + 1.25 + 2.0 + 1.125 * 16.667) would be Billy Bob's rating.
Note: The NFL takes the total of the individual values (3.625) and then multiplies it by 100 and then divides that by 6. In the formula above the 16.667 is derived from dividing the 100 by 6. This difference will create a small discrepancy of a few tens and hundredths of a point in the final number.


Advantages of Online Calculators.
Mathematical calculations are helped by a small device known as a calculator. Someone’s problem is reduced by the calculators that are of benefit to the human beings. The major role played by calculators helps the individuals and the society as a whole. Work is made easy with the help of online calculators that are useful to the people and are all over the country.
Online calculators are easy to find and simple to use them. The benefit that the online calculator has makes people prefer it. Online calculators have the compound interest formula making people do not worry.
People get the accurate result from the online calculators that can do any sum. The time of people is not wasted while doing the calculations because the online calculator helps. One needs to be keen while calculating compound interest because of the different numbers but the online calculators help people while doing the sums.
At ones comfort the online calculators can be reached from anywhere. The different designs and shapes of the online calculators make people love the service that it provides.
The online calculators are attractive and are labeled well-making people have easy access. Work is simplified by the online calculators that calculate the currency sum.
People get what they want in a fast way because the online calculators are updated. Many people can use the online calculator at the same time without having any effect. People use the online calculators because they may be consulted at any time and their accuracy that has a lot of formulas that what research shows. The different places and times that online calculators are used help people meet their goals and objectives.
The pressure of hard sums that makes people have stressed is reduced by the online calculators. People do not worry because the online calculators are affordable.
Online calculators are trusted by people and have no limits because one can use them as many times as he or she wants. People can access the online calculators any place and at any time. The fast and accurate online calculators help people enjoy them. The online calculators have the formulas that one may need making people do not worry.
The online calculators have no negative effect on the people making them benefit. The interest that people want is calculated by the compound interest calculator found in the online calculators.
Online calculators have the compounding interest calculator that helps in solving the issues that people may have. The compound growth calculator has big numbers that can confuse someone, but the online calculator helps in getting the accurate results.


The Vantage Point Bees Algorithm
MetadataShow full item record
CitationZEYBEK, Sultan & Ebubekir KOÇ. "The Vantage Point Bees Algorithm". 7th International Joint Conference on Computational Intelligence (IJCCI), 1 (2015): 340-345.
In this paper, an implementation of vantage point local search procedure for the Bees Algorithm (BA) in combinatorial domains is presented. In its basic version, the BA employs a local search combined with random search for both continuous and combinatorial domains. In this paper, a more robust local searching strategy namely, vantage point procedure is exploited along with random search to deal with complex combinatorial problems. This paper proposes a hybridization technique which involves the Bee Algorithm (BA) and a local search technique based on Vantage Point Tree (VPTs) construction. Following a description of the Vantage Point Bees Algorithm (VPBA), the paper presents the results obtained for several local search strategies for BA, demonstrating efficiency and robustness of the VPBA.
Source7th International Joint Conference on Computational Intelligence (IJCCI)


e with a Hint of Pi
About the piece:
While trying to look for ideas for a song, I found that the submission date was near March 14th , which some people celebrate as pi day. I have heard and seen many videos that transcribe the digits of pi onto a piano. The result is not too surprising as the notes tend to be in a random stream. Instead of having the digits correspond to a major scale, which has seven distinct notes compared to ten digits, I used two octaves of pentatonic scales which can create more consonances, even if the notes are random. I used Euler’s constant, e, to be the main notes in the piece. The high pitched bells play the digits of e while I slipped in some digits of pi, which is played by a more electronic voice 12 seconds in.


Math 6 - Lynne Lindsay
Value of the Month
Welcome May Flowers and Falcon Fun!
M.S. Longwood University
Lynne Lindsay teaches sixth grade math at Fuqua Middle School and serves as the Coordinator of Curriculum and Instruction PK-12. She joined Fuqua's staff in 2015 as a middle school math and history teacher. Lynne received her bachelors and masters degrees from Longwood University. She and her husband, Eric, have two children, Morgan (19) and Andrew (16).
Monday: Area and Perimeter HW 1
Tuesday: Complete Area Matrix
Wednesday: Area of Triangles and Quads
Quiz on Area of Triangles and Quads Thursday!
Monday, May 3: LS/MS BOGO Bookfair Begins!
Wednesday, May 5: JV Baseball and Softball @ Southampton Academy - Dismissal at 1:15
Friday, May 7: First Friday: Dress Down Day!
Upcoming Dates to Remember
May 10 - 15: Homecoming Week!
Monday: PJ Day
Tuesday: Country vs Country Club
Wednesday: Wacky Wednesday
Thursday: Twin Day
Friday: Falcon Frenzy
May 20: Grads Retracing Steps - LS/MS Campus 10am
May 21: Noon Dismissal - Dress Down Day
May 27: 6th Grade Pool Party - 1/2 Day - Dress Down
May 28: MS Class Competition - 1/2 Day - Last Day of School for the Year!!