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Section MO Matrix Operations
In this section we will back up and start simple. We begin with a definition of a totally general set of matrices, and see where that takes us.
Subsection MEASM Matrix Equality, Addition, Scalar Multiplication
Definition VSM Vector Space of $m\times n$ Matrices
The vector space $M_{mn}$ is the set of all $m\times n$ matrices with entries from the set of complex numbers.
Just as we made, and used, a careful definition of equality for column vectors, so too, we have precise definitions for matrices.
Definition ME Matrix Equality
The $m\times n$ matrices $A$ and $B$ are equal, written $A=B$ provided $\matrixentry{A}{ij}=\matrixentry{B}{ij}$ for all $1\leq i\leq m$, $1\leq j\leq n$.
So equality of matrices translates to the equality of complex numbers, on an entry-by-entry basis. Notice that we now have yet another definition that uses the symbol “=” for shorthand. Whenever a
theorem has a conclusion saying two matrices are equal (think about your objects), we will consider appealing to this definition as a way of formulating the top-level structure of the proof.
We will now define two operations on the set $M_{mn}$. Again, we will overload a symbol (`+') and a convention (juxtaposition for scalar multiplication).
Definition MA Matrix Addition
Given the $m\times n$ matrices $A$ and $B$, define the sum of $A$ and $B$ as an $m\times n$ matrix, written $A+B$, according to \begin{align*} \matrixentry{A+B}{ij}&=\matrixentry{A}{ij}+\matrixentry
{B}{ij} &&1\leq i\leq m,\,1\leq j\leq n \end{align*}
So matrix addition takes two matrices of the same size and combines them (in a natural way!) to create a new matrix of the same size. Perhaps this is the “obvious” thing to do, but it does not
relieve us from the obligation to state it carefully.
Our second operation takes two objects of different types, specifically a number and a matrix, and combines them to create another matrix. As with vectors, in this context we call a number a scalar
in order to emphasize that it is not a matrix.
Definition MSM Matrix Scalar Multiplication
Given the $m\times n$ matrix $A$ and the scalar $\alpha\in\complex{\null}$, the scalar multiple of $A$ is an $m\times n$ matrix, written $\alpha A$ and defined according to \begin{align*} \
matrixentry{\alpha A}{ij}&=\alpha\matrixentry{A}{ij}&& \quad 1\leq i\leq m,\,1\leq j\leq n \end{align*}
Notice again that we have yet another kind of multiplication, and it is again written putting two symbols side-by-side. Computationally, scalar matrix multiplication is very easy.
Subsection VSP Vector Space Properties
With definitions of matrix addition and scalar multiplication we can now state, and prove, several properties of each operation, and some properties that involve their interplay. We now collect ten
of them here for later reference.
Theorem VSPM Vector Space Properties of Matrices
Suppose that $M_{mn}$ is the set of all $m\times n$ matrices (Definition VSM) with addition and scalar multiplication as defined in Definition MA and Definition MSM. Then
• ACM Additive Closure, Matrices
If $A,\,B\in M_{mn}$, then $A+B\in M_{mn}$.
• SCM Scalar Closure, Matrices
If $\alpha\in\complex{\null}$ and $A\in M_{mn}$, then $\alpha A\in M_{mn}$.
• CM Commutativity, Matrices
If $A,\,B\in M_{mn}$, then $A+B=B+A$.
• AAM Additive Associativity, Matrices
If $A,\,B,\,C\in M_{mn}$, then $A+\left(B+C\right)=\left(A+B\right)+C$.
• ZM Zero Matrix, Matrices
There is a matrix, $\zeromatrix$, called the zero matrix, such that $A+\zeromatrix=A$ for all $A\in M_{mn}$.
• AIM Additive Inverses, Matrices
If $A\in M_{mn}$, then there exists a matrix $-A\in M_{mn}$ so that $A+(-A)=\zeromatrix$.
• SMAM Scalar Multiplication Associativity, Matrices
If $\alpha,\,\beta\in\complex{\null}$ and $A\in M_{mn}$, then $\alpha(\beta A)=(\alpha\beta)A$.
• DMAM Distributivity across Matrix Addition, Matrices
If $\alpha\in\complex{\null}$ and $A,\,B\in M_{mn}$, then $\alpha(A+B)=\alpha A+\alpha B$.
• DSAM Distributivity across Scalar Addition, Matrices
If $\alpha,\,\beta\in\complex{\null}$ and $A\in M_{mn}$, then $(\alpha+\beta)A=\alpha A+\beta A$.
• OM One, Matrices
If $A\in M_{mn}$, then $1A=A$.
For now, note the similarities between Theorem VSPM about matrices and Theorem VSPCV about vectors.
The zero matrix described in this theorem, $\zeromatrix$, is what you would expect — a matrix full of zeros.
Definition ZM Zero Matrix
The $m\times n$ zero matrix is written as $\zeromatrix=\zeromatrix_{m\times n}$ and defined by $\matrixentry{\zeromatrix}{ij}=0$, for all $1\leq i\leq m$, $1\leq j\leq n$.
Subsection TSM Transposes and Symmetric Matrices
We describe one more common operation we can perform on matrices. Informally, to transpose a matrix is to build a new matrix by swapping its rows and columns.
Definition TM Transpose of a Matrix
Given an $m\times n$ matrix $A$, its transpose is the $n\times m$ matrix $\transpose{A}$ given by \begin{equation*} \matrixentry{\transpose{A}}{ij}=\matrixentry{A}{ji},\quad 1\leq i\leq n,\,1\leq j\
leq m. \end{equation*}
It will sometimes happen that a matrix is equal to its transpose. In this case, we will call a matrix symmetric. These matrices occur naturally in certain situations, and also have some nice
properties, so it is worth stating the definition carefully. Informally a matrix is symmetric if we can “flip” it about the main diagonal (upper-left corner, running down to the lower-right corner)
and have it look unchanged.
Definition SYM Symmetric Matrix
The matrix $A$ is symmetric if $A=\transpose{A}$.
You might have noticed that Definition SYM did not specify the size of the matrix $A$, as has been our custom. That is because it was not necessary. An alternative would have been to state the
definition just for square matrices, but this is the substance of the next proof.
Before reading the next proof, we want to offer you some advice about how to become more proficient at constructing proofs. Perhaps you can apply this advice to the next theorem. Have a peek at Proof
Technique P now.
Theorem SMS Symmetric Matrices are Square
Suppose that $A$ is a symmetric matrix. Then $A$ is square.
We finish this section with three easy theorems, but they illustrate the interplay of our three new operations, our new notation, and the techniques used to prove matrix equalities.
Theorem TMA Transpose and Matrix Addition
Suppose that $A$ and $B$ are $m\times n$ matrices. Then $\transpose{(A+B)}=\transpose{A}+\transpose{B}$.
Theorem TMSM Transpose and Matrix Scalar Multiplication
Suppose that $\alpha\in\complex{\null}$ and $A$ is an $m\times n$ matrix. Then $\transpose{(\alpha A)}=\alpha\transpose{A}$.
Theorem TT Transpose of a Transpose
Suppose that $A$ is an $m\times n$ matrix. Then $\transpose{\left(\transpose{A}\right)}=A$.
Subsection MCC Matrices and Complex Conjugation
As we did with vectors (Definition CCCV), we can define what it means to take the conjugate of a matrix.
Definition CCM Complex Conjugate of a Matrix
Suppose $A$ is an $m\times n$ matrix. Then the conjugate of $A$, written $\conjugate{A}$ is an $m\times n$ matrix defined by \begin{equation*} \matrixentry{\conjugate{A}}{ij}=\conjugate{\matrixentry
{A}{ij}} \end{equation*}
The interplay between the conjugate of a matrix and the two operations on matrices is what you might expect.
Theorem CRMA Conjugation Respects Matrix Addition
Suppose that $A$ and $B$ are $m\times n$ matrices. Then $\conjugate{A+B}=\conjugate{A}+\conjugate{B}$.
Theorem CRMSM Conjugation Respects Matrix Scalar Multiplication
Suppose that $\alpha\in\complex{\null}$ and $A$ is an $m\times n$ matrix. Then $\conjugate{\alpha A}=\conjugate{\alpha}\conjugate{A}$.
Theorem CCM Conjugate of the Conjugate of a Matrix
Suppose that $A$ is an $m\times n$ matrix. Then $\conjugate{\left(\conjugate{A}\right)}=A$.
Finally, we will need the following result about matrix conjugation and transposes later.
Theorem MCT Matrix Conjugation and Transposes
Suppose that $A$ is an $m\times n$ matrix. Then $\conjugate{\left(\transpose{A}\right)}=\transpose{\left(\conjugate{A}\right)}$.
Subsection AM Adjoint of a Matrix
The combination of transposing and conjugating a matrix will be important in subsequent sections, such as Subsection MINM.UM and Section OD. We make a key definition here and prove some basic results
in the same spirit as those above.
Definition A Adjoint
If $A$ is a matrix, then its adjoint is $\adjoint{A}=\transpose{\left(\conjugate{A}\right)}$.
You will see the adjoint written elsewhere variously as $A^H$, $A^\ast$ or $A^\dagger$. Notice that Theorem MCT says it does not really matter if we conjugate and then transpose, or transpose and
then conjugate.
Theorem AMA Adjoint and Matrix Addition
Suppose $A$ and $B$ are matrices of the same size. Then $\adjoint{\left(A+B\right)}=\adjoint{A}+\adjoint{B}$.
Theorem AMSM Adjoint and Matrix Scalar Multiplication
Suppose $\alpha\in\complexes$ is a scalar and $A$ is a matrix. Then $\adjoint{\left(\alpha A\right)}=\conjugate{\alpha}\adjoint{A}$.
Theorem AA Adjoint of an Adjoint
Suppose that $A$ is a matrix. Then $\adjoint{\left(\adjoint{A}\right)}=A$.
Take note of how the theorems in this section, while simple, build on earlier theorems and definitions and never descend to the level of entry-by-entry proofs based on Definition ME. In other words,
the equal signs that appear in the previous proofs are equalities of matrices, not scalars (which is the opposite of a proof like that of Theorem TMA).
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The AiEdge+: Model Compression Techniques
Reducing the cost of Machine Learning
Today we dive into the subject of Model Compression. For widespread adoption and further development of generative ML, we first need to make those models more manageable to deploy and fine-tune. The
cost associated with those large models makes model compression one of the most critical research areas in Machine Learning today. We cover:
• Model Compression Methods
• Knowledge distillation
• Learn more about Model Compression Methods
Model Compression Methods
Not too long ago, the biggest Machine Learning models most people would deal with, merely reached a few GB in memory size. Now, every new generative model coming out is between 100B and 1T
parameters! To get a sense of the scale, one float parameter that's 32 bits or 4 bytes, so those new models scale between 400 GB to 4 TB in memory, each running on expensive hardware. Because of the
massive scale increase, there has been quite a bit of research to reduce the model size while keeping performance up. There are 5 main techniques to compress the model size.
• Model pruning is about removing unimportant weights from the network. The game is to understand what "important" means in that context. A typical approach is to measure the impact to the loss
function of each weight. This can be done easily by looking at the gradients that approximate the loss variation when varying the weights. In 1989, Yann LeCun established a classic method by
computing the second order derivative for improved results. Another way to do it is to use L1 or L2 regularization and get rid of the low magnitude weights. When removing weights across the
network following those methods, we reduce model size but the time complexity remains more or less the same due to the way matrix multiplications are parallelized. Removing whole neurons, layers
or filters is called "structured pruning" and is more efficient when it comes to inference speed.
• Model quantization is about decreasing parameter precision, typically by moving from float (32 bits) to integer (8 bits). That's 4X model compression. Quantizing parameters tends to cause the
model to deviate from its convergence point so it is typical to fine-tune it with additional training data to keep model performance high. We call this "Quantization aware training". When we
avoid this last step, it is called "Post training quantization", and additional heuristic modifications to the weights can be performed to help performance.
• Low-rank decomposition comes from the fact that neural network weight matrices can be approximated by products of low-dimension matrices. A N x N matrix can be approximately decomposed into a
product of 2 N x 1 matrices. That's an O(N^2) -> O(N) space complexity gain!
• Knowledge distillation is about transferring knowledge from one model to another. Typically from a large model to a smaller one. When the student model learns to produce similar output responses,
that is response-based distillation. When the student model learns to reproduce similar intermediate layers, it is called feature-based distillation. When the student model learns to reproduce
the interaction between layers, it is called relation-based distillation.
• Lightweight model design is about using knowledge from empirical results to design more efficient architectures. That is probably one of the most used methods in LLM research.
I would advise to read the following article to learn more about those methods: “Model Compression for Deep Neural Networks: A Survey“.
Knowledge distillation
In this new Machine Learning era dominated by LLMs, knowledge Distillation is going to be at the forefront of LLMOps. For widespread adoption and further development of generative ML, we first need
to make those models more manageable to deploy and fine-tune.
Keep reading with a 7-day free trial
Subscribe to The AiEdge Newsletter to keep reading this post and get 7 days of free access to the full post archives.
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SPbSU ITMO contest. Petrozavodsk training camp. Winter 2008
There is a game called "Yoga". On the game board there are 32 checkers, standing as shown on the picture. Each turn a checker jumps over another one and lands on a free cell — almost like in the
checker game, but vertically or horizontally, not diagonally. The checker which was jumped over is removed from the board.
We will look at endspiel, the last part of the game. Imagine that there is only one checker left. Given its location, find a possible sequence of turns that leads to this endspiel.
Let us introduce a coordinate system similar to the one that is used in the game of chess. The columns are numbered by Latin letters from A to G, the rows are numbered from 1 to 7. For example, a
cell with coordinates "D4" is the central cell.
The first line contains the coordinates of the last checker in the notation described above.
If it is possible to reach the specified endspiel, output the sequence of turns leading to it. Each turn should be printed in the following format: <start cell>–<finish cell>, where <start cell> is
the coordinates of a cell where the moving checker is located before the turn, and <finish cell> is the coordinates of its destination cell. There will always be 31 turns.
If it is impossible to find the necessary sequence, output the word "Impossible".
input output
D4 ...
D3 Impossible
Problem Source: SPbSU ITMO contest. Petrozavodsk training camp. Winter 2008.
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Fixed Point - (Algebraic Combinatorics) - Vocab, Definition, Explanations | Fiveable
Fixed Point
from class:
Algebraic Combinatorics
A fixed point in the context of permutations and group theory is an element that remains unchanged when a specific permutation is applied to it. This concept is crucial for understanding the
structure of permutations, especially in terms of cycle notation, where fixed points can simplify the representation of a permutation's action on a set.
congrats on reading the definition of Fixed Point. now let's actually learn it.
5 Must Know Facts For Your Next Test
1. In cycle notation, fixed points are typically omitted since they do not change positions during the permutation.
2. For a permutation with n elements, the number of fixed points can influence the overall structure and properties of the permutation.
3. In group theory, the existence of fixed points can help in determining whether a permutation is an even or odd permutation based on its cycle structure.
4. A fixed point can also be seen as a trivial cycle in cycle notation, represented as (i), where i is the element that stays in its original position.
5. Understanding fixed points can aid in analyzing the stability and behavior of various mathematical systems, including combinatorial structures and functions.
Review Questions
• How do fixed points affect the representation of permutations in cycle notation?
□ Fixed points are elements that remain unchanged under a permutation, and they are generally not shown in cycle notation to streamline the representation. This means that when writing out a
permutation using cycles, if an element does not move, it is omitted from the notation. Consequently, this can lead to simpler representations of complex permutations and allows for easier
analysis of their structures.
• Discuss the role of fixed points in determining whether a permutation is even or odd.
□ The presence of fixed points can impact whether a permutation is classified as even or odd. In general, a permutation's parity is determined by its cycle structure, where even permutations
can be expressed as a product of an even number of transpositions and odd permutations with an odd number. Fixed points do not contribute to this count directly, but they can influence how
cycles are formed and combined, ultimately affecting the overall classification.
• Evaluate how understanding fixed points contributes to analyzing group actions and their corresponding conjugacy classes.
□ Recognizing fixed points provides insight into group actions and helps to understand the relationships between elements within conjugacy classes. When studying how groups act on sets, fixed
points highlight invariant elements under those actions, which can reveal important properties about the structure of the group. Additionally, understanding which elements remain fixed can
assist in classifying elements into conjugacy classes based on their symmetrical behavior and how they interact under group operations.
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Shell Sort Algorithm
Introduction to Shell Sort
Shell sort is an in-place comparison-based sorting algorithm. It generalizes insertion sort by allowing the exchange of items that are far apart. The algorithm starts by sorting pairs of elements far
apart from each other, then progressively reducing the gap between elements to be compared. The efficiency of Shell sort depends on the gap sequence used, with time complexity varying from O(n^2) to
O(n log n) based on the sequence.
Step-by-Step Process
Consider an array with the following elements:
[12, 34, 54, 2, 3]
To sort this array using Shell sort, follow these steps:
1. Choose a Gap Sequence: Choose a gap sequence to determine the intervals for comparing elements. A common sequence is to start with n/2 and halve the gap until it reaches 1.
2. Sort Elements with Current Gap: For each gap, perform a gapped insertion sort on the array. Compare and swap elements that are 'gap' distance apart.
3. Reduce Gap and Repeat: Reduce the gap and repeat the process until the gap is 1, at which point the array is fully sorted using a standard insertion sort.
Let's apply these steps to the example array:
• Initial array: [12, 34, 54, 2, 3]
• Gap = 2: Compare and swap elements that are 2 positions apart
• Array after gap 2: [12, 3, 54, 2, 34]
• Gap = 1: Perform a standard insertion sort
• Final sorted array: [2, 3, 12, 34, 54]
Pseudo Code for Shell Sort
Below is the pseudo code for the Shell sort algorithm:
function shellSort(array)
n = length(array)
gap = n // 2
while gap > 0 do
for i = gap to n - 1 do
temp = array[i]
j = i
while j >= gap and array[j - gap] > temp do
array[j] = array[j - gap]
j = j - gap
end while
array[j] = temp
end for
gap = gap // 2
end while
end function
• Initialize: The shellSort function starts by determining the initial gap, usually set to half the length of the array.
• Gapped Insertion Sort: For each gap, the algorithm performs a gapped insertion sort, where elements that are 'gap' distance apart are compared and swapped if necessary.
• Reduce Gap: The gap is reduced by half after each iteration, and the process is repeated until the gap is 1, at which point a final insertion sort is performed to fully sort the array.
Python Program to Implement Shell Sort
def shell_sort(arr):
n = len(arr)
gap = n // 2
while gap > 0:
for i in range(gap, n):
temp = arr[i]
j = i
while j >= gap and arr[j - gap] > temp:
arr[j] = arr[j - gap]
j -= 1
arr[j] = temp
gap //= 2
return arr
# Example usage:
array = [12, 34, 54, 2, 3]
print("Original array:", array)
sorted_array = shell_sort(array)
print("Sorted array:", sorted_array) # Output: [2, 3, 12, 34, 54]
This Python program defines a function to perform Shell sort on an array. The function initializes the gap, performs a gapped insertion sort for each gap, and reduces the gap until the array is fully
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Recognize and generate simple equivalent fractions. – Seesaw Activity by KAVITA AUL
Recognize and generate simple equivalent fractions.
Grades: 3rd Grade, 5th Grade, 4th Grade
Subjects: Math
Student Instructions
Essential Skills Problem Solving - complete mathematical puzzles Recognize and generate simple equivalent fractions. Explain why the fractions are equivalent by using a visual fraction model.
Decompose a fraction into a sum of fractions with the same denominator. Compare two fractions with different numerators and different denominators. Recognize that comparisons are valid only when
fractions refer to the same whole. Make groups of equal value. Each group must contain a fraction, a decimal and a percent.
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Homework 4 – Banker’s algorithm Implement the Banker's algorithm for deadlock avoidance, with a given set of N processes (N<10, processes are P1, P - CUSTOM WRITING
Homework 4 – Banker’s algorithm Implement the Banker’s algorithm for deadlock avoidance, with a given set of N processes (N<10, processes are P1, P
Homework 4 – Banker’s algorithm
Implement the Banker’s algorithm for deadlock avoidance, with a given set of N processes
(N<10, processes are P1, P2, …, PN) and M resource types (M<10, resources are R1, R2, …, RM). Use Java or C/C++ for the implementation, with a simple interface, where the user only supplies the name
of the input file (text file, say “input.txt”). The program reads all the necessary input data from that file. You are free to choose the format of the input file, just make sure it contains all the
necessary data. The input data and the result of the algorithm must be displayed on the screen.
The pseudo code for the Greedy version for the Banker’s algorithm can be found in this module’s Commentary. We know that this algorithm only finds ONE solution (safe sequence of processes) assuming
there is one; otherwise reports there is no solution.
You must adjust this algorithm to find exactly TWO solutions instead of just one (assuming of course there are at least two solutions). If the algorithm only has one solution this must be reported as
such and the single solution must be displayed.
To resume, there are only three possible scenarios, for a correct input file:
1. There is no solution (no safe sequence). The program must report there is no solution.
2. There is EXACTLY ONE solution (safe sequence). The program must report there is exactly one solution. The solution must be displayed on screen.
3. There are TWO OR MORE solutions (safe sequences). The program must find EXACTLY TWO solutions. The solutions must be displayed on screen. If there are more than two solutions, it doesn’t
matter which ones are found, as long as they are exactly two.
Note 1: the solution must be based on the Greedy version for the Banker’s algorithm; in particular, using an approach for finding ALL solutions and then keeping and reporting just two of them is NOT
Note 2: The input file may be incorrect, for various reasons. This must be checked and reported by your program before any attempt of finding a solution.
– The input files should start with N and M, then the necessary matrices
– Work on your own computer or online using https://replit.com
1. The source code for your program, stored in a text file
2. One incorrect input file, “input0.txt” and one or more screenshots showing the results of your program for this incorrect input.
3. Three input files, “input1.txt” (N=4, M=3), “input2.txt” (N=4, M=3), “input3.txt” (N=4, M=3). Please choose the input data in such a way that the first input has no solution, the second input
has exactly one solution and the third input has at least two solutions.
4. Three screenshots showing the final results of your program’s execution for these three input files.
5. Store the screenshots in graphic files of your preferred format(“.png” or ”.gif” or “.pdf”, etc.).
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10 Times Table Chart & Printable PDF – Times Table Club
10 Times Table Chart
10 Times Table is the multiplication table of the number 10 provided here for students, parents and teachers. These times tables are helpful in solving math questions quickly and easily.
How to read the 10 times table chart?
One times ten is 10, two times ten is 20, three times ten is 30, ect.
How to memorise the 10 times tables chart orally?
Write the 10 times tables on a sheet of paper and read them aload repeatedly.
What is an example of a 10 times table chart maths question?
Maths Question: 10 customers come in to a shop every day. How many customers will come into the shop in 1 week (7 days)?
Solution: 10 customers come in to a shop per day. Therefore, using the 10 times table, the total number of customers in 7 days is: 10 × 7 = 70 customers.
│Tables 2 to 20 │Multiplication Chart │
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Uniqueness of tetration. Small lemma.
I try to simplify the proof of uniqueness of holomorphic tetration by bo198214.
Let \( u \in \mathbb{R} \) and \( v \in \mathbb{R} \) and \( v>0 \)
Let \( \mathcal{S}=\{z\in \mathbb{C} : u<\Re(z)<u+1+v} \)
Let \( \alpha \) be entire 1-periodic function.
Let \( \mathcal{D}=\alpha(\mathcal{S}) \).
According to the Picard’s Little Theorem [1],
either \( \alpha \) is trivial function (constant),
or \( \mathcal{D}=\mathbb{C} \) or \( \exists c \in \mathbb{C} \) such that \( \mathbb{C}=\mathcal{D} \cup c \).
Let \( \mu(z)=z-\alpha(z) \forall z \in \mathbb{C} \).
Let \( \mathcal{E}=\mu(\mathcal{S}) \)
Assume, \( \alpha \) is not constant.
How to prove that at least one of these two statements is true: \( -2\in \mathcal{E} \), \( -3\in \mathcal{E} \)
[1] Weisstein, Eric W. ”Picard’s Little Theorem.” From MathWorld–A Wolfram Web Resource.
11/04/2008, 12:00 PM (This post was last modified: 11/05/2008, 11:35 AM by bo198214.)
But if you had this lemma, how would you continue with the uniqueness of tetration?
11/05/2008, 02:56 AM
bo198214 Wrote:... if you had this lemma, how would you continue with the uniqueness of tetration?
Did not you found the tail01.tex, or do you just want to have it posted on the forum?
I go to copypast it here; I'll convert it from latex and copyedit a bit.
11/05/2008, 09:07 AM (This post was last modified: 11/05/2008, 11:35 AM by bo198214.)
Kouznetsov Wrote:
bo198214 Wrote:... if you had this lemma, how would you continue with the uniqueness of tetration?
Did not you found the tail01.tex, or do you just want to have it posted on the forum?
I go to copypast it here; I'll convert it from latex and copyedit a bit.
Dmitrii, a forum is not for private communaction, either post it that everybody has a chance to understand or dont post at all.
PS: I just saw that my previous post was nonsense, I changed it accordingly.
11/06/2008, 03:19 PM (This post was last modified: 11/10/2008, 09:47 AM by Kouznetsov.)
Lemma is not yet ready, but Bo asked me about pics of slightly modified tetration.
I consider the simplest possible modification of tetration
\( \mathrm{tem}(z)=\mathrm{tet}(J(z)) \)
where \( J(z)=z+a\cdot\sin(2 \pi z) \)
Such modified tetration satisfies the tetration equaitons,
\( \mathrm{tem}(z+1)=\exp(\mathrm{tem}(z)) \),
\( \mathrm{tem}(0)=1 \)
For \( a=10^{-9} \), the function \( J \) is plotted in the complex plane
It is almost identical function, but the defiation is seen, if the imaginary part becomes of larget than 3.
Now, the plot of modified tetration:
The grid shown occupies the range [-10,10], [-4,4] with step unity.
Levels of integer values of \( \Re(\mathrm{tem} \) and
Levels of integer values of\( \Im(\mathrm{tem} \) are plotted.
Below I show the zoom-in of the part of the previoous figure:
There are cutlines there. The structure in the upper part folloes the topology of tetration, but it is reduced in size and strongly deformed.
For highest harmonics like \( \sin(4\pi z) \) or \( 1\!-\!\cos(4\pi z) \), added to the argument of tetration in definition of mofified tetratio "tem", the
structures of cuts is smaller, denser and approach closer to the real axis.
If some function \( F \) satisfies equations
\( \mathrm{tem}(0)=1 \)
\( \mathrm{tem}(z\!+\!1)=\exp(\mathrm{tem}(z)) \forall z : |\Re(z)|<1, |\Im(z)|<4 \),
and for some \( x\in \mathbb{R} : -1\!<\!x\!<\! 0 \), the function differ from tetration for at least \( 10^{−9} \), id est,
\( |Fxz)\!-\!\mathrm{tet}(x)| > 10^{-9} \).
then function \( F \) is not holomorphic in
\( \{z\in \mathbb{C}: |\Re(z)|<1, |\Im(z)|<4 \} \)
In such a way, any deformation of tetration (even small) breaks its continuity.
See also
11/11/2008, 08:34 AM (This post was last modified: 11/11/2008, 08:36 AM by Kouznetsov.)
bo198214 Wrote:But if you had this lemma, how would you continue with the uniqueness of tetration?
First, I repeat my Lemma: Let
\( S = \{ z \in \mathbb{C} {\text \bf ~:~} |\Re(z)| {\!\le\!} 1 \} \)
\( M = \{ n \in \mathbb{N} {\text \bf ~:~} n {\!\le\!} -2 \} \)
\( h \) is entire 1-periodic funcrion
\( h\big(z^{*}\big) = h(z)^{*} ~ \forall~ z ~\in \mathbb{C} \)
\( h(0) = 0 \)
\( h(x) \ne 0 \) for some \( x \in \mathbb{R} \)
\( J(z) = z+h(z) ~~~ \forall~ z ~\in \mathbb{C} \)
Then \( \exists t \in S {\text \bf ~:~} J(t)\in M \)
(end of Lemma 1).
This lemma does not requre any knowledge about properties of tetration.
Now, how do I apply it:
Assume some fixed base \( b\!>\!0 \) and let \( F(z)=\mathrm{tet}_{b}(z) \) within the range of holomorphism of tetration, id est, in the complex plane except some set of measure zero.
This set includes one line \( z {\!\le\!}-2 \), and, at \( 1 \!<\! b \!<\! \exp(1/\mathrm{e}) \),
additional horizontal cut lines that correspond to the periodicity of tetration.
Assume there exist some function \( G(z) \) which is also holomorphic within some region \( s \),
which includes at least some vicinity of the segment
\( |\Re(z)| \! {\!\le\!} \! 1 \).
Assume also, that function \( f=G \) is holomorphic at least in \( \{z\in \mathbb{C} {\text \bf ~:~ \Re(z)>-2 \} \) and
\( f(z+1)=\exp\big( f(z) \big) \)
\( f(0)=1 \)
\( f\big(z^{*}\big)=f(z)^{*} \)
\( f^{\prime}(x) > 0 \forall x> -2 \)
and \( G \) is not tetration \( F \). (Tetration \( F \) satisfies the equations above)
Then, there exist function \( J \) such that in vicinity of the segment
\( [-1,1] \) function \( G \) can be expressed as follows:
\( G=F\big(J(z)\big) \)
and, in some vicinity of the same segment,
\( J(z)=F^{-1}\big( G(z) \big) \)
We need this expression only in vicinity of the segment \( [-1,1] \); therefore, we have no need to specify, which of \( G^{-1} \) we mean.
There exist only one holomorphic extension of a function in a domain of trivial topology.
Therefore, we can extend the function \( J \) outside of the range of definition.
Consider function
\( h(z)=J(z)-z \)
Assuming that \( F \) and \( G \) are not identical in the segment \( [-1,1] \), function \( h \) should not be identically zero. This function also allows the holomorphic extension. Consider
behavior of function \( h \) in the complex plane. This extension should be periodic, satisfying conditions of my Lemma.
Therefore, function \( J \) should take values from set \( M \) inside domain \( S \). Function \( F \) has singularities at these points.
Therefore, function \( G \) has singularities in the domain \( S \). With these singulatiries,
function \( G \) does not satisfy the criterion in the definition of tetration.
In such a way, there exist only one tetration, that is strictly increasing function in the segment \( [-1,1] \).
04/12/2009, 09:58 PM (This post was last modified: 04/12/2009, 11:25 PM by sheldonison.)
Now that I can understand it, this is an amazing graph, showing the fractal copies of the main tetration curve distorted and duplicated with singularities into each vertical strip of the function.
In the tetration curve, the imaginary value for f(z) at the real axis is equal to zero for all z>=-2. For the solution based on the fixed point, is the imaginary value of f(z) nonzero everywhere
outside of the real axis? Otherwise it would seem likely that there would be other singularities, outside of the real axis.
- Sheldon
04/13/2009, 04:42 AM
sheldonison Wrote:..
In the tetration curve, the imaginary value for f(z) at the real axis is equal to zero for all z>=-2. For the solution based on the fixed point, is the imaginary value of f(z) nonzero everywhere
outside of the real axis?
What does it mean, "based on the fixed point"? Do you mean the case of other base?
For base \( 1<b<\exp(1/e) \), there are two real fixed points.
We may use any of them to build-up the "real" super-exponential.
Both resulting superexponentials have imaginary periods.
If such a superexponential has zero somewhere (for example, at minus unity),
then it has countable set of zeros.
sheldonison Wrote:..
Otherwise it would seem likely that there would be other singularities, outside of the real axis.
Yes, at \( 1<b<\exp(1/e) \), the superexponential F such that F(0)=1 is periodic; hence, just translate the singularity at -2 for the period, and you get the singularity outside the real axis. See
the picture for b=sqrt(2) at
It shows one additional singularity and the corresponding cut line.
04/13/2009, 01:03 PM (This post was last modified: 04/13/2009, 07:19 PM by sheldonison.)
Kouznetsov Wrote:
sheldonison Wrote:..
In the tetration curve, the imaginary value for f(z) at the real axis is equal to zero for all z>=-2. For the solution based on the fixed point, is the imaginary value of f(z) nonzero
everywhere outside of the real axis?
Otherwise it would seem likely that there would be other singularities, outside of the real axis.
What does it mean, "based on the fixed point"? Do you mean the case of other base?
I was refering to the base e solution.
Khoustenov earlier wrote:
Kouznetsov Wrote:\( \mathrm{tem}(z)=\mathrm{tet}(J(z)) \)
where \( J(z)=z+a\cdot\sin(2 \pi z) \)
In your equation, you have the original tet(z) and you also have tem(z)=tet(J(z)). When I said "solution based on a fixed point", I was referring to tet(z), the "correct" solution, rather than the
modified tem(z) equation, with the extra singularities.
For tet(z), is the only contour line where \( \Im(\text{tet}(z))=0 \) at the real axis, for z>-2?
Just realized this cannot be true, since \( \Im(e^{(\pi*k*i)})=0 \). So when the imaginary part >=pi, contours where the imaginary component is zero arise.
For the modified equation tem(z)=tet(J(z)), the fractal copies of the real axis (including the singularities) occur where \( \Im(\text{tem}(z))=0 \), and these occur where there is a contour of \( \
Im(J(z))=0 \). Wherever there is a contour line for \( \Im(J(z))=0 \) outside of the real axis, a distorted copy of the original tet(z) real axis gets generated.
All of this seems to imply that the contours of \( \Im(\text{tem}(z))=0 \) (outside the real axis) have much more to do with the particular one-cyclic \( \theta(z) \) function, where \( J(z)=z+\theta
(z) \), then with tet(z) itself.
- Sheldon
04/14/2009, 01:20 AM
sheldonison Wrote:In the tetration curve, the imaginary value for f(z) at the real axis is equal to zero for all z>=-2. For the solution based on the fixed point, is the imaginary value of f(z)
nonzero everywhere outside of the real axis?
I was refering to the base e solution.
For tet(z), is the only contour line where \( \Im(\text{tet}(z))=0 \) at the real axis, for z>-2?
The part of the real axis is not the only contour where \( \Im(\text{tet}(z))=0 \).
There are many of them in the right hand side of the complex plane.
Both real and imaginary parts of \( \Im(\text{tet}(z)) \) have huge values in vicinity of the real axis. Many times the imaginary part passes through the values that are integer factors of \( \pi \)
Each line \( \Im(\text{tet}(z))=\pi n , n \in \mathbb{N} \), at the unity translation \( z \rightarrow z+1 \), produces the line \( \Im(\text{tet}(z))=0 \),
These lines form the complicated structure.
The fractal of lines \( \Im(\text{tet}(z))=0 \) is shown in fig.2 at
sheldonison Wrote:Just realized this cannot be true, since \( \Im(e^{(\pi*k*i)})=0 \). So when the imaginary part >=pi, contours where the imaginary component is zero arise.
sheldonison Wrote:For the modified equation tem(z)=tet(J(z)), the fractal copies of the real axis (including the singularities) occur where \( \Im(\text{tem}(z))=0 \), and these occur where there
is a contour of \( \Im(J(z))=0 \). Wherever there is a contour line for \( \Im(J(z))=0 \) outside of the real axis, a distorted copy of the original tet(z) real axis gets generated.
Yes. In the paper mentioned, the example with modified tetration is also considered.
However, we have no need to modify the tetration in order to have lines \( \Im(\text{tet}(z))=0 \)
outside the real axis, but these lines are at \( \Re(z)>2 \).
sheldonison Wrote:... the contours of \( \Im(\text{tem}(z))=0 \) (outside the real axis) have much more to do with the particular one-cyclic \( \theta(z) \) function, where \( J(z)=z+\theta(z)
\), then with tet(z) itself.
The multiple reproduction of reduced and deformed patterns is not a specific property of a tetration.
Take any other function with beautiful pattern in the complex plane, modify its argument with function J , and you get the fractal of the modified patterns at the plot.
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First thorough proteomics examination regarding amino acid lysine crotonylation inside
First thorough proteomics examination regarding amino acid lysine crotonylation inside
The phrase when it comes to diffusion coefficient offered in Eq. (34) is our primary outcome. This expression is a far more general efficient diffusion coefficient for thin 2D stations in the
presence of constant transverse power, which contains the well-known earlier results for a symmetric station acquired by Kalinay, in addition to the limiting cases whenever transverse gravitational
exterior field goes to zero and infinity. Eventually, we show that diffusivity can be explained because of the interpolation formula suggested by Kalinay, D_/[1+(1/4)w^(x)]^, where spatial
confinement, asymmetry, in addition to presence of a continuing transverse power can be encoded in η, which can be a function of channel width (w), station centerline, and transverse force. The
interpolation formula additionally reduces to well-known earlier outcomes, namely, those acquired by Reguera and Rubi [D. Reguera and J. M. Rubi, Phys. Rev. E 64, 061106 (2001)10.1103/
PhysRevE.64.061106] and by Kalinay [P. Kalinay, Phys. Rev. E 84, 011118 (2011)10.1103/PhysRevE.84.011118].We study a phase transition in parameter learning of hidden Markov models (HMMs). We do that
by generating sequences of observed symbols from offered discrete HMMs with consistently distributed transition probabilities and a noise level encoded in the result probabilities. We apply the
Baum-Welch (BW) algorithm, an expectation-maximization algorithm from the industry of device discovering. Using the BW algorithm we then you will need to calculate the parameters of every Iadademstat
investigated realization of an HMM. We study HMMs with n=4,8, and 16 says. By switching the amount of available discovering information together with sound level, we observe a phase-transition-like
improvement in the performance regarding the understanding algorithm. For larger HMMs and more learning data, the educational behavior improves immensely below a particular threshold within the sound
strength. For a noise degree above the limit, discovering isn’t possible. Furthermore, we use an overlap parameter applied to the outcome of a maximum a posteriori (Viterbi) algorithm to analyze the
precision symbiotic cognition of the concealed condition estimation round the stage transition.We think about a rudimentary model for a heat motor, referred to as the Brownian gyrator, that comes
with an overdamped system with two levels of freedom in an anisotropic temperature field. Whereas the sign of the gyrator is a nonequilibrium steady-state curl-carrying probability current that may
create torque, we explore the coupling for this all-natural gyrating movement with a periodic actuation possibility the purpose of extracting work. We show that path lengths traversed in the manifold
of thermodynamic states, assessed in a suitable Riemannian metric, express dissipative losings, while area integrals of a work thickness quantify work being extracted. Thus, the maximal amount of
work which can be extracted relates to an isoperimetric issue, trading off area against period of an encircling path. We derive an isoperimetric inequality that delivers a universal bound on the
efficiency of all cyclic running protocols, and a bound how fast a closed road can be traversed before it becomes impossible to draw out good work. The analysis presented provides guiding principles
for building independent engines that extract work from anisotropic fluctuations.The thought of an evolutional deep neural community (EDNN) is introduced for the answer of limited differential
equations (PDE). The parameters of this community are trained to portray the initial state regarding the system just and are usually consequently updated dynamically, without any additional
education, to give you an accurate prediction of this development of the PDE system. In this framework, the community variables tend to be treated as features according to the appropriate coordinate
as they are numerically updated using the governing equations. By marching the neural community weights into the parameter area, EDNN can anticipate state-space trajectories that are indefinitely
lengthy, which is problematic for various other neural system techniques. Boundary conditions associated with PDEs tend to be treated as tough limitations, tend to be embedded in to the neural
network, and are consequently exactly pleased through the entire option trajectory. Several applications including the heat equation, the advection equation, the Burgers equation, the Kuramoto
Sivashinsky equation, therefore the Navier-Stokes equations are fixed to show the versatility and precision of EDNN. The use of EDNN towards the incompressible Navier-Stokes equations embeds the
divergence-free constraint into the network Salivary microbiome design so the projection regarding the momentum equation to solenoidal room is implicitly achieved. The numerical results verify the
accuracy of EDNN solutions in accordance with analytical and benchmark numerical solutions, both for the transient characteristics and statistics of the system.We investigate the spatial and temporal
memory outcomes of traffic density and velocity in the Nagel-Schreckenberg cellular automaton design. We show that the two-point correlation purpose of automobile occupancy provides use of spatial
memory effects, such as headway, in addition to velocity autocovariance function to temporal memory effects such as traffic leisure some time traffic compressibility. We develop stochasticity-density
plots that permit determination of traffic density and stochasticity through the isotherms of first- and second-order velocity statistics of a randomly selected vehicle.
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5 Effective Ways to Replace All Zeros with Fives in a Python Program
π ‘ Problem Formulation: Python programmers often need to manipulate numerical data. A common task could be replacing specific digits in a number – for instance, replacing all occurrences of the
digit ‘0’ with ‘5’ in a given integer. For example, transforming the integer 1050 should result in 1555.
Method 1: Using Sting Conversion
This method leverages Python’s ability to convert numbers to strings for easy manipulation. The str.replace() function is then utilized to substitute ‘0’ with ‘5’, followed by a conversion back to an
Here’s an example:
number = 1050
replaced_number = int(str(number).replace('0', '5'))
Output: 1555
This method is straightforward and takes advantage of Python’s dynamic type conversion. First, the integer is converted to a string allowing the replace function to change all ‘0’s to ‘5’s. The
string is then converted back into a integer for the final result.
Method 2: Using Mathematical Manipulation
By decomposing a number mathematically, digits can be examined and replaced logically. This involves digit extraction, comparison, and construction of the new number.
Here’s an example:
def replace_zero_with_five(number):
result_num = 0
decimal_place = 1
while (number > 0):
digit = number % 10
if digit == 0:
digit = 5
result_num += digit * decimal_place
number //= 10
decimal_place *= 10
return result_num
Output: 1555
This mathematical approach inspects each digit and conditionally replaces zeros. It builds the new number from the rightmost digit, shifting it correctly according to its original place value. This
method is efficient but might be less intuitive for those unfamiliar with numeric algorithms.
Method 3: Using List Comprehension and Join
Python’s list comprehension provides a concise way to iterate over elements. In this method, a list of single characters is created from the original number, each character is replaced if needed, and
then concatenated back into a string.
Here’s an example:
number = 1050
replaced_number = int(''.join(['5' if digit=='0' else digit for digit in str(number)]))
Output: 1555
This method uses Python’s powerful list comprehension to create a list of characters, replacing ‘0’ with ‘5’. The join() method then merges these into a new string, which is converted back to an
integer. This is a more Pythonic approach and maintains readability.
Method 4: Using Recursion
Recursion is a way to break down a problem into smaller instances of itself. It’s a more elegant solution where the function calls itself with a simpler version of the original problem until it
arrives at a simple base case.
Here’s an example:
def replace_zero_with_five_recursive(number):
if number == 0:
return 0
digit = number % 10
if digit == 0:
digit = 5
return replace_zero_with_five_recursive(number // 10) * 10 + digit
Output: 1555
The recursive method works by reducing the number by a factor of ten each time and dealing with the last digit. If the digit is zero, it is replaced by five; otherwise, it remains unchanged. This
method keeps the stack depth proportional to the number of digits in the number, which could be a limitation for very large numbers.
Bonus One-Liner Method 5: Using Regular Expressions
Regular expressions are a powerful tool for pattern matching and string manipulation. Python’s re library can be used to replace characters in a string following a certain pattern, in this case,
Here’s an example:
import re
number = 1050
replaced_number = int(re.sub('0', '5', str(number)))
Output: 1555
This one-liner uses the re.sub() method from Python’s regular expression library to replace all zeros in a string representation of the number directly. It’s a compact and efficient method but
understanding regular expressions can be challenging for new programmers.
• Method 1: String Conversion. Simple and intuitive. Can be slow for very large numbers due to string manipulation overhead.
• Method 2: Mathematical Manipulation. Fast and doesn’t depend on string conversion. Less readable and could be complex for some users.
• Method 3: List Comprehension and Join. Pythonic and readable. It’s a middle ground between string and mathematical approaches.
• Method 4: Recursion. Elegant and divides the problem into subproblems. Not suitable for very large numbers due to stack overflow risks.
• Method 5: Regular Expressions. Compact one-liner approach. Requires understanding of regex patterns and might have performance considerations.
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Computer Modelling of Nanoscale Systems
Computer Modelling of Nanoscale Systems (UE S3-8)
This course introduces the principles underlying common methods of numerical simulations used in the nanosciences, going from the atomistic scale to the continuum. It discusses the appropriateness of
atomic scale and continuum modeling. One of the goals is to understand the principles of the models and algorithms used in standard codes.
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STATISTICAL MECHANICS | Phase space, Phase point, Phase density, Phase cells,Micro-state of a system, Macro-stale of a system, Accessible state, Postulate of equal a priory probability, Density of quantum state, Phase trajectory - akritinfo.com
STATISTICAL MECHANICS | Phase space, Phase point, Phase density, Phase cells,Micro-state of a system, Macro-stale of a system, Accessible state, Postulate of equal a priory probability, Density of
quantum state, Phase trajectory
Phase space
In classical mechanics , the instantaneous dynamical state of a particle is completely specified by its three position coordinates (x, y, z) and three corresponding momentum coordinates (Px, Py, Pz).
Thus six, co-ordinates are needed to specify a “one particle system” completely. These six coordinates (x, y, z, Px, Py, Pz) are marked along six mutually perpendicular axes in six-dimensional space.
This space is called “phase space” ( Γ space).
Phase point
The point in the phase space represents the instantaneous state of the particle is known as phase point.
Phase density
The number of phase points per unit volume of phase space is called phase density.
Phase cells
When a phase space (or position-momentum space) is divided into tiny six dimensional cells whose sides are
then such tiny cells are called “phase cells”.
Micro-state of a system
A system of atomic dimension or smaller size is called microscopic system.
If the state of a system of particles are specified by quoting the position and momentum of individual particles then it is called micro-state or microscopic state of the system.
Example : a molecule , an atom, an ion.
Macro-stale of a system
A system which is large enough to be observable in the ordinary sense, is called macroscopic system.
The macroscopic state or macro-state of a system specified by quoting the macroscopic parameters as like pressure, volume, temperature, energy, Chemical potential.
Example : Solid, liquid , gas.
Illustration of micro states and macro-state of a system
Let us consider a system of three molecules a, b, c which are distributed in two halves of a box ; left half-L and right half-R:
The distribution of molecules are shown as
Hence there are four possible distributions —
a) 3 molecules in L and O molecule in R i.e. (3,0).
b) 0 molecule in L and 3 molecules in R i.e. (0,3).
c) 2 molecules in L and 1 molecule in R i.e. (2,1).
d) 1 molecule in L and 2 molecules in R i.e. (1,2).
Total number of ways in which three molecules can occupy two halves of the box = 1+1+3+3 = 8 = 2^3 corresponding to four different distributions.
Each way of arrangement of molecules is a micro-state of the system while each different distribution of molecules is a macro-state.
Thus there are 8 micro-states and a macro-states in the above example.
Accessible state
The micro-states which are permitted, under the constraints imposed upon the system are called accessible states.
If there are n1 molecules in a cell 1; each having energy ε1,
n2 molecules in call 2 ; each having energy ε2
and so on, then constraints are –
The micro-states which are allowed under given restrictions are called accessible micro-states.
For example : In case of 3 molecules, a,b,c to be restricted between two halves of a box. If none of can be outside the box, then (a,b,c), (a,bc), (ac, b) are accessible micro-states while (a,b),
(b,c), (a,c) etc. are in accessible micro-states.
Postulate of equal a priory probability
Accessible micro-states corresponding to possible macro-states are equally probable. In other words this states that the probability of finding the phase point in any one region is identical with
that for any other regions which are equally well with the given conditions.
Thus, this postulate is known as postulate of equal a priory probability.
The probability of finding the phase point for a given system in any one region of phase space is identical with that for any other region of equal extension of volume. This postulate is known as the
postulate of equal a priory probability.
Density of quantum state
OR Show that for a particle in an enclosure of volume V, the number of states in which the particle has a momentum between p and p+dp
ans- For a single particle free to move in a three dimensional space. We have six dimensional phase space. Three position co-ordinate
and three momentum co-ordinates
specify the micro-state of the particle in the phase space ,
So, volume of phase cell
Total volume of the phase space
Volume of momentum space containing momentum between p and p+dp will be given by the volume of a spherical shell of radius p and thickness dp i.e.
So, Number of micro-states in the momentum range p and p+dp
Switching over the energy E from momentum p via the relation
We get, number of quantum state (or micro-state) in the energy range E and E+dE,
The function f(E) is called density of quantum states .
where f(E)=g(E)/V
For electron.
Since an electron has two independent directions of spin orientation, hence for electron above equation can be formed as
problem- A particle is moving in XY plane . What is the dimension corresponding phase space?
The particle may have position coordinates
and momentum coordinates
Dimension of corresponding phase space = 4
Phase trajectory
The locus of the phase points of a particle on the phase plane is called phase trajectory.
Actually, the curve Showing the variation of px with x represents the phase trajectory.
Determine the phase trajectory of a one dimensional linear harmonic oscillator of constant energy E moving along x axis.
Also find the region of states accessible to like oscillator.
Sol :
The energy of one dimensional linear harmonic oscillator moving along x axis
where, m is mass of the particle
Above equation represents en equation of ellipse, whose semi major length a = √2E/k and semi minor length b=√2mE. Thus the phase trajectory of the oscillator (i.e. x vs Px) in phase space is an
Region of the states accessible to the oscillator with energy E is equal to the area of the ellipse
Read more –
STATISTICAL MECHANICS | Ensembles | Equilibrium | Third law of thermodynamics
Leave a Comment
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Compute With Scientific Notation Worksheet
Compute With Scientific Notation Worksheet. Each sheet is scaffolder and has model problems explained step by step Express each of the following in standard form.
The Scientific Notation (Old) Math Worksheet in 2020 Scientific from www.pinterest.com
Write each of the measurements below using standard notation. Web worksheet # 1. Web free worksheets(pdf) and answer keys on scientific notation.
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Linear Equations-Class9
In this activity, you will use concepts learnt in the chapter "Linear Equations in Two Variables" to identify points of intersection of lines, equations of lines parallel to axes and plot graph of
linear equations in two variables.
Please type in your name in the box below:
TASK 1 The letter I is placed along lines parallel to the axes; identify these three lines and write their equations in the box below.
TASK 2:In the following GeoGebra applet, use input bar (on the left) to plot following equations: x = 3 , 2x +3y = 12
TASK 3: The two lines plotted in the TASK 2 , form a triangle with x-axis . Read the coordinates of the vertices from the graph and write down these coordinates in the box below.
TASK 4 Is ( 4, 1 ) a solution of the equation 2x +3y = 12? Justify your answer with reference to its graph plotted in the TASK 2.
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ZPEnergy.com - To B3 or not to B3 - ECE Theory
Myron W. Evans has corrected Einstein's error, and has derived all physics equations from Cartan geometry, without resorting to arbitrary constants.
Einstein's General Relativity cannot explain many things. One glaring example is the spiral shape of galaxies. In contrast, ECE Theory explains the spiral shape very simply, and the prediction
matches the observations.
This is a transformative, "Fair use in Reporting" video. Planet Earth will die, if clean energy devices are not manufactured, using the correct version of physics.
The B(3) magnetic field was discovered in 1991, from the Inverse Faraday Effect, which proved Einstein's General Relativity is incomplete, and forced the abandonment of Riemann geometry, as a model
for space-time.
Those who still uphold the "standard model" of General Relativity, are willfully ignorant of these major discoveries. Free Energy comes from "Spin Connection Resonance", and the conservation of
angular momentum of spacetime torsion (which does not exist in relativity).
OVERVIEW OF ECE THEORY / AIAS.US: When Einstein developed General Relativity, in 1915, he chose the only known mathematical model for spacetime, Riemann geometry. Now, we know that his choice was not
correct. Since he started with the wrong geometric model of spacetime, it is impossible for General Relativity to solve many of the mysteries in modern physics.
Einstein Cartan Evans (ECE) Theory is based on the correct geometric model of spacetime, Cartan geometry. Cartan geometry adds "spacetime torsion" to the standard model of physics. Spacetime torsion
represents electro-magnetism, which does not exist in the 1915 version of General Relativity. Without spacetime torsion, it is impossible for General Relativity to explain the natural world. In
contrast to the standard model of physics, ECE Theory is based only on geometry, with no arbitrary "fudge" factors added, like General Relativity requires. ECE Theory replaces General Relativity,
since ECE Theory produces the right answers, as verified by numerical calculations, and experimental data.
The simplest form of ECE Theory is an engineering model, consisting of eight equations, which replace the four Maxwell's Equations, from 150 years ago. Those equations explained the physics of
electricity and magnetism. ECE Theory adds four new equations, which are identical to Maxwell's, with the exception, that the added equations control gravity, and time. The four new ECE equations
show a "spin connection resonance" with the four older electro-magnetism equations. The four new equations are a good example of the simplicity, and beauty, of a true natural theory. These equations
are a mirror image of Maxwell's Equations, but apply to gravity and time. This simple, and beautiful, solution is what Einstein spent the last 40 years of his life searching for, but was not able to
By creating the conditions for spin connection resonance, new physics effects can be engineered. The full set of eight ECE equations will allow new carbon-free energy sources to be developed, in the
same way that present-day technology was engineered from Maxwell's four equations, when combustion was the only known energy source. Now, using ECE Theory, energy can be pulled directly from
space-time, similar to a hydro-electric turbine, but without the need for falling water.
The rapid development of new, spin connection resonance, carbon-free energy sources, is a matter of great importance, for the entire planet. ECE Theory was discovered by chemist, physicist, and
mathematician, Myron Wyn Evans (Doctor of Science, Royal Civil List Scientist), as a way to explain the Inverse Faraday Effect. His discovery of the "B3" field, in the early 1990's, showed a new
model of spacetime was needed to explain many mysterious physics phenomena.
For more than twenty years, Dr Evans has devoted nearly all of his time and energy to deriving, and proving, the correct version of physics. ECE Theory can now explain, with exact precision, many
physics mysteries which General Relativity, and Quantum Mechanics, cannot explain.
Dr Evans' website, describing the new Unified Field Theory:
Fair Use Notice
This video may contain copyrighted material the use of which has not always been specifically authorized by the copyright owner. We are making such material available in our efforts to advance the
understanding of humanity's problems and hopefully to help find solutions for those problems. We believe this constitutes a transformative 'fair use in reporting' of any such copyrighted material as
provided for in section 107 of the US Copyright Law. In accordance with Title 17 U.S.C. Section 107, the material in this document is distributed without profit to those who have expressed a prior
interest in receiving the included information for research and educational purposes.
Any historical account of modern developments in physics must include Einstein’s geometrical concept, which was the first paradigm shift in physics since Newton introduced his laws of motion, more
than two hundred years before. Attempts at experimental validation of Einstein’s general relativity have been rare, and have mainly concerned the Solar System, like the deflection of light by the Sun
and the precession of the orbit of Mercury. In spite of this limitation, the theory was taken as a basis for cosmology, from which the Big Bang and dark matter were later extrapolated.
Unfortunately, this approach to cosmology has introduced self-contradictory inconsistencies. For example, the concept that the speed of light is an absolute upper limit is incompatible with the fact
that, to explain the first expansive phase of the universe, one has to assume that this happened with an expansion velocity faster than the speed of light. This example is only one of the criticisms
of Einstein that have yet to be answered properly and scientifically.
Later, after Einstein’s death, the velocity curve of galaxies was observed by astronomers. They observed that stars in the outer arms of galaxies do not move according to Newton’s law of gravitation,
but have a constant velocity. However, Einstein’s theory of general relativity is not able to explain this behavior. Both of these theories break down in cosmic dimensions, and when a theory does not
match experimental data, the scientific method requires that the theory be improved or replaced with a better concept.
In the case of galactic velocity curves, however, it was “decided” that Einstein was right and that there had to be another reason why stars behave in this way. Dark matter that interacts through
gravity and is distributed in a way that accounts for observed orbits was then postulated. Despite an intensive search for dark matter, even on the sub-atomic level, nothing has been found that could
interact with ordinary matter through gravity, but not interact with observable electromagnetic radiation, such as light.
Although Einstein’s theory has become increasingly problematic, the scientific community has been reluctant to abandon it. Starting around 2001, the members of the AIAS institute, with Myron Evans as
the guiding researcher, developed a new theory of physics, Einstein-Cartan-Evans (ECE) theory, which overcomes the problems in Einstein’s general relativity. They were even able to unify this new
theory with electrodynamics and quantum mechanics. This enabled significant progress in several fields of physics, and the most important aspects are described in the ECE textbooks (available as ECE/
UFT Papers 438 and 448).
ECE theory is based entirely on geometry, as was Einstein’s general theory of relativity. Therefore, Einstein is included in the name of this new theoretical approach. Both theories take the geometry
of spacetime (three space dimensions, plus one time dimension) as their basis. While Einstein thought that matter curves spacetime and assumed matter to be a “source” of fields, ECE theory is based
entirely on the field concept and does not need to introduce external sources.
This idea of sources created a number of difficulties in Einstein’s theory. Another reason for these difficulties is that Einstein made an unavoidable but significant mathematical error in his
original theory (1905 to 1915), because all of the necessary information was not yet available.
Riemann inferred the metric around 1850, and Christoffel inferred the connection around the 1860s. The idea of curvature was introduced at the beginning of the twentieth century, by Levi-Civita,
Ricci, Bianchi and colleagues in Pisa. However, the concept of torsion was not fully developed until the early 1920s, when Cartan and Maurer introduced the first structure equation. Therefore, in
1915, when Einstein published his field equation, Riemann geometry did not include torsion, and there was no way of determining that the Christoffel connection must be antisymmetric. The connection
was assumed to be symmetric (probably for ease of calculation), and the inferences of Einstein’s theory ended up being based on incorrect geometry.
Setting torsion to zero and using a symmetric connection leads to a contradiction with significant consequences, as has been shown by the AIAS Institute. For details, please see the book Criticisms
of the Einstein Field Equation (ECE/UFT Paper 301, 548 pages, 2010).
Torsion (which is a twisting of space) turns out to be essential and inextricably linked to curvature, because if the torsion is zero then the curvature vanishes. In fact, torsion is even more
important than curvature, because the unified laws of gravitation and electrodynamics are basically physical interpretations of twisting, which is formally described by the torsion tensor. ECE theory
unifies physics by deriving all of it directly and deterministically from Cartan geometry, and does so without using adjustable parameters in the foundational axioms. The parameters that combine
geometry with physics are derived from experimental data and are thus not arbitrarily adjustable. Spacetime is completely specified by curvature and torsion, and ECE theory uses these underlying
fundamental qualities to derive all of physics from differential geometry, and to predict quantum effects without assuming them (as postulates) from the beginning. It is the first (and only)
generally covariant, objective and causal unified field theory.
Instead of using Einstein’s field equation, ECE theory uses the equations of Cartan geometry, which can be written in a form equivalent to Maxwell’s equations, for electrodynamics as well as for
gravitation. By comparing this form of the Cartan equations with Maxwell’s original equations (which include charges and currents), one can define charge and current terms that consist of field terms
combined with curvature and torsion terms. The same can be done for gravitation, and this allows unification to happen via geometry. If a charge is present, we have electromagnetism, if not, we have
only gravitation.
In ECE theory, there are no sources a priori, only fields. Matter is considered to be a “condensed field” of general relativity, and spacetime itself may be interpreted as a vacuum or aether field
that exists everywhere. Matter as condensed fields leads directly to quantum mechanics, and avoids extra concepts like quantum electrodynamics.
The same equations hold for electrodynamics, gravitation, mechanics and fluid dynamics, which places all of classical physics on common ground. Physics is extended to the microscopic level by
introducing canonical quantization and quantum geometry. The quantum statistics that is used is classically deterministic, because the Heisenberg uncertainty principle has been shown to not be valid
for all combinations of conjugated operators. Also, there is no need for renormalization and quantum electrodynamics, because of the intrinsic qualities of ECE quantum mechanics. All known effects,
up to and including the structure of the vacuum, can be explained within the ECE axioms, which are based on Cartan geometry.
Vacuum forces give rise to microscopic effects like the Lamb shift and vacuum fluctuations. ECE m- theory, when applied to quantum mechanics, leads to a unification of this subject with general
relativity, and enables new derived methodologies like a quantum force and quantum-Hamilton equations. Consequently, quantum mechanics has become a simpler and better understood subject.
The following two categorizations are provided as possible starting points for exploration of ECE theory.
Important achievements of ECE theory include:
• Refutation of the Einstein field equation, big bang and black hole theory.
• Discovery of the antisymmetry laws of electrodynamics and gravitation.
• Replacement of the Einstein de Broglie equations by R theory.
• Development of the first single particle fermion equation that is based only on the Pauli spin matrices.
• Refutation of the dogma of negative energy in quantum field theory.
• Demonstration that energy from spacetime does not violate conservation laws of physics.
• Refutation of the Heisenberg uncertainty principle.
• Discovery of the quantum Hamilton equations.
• Discovery of the quantum force equation and pure quantum force.
• Discovery of spin connection resonance in the laws of nature. (By creating enabling conditions for spin connection resonance, new physics effects can be engineered.)
Physical phenomena that can now be explained using ECE theory include:
• The spiral shape of galaxies (with stars on hyperbolic paths in spiral arms), and without the need for "dark matter".
• Large-scale motion of mass jets from the center of galaxies back to the outer edge of the disk.
• Explanation of star motion by generally-relativistic extensions of Lagrange and Hamilton dynamics.
• The Hubble red shift of light from distant galaxies.
• Why the Michelson/Morley experiment did not detect an "aether". [[pending clarification]]
• Why ring laser gyros would need aether to function. (The ring laser gyro is the Sagnac effect that is explained by the vector potential of rotation.)
• The quantum mechanical Lamb shift as a vacuum effect.
• Vacuum fluctuations as the origin of orbital precession.
• Spacetime as a potential source of energy (for example, by spin connection resonance).
• Explanation of Low Energy Nuclear Reactions (LENR) as an effect of general relativity.
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Trading strategy backtesting: calculating trade profit and loss
13371 Views
5 Replies
12 Total Likes
Trading strategy backtesting: calculating trade profit and loss
This is a snippet from a strategy backtesting system that I am currently building in Mathematica.
One of the challenges when building systems in WL is to avoid looping wherever possible. This can usually be accomplished with some thought, and the efficiency gains can be significant. But it can be
challenging to get one's head around the appropriate construct using functions like FoldList, etc, especially as there are often edge cases to be taken into consideration.
A case in point is the issue of calculating the profit and loss from individual trades in a trading strategy. The starting point is to come up with a FoldList compatible function that does the
necessary calculations:
CalculateRealizedTradePL[{totalQty_, totalValue_, avgPrice_, PL_,
totalPL_}, {qprice_, qty_}] :=
Module[{newTotalPL = totalPL, price = QuantityMagnitude[qprice],
newTotalQty, tradeValue, newavgPrice, newTotalValue, newPL},
newTotalQty = totalQty + qty;
tradeValue =
If[Sign[qty] == Sign[totalQty] || avgPrice == 0, price*qty,
If[Sign[totalQty + qty] == Sign[totalQty], avgPrice*qty,
price*(totalQty + qty)]];
newTotalValue =
If[Sign[totalQty] == Sign[newTotalQty], totalValue + tradeValue,
newavgPrice =
If[Sign[totalQty + qty] ==
Sign[totalQty], (totalQty*avgPrice + tradeValue)/newTotalQty,
newPL =
If[(Sign[qty] == Sign[totalQty] ), 0,
totalQty*(price - avgPrice)];
newTotalPL = newTotalPL + newPL;
{newTotalQty, newTotalValue, newavgPrice, newPL, newTotalPL}]
Trade P&L is calculated on an average cost basis (as opposed to FIFO or LIFO).
Note that the functions handle both regular long-only trading strategies and short-sale strategies, in which (in the case of equities), we have to borrow the underlying stock to sell it short. Also,
the pointValue argument enables us to apply the functions to trades in instruments such as futures for which, unlike stocks, the value of a 1 point move is typically larger than $1 (e.g. $50 for the
ES S&P 500 mini futures contract).
We then apply the function in two flavors, to accommodate both standard numerical arrays and timeseries (associations would be another good alternative):
CalculateRealizedPLFromTrades[tradeList_?ArrayQ, pointValue_ : 1] :=
Module[{tradePL =
Rest@FoldList[CalculateRealizedTradePL, {0, 0, 0, 0, 0},
tradePL[[All, 4 ;; 5]] = tradePL[[All, 4 ;; 5]]*pointValue;
CalculateRealizedPLFromTrades[tsTradeList_, pointValue_ : 1] :=
Module[{tsTradePL =
Rest@FoldList[CalculateRealizedTradePL, {0, 0, 0, 0, 0},
tsTradePL[[All, 4 ;; 5]] = tsTradePL[[All, 4 ;; 5]]*pointValue;
tsTradePL[[All, 2 ;;]] =
Quantity[tsTradePL[[All, 2 ;;]], "US Dollars"];
tsTradePL =
Join[Transpose@tsTradeList["Values"], Transpose@tsTradePL],
These functions run around 10x faster that the equivalent functions that use Do loops (without parallelization or compilation, admittedly)
Let's see how they work with an example:
tsAAPL = FinancialData["AAPL", "Close", {2020, 1, 2}]
Next, we'll generate a series of random trades using the AAPL time series, as follows (we also take the opportunity to convert the list of trades into a time series, tsTrades):
trades = Transpose@
20]]]], {RandomChoice[{-100, 100}, 20]}];
trades // TableForm
We are now ready to apply our Trade P&L calculation function, first to the list of trades in array form:
Flatten[#] & /@
CalculateRealizedPLFromTrades[trades[[All, 2 ;; 3]]]], 2],
TableHeadings -> {{}, {"Date", "Price", "Quantity", "Total Qty",
"Position Value", "Average Price", "P&L", "Total PL"}}]
The timeseries version of the function provides the output as a timeseries object in Quantity["US Dollars"] format and, of course, can be plotted immediately with DateListPlot (it is also convenient
for other reasons, as the complete backtest system is built around timeseries objects):
tsTradePL = CalculateRealizedPLFromTrades[tsTrades]
So far so good - but this only covers calculation of the realized profit and loss. I will leave the calculation of unrealized gains and losses for another post.
5 Replies
Hello Jonathan,
Thank you for sharing such a great WL code.
Using the exact same functions you wrote in your post, I've some weird P&L results. I'm using Mathematica 13.2.
test = {{1, 100, 1}, {2, 200, 1}, {3, 300, -1}}
Flatten[#] & /@
Riffle[test, CalculateRealizedPLFromTrades[test[[All, 2 ;; 3]]]],
2], TableHeadings -> {{}, {"Date", "Price", "Quantity",
"Total Qty", "Position Value", "Average Price", "P&L",
"Total PL"}}]
This command gives :
P&L should be 150 instead of 300.
On real track records, I have errors similar to the one I just wrote about.
I have not been able to detect an anomaly in the code you shared.
Can you help me ?
Thank you !
Hi Jonathan - I've been working in a similar vein for some time. I appreciate your approach with FoldList and, like you, have struggled to put the parts together - so use loops everywhere. Your post
helped me see how much of this simple trade accounting could be done in a FoldList and so I've merged some of yours with some of mine. In particular I've added more P&L accounting and expanded the
step-by-step (across the table) so a user can start at the left, read numbers while doing arithmetic in their head, and confirm the results. Mined you, I've not spent much time auditing so there
could be scenarios that need rework. In the code I've included comments regarding my preferences and various assumptions being make for these calculations.
Needless to say a full on trade accounting module is not a small project.
This is interesting but I am suspicious of Do loops being an order of magnitude slower for this purpose.
(1) Did you ascertain that both give the same result? (I assume so.)
(2) Did you do anything to locate possible bottlenecks? This is admittedly not so easy. I usually do pedestrian things like add Print and Timing in various places. These are blunt tools but
nevertheless can be effective. I have learned that often enough the problem is not what or where I would have expected.
Hi Daniel,
(1) yes, I cross-checked the results for several examples , in both Mathematica and Excel.
(2) No I did not investigate bottlenecks, other than to identify the Do loop version of this P&L calculation function as the chief bottleneck in a larger program. As I wrote, it could well be that
the code could have been streamlined, or accelerated with compilation or parallelization.
But I can report that the 10x speedup is indeed accurate, for the large dataset of trade transactions I am evaluating. Initially, the performance improvement was on the order of 100x. But then I had
to add some quite messy WL code to handle various "edge" cases, which slowed it down considerably. Still, the end result represents a significant speed improvement.
I was rather surprised by this as I realize WR has put a great deal of effort into speeding up the procedural functionality in WL code. But, in any case, I think that FoldList application is better
aligned with the WL paradigm and modern programming techniques.
I will say, however, that it usually takes me a long time to figure out the code logic when applying functions like FoldList, SequenceFoldList, etc. I suspect this is just because my early
programming experience was in Fortran, Algol and C. Mathematica users unburdened by such legacy languages and concepts are likely to a great deal faster than me!
Be respectful. Review our Community Guidelines to understand your role and responsibilities. Community Terms of Use
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Rida Laraki
• Paris Dauphine University, France
According to our database
, Rida Laraki authored at least 42 papers between 2001 and 2024.
Collaborative distances:
Book In proceedings Article PhD thesis Dataset Other
Online presence:
On csauthors.net:
New characterizations of strategy-proofness under single-peakedness.
Math. Program., January, 2024
Mathematical Optimization for Fair Social Decisions: A Tribute to Michel Balinski.
Math. Program., January, 2024
BAR Nash Equilibrium and Application to Blockchain Design.
Proceedings of the 23rd International Conference on Autonomous Agents and Multiagent Systems, 2024
On sustainable equilibria.
J. Econ. Theory, October, 2023
O'Neill's Theorem for Games.
CoRR, 2023
Grading and Ranking Large number of candidates.
CoRR, 2023
FSDA: Tackling Tail-Event Analysis in Imbalanced Time Series Data with Feature Selection and Data Augmentation.
Proceedings of the Fifth International Workshop on Learning with Imbalanced Domains: Theory and Applications, 2023
Majority Judgment vs. Approval Voting.
Oper. Res., 2022
An α-No-Regret Algorithm For Graphical Bilinear Bandits.
CoRR, 2022
Level-strategyproof Belief Aggregation Mechanisms.
Proceedings of the EC '22: The 23rd ACM Conference on Economics and Computation, Boulder, CO, USA, July 11, 2022
An $\alpha$-No-Regret Algorithm For Graphical Bilinear Bandits.
Proceedings of the Advances in Neural Information Processing Systems 35: Annual Conference on Neural Information Processing Systems 2022, 2022
Smooth Fictitious Play in Stochastic Games with Perturbed Payoffs and Unknown Transitions.
Proceedings of the Advances in Neural Information Processing Systems 35: Annual Conference on Neural Information Processing Systems 2022, 2022
Fictitious Play and Best-Response Dynamics in Identical Interest and Zero-Sum Stochastic Games.
Proceedings of the International Conference on Machine Learning, 2022
Proceedings of the 21st International Conference on Autonomous Agents and Multiagent Systems, 2022
EPTAS for stable allocations in matching games.
CoRR, 2021
Learning in nonatomic games, Part I: Finite action spaces and population games.
CoRR, 2021
Best Arm Identification in Graphical Bilinear Bandits.
Proceedings of the 38th International Conference on Machine Learning, 2021
Majority judgment vs. majority rule.
Soc. Choice Welf., 2020
Trade Selection with Supervised Learning and Optimal Coordinate Ascent (OCA).
Proceedings of the Mining Data for Financial Applications - 5th ECML PKDD Workshop, 2020
Deep Reinforcement Learning (DRL) for Portfolio Allocation.
Proceedings of the Machine Learning and Knowledge Discovery in Databases. Applied Data Science and Demo Track, 2020
NGO-GM: Natural Gradient Optimization for Graphical Models.
CoRR, 2019
Approachability of convex sets in generalized quitting games.
Games Econ. Behav., 2018
A discrete version of CMA-ES.
CoRR, 2018
A new approach to learning in Dynamic Bayesian Networks (DBNs).
CoRR, 2018
A continuity question of Dubins and Savage.
J. Appl. Probab., 2017
Online Learning and Blackwell Approachability in Quitting Games.
Proceedings of the 29th Conference on Learning Theory, 2016
Inertial Game Dynamics and Applications to Constrained Optimization.
SIAM J. Control. Optim., 2015
Higher order game dynamics.
J. Econ. Theory, 2013
Two-Person Zero-Sum Stochastic Games with Semicontinuous Payoff.
Dyn. Games Appl., 2013
A Continuous Time Approach for the Asymptotic Value in Two-Person Zero-Sum Repeated Games.
SIAM J. Control. Optim., 2012
Semidefinite programming for min-max problems and games.
Math. Program., 2012
Explicit formulas for repeated games with absorbing states.
Int. J. Game Theory, 2010
Informationally optimal correlation.
Math. Program., 2009
Semidefinite programming for N-Player Games
CoRR, 2008
The Value of Zero-Sum Stopping Games in Continuous Time.
SIAM J. Control. Optim., 2005
Continuous-time games of timing.
J. Econ. Theory, 2005
The Preservation of Continuity and Lipschitz Continuity by Optimal Reward Operators.
Math. Oper. Res., 2004
The splitting game and applications.
Int. J. Game Theory, 2002
Variational Inequalities, System of Functional Equations, and Incomplete Information Repeated Games.
SIAM J. Control. Optim., 2001
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interval - Factor Documentation
Factor handbook » The language » Numbers » Intervals
Prev: Properties of interval arithmetic
Next: interval? ( object -- ? )
intervalsClass description
An interval represents a set of real numbers between two endpoints; the endpoints can either be included or excluded from the interval.
slots store endpoints, represented as arrays of the shape
{ number included? }
Intervals are created by calling
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Vol. 30, No. 4, 2004
HOUSTON JOURNAL OF
Electronic Edition Vol. 30, No. 4, 2004
Editors: H. Amann (Zürich), G. Auchmuty (Houston), D. Bao (Houston), H. Brezis (Paris), J. Damon (Chapel Hill), K. Davidson (Waterloo), C. Hagopian (Sacramento), R. M. Hardt (Rice), J. Hausen
(Houston), J. A. Johnson (Houston), J. Nagata (Osaka), V. I. Paulsen (Houston), G. Pisier (College Station and Paris), S. W. Semmes (Rice)
Managing Editor: K. Kaiser (Houston)
Houston Journal of Mathematics
D.D. Anderson, Department of Mathematics, The University of Iowa, Iowa City, Iowa 52242-1419, U.S.A. (dan-anderson@uiowa.edu) and Tiberiu Dumitrescu, Facultatea de Matematică, Universitatea
Bucureşti, 14 Academiei Str., Bucharest, RO 70109, Romania (tiberiu@al.math.unibuc.ro).
Half Condensed Domains, pp. 929-936.
ABSTRACT. An integral domain D is condensed (resp., strongly condensed) if for each pair of ideals I, J of D, IJ={ij ; i in I, j in J} (resp., IJ=iJ for some i in I or IJ =Ij for some j in J). In
this paper we introduce and study the two related notions of a half condensed domain and a strongly half condensed domain. An integral domain D is half condensed if whenever a nonzero z is in IJ with
I, J ideals of D, there exist I', J' (invertible) ideals of D such that I' is a subset of I, J' is a subset of J, and zD=I'J'. And D is strongly half condensed if whenever I, J are nonzero ideals of
D, IJ=I[1]J for some invertible ideal I[1] that is a subset of I or IJ=IJ[1] for some invertible ideal J[1] that is a subset of J.
John Harding and Guram Bezhanishvili, Department of Mathematical Sciences, New Mexico State University, Las Cruces, NM 88003-0001, USA (jharding@nmsu.edu), (gbezhani@nmsu.edu).
MacNeille Completions of Heyting Algebras, pp. 937-952.
ABSTRACT. In this note we provide a topological description of the MacNeille completion of a Heyting algebra similar to the description of the MacNeille completion of a Boolean algebra in terms of
regular open sets of its Stone space. We also show that the only varieties of Heyting algebras that are closed under MacNeille completions are the trivial variety, the variety of all Boolean
algebras, and the variety of all Heyting algebras.
Amir Khosravi, Faculty of Mathematical Sciences and Computer Engineering, University For Teacher Education, 599 Taleghani Ave., Tehran 15614, IRAN, and Behrooz Khosravi, Dept. of Pure Math., Faculty
of Math. and Computer Science, Amirkabir University of Technolog (Tehran Polytechnic), 424, Hafez Ave., Tehran 15914, IRAN (khosravibbb@yahoo.com).
A New Characterization of Some Alternating and Symmetric Groups (II), pp. 953-967.
ABSTRACT. The order of every finite group G can be expressed as a product of coprime positive integers m[1],...,m[t] such that the set of prime numbers divided m[i] is a connected component of the
prime graph of G. The integers m[1],...,m[t] are called the order components of G. Order components of a finite group are introduced in Chen (J. Algebra 15 (1996) 184).
There exist some characterizations about alternating and symmetric groups. Some non-abelian simple groups are known to be uniquely determined by their order components. In this paper, we suppose that
p=2^a x^b+1>5 be a prime number, where a,b>0 are positive integers and x>3 is an odd prime number. Then by using the classification of finite simple groups, we proved that A[p], A[p+1], A[p+2], S[p],
S[p+1], are also uniquely determined by their order components. As corollaries of these results, the validity of a conjecture of J. G. Thompson and a conjecture of W. Shi and J. Bi both on A[n],
where n=p, p+1 or p+2 are obtained. Also we generalize these conjectures for the groups S[n], where n=p, p+1.
Coffman, Adam, Indiana University - Purdue University Fort Wayne, Fort Wayne, IN 46805 (http://www.ipfw.edu/math/Coffman/).
Analytic Normal Form for CR Singular Surfaces in C^3, pp. 969-996.
ABSTRACT. A real analytic surface inside complex 3-space with an isolated, non-degenerate complex tangent is shown to be holomorphically equivalent to a fixed real algebraic variety. The analyticity
of the normalizing transformation is proved using a rapid convergence argument. Real surfaces in higher dimensions are also shown to have an algebraic normal form.
Hao Fang, Courant Institute of Mathematical Sciences, New York University, New York 10012, USA and Changyou Wang, Department of Mathematics, University of Kentucky, Lexington, KY 40506, USA
On the Mean Curvature Flow for σ[k]-Convex Hypersurfaces, pp. 997-1007.
ABSTRACT. We obtain estimates on both size and dimensions of the singular set at the first blow-up time of the mean curvature flow of hypersurfaces whose initial data is σ[k]-convex.
Erdem, Sadettin, Middle East Technical University, 06531 Ankara, Turkey. (saerdem@fef.sdu.edu.tr ) or ( serdem@metu.edu.tr ).
J-Pseudo Harmonic Morphisms, Some Subclasses And Their Liftings To Tangent Bundles, pp. 1009-1038.
ABSTRACT. New subclasses of J- pseudo harmonic morphisms F of a (semi-) Rumanian manifold ( M, g) into a metric (para-) f-manifold (N,h,J ) are introduced, namely; nearly, quasi, semi homothetic
harmonic maps. On the way, some characterizations of its tension field of F are given. Also liftings of J-pseudo harmonic morphisms F to the tangent bundles TM and TN, with various type of lifted
metric (para-) f-structures are considered. Finally, some supporting examples are provided.
Garcia-Ferreira, S., Instituto de Matemáticas (UNAM), Apartado Postal 61-3, Xangari, 58089, Morelia, Michoacán, México (sgarcia@matmor.unam.mx), Sakai, S., Department of Mathematics, Kanagawa
University, Yokohama 221-8686, Japan (sakaim01@kanagawa-u.ac.jp), and Sanchis, M., Departament de Matemátiques, Universitat Jaume I, Campus Riu Sec, 12071, Castelló, Spain (sanchis@mat.uji.es).
Free topological groups over ω[μ]-metrizable spaces , pp 1039-1053.
ABSTRACT. Let be an uncountable regular cardinal. For a Tychonoff space X, we let A(X) and F(X) be the free Abelian topological group and the free topological group over X, respectively. In this
paper, we establish the next equivalences.
Theorem. Let X be a space. The following are equivalent.
1. (X,U[X]) is an -metrizable uniform space, where is the universal uniformity on X.
2. A(X) is topologically orderable and χ(A(X)) =ω[μ] .
3. The derived set is ω[μ]-compact and X is ω[μ]-metrizable.
Theorem. Let X be a non-discrete space. Then, the following are equivalent.
1. X is ω[μ]-compact and ω[μ]-metrizable.
2. (X,U[X]) is ω[μ]-metrizable and X is ω[μ]-compact.
3. F(X) is topologically orderable and χ(F(X)) =ω[μ] .
We also prove that an ω[μ]-metrizable uniform space (X,U) is a retract of its uniform free Abelian group A(X,U) and of its uniform free group F(X,U).
Naotsugu Chinen, University of Tsukuba, Ibraki 305-8571, Japan (naochin@math.tsukuba.ac.jp).
Sets of all ω-limit points for one-dimensional maps, pp. 1055-1068.
ABSTRACT. Let f be a continuous map from a graph G to itself and m the maximum of orders of all points of G. The main result of this paper is that a point c in G lies in the limit set of some point
of G if and only if every open neighborhood of c contains at least (m + 1) points of some trajectory. This shows that every set of all limit points for every graph map satisfies the analogue to the
Birkhoff theorem. But, the above does not holds for one-dimensional maps.
J.J. Charatonik, Instituto de Matematicas, UNAM, Cd. Universitaria, 04510 Mexico, D.F., Mexico (jjc@matem.unam.mx), W.J. Charatonik, Department of Mathematics and Statistics, University of
Missouri-Rolla, Rolla, MO 65409-0020, U.S.A. (wjcharat@umr.edu) and J.R. Prajs, Department of Mathematics, Idaho State University, Pocatello, ID 83209, U.S.A. (prajs@isu.edu)
Atriodic absolute retracts for hereditarily unicoherent continua, pp. 1069-1087.
ABSTRACT. Let X be an absolute retract for the class of hereditarily unicoherent continua that contains no simple triod. In the paper we prove that (a) X is atriodic; (b) X is either an arc or an
indecomposable continuum having only arcs as its proper subcontinua; (c) if X is either tree-like or circle-like, then it is arc-like.
Jonathan Hatch, Department of Mathematical Sciences, University of Delaware, Newark, DE 19716 (hatch@math.udel.edu).
On a characterization of W-sets, pp. 1089-1101.
ABSTRACT. A proper subcontinuum H of a continuum X is said to be a W-set provided for each continuous surjective function f from a continuum Y onto X, there exists a subcontinuum C of Y that maps
entirely onto H. Descriptions, definitions, and results concerning two new types of W-sets are given, as well as a new characterization of W-sets.
Louis Block, Department of Mathematics, University of Florida, Gainesville, FL 32611-8105, (block@math.ufl.edu) and James Keesling, Department of Mathematics, University of Florida, Gainesville, FL
32611-8105, (jek@math.ufl.edu).
Topological Entropy and Adding Machine Maps, pp. 1103-1113.
ABSTRACT. We prove two theorems which extend known theorems concerning periodic orbits and topological entropy in one-dimensional dynamics. Our first result may be described as follows. Given a
sequence of prime numbers, we form the corresponding adding machine map (also called the odometer map). We then determine the infimum of the topological entropies of all continuous maps of the
interval which contain a copy of the given adding machine map. Our second result deals with the following question. Suppose we are given a closed subset of the interval and a continuous map of this
closed subset to itself. How do we extend the given map to a map of the entire interval which has the smallest possible entropy?
Gady Kozma, Faculty of Mathematics, The Weizmann Institute of Science, Rehovot 76100, Israel (gadykozma@hotmail.com), (gadyk@wisdom.weizmann.ac.il).
On removing one point from a compact space, pp. 1115-1126.
ABSTRACT. If B is a compact space such that after removing one point x it is still Lindelof, then any power of B satisfies that after removing one point (namely the point all whose coordinates are x)
it is still star-Lindelof. If after removing one point B is still compact, then any power of B, after removing one point is still discretely star-Linedlof. In particular, this gives new examples of
Tychonoff, discretely star-Lindelof spaces with unlimited extent.
R. Lowen and S. Verwulgen, Department of Mathematics, University of Antwerp, Antwerp 2020, Belgium (rlow@ruca.ua.ac.be), (vwulgen@ruca.ua.ac.be).
Approach Vector Spaces, pp. 1127-1142.
ABSTRACT. In this paper we determine what properties an approach structure has to fulfil for it to concord well with a vector space structure. Not surprisingly these conditions are more subtle than
those for a topology. That the conditions we impose are the right ones follows mainly from the good categorical relationship among the different categories which play an important role in this
setting, namely topological vector spaces, completely regular spaces, metrizable vector spaces and of course approach vector spaces.
Boulabiar, Karim , IPEST, University of Carthage, BP 51, 2070-La Marsa, Tunisia 42101 (karim.boulabiar@ipest.rnu.tn).
Order Bounded Separating Linear Maps on Φ-Algebras, pp.1143-1155.
ABSTRACT. A Φ-algebra is an Archimedean lattice ordered algebra with a weak order unit. Let be A and B be Φ -algebras and let T be a separating linear map from A into B, that is, T is a linear map
such that T(f)T(g) = 0 in B whenever fg = 0 in A. It is proven by an order theoretical and purely algebraic method that there exist a 'weight' element w in B and a positive algebra homomorphism S
from A into the maximal ring of quotients Q(B) of B such that T(f) = wS(f) holds for all f in A. Both real and complex cases are considered. This result generalizes the following theorem proved by W.
Arendt in his paper [Spectral properties of Lamperti operators, Indiana Univ. J. Math., 32 (1983), 199-215]. Let C(X) and C(Y) be the Φ-algebras of all scalar-valued continuous functions on compact
Hausdorff topological spaces X and Y, respectively. Then for every separating linear map T from C(X) into C(Y) there exist a 'weight' function w in C(Y) and a function h from Y into X (continuous on
the cozero set of w) such that T(f)(y) = w(y)f(h(y)) holds for all f in C(X) and y in Y.
B.E. Forrest and L.W. Marcoux, Department of Pure Mathematics, University of Waterloo, Waterloo, Ontario, Canada, N2L 3G1 (beforres@math.uwaterloo.ca) , (LWMarcoux@math.uwaterloo.ca).
Second Order Cohomology of Triangular Banach Algebras, pp. 1157-1176.
ABSTRACT. Explicit calculations of the various cohomology groups of a Banach algebra I are often very difficult to obtain. In this paper, we will present an elementary method to describe the second
cohomology group H^2(I ,I) for a class of algebras called triangular Banach algebras. The techniques are then illustrated through a number of examples.
Zhai Fahui, Institute of Mathematics, Institute of Qingdao Chemical Technology, Qingdao 266042 , P.R. China, (fahuiz@163.com).
The Closures of (u+k)-Orbits of Class Essentially Normal Operators, pp. 1177-1194.
ABSTRACT. Let A , B be simply connected analytic cauchy domains, B be a subset of the closure of A , u and v be measures on the boundary of A and B* which are assumed to be equivalent to arc length
measures, respectively. M(A , u) and M(B*, v) are the multiplication operators on Hardy spaces of functions on the simply connected cauchy domain A and the simply connected cauchy domain B* ,
respectively. In this paper, we describe the closure of (u+k)-oribt of direct sum of the finitely direct sum M(A , u) with the finitely direct sum M(B*, v)*, furthermore, if A=D={z: |z|<1}, we also
describe the closure of (u+k)-oribt of finitely direct sum of a class essentially normal operator models.
Leo Livshits, Department of Mathematics and Computer Science, Colby College, Waterville, ME 04901 (llivshi@colby.edu), Sing-Cheong Ong, Department of Mathematics, Central Michigan University, Mount
Pleasant, MI 48859 (ong1s@cmich.edu ) and Sheng-Wang Wang Department of Mathematics, Nanjing Audit Institute, Nanjing 210029, China (wang2598@nju.edu.cn).
Schur Algebras Over Function Algebras, pp. 1195-1217.
ABSTRACT. The authors generalize results of L. Livshits, S.-C. Ong and S. W. Wang, Banach Space Duality in Absolute Schur Algebras, Integral Equations and Operator Theory. Vol. 41 (2001) 343-359.
Hem Raj Joshi, Department of Math and CS, Xavier University Cincinnati, OH 45207-4441 (joshi@xavier.edu) and Suzanne Lenhart, Department of Mathematics, University of Tennessee, Knoxville, TN
37996-1300 (lenhart@math.utk.edu).
Solving a Parabolic Identification Problem by Optimal Control Methods, pp. 1219-1242.
ABSTRACT. An unknown coefficient of the interaction term of a parabolic system with a Neumann boundary condition in a multi-dimensional bounded domain is identified. The solution of the system
represents the concentrations of prey and predator populations. Given partial (perhaps noisy) observations of a true solution in a subdomain, we seek to ``identify" the coefficient of the interaction
term using an optimal control technique, involving Tikhonov's regularization. The existence and uniqueness of the optimal control approximating the desired coefficient are obtained, an optimality
system is derived, the identification problem is discussed and an example illustrating how to find a solution numerically is presented.
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Conversion and difference kilobyte to kibibyte, megabyte to mebibyte
IEC units
0 KiB
0 MiB
0 GiB
0 TiB
0 PiB
SI units
0 KB
0 MB
0 GB
0 TB
0 PB
Difference in %
How is it possible that "kilobyte" may represent two different values (1000 Bytes and 1024 Bytes)?
Up to the year 2000, if someone was to say "one kilobyte". By that they meant of value 2^10 or 1024. We can say that there was a good reason for that. Computers are based on binary system, base of
the binary system is 2 since it has only two values 0 and 1. Base of decimal system, system we use in our every day life is 10. We know there are standardized values of prefixes that represent bigger
values so we don't have to write all those zeroes at the end for large numbers. Now, if we are to get the value of the first prefix, kilo, we can write 10^3 or 1000 in decimal system. If we are to
calculate that prefix value in binary system, the closest value to 1000 would be 1024. For example, 2^9 is 512, 2^10 is 1024. Which value of these two is closer to 1000? It is 1024. So 1024 or 2^10
was taken to represent the value of kilo in computer systems. That is reason why it was agreed at beginning for kilobyte to have value of 1024 bytes.
To fix the issue, in year 1999 IEC (International Electrotechnical Commission) suggested to change the names of units based on binary system. Base of binary system is 2, word that represent the value
of two is "bi". So IEC suggested to change last two letters in standard prefix name with "bi". So, we got kilo as kibi. Kilobyte became kibibyte, megabyte became mebibyte and so on. These values have
it's own abbreviations as well, kibibyte is KiB, megabyte is MiB. Notice the letter i in the abbreviated name. So, old value "10 KB" became "10 KiB".
Table bellow is showing some of these new values. These new values should be used.
Name of unit Simbol Value
1 kilobyte 1 kB 10^3 = 1000 bytes
1 megabyte 1 MB 10^6 = 1000000 bytes
1 gigabyte 1 GB 10^9 = 1000000000 bytes
1 terabyte 1 TB 10^12 = 1000000000000 bytes
1 petabyte 1 PB 10^15 = 1000000000000000 bytes
1 kibibayte 1 KiB 2^10 = 1024 bytes
1 mebibyte 1 MiB 2^20 = 1048576 bytes
1 gibibyte 1 GiB 2^30 = 1073741824 bytes
1 tebibyte 1 TiB 2^40 = 1099511627776 bytes
1 pebibyte 1 PiB 2^50 = 1125899906842624 bytes
We still have problems related to these notations, majority of people and even school books are still using those wrong values with base 2 when they write prefixes kilo, mega, giga and so on.
It seems like we are stack between values for "megabyte". Is it 1 000 000 bytes or 1 048 576 bytes? Since there is a huge difference between 'SI' and 'IEC' units, the confusion rises even bigger with
larger capacity of disks. For example gibibyte is 7,4% larger than gigabyte, tebibyte is 10% larger than terabyte, and that is just a huge difference. Many people complain when they for example buy
hard disk of 250 GB their OS (operating system) is showing only "232 GB". This is not manufacturer fault, and it is not a marketing trick. It is true that 250 gigabytes is equal to 232, but 232
gibibytes. We need to blame OS here, because it is still using that wrong notation.
Some OS's like windows are still using obsolete notation without proper IEC names for units, that confuses users. But, on the other hand some OS's like Linux are using the right notation, because of
it's "open source" code policies.
All the manufacturers should use these new, corrected values with new names with Si abbreviations 'kB', 'MB' or IEC 'KiB', 'MiB' etc. We should write small k in kilo abbreviation, because uppercase K
is Kelvin (so kB is kilobyte and KB would represent to some people kelvinbyte, and that is just wrong unit).
The fact that IEC units "sound weird" isn't good enough reason to stick with SI units and prefixes with binary notation (base 2). If you don't like the name gibibyte than use something that won't be
linked to real value of gigabyte, like "binary gigabyte".
Why should we decide which notation to use? Imagine that you are going to warehouse to buy some wooden planks of exact length of 10 'units', because you want them to fit exactly to what you are
building. But, when you come back with planks, you notice that there is deviation in length of 7% (7% longer than you needed). Than you spend some time fixing these planks by making them shorter and
than you realize you just bought one extra plank. Next day you go to another warehouse because first one from yesterday was closed. But, this time you are smarter and you buy planks of length 9.3
'units'. You arrive home and notice that now these are shorter than you hoped to be, because second warehouse was using different values. Now you have this ugly empty part in your built because of
shorter planks. Fortunately, this is not happening in real life because units for length are defined. Why shouldn't this be case with memory units for capacity in computer systems?
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Formal description of TL combinators
Formal declaration of TL combinators
Main article: Formal description of TL. See also TL Language.
Combinators in TL are declared as follows:
combinator-decl ::= full-combinator-id { opt-args } { args } = result-type ;
full-combinator-id ::= lc-ident-full | _
combinator-id ::= lc-ident-ns | _
opt-args ::= { var-ident { var-ident } : [excl-mark] type-expr }
args ::= var-ident-opt : [ conditional-arg-def ] [ ! ] type-term
args ::= [ var-ident-opt : ] [ multiplicity *] [ { args } ]
args ::= ( var-ident-opt { var-ident-opt } : [!] type-term )
args ::= [ ! ] type-term
multiplicity ::= nat-term
var-ident-opt ::= var-ident | _
conditional-arg-def ::= var-ident [ . nat-const ] ?
result-type ::= boxed-type-ident { subexpr }
result-type ::= boxed-type-ident < subexpr { , subexpr } >
We shall clarify what all this means.
• A combinator identifier is either an identifier starting with a lowercase Latin letter (lc-ident), or a namespace identifier (also lc-ident) followed by a period and another lc-ident. Therefore,
cons and lists.get are valid combinator identifiers.
• A combinator has a name, also known as a number (not to be confused with the designation) -- a 32-bit number that unambiguously determines it. It is either calculated automatically (see below) or
it is explicitly assigned in the declaration. To do this, a hash mark (#) and exactly 8 hexadecimal digits -- the combinator's name -- are added to the identifier of the combinator being defined.
• A combinator's declaration begins with its identifier, to which its name (separated by a hash mark) may have been added.
• After the combinator identifier comes the main part of the declaration, which consists of declarations of fields (or variables), including an indication of their types.
• First come declarations of optional fields (of which there may be several or none at all). Then there are the declarations of the required fields (there may not be any of these either).
• Any identifier that begins with an uppercase or lowercase letter and which does not contain references to a namespace can be a field (variable) identifier. Using uc-ident for identifiers of
variable types and lc-indent for other variables is good practice.
• Next a combinator declaration contains the equals sign (=) and the result type (it may be composite or appearing for the first time). The result type may be polymorphic and/or dependent; any
fields of the defined constructor's fields of type Type or # may be returned (as subexpressions).
• A combinator declaration is terminated with a semicolon.
In what follows, a constructor's fields, variables, and arguments mean the same thing.
Optional field declarations
• These have the form { field_1 ... field_k : type-expr }, where field_i is a variable (field) identifier that is unique within the scope of the combinator declaration, and type-expr is a type
shared by all of the fields.
• If k>1, this entry is functionally equivalent to { field_1 : type-expr } ... { field_k : type-expr }.
• All optional fields must be explicitly named (using _ instead of field_i is not allowed).
• Moreover, at present the names of all optional fields must share the combinator's result type (possibly more than once) and themselves be of type # (i,e., nat) or Type. Therefore, if the exact
result type is known, it is possible to determine the values of all of the combinator's implicit parameters (possibly obtaining a contradiction of the form 2=3 in doing so, which means that the
combinator is not allowed in the context).
Required field declarations
• These may have the form ( field_1 ... field_k : type-expr ), similar to an optional field declaration, but with parentheses. This entry is equivalent to ( field_1 : type-expr ) ... ( field_k :
type-expr ), where the fields are defined one at a time.
• The underscore sign (_) can be used as names of one or more fields (field_i), indicating that the field is anonymous (the exact name is unimportant).
• One field may be declared without outer parentheses, like this: field_id : type-expr. Here, however, if type-expr is a complex type, parentheses may be necessary around type-expr (this is
reflected in BNF).
• Furthermore, one anonymous field may be declared using a type-expr entry, functionally equivalent to _ : type-expr.
• Required field declarations follow one after another, separated by spaces (by any number of whitespace symbols, to be more precise).
• The declared field's type (type-expr) may use the declared combinator's previously defined variables (fields) as subexpressions (i.e. parameter values). For example:
nil {X:Type} = List X; cons {X:Type} hd:X tl:(list X) = List X; typed_list (X:Type) (l : list X) = TypedList;
• These may only exist among required parameters. They have the form [ field-id : ] [ multiplicity * ] [ args ], where args has the same format as the combinator's declaration of (several) required
fields, except that all of the enclosing combinator's previously declared fields may be used in the argument types.
• The name of a field of an enclosing combinator that receives a repetition as a value may be specified (field-id), or bypassed, which is equivalent to using the underscore sign as a field-id.
• The multiplicity field is an expression of the type # (nat), which can be a real constant, the name of a preceding field of type #, or an expression in the form ( c + v ), where c is a real
constant and v is the name of a field of type #. The sense of the multiplicity field is to provide the length of the (repetition) vector, each element of which consists of values of the types
enumerated in args.
• The multiplicity field may be bypassed. In this case, the last preceding parameter of type # from the enclosing combinator is used (it must be).
• Functionally, the repetition field-id : multiplicity * [ args ] is equivalent to the declaration of the single field ( field-id : %Tuple %AuxType multiplicity ), where aux_type is an auxiliary
type with a new name defined as aux_type *args* = AuxType. If any of the enclosing type's fields are used within args, they are added to the auxiliary constructor aux_type and to its AuxType
result type as the first (optional) parameters.
• If args consists of one anonymous field of type some-type, then some-type can be used directly instead of %AuxType.
• If during implementation the repetitions are rewritten as indicated above, it is logical to use instead of aux_type and AuxType, some identifiers that contain the name of the outer combinator
being defined and the repetition's index number inside its definition.
matrix {m n : #} a : m* [ n* [ double ] ] = Matrix m n;
is functionally equivalent to
aux_type {n : #} (_ : %Tuple double n) = AuxType n;
matrix {m : #} {n : #} (a : %Tuple %(AuxType n) m) = Matrix m n;
Moreover, the built-in types Tuple and Vector could be defined as:
tnil {X : Type} = Tuple X 0;
tcons {X : Type} {n : #} hd:X tl:%(Tuple X n) = Tuple X (S n);
vector {X : Type} (n : #) (v : %(Tuple X n)) = Vector X;
Actually, the following equivalent entry is considered the definition of Vector (i.e. it is specifically this entry that is used to compute the name of the vector constructor and its partial
vector {t : Type} # [ t ] = Vector t;
If we expand it using Tuple, we obtain the previous definition exactly.
Conditional fields
The construction
args ::= var-ident-opt : [ conditional-arg-def ] [ ! ] type-term
conditional-arg-def ::= var-ident [ . nat-const ] ?
permits assigning fields which are only present if the value of a preceding mandatory or optional field of type # is not null (or if its chosen bit is not zero if the special binary bit-selection
operator . is applied). Example:
user {fields:#} id:int first_name:(fields.0?string) last_name:(fields.1?string) friends:(fields.2?%(Vector int)) = User fields;
get_users req_fields:# ids:%(Vector int) = Vector %(User req_fields)
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A versatile three-dimensional ray tracing computer program for radio waves in the ionosphere
This report describes an accurate, versatile FORTRAN computer program for tracing rays through an anisotropic medium whose index of refraction varies continuously in three dimensions. Although
developed to calculate the propagation of radio waves in the ionosphere, the program can be easily modified to do other types of ray tracing because of its organization into subroutine. The program
can represent the refractive index by either the Appleton-Hartree or the Sen-Wyller formula, and has several ionospheric models for electron density perturbations to the electron density
(irregularities), the earth's magnetic field and electron collision frequency. For each path, the program can calculate group path length, phase path length, absorption, Doppler shift due to a
time-varying ionosphere, and geometrical path length.
NASA STI/Recon Technical Report N
Pub Date:
October 1975
□ Computer Programs;
□ Earth Ionosphere;
□ Radio Waves;
□ Atmospheric Models;
□ Doppler Effect;
□ Fortran;
□ Hartree Approximation;
□ Ionospheric Electron Density;
□ Numerical Integration;
□ Partial Differential Equations;
□ Radio Transmission;
□ Communications and Radar
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Returning multiple Variables in Python - CodeKyro
Returning multiple Variables in Python
In the previous article saw how parameters are passed and handled in Python, now we will look at how we can return multiple variables in Python. Here is an example program that returns two values:
def add_multiply(x,y):
return sum,mul
When we call the function we receive the two values from add_multiply, later the program prints out the value of both the variables which are initialized with the values from add_multiply. The
function stores values in the form of tuple by default.
Working with returning multiple Variables in Python
We can modify the earlier program to accept further values or to perform various different operations. Here is our modified program, it returns three values and perform additional tasks with
def add_multiply_subtract(x,y):
return sum,mul,sub
Rather than as a tuple we could also choose to return it as a list or dictionary. In the below code we store elements as in list structure and return list elements with just the simple addition of
square brackets.
def add_multiply_subtract(x,y):
return [sum,mul,sub]
So there are multiple ways of returning multiple variables in Python but the most straightforward way is to separate them through the use of commas. In the next article we will look at Global
Variables in Python.
Leave a Comment
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Testing New Operators using the MOEADr
Testing New Operators using the MOEADr Package
Claus Aranha
Let us show how to prepare a new component for the MOEADr framework, add it to an existing algorithmic composition, and perform some basic comparisons. For this example, we will create a simple
variation operator that does not exist in the package in its current state, but the same principle works for adding components of other classes.
Creating a new operator
Consider the following “Gaussian Mutation” operator: given a set \(X\) of solutions \(\mathbf{x_i} \in X\) we add, with probability \(p\), a gaussian noise \(r_{ij} \sim \mathcal{N}(\mu, \sigma)\) to
each \(x_{ij} \in \mathbf{x_i} \in X\). The R code for this operator is as follows:
variation_gaussmut <- function(X, mean = 0, sd = 0.1, p = 0.1, ...) {
# You want to do some error checking on the parameters here
# But for the sake of brevity in this case study, we are skipping it.
R <- rnorm(length(X), # vector of normally distributed values,
mean = mean, # length(R) = nrow(X) * ncol(X)
sd = sd)
R <- R * (runif(length(X)) <= p) # Apply binary mask, probability = p
return (X + R) # Add mutations to the solution matrix
We would like to highlight a few characteristics of the code sample above. First, the name of the function must be in the form variation_[functionname]. The MOEADr package uses the function name
prefixes to perform some automated functions such as listing existing components and error checking. The list of current function prefixes and their meaning is:
• constraint_: Constraint handling components
• decomposition_: Decomposition functions
• ls_: Local search operators
• scalarization_: Scalarization functions
• stop_: Stop criteria components
• uptd_: Update components
• variation_: Variation operators
Second, the parameters in the function definition must include: the solution set matrix \(X\), the local parameters for the function, and finally an ellipsis argument to catch any other inputs passed
by the main moead() call, such as objective values and former solution sets. If you want examples of using these parameters please look at the source code for some of the variation operators
included, such as the Binomial Recombination operator (variation_binrec)
Other component classes follow similar rules. Please look at existing implementations as basis for new ideas.
Testing the new operator
The MOEADr package first searches for operators in the base R environment. Therefore, if you have named your component correctly, all you need to do is add it to the appropriate parameter in the
moead() call.
For example, let us replace the variation stack of the original MOEA/D by our Gaussian Mutation operator, followed by simple truncation, and test it on a standard benchmark function (performing a
qualitative graphical comparison against the original MOEA/D just for fun):
ZDT1 <- make_vectorized_smoof(prob.name = "ZDT1",
dimensions = 30)
problem.zdt1 <- list(name = "ZDT1",
xmin = rep(0, 30),
xmax = rep(1, 30),
m = 2)
myvar <- list() # Initialize variation stack
myvar[[1]] <- list(name = "gaussmut", p = 0.5) # Our new operator
myvar[[2]] <- list(name = "truncate") # Truncation repair operator
results.orig <- moead(problem = problem.zdt1,
preset = preset_moead("original"),
showpars = list(show.iters = "none"),
seed = 42)
results.myvar <- moead(problem = problem.zdt1,
preset = preset_moead("original"),
variation = myvar,
showpars = list(show.iters = "none"),
seed = 42)
Since the function that implements the Gaussian Mutation operator was defined in the main environment, and has the required variation_ prefix, all that we need to do to use it in the moead() function
is to add the necessary parameters to the variation stack. The figures below show the final Pareto front for both the standard MOEA/D and the MOEA/D with a Gaussian Mutation operator. From these
images, it seems (rather unsurprisingly) that the new operator still needs some work.
We welcome new contributions to MOEADr’s component library: users are invited and encouraged to contact us to add their published contributions to the package (or simply clone the GitHub repository
and submit his or her contribution as a pull request).
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The co-end justifies the co-means
Add to your list(s) Download to your calendar using vCal
If you have a question about this talk, please contact Philip Saville.
Ends and coends are generalisations of limits and colimits: if you think of a colimit as a generalised `sum’, then the corresponding coend is a kind of `integral’. They also have close connections to
the Yoneda lemma and Kan extensions. I will recap the definitions of (co)limits, then introduce (co)ends and their basic theory. In particular, I will show how every limit is an end, and every end is
a limit. Finally I will try to present examples of ends and coends at work, and show how they can be extremely useful tools for reasoning in category theory.
• recap of definition of (co)limits,
• definition of (co)ends,
• basic theory of (co)ends: relationship to limits, Fubini theorem
• (co)ends in the wild: Kan extensions, Yoneda, other examples
• basic category theory: functors, natural transformations
• I will cover the definition of limits and colimits, having some intuition already will be useful (eg. as covered in Awodey’s `Category Theory’)
This talk is part of the Logic & Semantics for Dummies series.
This talk is included in these lists:
Note that ex-directory lists are not shown.
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Loài Danio Rerio.pdf (.docx) | Tải miễn phí với 1 click
Tài liệu, văn bản bạn tìm kiếm
Unstructured triangular meshes, The Green's theorem technique, Random observation, Cube Summing, Dynamic Programming without Semirings, Treat high-dimensional problems, Tensor representation,
Hierarchical representation, Low-rank approximation of matrices, Data-sparse representations, Mass spring’s system, Hybrid speed bumps application, Mechanical energy harvesting, Real space approach
for the electronic calculation, The mass-spring, The electronic calculation of twisted bilayer graphene, The hybrid system, The orthogonal polynomial technique, The associated time auto-correlation
function, Atomic structure is quasi-crystalline, User-Defined functions, Function files, Three dimensional plots, Symbolic math, Rank product statistic, Post-genomic data analysis, Molecule
selection, Gamma approximations, Chapter 12: Finite element programming with MALTAB, Finite element programming with MALTAB, Quantitative analysis of fresh tomatoes, Features of Matlab, Rrace of
pesticide residues, Collection of finite element programs, basics of using Matlab, General one-dimensional problems, Markets in Peshawar, error messaObject-Oriented Programming, High performance thin
liquid chromatography technique, Java Programme, Language Basics, Some Syntax, Semanticsges, scalars, Ground Block, Lateral epicondylitis, Derivative Block, Wrist extension, Transfer Fcn Block,
Nirschl technique, Fundamental Types, Facilitating organisational sustainability, KANTBP 4M, Direct Word Sense Matching, Photoleasticity image, MATLAB for dummie, Rhinoceros, State-of-the-art NLP,
Learning to Follow, A CLASS-BASED APPROACH, Approaches to Zero, Clinical Approach, Constituent-Based Morphological Parsing, PhC devices in SOI, A TWO-WAY APPROACH, Expert investment systems, Lexical
Substitution, Data Enveronment, Arrays and Pointers, Coral foundation, The population dynamics of file sharing Peer-to-Peer networks, Introducing MATLAB, Calculation of fracture mechanic parameters
via fem, cấu trúc lập trình C, Working with Arrays Benefits of arrays, Approaches to Coreference Resolution, IPython, Beginning programming with Java for dummies, Navigational Directions, Finite
dimensional vector spaces, Introducing Arrays, LEXICAL DISCOVERY, Adnominal Recognition, A New Approach to the Problem, STRUCTURAL TRANSFER IN MT, Human capital analysis, Scripts, Self-adjoint
system, Mechanical adaptive design, Defining Arrays, Stress wave, Dynamic interaction, Plotting Data in MATLAB, File sharing Peer-to-Peer networks, Cracked plates under different loads, Sequential
arrays, Theory and Practical Recipes, Controlling the flow, Array Concepts, Adam Vogel and Dan Jurafsky, Mitsuko Yamura-Takei, Declaring Array Variables, Maxwell relations, Word-Recognition, Rebecca
Root, tính đa hình trong lập trình hướng đối tượng, Symmetric quadratic functional, Finite dimensional vector, Control of knee continuous passive motion machine, using shell scripting, Transparent
materials, Streamlining MATLAB, Finance industry, Multidimensional lists, Zero-finding, Using program units, Creating Arrays, Peer-to-peer file sharing systems, Analyzing cracked plates via FEM,
Nonlinear dynamics, Surface roughness of laminated glass cut, JavaServer Pages, Modern mathematic, Connecting NumPy, Knee physical training, Accessing MATLAB, Linux expert, Learning to program with
Matlab, Quality of simulated images, Using SQLite, Functions Basic PHP Functions, Gradle Effective, Creating Associative Arrays, Functions of vectors, The Length of an Array, MATLAB Techniques,
Abrasive water jet, Slip element, Linux essentials chapters, scripting language, Thigh and device's linkages, Building GUI Tools, Using loops and arrays, Implementation Guide, Write your own
functions, Array Initializers, Universal Functions, The suitability of the model, Jay A. Kreibich, Weather problems, The Matlab programming, Programming with objects and classes, Manhattan plot,
Colon Notation, Using External Script Files, Command line skills, cơ bản về gỗ lũa, A GUI tool, Data plotting, The abs Functionabs, Buckingham’s π –method, Hubert Klein Ikkink, scripting HTML,
Control Flow Statements, The Plotting vector, Data interaction, The sqrt Function, Experimental average Ra, More GUI techniques, vi phân thương, lệnh DOS, Lý thuyết Floquet, Modified tabu search
algorithm, Basic mechanisms, The single-machine scheduling problem, Integrated review genetics, Multicomponent material, animal anatomy basic body plan, Additive manufacturing technology, Composite
porous cage, Elsevier's integrated review genetics, New meta-heursitic, Supply chain disintermediation, cơ chế cơ khí, Summary of Material science thesis, Porcine study, Chromosomes in the cell, cách
tạo hình chiếu 3D bằng lệnh, tạo hình chiếu từ mô hình 3D, Fem-model the conductivity, lợi ích nước mưa, The tabu-search algorithms, Influence of technological parameters, Means of demand
uncertainty, Mechanisms of inheritance, kiểm soát motion, xây dựng Ryogoku Kokigikan, Multi-component materials, Technological parameters on performance, Additive layer manufacturing, robert a
sterndale, The Semantics of Grammar, Hematologic genetics, nguồn nước dự trữ, Aerospace and Defense, Formalisms Seen as Computer Languages, the denizens of the jungle, Fernando C. N. Pereira and
Stuart M. Shieber, Key Concepts in Communication, ANALYSIS OF SETTINGS, The Design of a Computer Language, Cultural Student, Ecological Orientation, General editor’s preface, Likelihoodism, Design
and analysis of ZVZCS converter with active clamping, Preface to the Second Edition, landscaping, ecological transitions, Projection theory, Reasoning about coincidences, ZVZCS converter with active
clamping, environmental events, bardic function, horizontal, Critias, C++ AMP, intelligent design, Perspective projection, Several tens kilo-watts, Timaeus, probabilities to likelihoods, Accelerated
Massive Parallelism, Projection plane, Partial Parsing, Evaluation of functional modest designs, Plato, criterion of falsifiability, Bitext Projections, A Graph-based Cross-lingual Projection
Approach, Perspective projection characteristics, Women bath wear, Kate Gregory Ade Miller, Benjamin Jowett, total evidence, Weakly Supervised Relation Extraction, Prashanth Mannem and Aswarth Dara,
Oblique projections, Dependency Grammar Induction, Perspective sketch, Modesty features prime importance, Gary Geunbae Lee, Oblique sketches, Bitext Projection Constraints, Socrates' ProposalPlato,
Freehand sketching, Vanishing points, Clothing needs identified, verification format project, Optimal Constituent Alignment, Kuzman Ganchev and Jennifer Gillenwater and Ben Taskar, Oblique projection
geometry, Fracture-related infection, Socrates' Proposal, project ODA, Five-point continuum scale, Edge Covers, Parsing Aligned Parallel Corpus, Oblique projection angle, Tibial shaft fracture,
project start tư, Semantic Projection, Learning Multilingual Subjective Language, Projecting Syntactic Relations, Cross-Lingual Projections, Observational registry, Annotated Source Corpus, hoach
format project, Rada Mihalcea and Carmen Banea, administrator project, Parallel organization, Combine Curves, collective effort, Interactive simulations, projection, Assembly language projects,
Extremum, Reflect Line, Intersection, Parallel Curve, Insights into Non-projectivity in Hindi, Project name Database Model, Prashanth Mannem, Database Model, Himani Chaudhry, Persistent data, Akshar
Bharati, Storage perspective, Multi-Crop, Perspective of the system, Data Model Detail, Manually operated multi-crop inclined plate planter, Bevel gears, Parametric vibration of the prismatic shaft,
Hereditary and nonlinear geometry, The prismatic shaft, Aboriginal Fishing, Rock Lobster Fishery, The holder of a recreational rock, More information on Aboriginal, Aborigines engaged in aboriginal,
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Code, Technical Notes, Design Standards in C#, GHG emission, general Series, STRUCTURAL STEEL DESIGN, versions of .NET and beyond, empirical-experimental, Readability, TENSION MEMBERS, Free-body
model, tài liệu về ram, terminologist and professional, PLASTIC DESIGN, Overlong, Modeling Macroscopic, DESIGNING ILLUSTRATED TEXTS, Elliptic and hyperelliptic curve cryptography, The force free-body
diagrams, iram10up60A, scientific and technical, Declaring tolerances, RESISTANCE FACTOR METHOD, HOW LANGUAGE PRODUCTION IS INFLUENCED, in Porous Media, Hyperelliptic curve cryptography, The Digital
Signature Standard, Technical Writing Made Easier, Limit tolerances, The functional equations’ sensitivities, tài liệu iram10up60A, Statistical Inferenc, Cohomological background on point counting,
HANGERS, Natural Formations, controlled language, GRAPHICS GENERATION, Definition of a Signature Scheme, The geometrical tolerances, Background on pairings, Novel two step random colored grid
process, CONNECTORS, Hydrodynamic Dispersion, Plus tolerances, curvature measures, Computers for individual use, Classification of Attacks, Mathematical Preliminarie, Background on weil descent,
Minus tolerances, WIND-STRESS ANALYSIS, The manufacturing signature, Elementary Stochastic Calculus, Graphical password authentication system, Notebook computers, Geometry and Computing, ESE2000 Q,
Sample Location Problem, Linux Pocket Guide, A Quick Overview, Unix systems, english for technical students, Traditional desktop systems, text editing, PowerShell Pocket Reference, Michael W. Trosset
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Graphing lines with negative slopes in context of two point slope
27 Aug 2024
Title: Graphing Lines with Negative Slopes: A Two-Point Perspective
This article explores the concept of graphing lines with negative slopes using the two-point slope formula. We will delve into the mathematical framework that underlies this process, providing a
comprehensive overview of the formula and its applications.
Graphing lines is a fundamental skill in mathematics, and understanding how to do so with negative slopes is crucial for solving various problems in algebra and geometry. The two-point slope formula
provides a powerful tool for graphing lines, allowing us to determine the equation of a line given two points on the line. In this article, we will focus on applying this formula to graph lines with
negative slopes.
The Two-Point Slope Formula:
The two-point slope formula is given by:
m = (y2 - y1) / (x2 - x1)
where (x1, y1) and (x2, y2) are the coordinates of two points on the line. The slope m represents the rate of change between the two points.
Graphing Lines with Negative Slopes:
When graphing lines with negative slopes, we can use the same formula as above. However, we need to be mindful of the direction in which the line is sloping. A negative slope indicates that the line
is sloping downward from left to right.
To graph a line with a negative slope, we can follow these steps:
1. Identify the two points on the line.
2. Calculate the slope m using the two-point slope formula.
3. Determine the direction of the slope (downward from left to right).
4. Plot one of the points and draw a line through it with the given slope.
In this article, we have explored the concept of graphing lines with negative slopes using the two-point slope formula. By understanding how to apply this formula, students can develop a deeper
appreciation for the mathematical framework that underlies graphing lines. This knowledge will enable them to tackle a wide range of problems in algebra and geometry.
• [Insert relevant references here]
ASCII Formula:
m = (y2 - y1) / (x2 - x1)
Related articles for ‘two point slope ‘ :
• Reading: **Graphing lines with negative slopes in context of two point slope **
Calculators for ‘two point slope ‘
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Math Is Fun Forum
Registered: 2005-06-22
Posts: 4,900
Prime generation method
I came across this method of generating prime numbers recently.
It can be used to list every prime there is, in order, with nothing else.
It's too slow to have useful applications, but I thought it was really interesting, and my mind boggles at the fact that it works at all.
Here is a list of fractions that makes it work:
Using this list, you generate a sequence in the following way:
The first element of the sequence is 2.
To get from one element of the sequence to the next, you start at the top of the list of fractions, and try multiplying by each one in turn. The next member of the sequence is the first integer you
get as a result.
So, all of the fractions from 17/91 down to 13/11 will not produce integers when multiplied by 2.
However, the next one produces 15, and so that is the next element in the sequence.
The sequence starts like this:
2, 15, 875, 725, 1925, 2275, 425, ...
The interesting part is that every so often, an element of the sequence will be a power of 2.
If you ignore all elements of the sequence other than powers of 2 (and also ignore the first 2), you get this:
2^2, 2^3, 2^5, 2^7, 2^11, ...
That is, the powers of two that are produced are precisely the list of prime numbers.
Like I said, this works far too slowly to have any actual use.
Generating the first thousand terms of this sequence will just get you 2, 3, 5, 7.
Generating the first million will list the primes up until 89.
I still think it's neat though.
Why did the vector cross the road?
It wanted to be normal.
Registered: 2005-01-21
Posts: 7,713
Re: Prime generation method
Dare I ask why it works?
"The physicists defer only to mathematicians, and the mathematicians defer only to God ..." - Leon M. Lederman
Registered: 2005-08-29
Posts: 3,588
Re: Prime generation method
...so if we do this in binary, the power of 2 will be easier to detect as it will be a 1 followed by several zeros, and the number of zeros will be the prime number, is that correct?
igloo myrtilles fourmis
Registered: 2005-08-29
Posts: 3,588
Re: Prime generation method
...also I computed the 2, 15, 825, and 725 by looping around the list of fractions...
Are you suppossed to loop around continuously, or should you jump to the first fraction
after you find an integer and skip the remaining fractions below in the list?
Also do you know the author of this creation?
igloo myrtilles fourmis
Registered: 2005-06-22
Posts: 4,900
Re: Prime generation method
I found that it was easier to write numbers in terms of their prime factors.
When I wrote my code to write the first million terms, I used vectors.
The first term is [1,0,0,0,0,0,0,0,0,0].
Then it goes to [0,1,1,0,0,0,0,0,0,0]
Then [0,1,2,0,1,0,0,0,0,0], and so on.
The nth element of the vector means that the number contains that many powers of the nth prime.
You can tell by looking that no number in the sequence will ever have a prime factor higher than 29, which is why we can have length-10 vectors.
A number in the sequence will be significant if and only if all elements of the vector except the first are equal to 0. [In the code, I used A(1) == sum(A)]
To answer your other questions, you should always jump back to the top of the list. So if a number is divisible by 91, you should always multiply it by 17/91.
And the method was designed by John Conway.
I probably should have given my source in the first place, so here it is.
(The closest I am to solving that is figuring out a likely order of magnitude though)
Why did the vector cross the road?
It wanted to be normal.
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Quantum-theory Sentence Examples
• The Heisenberg uncertainty principle in quantum mechanics implies that vacuum fluctuations are present in every quantum theory.
• The main problem of quantum cosmology is the lack of a quantum theory of gravity.
• Einstein quantum theory COMPLEXITY AT LOW REYNOLDS NUMBER.
• In 1925 Heisenberg had developed the first coherent mathematical formalism for quantum theory (Heisenberg, 1925 ).
• By comparison, he seemed to put little value on arguments starting from the mathematical formalism of quantum theory.
• It started with quantum theory and its radical indeterminacy.
• Some of the most important like quantum theory have required great imaginative leaps.
• All the standard interpretations of quantum theory posit that the cosmos changes in accordance with causal principles that are stochastic or probabilistic.
• I didn't understand quantum theory before I entered the cinema and sure as hell didn't when I left.
• String theory is a prime candidate for a quantum theory of gravity.
• I did n't understand quantum theory before I entered the cinema and sure as hell did n't when I left.
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Understanding the Difference Between Accumulators and Counters in Programming
Understanding the Difference Between Accumulators and Counters in Programming
Accumulator and counter are two terms commonly used in the field of computer science and mathematics. While they might seem similar, there is a clear divergence between their meanings and
An accumulator is a device or register that stores the result of a computation or operation. It is used to keep track of intermediate values during a process. Think of it as a bank where you can
deposit and withdraw values as needed. It is commonly used in arithmetic and logical operations, where the result of each operation is accumulated and stored for future use.
On the other hand, a counter is a device or register that keeps a tally or count of something. It is used to keep track of the number of occurrences or repetitions of an event. For example, a counter
can be used to count the number of times a button is pressed or the number of cycles performed by a processor. Unlike an accumulator, a counter is mainly used for counting and does not perform
arithmetic or logical computations.
In summary, the key distinction between an accumulator and a counter lies in their purpose and usage. While an accumulator is used for computation and storing intermediate values, a counter is used
for counting and keeping a record of occurrences. They are both important components of a system, but they serve different functions and play different roles. Understanding the disparity between
these two terms is essential for anyone working in the fields of computer science or mathematics.
Battery and tally disparity
One key distinction between a battery and a tally counter lies in their purpose and function. While both devices are used to count and keep track of numerical values, there is a clear divergence in
how they achieve this task.
A battery, also known as a power reserve or bank, is designed to store and release electrical energy. It is commonly used to power various electronic devices, such as smartphones, laptops, and cars.
The primary function of a battery is to provide a steady source of power to these devices, enabling them to function properly.
On the other hand, a tally counter is a mechanical device used to count and keep track of numerical values. It typically consists of a handheld device with a button or lever that is pressed each time
a count needs to be recorded. Tally counters are commonly used in various industries, such as sports, event management, and inventory tracking. Their purpose is to ensure accurate counting and
recording of quantities.
The disparity between a battery and a tally counter is evident in their design and usage. While a battery stores and releases electrical energy, a tally counter physically records and keeps track of
counts. They serve different functions and cater to different needs, making them distinct devices in their own right.
Reserve and computation distinction
Reserve and accumulator are both terms used in the field of electronics, specifically in relation to the concepts of count and tally. While they may sound similar, there is a crucial difference
between the two.
An accumulator is like a battery that stores and collects data or values over time. It can be thought of as a container that continuously adds up the values it receives, keeping a running total or
sum. In this sense, an accumulator is used for computation, performing calculations based on the values it holds.
A reserve, on the other hand, is more like a bank or a savings account. It also stores data or values, but it does not perform any calculations. Instead, it holds the values in a static or unchanging
state, ready to be accessed or used when needed. It serves as a disparity to an accumulator, simply providing a reserve of values without any active computation.
So, the distinction between an accumulator and a reserve lies in the presence or absence of computation. An accumulator performs calculations based on the values it holds, while a reserve simply
holds values without performing any calculations.
Bank and count divergence
In the context of the difference between accumulators and counters, one important distinction to consider is the divergence between bank and count operations. While both involve keeping track of
numbers, they serve different purposes and operate in different ways.
A bank, similar to an accumulator, is a device used to store and reserve a quantity of something. It acts as a repository or a battery, collecting and storing the amount of something over time. The
emphasis here is on accumulation and preservation of a value.
On the other hand, a count, similar to a counter, is focused on keeping track of the quantity of something in a tally-like manner. Its purpose is to increment or decrement a number based on specific
conditions or events. It serves as a means to count and keep track of occurrences or changes.
The main disparity between a bank and count lies in their primary objective. While a bank seeks to accumulate and hold a reserve, a count emphasizes the act of counting and keeping track of a
numerical value. This divergence is crucial in understanding the different functionalities and applications of accumulators and counters.
Question and Answer:
What is the difference between an accumulator and a counter?
An accumulator is a type of register that stores the result of arithmetic and logical operations, while a counter is a circuit that continuously counts and records the number of input pulses.
What is the disparity between a battery and a tally?
A battery is a device that stores electrical energy for later use, while a tally is a system of counting and keeping track of numerical quantities.
How does a bank differ from a count?
A bank is a financial institution that offers various financial services, such as personal loans and savings accounts, while a count refers to the action of determining the number of something.
What is the distinction between a reserve and a computation?
A reserve is a stock or supply of something that is held in reserve for future use, while a computation refers to the process of calculating or determining a mathematical result.
Can you explain the disparity between an accumulator and a counter in more detail?
Of course! An accumulator is a type of register that stores the intermediate and final results of arithmetic and logical operations performed by a computer’s arithmetic logic unit (ALU). It is used
to hold data for future processing or for storage. On the other hand, a counter is a circuit that continuously counts and records the number of input pulses it receives. Counters are commonly used to
keep track of events or occurrences, such as the number of times a button is pressed or the number of rotations of a motor. They can be implemented using various technologies, such as binary,
modulo-n, or decimal counters. So, while both an accumulator and a counter are used in computing systems, they serve different purposes and have different functionalities.
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Identity Matrix - (Mathematical Methods for Optimization) - Vocab, Definition, Explanations | Fiveable
Identity Matrix
from class:
Mathematical Methods for Optimization
An identity matrix is a square matrix that has ones on the diagonal and zeros elsewhere, effectively acting as the multiplicative identity in matrix algebra. This means that when any matrix is
multiplied by the identity matrix, it remains unchanged, similar to how multiplying a number by one keeps it the same. The identity matrix plays a crucial role in various mathematical processes,
including solving systems of equations and optimization algorithms.
congrats on reading the definition of Identity Matrix. now let's actually learn it.
5 Must Know Facts For Your Next Test
1. The identity matrix for a 2x2 matrix is represented as: $$I = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$$.
2. In an n-dimensional space, the identity matrix is an n x n square matrix with ones on the diagonal and zeros elsewhere.
3. When performing row operations during pivoting, the identity matrix helps track transformations applied to other matrices.
4. In quasi-Newton methods, the identity matrix can be used as an initial approximation of the Hessian matrix, aiding in the convergence of optimization algorithms.
5. The identity matrix remains unchanged when multiplied by itself or any compatible matrix, making it a fundamental concept in linear algebra.
Review Questions
• How does the identity matrix facilitate the process of solving systems of equations through pivoting?
□ The identity matrix plays a key role in the row reduction process used to solve systems of equations. During pivoting, elementary row operations are applied to transform a given augmented
matrix into reduced row echelon form. The presence of the identity matrix allows us to maintain the equivalence of the original system while simplifying it, ensuring that solutions can be
easily derived from this structured format.
• Discuss how the identity matrix is utilized in quasi-Newton methods like BFGS and DFP updates.
□ In quasi-Newton methods such as BFGS and DFP, the identity matrix serves as a starting point for approximating the inverse Hessian. This initial approximation helps guide optimization
algorithms toward finding local minima efficiently. As iterations progress, this approximation is updated using gradient information and previous iterations' outcomes, allowing for improved
estimates that lead to faster convergence.
• Evaluate the impact of using an identity matrix as an initial guess in optimization algorithms on overall computational efficiency.
□ Using an identity matrix as an initial guess in optimization algorithms can significantly enhance computational efficiency. By starting with a simple and well-understood structure, algorithms
can avoid more complex initial conditions that may lead to slower convergence or divergence. This approach streamlines calculations in methods like BFGS and DFP, allowing for quicker updates
and refinements to Hessian approximations based on gradient information, ultimately leading to faster solutions in optimization problems.
© 2024 Fiveable Inc. All rights reserved.
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Fundamental Analysis of TASEKO MINES LTD(NYSEARCA:TGB) stock
Taking everything into account, TGB scores 6 out of 10 in our fundamental rating. TGB was compared to 157 industry peers in the Metals & Mining industry. While TGB has a great profitability rating,
there are quite some concerns on its financial health. TGB is evaluated to be cheap and growing strongly. This does not happen too often!
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Multiplication Chart 1 12 Answers 2024 - Multiplication Chart Printable
Multiplication Chart 1 12 Answers
Multiplication Chart 1 12 Answers – You can get a blank Multiplication Chart if you are looking for a fun way to teach your child the multiplication facts. This may let your little one to complete
the important points on their own. You can find blank multiplication graphs for different product or service ranges, including 1-9, 10-12, and 15 items. You can add a Game to it if you want to make
your chart more exciting. Here are some tips to get the little one started off: Multiplication Chart 1 12 Answers.
Multiplication Charts
You should use multiplication graphs in your child’s college student binder to enable them to remember arithmetic information. Although many kids can commit to memory their mathematics facts
naturally, it requires many others time to do so. Multiplication maps are an excellent way to strengthen their learning and boost their assurance. In addition to being academic, these charts could be
laminated for additional toughness. Listed below are some useful strategies to use multiplication graphs. You can even have a look at websites like these for helpful multiplication fact assets.
This training includes the basics from the multiplication kitchen table. As well as discovering the rules for multiplying, pupils will understand the thought of elements and patterning. Students will
be able to recall basic facts like five times four, by understanding how the factors work. They may also be able to utilize your property of one and zero to fix more advanced items. By the end of the
lesson, students should be able to recognize patterns in multiplication chart 1.
As well as the standard multiplication graph or chart, college students might need to develop a graph or chart with increased variables or fewer aspects. To produce a multiplication graph or chart
with a lot more variables, students need to create 12 desks, every with 12 rows and a few posts. All 12 desks must fit in one sheet of papers. Collections should be driven using a ruler. Graph pieces
of paper is the best for this undertaking. Students can use spreadsheet programs to make their own tables if graph paper is not an option.
Video game concepts
Whether you are educating a newbie multiplication lesson or concentrating on the competence of the multiplication table, you are able to think of exciting and engaging activity tips for
Multiplication Graph 1. A few exciting tips are the following. This game demands the individuals to be in pairs and work about the same dilemma. Then, they are going to all hold up their credit cards
and go over the answer to get a min. If they get it right, they win!
When you’re teaching kids about multiplication, among the best resources you may provide them with is a printable multiplication graph or chart. These printable bedding can come in a variety of
patterns and might be printed on a single web page or several. Children can understand their multiplication information by copying them through the memorizing and chart them. A multiplication graph
may help for a lot of motives, from helping them discover their math concepts specifics to instructing them using a calculator.
Gallery of Multiplication Chart 1 12 Answers
Times Table Grid To 12×12
10 Best Printable Multiplication Tables 0 12 Printablee
Multiplication Table With Answers Printable Printable Multiplication
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owerpoint presentation
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The Harris Poll asked a sample of 1009 adults which causes of death they thought would become more common in the future. Gun violence topped the list: 706 members of the sample thought deaths from guns would increase. ? | Socratic
The Harris Poll asked a sample of 1009 adults which causes of death they thought would become more common in the future. Gun violence topped the list: 706 members of the sample thought deaths from
guns would increase. ?
(a) how to explain what the population proportion p is for this poll ?
(b) How to find a 95% confidence interval for p ?
(c) Harris announced a margin of error of plus or minus three percentage points for this poll result. Is there any effect on answer (b) ?
(d) If the confidence level is 90%, what is the difference between 95% and 90% confidence level ?
1 Answer
a) the population proportion cannot be determined as that represents everyone. The sample, however, allows us to determine a point estimate, $\hat{p}$, which we can assume is close the actual.
The Point estimate is,
$\hat{p} = \left(\text{Everyone with a particular characteristic")/("Everyone in the sample}\right)$
or in this case
$\hat{p} = \frac{706}{1009}$
b) to determine a confidence interval we use the equation,
$\hat{p} - z \cdot \sigma < p < \hat{p} + z \cdot \sigma$
where z is the z-score for your confidence level and $\sigma$ is your standard deviation.
For a 95% confidence z= 1.959963985 $\approx$1.96
$\sigma = \sqrt{\frac{\hat{p} \left(1 - \hat{p}\right)}{n}}$
$\sigma = \sqrt{\frac{\frac{706}{1009} \left(1 - \frac{706}{1009}\right)}{1009}}$
$\sigma = 0.014431$
so subbing into,
$\hat{p} - z \cdot \sigma < p < \hat{p} + z \cdot \sigma$
$\frac{706}{1009} - 1.96 \cdot 0.014431 < p < \frac{706}{1009} + 1.96 \cdot 0.014431$
$0.6714 < p < 0.728$
c) not sure exactly what you mean.
was think of using this equation but I'm not sure what you want to find out.
$n = \frac{{z}^{2} p \left(1 - p\right)}{e} ^ 2$ with e=0.03
d) a smaller confidence interval 95% $\implies$ 90% contains less values that the actual population could represent (remember that $\hat{p}$ is just an estimate of this value)
A 95% confidence interval means that 95% of sample means taken will lie within this range. A 90% confidence interval means that 90% will lie within its range, which is smaller. For example,
95% $\implies$$0.6714 < p < 0.728$
90% $\implies$$0.676 < p < 0.7234$
The 90% confidence interval is smaller and thus we are less confident that the actual population mean lies within its range.
Impact of this question
3887 views around the world
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Visual Basic 2012 Lesson 14- The Math Functions - Visual Basic Tutorial
Visual Basic 2012 Lesson 14- The Math Functions
In this lesson, you will learn how to use the built-in math functions in Visual Basic 2012. There are numerous built-in math functions in Visual Basic 2012. Let’s examine them one by one.
14.1 The Abs function
The Abs function returns the absolute value of a given number.
The syntax is
Math. Abs (number)
* The Math keyword indicates that the Abs function belong to the Math class.
14.2 The Exp function
The Exp of a number x is the exponential value of x, i.e. e^x . For example, Exp(1)=e=2.71828182
The syntax is
Math.Exp (number)
Example 14.1
Private Sub Button1_Click(ByVal sender As System.Object, ByVal e As System.EventArgs) Handles Button1.Click
Dim num1, num2 As Single
num1 = TextBox1.Text
num2 = Math.Exp(num1)
Label1.Text = num2
End Sub
14.3 The Fix Function
The Fix function truncates the decimal part of a positive number and returns the largest integer smaller than the number. However, when the number is negative, it will return smallest integer larger
than the number. For example, Fix(9.2)=9 but Fix(-9.4)=-9
Example 14.2
Private Sub Button1_Click(ByVal sender As System.Object, ByVal e As System.EventArgs) Handles Button1.Click
Dim num1, num2 As Single
num1 = TextBox1.Text
num2 = Fix(num1)
Label1.Text = num2
End Sub
14.4 The Int Function
The Int is a function that converts a number into an integer by truncating its decimal part and the resulting integer is the largest integer that is smaller than he number. For example
Int(2.4)=2, Int(6.9)=6 , Int(-5.7)=-6, Int(-99.8)=-100
14.5 The Log Function
The Log function is the function that returns the natural logarithm of a number. For example, Log(10)=2.302585
Example 14.3
Private Sub Button1_Click(ByVal sender As System.Object, ByVal e As System.EventArgs) Handles Button1.Click
Dim num1, num2 As Single
num1 = TextBox1.Text
num2 = Math.Log(num1)
Label1.Text = num2
End Sub
* The logarithm of num1 will be displayed on label1
14.6 The Rnd( ) Function
Rnd is a very useful function in Visual Basic 2012 . We use the Rnd function to write code that involves chance and probability. The Rnd function returns a random value between 0 and 1. Random
numbers in their original form are not very useful in programming until we convert them to integers. For example, if we need to obtain a random output of 6 integers ranging from 1 to 6, which makes
the program behave like a virtual dice, we need to convert the random numbers to integers using the formula Int(Rnd*6)+1.
Example 14.4
Private Sub Button1_Click(ByVal sender As System.Object, ByVal e As System.EventArgs) Handles Button1.Click
Dim num as integer
End Sub
In this example, Int(Rnd*6) will generate a random integer between 0 and 5 because the function Int truncates the decimal part of the random number and returns an integer. After adding 1, you will
get a random number between 1 and 6 every time you click the command button. For example, let say the random number generated is 0.98, after multiplying it by 6, it becomes 5.88, and using the
integer function Int(5.88) will convert the number to 5; and after adding 1 you will get 6.
14.7 The Round Function
The Round function is a Visual Basic 2012 function that rounds up a number to a certain number of decimal places. The Format is Round (n, m) which means to round a number n to m decimal places. For
example, Math.Round (7.2567, 2) =7.26
Example 14.5
Private Sub Button1_Click(ByVal sender As System.Object, ByVal e As System.EventArgs) Handles Button1.Click
Dim num1, num2 As Single
num1 = TextBox1.Text
num2 = Math.Round(num1, 2)
Label1.Text = num2
End Sub
* The Math keyword here indicates that the Round function belong to the Math class.
[Lesson 13] << [CONTENTS] >> [Lesson 15]
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Managing “all” functions in DAX: ALL, ALLSELECTED, ALLNOBLANKROW, ALLEXCEPT - SQLBI
Because the topic of this article is somewhat intricate, it is a good idea to start with basic DAX theory reminders that will be useful later.
In DAX, these two measures are totally equivalent:
RedSalesCompact :=
Product[Color] = "Red"
RedSalesExtended :=
FILTER ( ALL ( Product[Color] ), Product[Color] = "Red" )
Indeed, the compact syntax (also referred to as a Boolean filter) is translated into the extended syntax by the engine as part of the evaluation of an expression. CALCULATE filters are tables. Even
though they can be written as Boolean expressions, they are always interpreted as tables.
A common practice in computing percentages is to divide a given measure by the same measure where certain filters are removed. For example, a proper expression for the percentage of sales against all
colors would look like this:
PctOverAllColors :=
DIVIDE (
CALCULATE (
ALL ( Product[Color] )
This formula reads as follows:
ALL returns a table containing all the colors; this table represents the valid colors to be used in the new filter context of CALCULATE. Forcing all the colors to be visible is equivalent to
removing any and all filters from the Color column.
This description consists of two sentences: both are wrong. This is not to say that the description is completely wrong. It is accurate most of the time, but not always. The correct description of
the behavior of ALL in the PctOverColors measure above is much simpler indeed:
ALL removes any active filters from the Color column.
In the correct description there is no statement about the result of ALL – in fact, it does not return anything – and there is no equivalence between a table with all values and the removal of a
filter. The reality is much simpler: filters are removed. At first sight, it looks like a very pedantic detail. However, this small difference may yield very different results when used in more
complex DAX expressions.
As an example, let us consider these two measures: NumOfProducts computes the total number of products, whereas NumOfProductsSold only counts products which have been sold, by leveraging table
NumOfProducts :=
DISTINCTCOUNT ( Product[ProductName] )
NumOfProductsSold :=
NumOfProducts is straightforward, whereas NumOfProductsSold requires additional DAX knowledge because it is based on table expansion. Because a table is being used as a filter parameter in CALCULATE,
the filter context contains all the columns of the expanded version of Sales. If you are not familiar with expanded tables, you will find additional resources in Chapter 10 of the book, The
Definitive Guide to DAX.
Consider the query:
MEASURE Sales[NumOfProducts] =
DISTINCTCOUNT ( Product[Product Name] )
ROW (
"NumOfProducts", [NumOfProducts],
"NumOfProductsSold", CALCULATE ( [NumOfProducts], Sales )
The result is:
• NumOfProducts: 2,517
• NumOfProductsSold: 1,170
In presence of a filter context, both measures restrict their calculation to the current filter context. For example, by adding an outer CALCULATETABLE that filters red products, the query becomes:
MEASURE Sales[NumOfProducts] =
DISTINCTCOUNT ( Product[Product Name] )
ROW (
"NumOfProducts", [NumOfProducts],
"NumOfProductsSold", CALCULATE ( [NumOfProducts], Sales )
'Product'[Color] = "Red"
And the result is:
• NumOfProducts: 99
• NumOfProductsSold: 51
So far, everything works exactly as expected. What happens if there is the need to compute the value in the current context against the grand total? For example, the number of red products divided by
the total number of products, and the number of red products sold against the total number of products sold, producing this report:
One might author the code this way:
PercOfProducts =
DIVIDE (
[NumOfProducts], -- Number of products
CALCULATE (
[NumOfProducts], -- Number of products
ALL ( Sales ) -- filtered by ALL Sales
PercOfProductsSold =
DIVIDE (
CALCULATE (
[NumOfProducts], -- Number of products
Sales -- filtered by Sales
CALCULATE (
[NumOfProducts], -- Number of products
ALL ( Sales ) -- filtered by ALL Sales
Surprisingly, this code does not produce the report above. Instead, the result looks like that:
In the PercOfProductsSold column, the percentage for red products is wrong. Here’s an explanation. First, an understanding of the subtle difference between using ALL as a filter remover and using ALL
as a table function is crucial. Let us start from the beginning:
ALL is a table function that returns all the rows of a table or of a set of columns. This is the correct behavior of ALL whenever that result is actually required. In the very specific case of
CALCULATE filters – and only in this specific case – ALL is not used to retrieve values from a table. Instead, ALL is used to remove filters from the filter context. Though the function name is the
same, the semantics of the function is completely different.
ALL, when used as a CALCULATE filter, removes a filter. It does not return a table result.
Using a different name for the different semantics of ALL would have been a good idea. A very reasonable name would have been REMOVEFILTER.
Indeed, Microsoft introduced the REMOVEFILTERS function in Analysis Services 2019 and in Power BI since October 2019.
REMOVEFILTERS is like ALL, but it can only be used as a filter argument in CALCULATE. While REMOVEFILTERS can replace ALL, there is not replacement for ALLEXCEPT and ALLSELECTED used as CALCULATE
Let us understand it better, by examining the denominator of PercOfSoldProducts:
PercOfProductsSold =
DIVIDE (
CALCULATE (
[NumOfProducts], -- Number of products
Sales -- filtered by Sales
CALCULATE (
[NumOfProducts], -- Number of products
ALL ( Sales ) -- filtered by ALL Sales
In this case, ALL is a filter parameter of CALCULATE. As such, it acts as a REMOVEFILTERS, not as an ALL. When CALCULATE evaluates the filter in the denominator, it finds ALL. ALL requires the
removal of any filters from the expanded Sales table, which includes Product[Color]. Thus, the filter is removed but no result is ever returned to CALCULATE.
Because no result is returned, the expanded Sales table is not used as a filter by CALCULATE. At the risk of being pedantic, here is the same code with the REMOVEFILTERS function instead of ALL:
PercOfProductsSold =
DIVIDE (
CALCULATE (
[NumOfProducts], -- Number of products
Sales -- filtered by Sales
CALCULATE (
[NumOfProducts], -- Number of products
REMOVEFILTERS ( Sales ) -- with filters removed by Sales
Using ALL ( Sales ) does not mean, “filter using all the rows in Sales”. It means, “remove any filters from Sales”. With this small change in how the formula reads, it is now clear that the number of
products is the total number of products if no filter is ever applied. Thus, the denominator always computes 2,517 instead of 1,170. This explains why the percentage goes from 4.36% to 2.03%.
This behavior definitely seems strange. Nevertheless, as is often the case with DAX, the behavior is not strange at all, that’s just the way it is. If it does not meet our expectations – then the
problem is our limited knowledge, not the behavior itself.
At this point, it is interesting to look at how to properly write the formula. As shown, ALL is not enough because it does not return its value, it only removes filters. An option is to still use ALL
, but move it inside an outer CALCULATETABLE. By doing this, ALL still behaves like a REMOVEFILTERS, but CALCULATETABLE forces the result back:
PercOfProductsSold =
DIVIDE (
CALCULATE (
CALCULATE (
CALCULATETABLE ( ALL ( Sales ) )
Using CALCULATETABLE outside of ALL looks like a trick, but it is not. It actually changes the semantics of the formula, making it explicit that the result of ALL ( Sales ) is needed in order to
filter the formula. A similar behavior can be obtained with a less elegant formula:
PercOfProductsSold =
DIVIDE (
CALCULATE (
CALCULATE (
FILTER ( ALL ( Sales ), 1 = 1 )
In this case it is FILTER that forces the result of ALL ( Sales ) to be returned, by using a dummy filter with a condition that always evaluates to TRUE.
It is worth noting that all the tables used as filter arguments are, indeed, expanded tables. Therefore, the action of removing filters impacts not only the base table but the entire expanded table.
ALL ( Sales ) acts as REMOVEFILTERS on the expanded version of Sales, removing filters from the table and from all related dimension.
This behavior is particularly important in the case of ALLEXCEPT. Consider the following measure:
NoFilterOnProduct =
CALCULATE (
[Sales Amount],
ALLEXCEPT ( Sales, Sales[ProductKey] )
One might think that ALLEXCEPT removes all filters from the columns in the Sales table except for the ProductKey column. However, the behavior is noticeably different. ALLEXCEPT removes filters from
the expanded version of Sales, which includes all the tables that can be reached through a many-to-one relationship starting from Sales. This includes customers, dates, stores, and any other
dimension table.
The following syntax prevents ALLEXCEPT from removing filters from a specific table:
NoFilterOnProduct =
CALCULATE (
[Sales Amount],
ALLEXCEPT (
In this example, ALLEXCEPT uses one column and three tables as arguments. One can use any table or column that is contained in the expanded version of the table used as the first argument.
The behavior shown in this article applies to four functions: ALL, ALLNOBLANKROW, ALLEXCEPT and ALLSELECTED. They are usually referred to as the ALLxxx functions. Importantly, ALL and ALLNOBLANKROW
hide no other surprises, whereas ALLSELECTED is a very complex function. ALLSELECTED is thoroughly covered in the article, The definitive guide to ALLSELECTED. ALLSELECTED merges two of the most
complex behaviors of DAX in a single function: shadow filter contexts and acting as REMOVEFILTERS instead of a regular filter context intersection.
For anyone wondering what the most complex DAX function is, now there is a clear winner: it is ALLSELECTED.
Context transition
Evaluates an expression in a context modified by filters.
CALCULATE ( <Expression> [, <Filter> [, <Filter> [, … ] ] ] )
CALCULATE modifier
Returns all the rows in a table, or all the values in a column, ignoring any filters that might have been applied.
ALL ( [<TableNameOrColumnName>] [, <ColumnName> [, <ColumnName> [, … ] ] ] )
Context transition
Evaluates a table expression in a context modified by filters.
CALCULATETABLE ( <Table> [, <Filter> [, <Filter> [, … ] ] ] )
CALCULATE modifier
Clear filters from the specified tables or columns.
REMOVEFILTERS ( [<TableNameOrColumnName>] [, <ColumnName> [, <ColumnName> [, … ] ] ] )
CALCULATE modifier
Returns all the rows in a table except for those rows that are affected by the specified column filters.
ALLEXCEPT ( <TableName>, <ColumnName> [, <ColumnName> [, … ] ] )
CALCULATE modifier
Returns all the rows in a table, or all the values in a column, ignoring any filters that might have been applied inside the query, but keeping filters that come from outside.
ALLSELECTED ( [<TableNameOrColumnName>] [, <ColumnName> [, <ColumnName> [, … ] ] ] )
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Lesson 13
Use Bar Graphs to Compare
Warm-up: True or False: Make Ten with 9 (10 minutes)
The purpose of this warm-up is to activate students’ previous experiences in which they looked for ways to make a ten—specifically, when one addend is 9. The ability to make a ten will help students
develop fluency within 20 and will be helpful later in this lesson and in upcoming lessons when students add and subtract within 20.
When students look for ways to make a ten and notice similarities in the addends and values in each of the expressions, they look for and make use of the structure of whole numbers and the properties
of operations (MP7).
• Display one statement.
• “Give me a signal when you know whether the statement is true and can explain how you know.”
• 1 minute: quiet think time
• Share and record answers and strategy.
• Repeat with each statement.
Student Facing
Decide if each statement is true or false. Be prepared to explain your reasoning.
• \(9 + 4 = 9 + 1 + 3\)
• \(9 + 4 = 10 + 3\)
• \(9 + 5 = 10 + 6\)
Activity Synthesis
• "How can you justify your answer without adding?" (I see that 9 + 4 and 9 + 1 + 3 are the same because 1 + 3 = 4.)
Activity 1: What’s the Difference? (15 minutes)
The purpose of this activity is for students to use a bar graph to compare two quantities and describe the methods they use to find the unknown difference. Monitor for students who draw on the graph
and describe ways of finding the difference by counting on or counting back. If students draw on their graph or do not discuss both counting methods during the activity, create and display work so
that each method can be discussed during the synthesis. The discrete segments of the bar graph are used to elicit these counting methods (MP5), however, some students may use addition or subtraction,
including known sums, to find the difference. Encourage these students to connect their methods to the counting methods shared in the synthesis.
MLR8 Discussion Supports. Provide all students with an opportunity to speak and to practice using comparison statements. Invite students to chorally repeat statements that use “more” and “fewer” in
unison 1–2 times.
Advances: Speaking, Listening
• Groups of 2
• Display the image (graph with no scale).
• “A group of third grade students were asked, ‘What pets do you have?’ Their responses are shown in the bar graph."
• "What do you notice about the data in the graph? What do you wonder?”
• 1 minute: quiet think time
• 1 minute: partner discussion
• Monitor for students to say: “More students have cats than have rabbits,” or wonder, “How many more students have cats than have rabbits?”
• Share and record student responses.
• If students do not make statements using “more” or “fewer,” display:
• “There are more _____ than _____.”
• “There are fewer _____ than _____.”
• “How could you complete the sentences to make them true statements about the graph?”
• 1 minute: quiet think time
• 1 minute: partner discussion
• Share responses.
• “You noticed that more students have cats than have rabbits. Your job now is to figure out how many more students have cats than have rabbits. Think about two different ways you can find the
answer and record them.”
• 3–4 minutes: independent work time
• 2–3 minutes: partner discussion
• Monitor for a student who uses a counting on method and a student who uses a counting back method.
Student Facing
A group of third grade students were asked, "What pets do you have?" Their responses are shown in the bar graph.
What do you notice? What do you wonder?
Their responses are also shown in this bar graph.
How many more students have cats than have rabbits? Show two ways to find the difference.
Activity Synthesis
• Invite previously identified students to share how they found the difference. If methods are only explained verbally, consider asking, “Can you show us on the bar graph how you could count on (or
count back) to find how many more?”
• Record student methods with an equation. For example, for students who describe counting on, write “8 + 9 = 17.”
• "How are their methods similar? How are their methods different?" (One method starts with the smaller number of pets and counts on to the larger number, like adding 9 more. The other starts with
the larger number of pets and counts back to the smaller number, like subtracting 9.)
Activity 2: Dogs in the Park (20 minutes)
The purpose of this activity is for students to use graphs to make comparison statements and solve Compare problems. Students represent their comparisons with equations. In the synthesis, students
connect the graph, their comparison statements, and their equations.
When students describe how they see their equations in the graph and how their equations relate to the context, they think abstractly and quantitatively (MP2).
Representation: Access for Perception. Display information in a flexible format to facilitate comparisons. Create a display of the graph with detachable bars, so that types of dogs that do not appear
next to each other in the original graph can be more easily compared. For example, number of poodles and number of pugs.
Supports accessibility for: Visual-Spatial Processing, Conceptual Processing
• Groups of 2
• Display the Dogs in the Park graph.
• “What statement can you make that compares the number of huskies to the number of bulldogs?” (There are more bulldogs than huskies. There are fewer huskies than bulldogs. There are 6 more
bulldogs than huskies.)
• 1 minute: partner discussion
• 1 minute: whole-class discussion
• Share statements that use “more” and “fewer.”
• “Now you are going to write some statements using ‘more’ and ‘fewer’ and write equations to show how to find the difference.”
• 5 minutes: independent work time
• “Now check with your partner. Are their statements true?”
• 3 minutes: partner discussion
• Monitor for different equations students use for each comparison.
Student Facing
Kiran and Lin counted the types of dogs they saw in a park. Their data is shown in the bar graph.
1. Make this statement true: There are more
__________________________ than ___________________________.
2. Write an addition and a subtraction equation to show how many more.
3. Make this statement true: There are fewer
___________________________ than ____________________________.
4. Write an addition and subtraction equation to show how many fewer.
Advancing Student Thinking
If students write statements that are not true, consider asking:
• "How did you decide which group had fewer?"
• "Where do you see this on the graph?"
Activity Synthesis
• Invite 1–2 previously identified students to share their statements using fewer and one equation. (Sample response: There are fewer pugs than bulldogs. There are 12 fewer. 8 + 12 = 20)
• Record equation.
• “What other equations could we write to show how many fewer for _____’s statement?” (Sample response: We could use subtraction. 20 – 12 = 8 shows you can count back 12 from 20 to get to 8.)
• Record equations.
• For each equation, ask, “What does each number represent in the equation?” (20 represents the number of bulldogs. 8 represents the number of pugs. 12 represents how many fewer pugs.)
Lesson Synthesis
“Today, we learned that there are different ways we can talk about comparisons and write equations to represent them.”
Display graph from Activity 2.
Display: \(14 + 6 = 20\)
“6 is the answer. What is the question?” (Sample response: How many more bulldogs are there than huskies?”)
Consider asking: “How did you use the graph? What did you look for?”
If time permits (or if \(14 + 6 = 20\) was discussed in Activity 2 Synthesis):
Display: \(8 - 2 = 6\)
“2 is the answer. What is the question?” (Sample response: How many fewer poodles are there than pugs?)
Consider asking: “How did you use the graph? What did you look for?”
Cool-down: Second Grade Absences (5 minutes)
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CS267 Handout 1
CS267 Handout 1: Class Survey
Please fill this out, electronically if possible, and turn it in to the TA (fredwong@cs).
Full Name:
Email, URL (if you have one):
Enrolled, auditting, or undecided? (Auditors will be strongly encouraged to take the class P/NP, where a P will be awarded for satisfactory class participation over the course of the semester.)
When would you be free for a weekly 1 hour discussion section? Please include 5-6pm in your schedule, since we want to have some common time with CS258, and it may only be possible to find a large
enough room after 5pm.
Department, year in grad school:
Phone, campus mail address, email address:
Do you have access to a UNIX workstation with internet access?
Relevant computing background (machines, languages used):
Relevant mathematics background (numerical analysis, engineering, modeling, physics, etc.):
Briefly describe your most ambitious (or relevant) programming project.
Why do you want to take this class?
Do you have a particular problem/application you'd like to parallelize?
Please fill in the following table, indicating your familiarity with the listed topics. We will cover some or all of these topics during the class, but I want to know what people know.
Quite Somewhat Know what Unfamilar
familiar familiar it is
UNIX (any flavor)
Computer Block Diagram
Memory Hierarchy
Race Condition
Graph algorithms
Traveling Salesman Problem
Sorting algorithms
Numerical Stability
Matrix Multiplication
Gaussian Elimination
Eigenvalues and Eigenvectors
Newton's Laws of Motion
Laplaces's or Poisson's equation
Heat equation
Wave equation
Finite difference methods
Successive Overrelaxation
Fast Fourier Transform
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9 Times Table Multiplication Chart – Times Table Club
9 Times Table
9 Times Table is the multiplication table of the number 9 provided here for students, parents and teachers. These times tables are helpful in solving math questions quickly and easily.
How to read the 9 times table?
One times nine is 9, two times nine is 18, three times nine is 27, ect.
How to memorise the 9 times tables orally?
Write the 9 times tables on a sheet of paper and read them aload repeatedly.
What is an example math question using the 9 times tables?
Maths Question: A girl picks 9 flowers from her garden every day. How many flowers will the girl pick in 1 week (7 days)?
Solution: The girl picks 9 flowers per day. Therefore, using the 9 times table chart, the total number of flowers picked by the girl in 1 week is 9 × 7 = 63 flowers.
│Tables 1 to 20 │Multiplication Table │
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Variable-Density Groundwater Flow
It can be seen that the density does not vary between the screens, so this means the point water hydraulic heads can be used directly in the calculation of q[z ]with Darcy’s law. Given the negative
numbers, the smallest absolute value is the highest hydraulic head so the hydraulic head increases with depth, thus flow is upward from screen 3 to screen 2. The factor of 1000 in the
equation converts meters to millimeters.
While from screen 2 to 1 it is
For the more general case where the density varies, one would first convert the hydraulic heads to freshwater heads using q[z]):
Screen h[f]
1 −2.815
2 −2.679
3 −2.542
And for the flow between piezometer 2 and 1
Which is the same answer as before (note that the unrounded result has a 0.43 percent error because K[f] was equated to the hydraulic conductivity).
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How Many Meters In A Kilogram - 666how.com
How Many Meters In A Kilogram
A kilogram (kg) is a unit of mass in the metric system, which is the official system of measurement used by most countries around the world. It is equal to 1,000 grams and is sometimes referred to as
the “kilogram-force.” The kilogram is a base unit in the International System of Units (SI), which is the modern form of the metric system.
The metric system uses prefixes to indicate values that are larger or smaller than one thousand. For example, “kilo” means one thousand, so a kilogram is equal to 1,000 grams. Similarly, “milli”
means one thousandth, so a milligram is equal to 0.001 grams.
When it comes to measuring length, however, the metric system does not use prefixes like “kilo” or “milli.” Instead, it uses different units for different lengths. The basic unit for length in the
metric system is the meter (m), and all other units are defined as fractions or multiples of this basic unit.
So how many meters are in a kilogram? The answer is simple: zero! A kilogram is a unit of mass and a meter is a unit of length; they are two completely different measurements that cannot be directly
To make matters more confusing, there are actually two different kinds of meters: scientific meters and everyday meters. Scientific meters measure very small distances such as those between atoms or
molecules; everyday meters measure much larger distances such as those between cities or countries.
In other words, even if you could convert from kilograms to meters, you would still need to specify which type of meter you were talking about in order to get an accurate answer. This can be
confusing for people who are not familiar with the metric system or scientific measurements.
In short, there is no direct conversion from kilograms to meters because they are two completely different measurements with no direct relationship between them. However, if you wanted to convert
from one unit of mass (such as kilograms) to another unit of length (such as meters), you would need to know what kind of meter you were converting to and then use specific conversion factors based
on that particular kind of meter.
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The Critical Role of Mathematics in Economic Analysis
(This appendix should be consulted after first reading Welcome to Economics!) Economics is not math. There is no important concept in this course that cannot be explained without mathematics. That
said, math is a tool that can be used to illustrate economic concepts. Remember the saying a picture is worth a thousand words? Instead of a picture, think of a graph. It is the same thing.
Economists use models as the primary tool to derive insights about economic issues and problems. Math is one way of working with (or manipulating) economic models.
There are other ways of representing models, such as text or narrative. But why would you use your fist to bang a nail, if you had a hammer? Math has certain advantages over text. It disciplines your
thinking by making you specify exactly what you mean. You can get away with fuzzy thinking in your head, but you cannot when you reduce a model to algebraic equations. At the same time, math also has
disadvantages. Mathematical models are necessarily based on simplifying assumptions, so they are not likely to be perfectly realistic. Mathematical models also lack the nuances which can be found in
narrative models. The point is that math is one tool, but it is not the only tool or even always the best tool economists can use. So what math will you need for this book? The answer is: little more
than high school algebra and graphs. You will need to know:
In this text, we will use the easiest math possible, and we will introduce it in this appendix. So if you find some math in the book that you cannot follow, come back to this appendix to review. Like
most things, math has diminishing returns. A little math ability goes a long way; the more advanced math you bring in, the less additional knowledge that will get you. That said, if you are going to
major in economics, you should consider learning a little calculus. It will be worth your while in terms of helping you learn advanced economics more quickly.
Often economic models (or parts of models) are expressed in terms of mathematical functions. What is a function? A function describes a relationship. Sometimes the relationship is a definition. For
example (using words), your professor is Adam Smith. This could be expressed as Professor = Adam Smith. Or Friends = Bob + Shawn + Margaret.
Often in economics, functions describe cause and effect. The variable on the left-hand side is what is being explained (â the effectâ ). On the right-hand side is what is doing the explaining (â
the causesâ ). For example, suppose your GPA was determined as follows: GPA = 0.25 Ã combined_SAT + 0.25 Ã class_attendance + 0.50 Ã hours_spent_studyingGPA = 0.25 Ã combined_SAT + 0.25 Ã
class_attendance + 0.50 Ã hours_spent_studying
This equation states that your GPA depends on three things: your combined SAT score, your class attendance, and the number of hours you spend studying. It also says that study time is twice as
important (0.50) as either combined_SAT score (0.25) or class_attendance (0.25). If this relationship is true, how could you raise your GPA? By not skipping class and studying more. Note that you
cannot do anything about your SAT score, since if you are in college, you have (presumably) already taken the SATs.
Of course, economic models express relationships using economic variables, like Budget = money_spent_on_econ_books + money_spent_on_music, assuming that the only things you buy are economics books
and music.
Most of the relationships we use in this course are expressed as linear equations of the form: y = b + mxy = b + mx
Graphs are useful for two purposes. The first is to express equations visually, and the second is to display statistics or data. This section will discuss expressing equations visually.
To a mathematician or an economist, a variable is the name given to a quantity that may assume a range of values. In the equation of a line presented above, x and y are the variables, with x on the
horizontal axis and y on the vertical axis, and b and m representing factors that determine the shape of the line. To see how this equation works, consider a numerical example: y = 9 + 3xy = 9 + 3x
In this equation for a specific line, the b term has been set equal to 9 and the m term has been set equal to 3. Table A1 shows the values of x and y for this given equation. Figure A1 shows this
equation, and these values, in a graph. To construct the table, just plug in a series of different values for x, and then calculate what value of y results. In the figure, these points are plotted
and a line is drawn through them.
This example illustrates how the b and m terms in an equation for a straight line determine the shape of the line. The b term is called the y-intercept. The reason for this name is that, if x = 0,
then the b term will reveal where the line intercepts, or crosses, the y-axis. In this example, the line hits the vertical axis at 9. The m term in the equation for the line is the slope. Remember
that slope is defined as rise over run; more specifically, the slope of a line from one point to another is the change in the vertical axis divided by the change in the horizontal axis. In this
example, each time the x term increases by one (the run), the y term rises by three. Thus, the slope of this line is three. Specifying a y-intercept and a slopeâ that is, specifying b and m in the
equation for a lineâ will identify a specific line. Although it is rare for real-world data points to arrange themselves as an exact straight line, it often turns out that a straight line can offer
a reasonable approximation of actual data.
The concept of slope is very useful in economics, because it measures the relationship between two variables. A positive slope means that two variables are positively related; that is, when x
increases, so does y, or when x decreases, y decreases also. Graphically, a positive slope means that as a line on the line graph moves from left to right, the line rises. The length-weight
relationship, shown in Figure A3 later in this Appendix, has a positive slope. We will learn in other chapters that price and quantity supplied have a positive relationship; that is, firms will
supply more when the price is higher.
A negative slope means that two variables are negatively related; that is, when x increases, y decreases, or when x decreases, y increases. Graphically, a negative slope means that, as the line on
the line graph moves from left to right, the line falls. The altitude-air density relationship, shown in Figure A4 later in this appendix, has a negative slope. We will learn that price and quantity
demanded have a negative relationship; that is, consumers will purchase less when the price is higher.
A slope of zero means that there is no relationship between x and y. Graphically, the line is flat; that is, zero rise over the run. Figure A5 of the unemployment rate, shown later in this appendix,
illustrates a common pattern of many line graphs: some segments where the slope is positive, other segments where the slope is negative, and still other segments where the slope is close to zero.
The slope of a straight line between two points can be calculated in numerical terms. To calculate slope, begin by designating one point as the â starting pointâ and the other point as the â end
pointâ and then calculating the rise over run between these two points. As an example, consider the slope of the air density graph between the points representing an altitude of 4,000 meters and an
altitude of 6,000 meters:
Rise: Change in variable on vertical axis (end point minus original point) = 0.100 â 0.307 = â 0.207 = 0.100 â 0.307 = â 0.207
Run: Change in variable on horizontal axis (end point minus original point) = 6,000 â 4,000= 2,000= 6,000 â 4,000= 2,000
Thus, the slope of a straight line between these two points would be that from the altitude of 4,000 meters up to 6,000 meters, the density of the air decreases by approximately 0.1 kilograms/cubic
meter for each of the next 1,000 meters.
Suppose the slope of a line were to increase. Graphically, that means it would get steeper. Suppose the slope of a line were to decrease. Then it would get flatter. These conditions are true whether
or not the slope was positive or negative to begin with. A higher positive slope means a steeper upward tilt to the line, while a smaller positive slope means a flatter upward tilt to the line. A
negative slope that is larger in absolute value (that is, more negative) means a steeper downward tilt to the line. A slope of zero is a horizontal flat line. A vertical line has an infinite slope.
Suppose a line has a larger intercept. Graphically, that means it would shift out (or up) from the old origin, parallel to the old line. If a line has a smaller intercept, it would shift in (or
down), parallel to the old line.
Economists often use models to answer a specific question, like: What will the unemployment rate be if the economy grows at 3% per year? Answering specific questions requires solving the â systemâ
of equations that represent the model.
Suppose the demand for personal pizzas is given by the following equation: Qd = 16 â 2PQd = 16 â 2P
where Qd is the amount of personal pizzas consumers want to buy (i.e., quantity demanded), and P is the price of pizzas. Suppose the supply of personal pizzas is: Qs = 2 + 5PQs = 2 + 5P
Finally, suppose that the personal pizza market operates where supply equals demand, or Qd = QsQd = Qs
We now have a system of three equations and three unknowns (Qd, Qs, and P), which we can solve with algebra:
Since Qd = Qs, we can set the demand and supply equation equal to each other: Qd = Qs16 â 2P = 2 + 5PQd = Qs16 â 2P = 2 + 5P
Subtracting 2 from both sides and adding 2P to both sides yields: 16 â 2P â 2 = 2 + 5P â 214 â 2P = 5P14 â 2P + 2P = 5P + 2P14 = 7P147 = 7P72 = P16 â 2P â 2 = 2 + 5P â 214 â 2P =
5P14 â 2P + 2P = 5P + 2P14 = 7P147 = 7P72 = P
In other words, the price of each personal pizza will be $2. How much will consumers buy?
Taking the price of $2, and plugging it into the demand equation, we get: Qd = 16 â 2P = 16 â 2(2) = 16 â 4 = 12Qd = 16 â 2P = 16 â 2(2) = 16 â 4 = 12
So if the price is $2 each, consumers will purchase 12. How much will producers supply? Taking the price of $2, and plugging it into the supply equation, we get: Qs = 2 + 5P = 2 + 5(2) = 2 + 10 =
12Qs = 2 + 5P = 2 + 5(2) = 2 + 10 = 12
So if the price is $2 each, producers will supply 12 personal pizzas. This means we did our math correctly, since Qd = Qs.
If algebra is not your forte, you can get the same answer by using graphs. Take the equations for Qd and Qs and graph them on the same set of axes as shown in Figure A2. Since P is on the vertical
axis, it is easiest if you solve each equation for P. The demand curve is then P = 8 â 0.5Qd and the supply curve is P = â 0.4 + 0.2Qs. Note that the vertical intercepts are 8 and â 0.4, and the
slopes are â 0.5 for demand and 0.2 for supply. If you draw the graphs carefully, you will see that where they cross (Qs = Qd), the price is $2 and the quantity is 12, just like the algebra
We will use graphs more frequently in this book than algebra, but now you know the math behind the graphs.
Growth rates are frequently encountered in real world economics. A growth rate is simply the percentage change in some quantity. It could be your income. It could be a businessâ s sales. It could be
a nationâ s GDP. The formula for computing a growth rate is straightforward: Percentage change = Change in quantityQuantityPercentage change = Change in quantityQuantity
Suppose your job pays $10 per hour. Your boss, however, is so impressed with your work that he gives you a $2 per hour raise. The percentage change (or growth rate) in your pay is $2/$10 = 0.20 or
To compute the growth rate for data over an extended period of time, for example, the average annual growth in GDP over a decade or more, the denominator is commonly defined a little differently. In
the previous example, we defined the quantity as the initial quantityâ or the quantity when we started. This is fine for a one-time calculation, but when we compute the growth over and over, it
makes more sense to define the quantity as the average quantity over the period in question, which is defined as the quantity halfway between the initial quantity and the next quantity. This is
harder to explain in words than to show with an example. Suppose a nationâ s GDP was $1 trillion in 2005 and $1.03 trillion in 2006. The growth rate between 2005 and 2006 would be the change in GDP
($1.03 trillion â $1.00 trillion) divided by the average GDP between 2005 and 2006 ($1.03 trillion + $1.00 trillion)/2. In other words: = $1.03 trillion â $1.00 trillion($1.03 trillion + $1.00
trillion) / 2 = 0.031.015 = 0.0296 = 2.96% growth = $1.03 trillion â $1.00 trillion($1.03 trillion + $1.00 trillion) / 2 = 0.031.015 = 0.0296 = 2.96% growth
Note that if we used the first method, the calculation would be ($1.03 trillion â $1.00 trillion) / $1.00 trillion = 3% growth, which is approximately the same as the second, more complicated
method. If you need a rough approximation, use the first method. If you need accuracy, use the second method.
A few things to remember: A positive growth rate means the quantity is growing. A smaller growth rate means the quantity is growing more slowly. A larger growth rate means the quantity is growing
more quickly. A negative growth rate means the quantity is decreasing.
The same change over times yields a smaller growth rate. If you got a $2 raise each year, in the first year the growth rate would be $2/$10 = 20%, as shown above. But in the second year, the growth
rate would be $2/$12 = 0.167 or 16.7% growth. In the third year, the same $2 raise would correspond to a $2/$14 = 14.2%. The moral of the story is this: To keep the growth rate the same, the change
must increase each period.
For centuries, economics was largely a verbal discourse, with theories explained through logical arguments and inferences. However, the field has evolved to rely extensively on mathematical analysis.
Today, mathematics plays an indispensable role in how economists develop, test, and apply theories to explain real-world phenomena.
In this comprehensive guide, we’ll explore the emergence of mathematical economics, its advantages and criticisms, and the many ways math enables economic insights that drive policies and decisions.
A Brief History of Math in Economics
Up until the 19th century economics lacked mathematical rigor. Theories were expressed verbally and economists used situational logic, anecdotes, and metaphors to describe economic principles.
There were obvious limitations to this approach. Verbal arguments could easily become vague or ambiguous, and different models could seemingly explain the same phenomenon. There was no definitive way
to analyze or measure economic changes.
Pioneers like William Stanley Jevons, Léon Walras, Francis Ysidro Edgeworth, and Vilfredo Pareto spearheaded the “marginal revolution” in the late 1800s. They proposed using calculus and mathematical
equations to model utility, prices, production, and other economic variables.
The mathematical approach enabled economists to quantify changes, make empirical predictions and test hypotheses against real data. It quickly became an essential part of formulating and proving
economic theories.
Econometrics emerged in the 20th century to fuse mathematical models with statistical analysis of economic data. With the rise of computing power, econometric and mathematical techniques now dominate
modern economics research and policy decisions.
Key Benefits of the Mathematical Approach
There are good reasons mathematics has become so indispensable to economics:
• Precision: Math enables economists to define assumptions and relationships exactly, avoiding ambiguity.
• Logic: Mathematical proofs apply deductive logic to derive conclusions from core assumptions.
• Measurement: Math allows quantifying economic changes and the impact of variables.
• Testing: Models can be empirically tested using real-world economic data.
• Prediction: Validated mathematical models can forecast future economic developments.
• Optimization: Math enables finding ideal solutions, like maximizing utility or profits.
Without mathematics, economics would lack rigor and predictive power. Math brings clarity, rigor, and decisiveness to assessing economic theories against empirical facts.
Major Areas of Mathematical Economics
Mathematical techniques now permeate nearly every area of economics. Here are some of the key fields:
• Microeconomics: Analyzes decision making by individuals and firms using calculus and optimization. Clearly defines concepts like utility.
• Macroeconomics: Uses mathematical models to understand economic growth, business cycles, unemployment, inflation, and the impact of monetary and fiscal policies.
• Econometrics: Combines statistics, mathematics, and software to estimate economic relationships, test hypotheses, and forecast trends.
• Game Theory: Studies strategic decision making using mathematical models to examine interactions between parties and predict outcomes.
• International Economics: Quantifies global trade and capital flows using mathematical finance models and statistical analysis.
• Labor Economics: Measures factors impacting employment, wages, and productivity based on mathematical functions and empirical data analysis.
• Public Economics: Models government taxation, spending, and social welfare policies mathematically to predict impacts.
Essentially, math touches every economic subfield in some way. It enables turning conceptual models into testable, quantifiable representations of the real economy.
Mathematical Tools Used in Economics
Economists employ a vast arsenal of mathematical concepts and methods:
• Calculus: Used ubiquitously to model dynamic economic processes involving rates of change. Enables optimization problems.
• Linear Algebra: Applied to model multiple interacting economic variables and solve systems of equations. Allows econometric analysis.
• Differential Equations: Represent economic relationships evolving over time. Enable analyzing growth, business cycles, and other dynamical systems.
• Vector Analysis: Used in spatial economics to study geographic distribution effects. Also used in general equilibrium theory.
• Matrices: Help model multiple interrelated markets and complex relationships between economic variables.
• Game Theory: Provides mathematical representations of strategic interactions between economic agents.
• Probability & Statistics: Enable measurement of economic data and uncertainties. Used in econometrics.
• Numerical Analysis: For complex models not conducive to analytical solutions. Allows economic simulations.
Economists also increasingly rely on sophisticated software tools that implement mathematical models and methods.
Criticisms of Mathematical Economics
Despite its wide use, mathematical economics has drawbacks:
• Unrealistic assumptions: Simplifying human behavior into mathematical terms is inherently limited.
• False precision: Formulaic models imply greater accuracy than possible for social sciences.
• Data limitations: Available economic data is often incomplete, low quality, or insufficient for techniques used.
• Ambiguity: Relationships without clear mathematical definitions create uncertainty in interpretations.
• Overreliance on math: Focus on technical elegance over practical application to the real economy.
• Excessive complexity: Some techniques lack intuitive comprehension and oversight.
• Misapplication: Inappropriate methods yield misleading or nonsensical results.
• Unclear communication: Mathematical economics papers are often inaccessible to lay policymakers.
Essentially, math should complement economics, not entirely subsume it. Human judgment is still crucial to develop reasonable assumptions and interpret results appropriately.
Using Math as an Economist
For working economists, math fluency is mandatory. Here are some key applications:
• Develop theoretical economic models using mathematical definitions, logic, and proofs.
• Quantify relationships between economic variables based on empirical data.
• Forecast future economic trends through modeling and statistical analysis.
• Evaluate the impacts of potential policy decisions through simulations.
• Research and develop new econometric techniques to apply in studies.
• Collaborate with software engineers to implement models computationally.
• Clearly communicate mathematical analysis through reports, policy briefs, and recommendations.
A strong foundation in mathematics, statistics, and programming is essential for today’s economists. Specialized skills in econometrics, financial modeling, or other techniques offer a competitive
The Future of Math in Economics
If anything, mathematics will play an even greater role in economics going forward. Some key trends:
• Ever-larger datasets require more advanced econometric methods to analyze.
• Computational power enables more complex models previously impossible.
• Data science and machine learning techniques may complement traditional econometrics.
• Economics is becoming increasingly interdisciplinary, incorporating new fields like behavioral economics, neuroeconomics, and complexity economics.
• Specialized software lowers barriers for applying mathematical models and running simulations.
Economics will continue relying on mathematics to adapt to emerging realities and create vital insights for economic prosperity.
Tracing its origins to pioneering work in the 19th century, mathematical economics has become essential to economic science. Math enables economists to move beyond verbal reasoning to create precise,
testable models with quantifiable predictions. As economics continues maturing as a data-driven field, mathematics and statistics will play an ever-greater role in shaping economic ideas, policies,
and outcomes.
Displaying Data Graphically and Interpreting the Graph
Graphs are also used to display data or evidence. Graphs are a method of presenting numerical patterns. They condense detailed numerical information into a visual form in which relationships and
numerical patterns can be seen more easily. For example, which countries have larger or smaller populations? A careful reader could examine a long list of numbers representing the populations of many
countries, but with over 200 nations in the world, searching through such a list would take concentration and time. Putting these same numbers on a graph can quickly reveal population patterns.
Economists use graphs both for a compact and readable presentation of groups of numbers and for building an intuitive grasp of relationships and connections.
Three types of graphs are used in this book: line graphs, pie graphs, and bar graphs. Each is discussed below. We also provide warnings about how graphs can be manipulated to alter viewersâ
perceptions of the relationships in the data.
Line Graphs
The graphs we have discussed so far are called line graphs, because they show a relationship between two variables: one measured on the horizontal axis and the other measured on the vertical axis.
Sometimes it is useful to show more than one set of data on the same axes. The data in Table A2 is displayed in Figure A3 which shows the relationship between two variables: length and median weight
for American baby boys and girls during the first three years of life. (The median means that half of all babies weigh more than this and half weigh less.) The line graph measures length in inches on
the horizontal axis and weight in pounds on the vertical axis. For example, point A on the figure shows that a boy who is 28 inches long will have a median weight of about 19 pounds. One line on the
graph shows the length-weight relationship for boys and the other line shows the relationship for girls. This kind of graph is widely used by healthcare providers to check whether a childâ s
physical development is roughly on track.
Boys from Birth to 36 Months Girls from Birth to 36 Months
Length (inches) Weight (pounds) Length (inches) Weight (pounds)
20.0 8.0 20.0 7.9
22.0 10.5 22.0 10.5
24.0 13.5 24.0 13.2
26.0 16.4 26.0 16.0
28.0 19.0 28.0 18.8
30.0 21.8 30.0 21.2
32.0 24.3 32.0 24.0
34.0 27.0 34.0 26.2
36.0 29.3 36.0 28.9
38.0 32.0 38.0 31.3
Not all relationships in economics are linear. Sometimes they are curves. Figure A4 presents another example of a line graph, representing the data from Table A3. In this case, the line graph shows
how thin the air becomes when you climb a mountain. The horizontal axis of the figure shows altitude, measured in meters above sea level. The vertical axis measures the density of the air at each
altitude. Air density is measured by the weight of the air in a cubic meter of space (that is, a box measuring one meter in height, width, and depth). As the graph shows, air pressure is heaviest at
ground level and becomes lighter as you climb. Figure A4 shows that a cubic meter of air at an altitude of 500 meters weighs approximately one kilogram (about 2.2 pounds). However, as the altitude
increases, air density decreases. A cubic meter of air at the top of Mount Everest, at about 8,828 meters, would weigh only 0.023 kilograms. The thin air at high altitudes explains why many mountain
climbers need to use oxygen tanks as they reach the top of a mountain.
Altitude (meters) Air Density (kg/cubic meters)
0 1.200
500 1.093
1,000 0.831
1,500 0.678
2,000 0.569
2,500 0.484
3,000 0.415
3,500 0.357
4,000 0.307
4,500 0.231
5,000 0.182
5,500 0.142
6,000 0.100
6,500 0.085
7,000 0.066
7,500 0.051
8,000 0.041
8,500 0.025
9,000 0.022
9,500 0.019
10,000 0.014
The length-weight relationship and the altitude-air density relationships in these two figures represent averages. If you were to collect actual data on air pressure at different altitudes, the same
altitude in different geographic locations will have slightly different air density, depending on factors like how far you are from the equator, local weather conditions, and the humidity in the air.
Similarly, in measuring the height and weight of children for the previous line graph, children of a particular height would have a range of different weights, some above average and some below. In
the real world, this sort of variation in data is common. The task of a researcher is to organize that data in a way that helps to understand typical patterns. The study of statistics, especially
when combined with computer statistics and spreadsheet programs, is a great help in organizing this kind of data, plotting line graphs, and looking for typical underlying relationships. For most
economics and social science majors, a statistics course will be required at some point.
One common line graph is called a time series, in which the horizontal axis shows time and the vertical axis displays another variable. Thus, a time series graph shows how a variable changes over
time. Figure A5 shows the unemployment rate in the United States since 1975, where unemployment is defined as the percentage of adults who want jobs and are looking for a job, but cannot find one.
The points for the unemployment rate in each year are plotted on the graph, and a line then connects the points, showing how the unemployment rate has moved up and down since 1975. The line graph
makes it easy to see, for example, that the highest unemployment rate during this time period was slightly less than 10% in the early 1980s and 2010, while the unemployment rate declined from the
early 1990s to the end of the 1990s, before rising and then falling back in the early 2000s, and then rising sharply during the recession from 2008â 2009.
Pie Graphs
A pie graph (sometimes called a pie chart) is used to show how an overall total is divided into parts. A circle represents a group as a whole. The slices of this circular â pieâ show the relative
sizes of subgroups.
Figure A6 shows how the U.S. population was divided among children, working age adults, and the elderly in 1970, 2000, and what is projected for 2030. The information is first conveyed with numbers
in Table A4, and then in three pie charts. The first column of Table A4 shows the total U.S. population for each of the three years. Columns 2â 4 categorize the total in terms of age groupsâ from
birth to 18 years, from 19 to 64 years, and 65 years and above. In columns 2â 4, the first number shows the actual number of people in each age category, while the number in parentheses shows the
percentage of the total population comprised by that age group.
Year Total Population 19 and Under 20â 64 years Over 65
1970 205.0 million 77.2 (37.6%) 107.7 (52.5%) 20.1 (9.8%)
2000 275.4 million 78.4 (28.5%) 162.2 (58.9%) 34.8 (12.6%)
2030 351.1 million 92.6 (26.4%) 188.2 (53.6%) 70.3 (20.0%)
In a pie graph, each slice of the pie represents a share of the total, or a percentage. For example, 50% would be half of the pie and 20% would be one-fifth of the pie. The three pie graphs in Figure
A6 show that the share of the U.S. population 65 and over is growing. The pie graphs allow you to get a feel for the relative size of the different age groups from 1970 to 2000 to 2030, without
requiring you to slog through the specific numbers and percentages in the table. Some common examples of how pie graphs are used include dividing the population into groups by age, income level,
ethnicity, religion, occupation; dividing different firms into categories by size, industry, number of employees; and dividing up government spending or taxes into its main categories.
Bar Graphs
A bar graph uses the height of different bars to compare quantities. Table A5 lists the 12 most populous countries in the world. Figure A7 provides this same data in a bar graph. The height of the
bars corresponds to the population of each country. Although you may know that China and India are the most populous countries in the world, seeing how the bars on the graph tower over the other
countries helps illustrate the magnitude of the difference between the sizes of national populations.
Country Population
China 1,369
India 1,270
United States 321
Indonesia 255
Brazil 204
Pakistan 190
Nigeria 184
Bangladesh 158
Russia 146
Japan 127
Mexico 121
Philippines 101
Bar graphs can be subdivided in a way that reveals information similar to that we can get from pie charts. Figure A8 offers three bar graphs based on the information from Figure A6 about the U.S. age
distribution in 1970, 2000, and 2030. Figure A8 (a) shows three bars for each year, representing the total number of persons in each age bracket for each year. Figure A8 (b) shows just one bar for
each year, but the different age groups are now shaded inside the bar. In Figure A8 (c), still based on the same data, the vertical axis measures percentages rather than the number of persons. In
this case, all three bar graphs are the same height, representing 100% of the population, with each bar divided according to the percentage of population in each age group. It is sometimes easier for
a reader to run their eyes across several bar graphs, comparing the shaded areas, rather than trying to compare several pie graphs.
Figure A7 and Figure A8 show how the bars can represent countries or years, and how the vertical axis can represent a numerical or a percentage value. Bar graphs can also compare size, quantity,
rates, distances, and other quantitative categories.
Comparing Line Graphs with Pie Charts and Bar Graphs
Now that you are familiar with pie graphs, bar graphs, and line graphs, how do you know which graph to use for your data? Pie graphs are often better than line graphs at showing how an overall group
is divided. However, if a pie graph has too many slices, it can become difficult to interpret.
Bar graphs are especially useful when comparing quantities. For example, if you are studying the populations of different countries, as in Figure A7, bar graphs can show the relationships between the
population sizes of multiple countries. Not only can it show these relationships, but it can also show breakdowns of different groups within the population.
A line graph is often the most effective format for illustrating a relationship between two variables that are both changing. For example, time series graphs can show patterns as time changes, like
the unemployment rate over time. Line graphs are widely used in economics to present continuous data about prices, wages, quantities bought and sold, the size of the economy.
How Graphs Can Be Misleading
Graphs not only reveal patterns; they can also alter how patterns are perceived. To see some of the ways this can be done, consider the line graphs of Figure A9, Figure A10, and Figure A11. These
graphs all illustrate the unemployment rateâ but from different perspectives.
Suppose you wanted a graph which gives the impression that the rise in unemployment in 2009 was not all that large, or all that extraordinary by historical standards. You might choose to present your
data as in Figure A9 (a). Figure A9 (a) includes much of the same data presented earlier in Figure A5, but stretches the horizontal axis out longer relative to the vertical axis. By spreading the
graph wide and flat, the visual appearance is that the rise in unemployment is not so large, and is similar to some past rises in unemployment. Now imagine you wanted to emphasize how unemployment
spiked substantially higher in 2009. In this case, using the same data, you can stretch the vertical axis out relative to the horizontal axis, as in Figure A9 (b), which makes all rises and falls in
unemployment appear larger.
A similar effect can be accomplished without changing the length of the axes, but by changing the scale on the vertical axis. In Figure A10 (c), the scale on the vertical axis runs from 0% to 30%,
while in Figure A10 (d), the vertical axis runs from 3% to 10%. Compared to Figure A5, where the vertical scale runs from 0% to 12%, Figure A10 (c) makes the fluctuation in unemployment look smaller,
while Figure A10 (d) makes it look larger.
Another way to alter the perception of the graph is to reduce the amount of variation by changing the number of points plotted on the graph. Figure A10 (e) shows the unemployment rate according to
five-year averages. By averaging out some of the year-to-year changes, the line appears smoother and with fewer highs and lows. In reality, the unemployment rate is reported monthly, and Figure A11
(f) shows the monthly figures since 1960, which fluctuate more than the five-year average. Figure A11 (f) is also a vivid illustration of how graphs can compress lots of data. The graph includes
monthly data since 1960, which over almost 50 years, works out to nearly 600 data points. Reading that list of 600 data points in numerical form would be hypnotic. You can, however, get a good
intuitive sense of these 600 data points very quickly from the graph.
A final trick in manipulating the perception of graphical information is that, by choosing the starting and ending points carefully, you can influence the perception of whether the variable is rising
or falling. The original data show a general pattern with unemployment low in the 1960s, but spiking up in the mid-1970s, early 1980s, early 1990s, early 2000s, and late 2000s. Figure A11 (g),
however, shows a graph that goes back only to 1975, which gives an impression that unemployment was more-or-less gradually falling over time until the 2009 recession pushed it back up to its â
originalâ levelâ which is a plausible interpretation if one starts at the high point around 1975.
These kinds of tricksâ or shall we just call them â presentation choicesâ â are not limited to line graphs. In a pie chart with many small slices and one large slice, someone must decide what
categories should be used to produce these slices in the first place, thus making some slices appear bigger than others. If you are making a bar graph, you can make the vertical axis either taller or
shorter, which will tend to make variations in the height of the bars appear more or less.
Being able to read graphs is an essential skill, both in economics and in life. A graph is just one perspective or point of view, shaped by choices such as those discussed in this section. Do not
always believe the first quick impression from a graph. View with caution.
Answer: Is economics becoming mathematics?
What is the role of mathematics in economics?
Although mathematics has a role in all types of economics, it’s most common in mathematical economics, where it’s a core component. In mathematical economics, economists apply mathematical principles
to economic theory.
Why do economists need math skills?
As an economist, you are likely to use your math skills throughout the process of creating an economic model. Accurate math provides reliable data you can use in constructing a model, which can
increase the value of the model provides upon completion. Economic projections provide predictions of future economic behavior and patterns.
What is mathematical economics?
Mathematical economics is a form of economics that relies on quantitative methods to describe economic phenomena. Although the discipline of economics is heavily influenced by the bias of the
researcher, mathematics allows economists to precisely define and test economic theories against real world data.
Does mathematical economics deserve support?
Some economists state that mathematical economics deserves support just like other forms of mathematics, particularly its neighbors in mathematical optimization and mathematical statistics and
increasingly in theoretical computer science.
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Discovering Conic Sections in the Motion of Heavenly Bodies
Sam H. Jones
It is generally believed that Conic sections were first studied, in the abstract, by Euclid (around 300 BC) and later extended by Apollonius of Perga (around 200 BC) for no apparent practical
purpose. Apollonius gave us the names of conic sections, which we still use today, ellipse, parabola, and hyperbola. Each is a cross-section of a cone (much like a paper cone for water at the
doctor's office) which is sliced. Each curve, or cross-section, results from the intersection of a plane with a cone as if the plane is slicing the cone from varying angles as can be seen in Figure
^ 2
(image available in print form)
Figure 1. source: Wikipedia
However the fact that Apollonius used his theories of conic sections to create more accurate sundials suggests that the following scenario may be more likely. During its daily course above the
horizon the Sun appears to describe a circular arc. Supplying in his mind's eye the missing portion of the daily circle, the Greek astronomer could imagine that his real eye was at the apex of a
cone, the surface of which was defined by the Sun's rays at different times of the day and the base of which was defined by the Sun's apparent diurnal course. Our astronomer, using the pointer of a
sundial, known as a gnomon, as his eye, would generate a second, shadow cone spreading downward. The intersection of this second cone with a horizontal surface, such as the face of a sundial, would
give the trace of the Sun's image (or shadow) during the day as a plane section of a cone. (The possible intersections of a plane with a cone, known as the conic sections, are the circle, ellipse,
point, straight line, parabola, and hyperbola.)
However, compilers of the ideas in the history of the philosophy of science (known as doxographers) ascribe the discovery of conic sections to Menaechmus (mid-4th century BC), a student of Eudoxus,
who used them to solve the problem of duplicating the cube. His restricted approach to conics--he worked with only right circular cones and made his sections at right angles to one of the straight
lines composing their surfaces--was standard down to Archimedes' era.
A right circular cone is defined as a cone with a circle as its base and the apex is centered directly over the center of the circle so that the height from the base to the apex is at a right angle
from the center of the circle to the apex. Figure 2 shows both a right circular cone (on the left) and an oblique circular cone (on the right). A cone may also have a non-circular base.
(image available in print form)
Figure 2. source: Wikipedia
Euclid adopted Menaechmus's approach in his lost book on conics, and Archimedes followed suit. Doubtless, however, both knew that all the conics can be obtained from the same right cone by allowing
the section at any angle. Whereas his predecessors had used finite right circular cones, Apollonius considered arbitrary (oblique) double cones that extend indefinitely in both directions.
The reason that Euclid's treatise on conics perished is that Apollonius of Perga (c. 262-c. 190 BC) did to it what Euclid had done to the geometry of Plato's time. Apollonius reproduced known results
much more generally and discovered many new properties of the figures. He first proved that all conics are sections of any circular cone, right or oblique. Apollonius introduced the terms ellipse ,
hyperbola , and parabola for curves produced by intersecting a circular cone with a plane at an angle less than, greater than, and equal to, respectively, the opening angle of the cone.
For this section there will be a lesson with a hands on activity where students will be constructing and also slicing cones. A sundial will also be demonstrated.
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Luck, logic, and white lies: The mathematics of games, second edition, by Jörg Bewersdorff
Luck, logic, and white lies: The mathematics of games, second edition, by Jörg Bewersdorff – book review
Luck, logic, and white lies: The mathematics of games, second edition, by Jörg Bewersdorff
CRC Press, 2021, 568 pages, ISBN 9780367548414, Number of chapters 51
review by Catalin Barboianu, PhD
(published in International Gambling Studies, DOI: 10.1080/14459795.2021.1965184)
As the title indicates, Bewersdorff’s book is intended to span the mathematics of games in general – not only games of chance but also including strategic and skill games. The author covers all the
big categories of games – casino, tournament, and house or social games. In fact, the skill-strategic dimension of the games balanced with the chance-uncertainty dimension is the central element
around which the author presents games as an important field of application of mathematics; he takes them as a good opportunity to advocate for the beauty and power of mathematics. To that point, the
book is written so as to be both popular and scholarly, and these attributes are not at all inconsistent with each other for such a general topic, content, and style.
The popular attribute is mainly provided by the problem-oriented selection of topics and sub-topics and the organization of the content: Each chapter starts with a nice relevant problem, general
question, or curiosity that mathematics is called upon to solve and can be read independently, like a collection of articles in a mathematics magazine. The presentation is very often in the style of
Martin Gardner’s recreational mathematics; in fact, some of the games analysed in the current book (Tic Tac Toe, Hex, Go) are also present in Martin Gardner’s influential book The Scientific American
Book of Mathematical Puzzles and Diversions (1959), of course using more advanced mathematical concepts than the latter. The historical approach, present in almost all topics, whether we talk about
the games themselves or the mathematical concepts or theories dealing with them, also counts toward the popular attribute.
The scholar attribute is given firstly by the general principle driving the problem-solving approach throughout the book, namely to show how in this field of application the three main branches of
mathematics – game theory, probability theory, and combinatorics – work together in describing the games, strategies, and optimal play. Further, some concepts and theoretical results (especially from
game theory) are relatively advanced and usually not employed in most of the published books on mathematics of gambling. Adding to these elements is that the game sections benefit by relevant
citations and are packed with further-reading sections and chapter notes. What we have here, then, is a kind of textbook on a popular subject addressed in a popular style.
In the preface, the author provides a basic taxonomy of games with respect to the three elements he identifies as causing the uncertainty and driving the mechanism of every game: chance, the large
number of combinations of different moves, and different levels of information among the individual players. These elements yield three basic categories of games, namely combinatorial games, games of
chance, and games of skill – which are not mutually exclusive – there are games that do fall in two categories or even in all three (like poker and Skat). Still in the preface, the author describes
the relationship between mathematics and games in terms of both mathematical applications and the interests and behavior of the players, and emphasizes the roles of game theory, probability theory,
and combinatorics. It is remarkable that in this section the author talks about the central notion of probability as being a measure (of the certainty with which a random event occurs) in the common
sense, while explaining from the beginning that probability is not all that counts toward strategy and optimal play. He further introduces a definition for the Laplacian probability in the first game
section (Dice and Probability).
Unlike the customary structure of a gaming mathematics book, this book does not have a systematically organized supporting-mathematics chapter preceding the descriptions and analyses of the games. In
fact, none of the mathematics presented is systematized as for a student’s or player’s study or use; it is simply applied for the gaming-related problems and questions posed across the game sections.
The mathematical level differs among the different chapters, but not increasingly. A new concept is introduced when needed and references back to previous concepts and explanations are present across
the chapters. This does not mean that chapters cannot be read quite independently, but definitely not by a non-mathematical reader.
The Laplacian probability and the associated basic notions are introduced in the first three chapters (Dice and Probability, Waiting for a Double 6, and Tips on Playing the Lottery: More Equal than
the Equal?), by using combinatorial examples from dice, lottery, and poker games. These examples are also used to explain the rules of combinatorial calculations (including the use of the Pascal
triangle) and to show the average reader how the multiplication power of combinations affects the probability of the various events in combinatorial games of chance. This is an important lesson for
gamblers in regard to the erroneous or subjective quick estimations of probabilities of winning (usually overestimated). Of course, the author chose the best game for this didactical purpose, namely
lottery, and made a comparison between the lottery combinations of numbers and card combinations in poker. Conditional probability is introduced in Chapter 9.
In the historical side of this introductory part, the author does not miss the opportunity to tell the reader the well-known story about the birth of probability theory, originating in a problem from
a game of chance (De Méré problem), debated in a correspondence between Pascal and Fermat at the middle of the 17^th century.
The law of large numbers (LLN) is introduced in Chapter 5 and explained through experimental examples, with reference to its history (a first version of LLN was formulated and proved in 1690, before
probability theory was crystallized!) and to the strong law of large numbers. What is remarkable is that the author, still in an experimental-example framework, brings into discussion the sensible
issue of the stabilization of the relative frequencies, which is a core question in philosophy of probability, linking the empirical-experimental and the abstract-theoretical (What makes the relative
frequency become stabilized around mathematical probability?)[1] This is not the only philosophical-foundational aspect touched on in this book in regard to probability theory. The issue of a
mathematical objective description of randomness and of the equally-possible attribute in the Laplacian definition of probability, the conciliation between chance (as the object of investigation of
probability theory) and mathematical certainty are discussed in the same historical style, in the context of physics. This contextual choice (in Chapter 8) in a book explaining the mathematics of
games is not at all something beyond the topic. From kinetic gas theory in the 19^th century to quantum mechanics in the 20^th century, physics has shown that the non-deterministic approach makes
randomness objectively present in our theories, since all such theories were tested true in physics, both theoretically and experimentally. As such, randomness is not just an expression of our
ignorance or lack of knowledge, as reflected in the Laplacian definition of probability, but a theoretical necessity. Indeed, since the entirety of the state parameters determining a theoretical
model of a system can never be precisely measured, predictions about the future behaviour of that system are always subject to randomness. Still, the laws of averages that the probabilistic
approach provides turn into a mathematical description of an “average behaviour” of the system that can be accommodated with the deterministic approach. On the other side, mathematics has shown that
a model for probability can be constructed without a mathematical definition of randomness, from Bohlmann’s to Kolmogorov’s axiomatic system on which standard probability theory relies. But, one may
ask, what do games have to do with kinetic gas theory and quantum mechanics? The author explains this relationship by introducing the theoretical concept of a chaotic system as an intermediary
concept between the physics and mathematics of random processes and phenomena: By placing the games in the realm of classical mechanics where unpredictable outcomes are physical, games can be
characterized by the fact that causal relations are highly sensitive to small perturbations that cannot be measured (“small causes, large effects”) – in other words, they are chaotic systems – and as
such, “physical” randomness becomes a sort of mathematical randomness. Put this way, the certainty of mathematics and the uncertainty of randomness and probability no longer appear to exclude each
other radically. It is an original preamble for the introduction of the Kolmogorovian axiomatic system defining probability (also in Chapter 8) Although the chapter ends with an elementary
description of this axiomatic system, I would say that this is “the most” scholarly chapter in this book, seemingly addressed to graduates in philosophy and history of science. However, this topic is
an ideal completion for the association of the terms luck and logic (even from the book’s title) in the strategic discussions in regard to games, one of the main focuses of this work.
Sometimes classical problems from recreational mathematics are discussed, apparently not related to the described games (e.g., Buffon’s Needle problem or the three-door problem). The goal seems to be
that of making the reader aware of how tricky or counterintuitive probability theory may be sometimes, with respect to both its application and interpretation, and how the same concepts (both
objective and subjective) of randomness and equiprobable are responsible for that.
The author does not limit the presentation of probability theory to elementary notions, foundation, and discrete probability, but extends it to advanced concepts when a description of a game requires
it. Markov chains are introduced in the discussion of the game of Monopoly. The transition from one square to another in Monopoly provides a good elementary example to start with when teaching
someone about a Markov chain as a sequence of random trials with finite sets of outcomes in which the probability that an event occurs on the (n + 1)-th trial depends only on the event of the n-th
trial. The Monopoly example also allows an easy generalization to other less popular games, namely Snakes and Ladders, and is applied as well to the classical Ruin Problem.
At the end of Part I (Games of Chance), Blackjack has the lion’s share. It is to the author’s credit that he has comprised a typical 200-page book on blackjack mathematics and strategy into a 20-page
chapter with essential information focused on optimal play. By using only the basic notions of probability, conditional probability, and expectation, the author explains the rules of the optimal play
in blackjack and the mathematics behind the High-Low count of cards. The numerical results of the most frequent gaming situations (probabilities, expectations, and criteria for optimal play) are
grouped efficiently in general tables in a much smaller number than in other books dedicated to blackjack.
We can say fairly that this is a chapter that exclusively addresses the gambler – the mathematics is elementary and easy to follow, and the aspects of optimal play and strategy are discussed in the
player’s favour.
Still in the part dedicated to games of chance, the author presents the Monte Carlo Method of simulating plays where chance is involved, as a long series of trials, as a convenient alternative to the
mathematical deduction of excessively complex formulas for probabilities and statistical indicators. It is a good opportunity for making the connection with programming, which is more involved in the
parts dedicated to combinatorial and strategic games, some of them being discussed through an algorithmic approach. The advantages of the Monte Carlo Method in analysis of results and decision-making
are clearly specified, and the brief history of this method is presented, as we have come to expect by now. In the same chapter, the author talks about the generation of random numbers – necessary
for the Monte Carlo simulations and for the contemporary electronic games of chance – and reveals the mathematical reasons for the accuracy and efficiency of such “algorithmic randomness” which
relates to the primes and their properties. Still related to the Monte Carlo Method, the notion of sample function is introduced at the end of this chapter with the simple example of die throwing.
In Part II (Combinatorial Games), chess is given (unexpectedly) only ten pages, but it is discussed only as related to Zermelo’s theorem (applicable to chess and other comparable games), which stands
as the main principle of chess programming. It is the most general (and shortest) mathematical description of chess, and the author recalls this theorem (also called the minimax theorem) in Chapter
27, dedicated to chess programming. By contrast, Go is given the largest space and is discussed in terms of advanced concepts of combinatorial game theory from the works of Conway, Milnor, and
Hanner. It is an excellent example of how a general mathematical theory was developed by analysing old classical positional games like Go.
In Part III (Strategic Games) a focus is given to poker as a game with imperfect information, characterized by both chance and skill, a perfect example to analyse mathematically whether the mixed
strategies are possible. While not getting into the details of poker and its explicit strategies, the author answers fundamental questions regarding optimal play by citing the works of von Neumann,
Morgenstern, and Farkas. The conclusion is that optimal play in poker is possible, according to the available mathematical models, only under the assumption (idealization) that strategies chosen by
players are random and the probabilities with which the various strategies are chosen determine the degree of success that a player can expect on average. Within this theoretical framework, the
psychology of poker (including bluffing) is reflected only as a probabilistic element and not as a deterministic one. The elaboration and calculation of minimax strategies for poker require universal
algorithms that are introduced in the next two chapters 36 and 37 – linear optimization, the simplex algorithm, and the rectangle rule.
The analysis of poker is resumed in Part IV, which is elaborated as an epilogue to the discussion on the relationships between chance, skill, and symmetry with respect to strategy and optimal play.
The author’s general conclusion, supported by the results of game theory, is that only the combinatorial elements of a game bring about a causality that can be influenced by skill, and the
chance-logic-bluff classification of the game elements does not count in giving skill roles that can be reflected or quantified in the mathematical models we use under idealized conditions for
deriving optimal play. As such, pure games of skill are without exception combinatorial games. Still, the author further addresses fundamental questions regarding skill such as “How can a system for
the influence of skill in a game be defined?” (the topic of Chapter 49); “What suggestions have been made hitherto for measuring the influence of skill on a game?” (the topic of Chapter 50); and the
hotly debated popular question regarding poker “Is poker a game of chance?”[2], which remains open and concludes the book.
Epiloguing in this way, the book leaves the impression of its author’s being a skilled advocate of the unlimited power of mathematics, shown through the examples of games. Not only is mathematics
able to describe the games and the way we play them, but it is entitled to address fundamental questions beyond the problem-solving aspects of games and gaming. It is mainly game theory and
probability theory that grant mathematics such a virtue. In particular, game theory provides the kind of generality that allows the mathematical treatment of all categories of games with respect to
strategy and optimal play, and it is the author’s merit for achieving such coverage in only 561 pages, including the math and math history lessons. Although the chapters can mostly be read
independent of each other, and the mathematical content is not systematized throughout the book, the mathematically-inclined reader can put things together to have an objective overview of one of the
most interesting fields in application of mathematics – games – which themselves shaped the development of mathematics. As for the average player, even the title itself suggests that the ingredients
of a game and any strategy of playing it are of radically different natures, and mathematics can deal with them together. Passing from the abstract mathematics of games to the real play in the form
of application is not so much a matter of mathematical skill as of interpreting empirically the mathematical concepts and theoretical results. In the realm of games, such empirical interpretation is
many times at least subjective, and it is not the object of investigation of pure mathematics.
[1] In other non-Kolmorogovian formal accounts of probability, the stabilization of the relative frequency is an axiom/postulate, conventionally taken to be true.
[2] Poker is lying in author’s taxonomy of games in the preface in a common zone for combinatorial games, strategic games, and games of chance; however, as per the arguments in the last chapter, this
placement may change with the variations of poker and their rules.
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equation of circle with radius 1
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equation of a circle: general form; standard form; parametric form; polar form. Finding the Equation of Circle passing through Three Points Stack Overflow en espaol es un sitio de preguntas y
respuestas para programadores y profesionales de la informtica Then combine the height and width properties with a matching value actually I know a lot of data That gives the point where the two
straight lines cross (4 That gives the point where the two straight Equivalently, it is the curve traced out by a point that moves in a plane so that its distance A circle is a shape consisting of
all points in a plane that are at a given distance from a given point, the centre.Equivalently, it is the curve traced out by a point that moves in a plane so that its distance from a given point is
constant.The distance between any point of the circle and the centre is called the radius.Usually, the radius is required to be a positive number. Given a great circle passing through (10N,13E) and
proceeding on an azimuth of 10, where does it intersect with a great As a simple example, consider the intersection points of two meridians, which are just great circles with azimuths of 0 or 180
Example 5: Point vs circle P: point being tested C: circle center If point is at coordinates 1,1 only other points at the same Explanation: . So the centre of the circle is: (0,a2 + 1 2) = (0,(a2 + 1
4) + 1 4) = (0, 5 4) The standard form of the equation of a circle is: (x h)2 + (y h)2 = When a predictor is categorical, the ROC The video provides two example problems for finding the radius of a
circle given the arc length Determine derivatives and equations of tangents for parametric curves A curve is given by the parametric equations: #x=cos(t) , y=sin(2t)#, how do you find the cartesian
equation? Center (4,3) Radius = 5 units. All of these values are related through the mathematical constant , or pi, which is the ratio of a circle's circumference to its diameter, and is
approximately 3.14159. is an irrational number meaning that it Solution for Find the equation of the circle with radius 1 and center (4, 1). A semicircle is simply half of a circle. The center was
given as (-1, 3), so h = -1 and k = 3. In this lesson well look at how to write the equation of a circle in standard form in order to find the center and radius of the circle. Equation of a Circle
The general equation of any type of circle is represented by: x2 + y2 + 2gx + 2fy + c = 0. Radius and center for a circle equation in standard Substituting the values of centre and radius, (x 2) 2 +
(y 3) 2 = 1 2. x 2 4x + 4 + y 2 6y + 9 = 1. x 2 + y 2 4x 6y + 10 Completing the square to write equation in standard form of a circle. If your answer is not an integer, express it in radical form.
Example: Convert the polar equation of a circle r = - 4 cos q into Cartesian coordinates. For example write the equation of a circle with centre (2, Give the equation of the Example. Stack Exchange
network consists of 180 Q&A communities including Stack Overflow, and how is it possible to solve for the equation, center, and radius of that circle? Step 1: Find the gradient of the radius of the
circle. c.Write an equation for the circle. Solution The radius is 3 and the center is at the origin. Area and Circumference Formula. Find the center and the radius of the circle $x^2 + y^2 + 2x - 3y
- \frac{3}{4} = 0 $ example 3: ex 3: Find the equation of a circle in standard form, with a center at $C(-3,4)$ and passing through the point A x 2+y 2+2 3 X2=0 B x 2+y 22 3 Y+2=0 C x 2+y 2+2 3 y+2=0
D x 2+y 2+2 3 x+2=0 Medium Solution Step 2: Put Euclidean geometry = where C is the circumference of a circle, d is the diameter.More generally, = where L and w are, respectively, the perimeter and
the width of any curve of constant width. Given: circle with center (0,1) and r = 2. Step 2: Now click to get the circles equation by clicking the Find Equation of Circle button. Step 3: Complete the
equation of the circle with the radius found in step 2.
Find the equation of the circle with radius 1 and center C (1, -2). a.Find the center of the circle. A. (3) = sin r + d d s ( tan 1 r r ) where is angle to x-axis, is between arc and radius Solution:
As, r = - 4 cos q. then r2 = - 4 r cos q, and by using polar to Cartesian conversion formulas, r2 = x2 + y2 and x = r cos q. obtained is x2 + y2 = - 4 x. That is, if the point satisfies the equation
of the circle, it lies on the circle's circumference. Find The Solution : |z - 2 - i| = 3 |z - (2 + i)| = 3. The above can be derived from intrinsic/natural differential equation of a circle is. 72
times. This means that, using Pythagoras theorem, the equation of a circle with radius r and centre (0, 0) is given by the formula \(x^2 + y^2 = r^2\). The equation given below is the general form of
equation of a circle. So, let parametric curve is defined by equations x=f(t) and Parametric Representation of a Parabola Parametric equations x = 2ap (1) y = ap 2 (2) A variable point on the
parabola is given by (2ap, ap 2 ), for constant a and parameter p or respectively The equation above is the parametric form of a circle with a radius of Find The Outward Normal N Of The Determine the
equation of a circle with radius {eq}\frac{1}{3} {/eq}, centered at (-4, 6). How to express the standard form equation of a circle of a given radius. Textbook Solutions 14511. Write the equation of a
circle with a radius of 2 units and a center at (-1, 3). Step 3: In the new window, the equation Important Solutions 9. x 21y25 9 Simplify. 2 Ways To Graph A Circle Dummies. His work is shown below.
Hint: In this problem, we are not given the center or radius however we can find the length of the diameter using the distance formula (Phythagoras) and then divide it by 2. c The equation of the
circle is x21y25 9. We can find the equation of any circle, given the coordinates of the center and the radius of the circle by applying the equation of circle formula. The calculator will generate a
step by step explanations and circle graph. Equation of a circle. You can play them alone, with your friends, or in teams Build lifelong math skills with these math worksheets Advanced Test This is
just one of 12 free GRE math problem solving tests available on majortests Each Basic Geometry Practice Test consists of ten to fifteen geometry problems Each Basic Geometry Practice Test Question:
Circle 1 has the equation . Step 2: Identify the radius of the circle, and let r equal this value. The general equation of a circle is (x h) 2 + (y k) 2 = r 2, where (h, k) represents the location of
the circle's center, and r represents the length of its radius. 7 months ago. x 2 + y 2 = 1 (1) Standard Form. In SI unit the Bohr radius is 5.29x10-11 m, in US units it is 2.08x10-9 in, while in
natural units it is 2.68x10-4 /eV or 3.27x10 24 l..
equation of circle with radius 1
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Question ID - 104183 | SaraNextGen Top Answer
The locus of the points
a) A straight line b) A circle c) A parabola d) None of these
Question ID - 104183 | SaraNextGen Top Answer
The locus of the points
a) A straight line b) A circle c) A parabola d) None of these
« Previous Next Question / Solution »
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English National Curriculum, Programme Of Study For Key Stage 3 Mathematics
Here are some specific activities, investigations or visual aids we have picked out. Click anywhere in the grey area to access the resource.
Here is an exam-style questions on this statement:
Click on a topic below for suggested lesson Starters, resources and activities from Transum.
How do you teach this topic? Do you have any tips or suggestions for other teachers? It is always useful to receive feedback and helps make these free resources even more useful for Maths teachers
anywhere in the world. Click here to enter your comments.
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245 research outputs found
A reasonable representation of large scale structure, in a closed universe so large it's nearly flat, can be developed by extending the holographic principle and assuming the bits of information
describing the distribution of matter density in the universe remain in thermal equilibrium with the cosmic microwave background radiation. The analysis identifies three levels of self-similar large
scale structure, corresponding to superclusters, galaxies, and star clusters, between today's observable universe and stellar systems. The self-similarity arises because, according to the virial
theorem, the average gravitational potential energy per unit volume in each structural level is the same and depends only on the gravitational constant. The analysis indicates stellar systems first
formed at z\approx62, consistent with the findings of Naoz et al, and self-similar large scale structures began to appear at redshift z\approx4. It outlines general features of development of
self-similar large scale structures at redshift z<4. The analysis is consistent with observations for angular momentum of large scale structures as a function of mass, and average speed of
substructures within large scale structures. The analysis also indicates relaxation times for star clusters are generally less than the age of the universe and relaxation times for more massive
structures are greater than the age of the universe.Comment: Further clarification of assumptions underlying the analysi
The reason for baryon asymmetry in our universe has been a pertinent question for many years. The holographic principle suggests a charged preon model underlies the Standard Model of particle physics
and any such charged preon model requires baryon asymmetry. This note estimates the baryon asymmetry predicted by charged preon models in closed inflationary Friedmann universes.Comment: 5 pages, no
figures, clarified discussion of comparison with observation
A closed vacuum-dominated Friedmann universe is asymptotic to a de Sitter space with a cosmological event horizon for any observer. The holographic principle says the area of the horizon in Planck
units determines the number of bits of information about the universe that will ever be available to any observer. The wavefunction describing the probability distribution of mass quanta associated
with bits of information on the horizon is the boundary condition for the wavefunction specifying the probability distribution of mass quanta throughout the universe. Local interactions between mass
quanta in the universe cause quantum transitions in the wavefunction specifying the distribution of mass throughout the universe, with instantaneous non-local effects throughout the universe.Comment:
4 pages, no figures, to be published in Int. J. Theor. Phys, references correcte
Cosmic ray flux measurements using calorimeter
A simple and surprisingly realistic model of the origin of the universe can be developed using the Friedmann equation from general relativity, elementary quantum mechanics, and the experimental
values of h, c, G and the proton mass. The model assumes there are N space dimensions (with N > 6) and the potential constraining the radius r of the invisible N -3 compact dimensions varies as r^4.
In this model, the universe has zero total energy and is created from nothing. There is no initial singularity. If space-time is eleven dimensional, as required by M theory, the scalar field
corresponding to the size of the compact dimensions inflates the universe by about 26 orders of magnitude (60 e-folds). If the Hubble constant is 65 km/sec Mpc, the energy density of the scalar field
after inflation results in Omega-sub-Lambda = 0.68, in agreement with recent astrophysical observations.Comment: To be published in General Relativity and Gravitation, August 200
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How Many Minutes Is 483.5 Days?
How many minutes in 483.5 days?
483.5 days equals 696240 minutes
Unit Converter
Conversion formula
The conversion factor from days to minutes is 1440, which means that 1 day is equal to 1440 minutes:
1 d = 1440 min
To convert 483.5 days into minutes we have to multiply 483.5 by the conversion factor in order to get the time amount from days to minutes. We can also form a simple proportion to calculate the
1 d → 1440 min
483.5 d → T[(min)]
Solve the above proportion to obtain the time T in minutes:
T[(min)] = 483.5 d × 1440 min
T[(min)] = 696240 min
The final result is:
483.5 d → 696240 min
We conclude that 483.5 days is equivalent to 696240 minutes:
483.5 days = 696240 minutes
Alternative conversion
We can also convert by utilizing the inverse value of the conversion factor. In this case 1 minute is equal to 1.4362863380444E-6 × 483.5 days.
Another way is saying that 483.5 days is equal to 1 ÷ 1.4362863380444E-6 minutes.
Approximate result
For practical purposes we can round our final result to an approximate numerical value. We can say that four hundred eighty-three point five days is approximately six hundred ninety-six thousand two
hundred forty minutes:
483.5 d ≅ 696240 min
An alternative is also that one minute is approximately zero times four hundred eighty-three point five days.
Conversion table
days to minutes chart
For quick reference purposes, below is the conversion table you can use to convert from days to minutes
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Scoring in Gin
The winning score for Gin can be determined by the two players involved in the game or the game can be played until one player reaches a score of 100 or more points. If you are playing an online or
computer software version of Gin, the winning score will be predetermined. Aside from the winning score, the point values throughout the game of Gin are fairly standard and are described below.
Card Values
In Gin the Ace is a low card, meaning that it counts as a 1, preceding the 2 and 3 of its suit, rather than acting as a high card that succeeds the Queen and King. All Aces in Gin are worth 1 point.
Face cards, which refer to the Jack, Queen and King of each suit, are all worth 10 points each. The remaining cards, 2-10, are worth the value of their rank. So, the 2 is worth 2 points, and the 10
is worth 10 points.
Knocking a player
Beyond this, there are a couple of other scoring considerations for Gin. The strategy of knocking is a key part of the winning strategy in Gin. When a player knocks his opponent, both players must
lay their cards on the table, revealing their sets, runs and remaining cards should they have them. The knocker's remaining cards must equal no more than 10 points, and the opponent can add to these
points by creating sets or runs with their remaining cards and the knocker's remaining cards.
If the knocker's opponent is unable to do the latter, the knocker will get the difference between his opponents remaining cards and his remaining cards. For example, if the knocker's remaining cards
total 5 points (perhaps an Ace of Queens and a 4 of Clubs), and the opponent's remaining cards total 11 points, the knocker will get an additional 6 points added to his score.
Another possibility is that the knocker's opponent has remaining cards that total the value of the knocker's remaining cards or is less that the value of the knocker's remaining cards. For example,
the knocker's remaining cards equal 7 and the opponent's remaining cards equal 7 or the opponent's remaining cards equal 5. If this happens the knocker has been undercut, and the opponent will
receive the difference, if any, between the knocker's remaining cards and his remaining cards as well as a 10 to 25 point bonus.
If the knocker has no remaining cards the game is automatically won; this is called "going gin." If the knocker goes gin he will receive a 20 to 25 point bonus as well as the value of the opponent's
remaining cards. In addition, there is a 100 point bonus for whichever player reaches the game's winning score first. An additional 100 points can be won by the game's first player to reach the
winning score if his opponent has managed to score no points at all. A final scoring possibility in Gin is the "line bonus." This refers to the additional 20 points a player receives per hand he has
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SquareCirclez Sitemap - Interactive MathematicsSquareCirclez Sitemap
SquareCirclez Sitemap
By Murray Bourne, 13 Oct 2006
2 Comments on “SquareCirclez Sitemap”
1. Rick Patterson says:
9 Dec 2014 at 10:15 am [Comment permalink]
I'm not getting a full web page - only menu tabs at the top of the page and a blank screen. I have tried loading Flash, Shockwave and manually typing the URL into my browser window.
Any advice or suggestions will be greatly appreciated.
2. Murray says:
9 Dec 2014 at 3:59 pm [Comment permalink]
Hello Rick
Sorry to hear there is a problem. Do you mean this Sitemap page (which is loading OK for me), or some other page?
Comment Preview
HTML: You can use simple tags like <b>, <a href="...">, etc.
To enter math, you can can either:
1. Use simple calculator-like input in the following format (surround your math in backticks, or qq on tablet or phone):
`a^2 = sqrt(b^2 + c^2)`
(See more on ASCIIMath syntax); or
2. Use simple LaTeX in the following format. Surround your math with \( and \).
\( \int g dx = \sqrt{\frac{a}{b}} \)
(This is standard simple LaTeX.)
NOTE: You can mix both types of math entry in your comment.
Search IntMath, blog and Forum
• Tips
• Categories
• Most Commented
• Recent Trackbacks
(External blogs linking to IntMath)
SquareCirclez is a "Top 100" Math Blog
From Math Blogs
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The arguments ... should be pixel images (objects of class "im"). Their spatial domains must overlap, but need not have the same pixel dimensions.
These functions compute the covariance or correlation between the corresponding pixel values in the images given.
The pixel image domains are intersected, and converted to a common pixel resolution. Then the corresponding pixel values of each image are extracted. Finally the correlation or covariance between the
pixel values of each pair of images, at corresponding pixels, is computed.
The result is a symmetric matrix with one row and column for each image. The [i,j] entry is the correlation or covariance between the ith and jth images in the argument list. The row names and column
names of the matrix are copied from the argument names if they were given (i.e. if the arguments were given as name=value).
Note that cor and cov are not generic, so you have to type cor.im, cov.im.
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Zero (0) is a number signifying the absence of the thing we count. It's the mathematical way of saying "nothing". Among integers it precedes 1 and follows -1.
Some properties of and facts about the number zero follow:
• It is even.
• It is neither positive nor negative, it lies exactly on the boundary of positive and negative numbers. However in some computer numeric encodings (such as one's complement) there exist two
representations of zero and so we may hear about a positive and negative zero, even though mathematically there is no such a thing.
• It is a whole number, a natural number, a rational number, a real number and a complex number.
• It is NOT a prime number.
• It is an additive identity, i.e. adding 0 to anything has no effect. Subtracting 0 from anything also has no effect.
• Multiplying anything by 0 gives 0, zero needs no units. Zero of anything is the same as zero of anything else: zero elephants is the same as zero frogs.
• Its graphical representation in all traditional numeral systems is the same: 0. To distinguish it graphically from the letter O it's sometimes crossed over with a line, or a dot is put in the
• 0^x (zero to the power of x), for x not equal to 0, is always 0.
• x^0 (x to the power of 0), for x not equal to 0, is always 1.
• 0^0 (0 to the power of 0) is generally not defined! However sometimes it's convenient to define it as equal to 1.
• In programming we start counting from 0 (unlike in real life where we start with 1), so we may encounter the term zeroth item (we say such system of counting is zero-based). We count from 0
because we normally express offsets from the first item, i.e. 0 means "0 places after the initial item".
• It is, along with 1, one of the symbols used in binary logic and is normally interpreted as the "off"/"false"/"low" value.
• Its opposite is most often said to be the infinity, even though it depends on the angle of view and the kind of infinity we talk about. Other numbers may be seen as its opposite as well (e.g. 1
in the context of probability).
• As it is one of the most commonly used numbers in programming, computers sometimes deal with it in special ways, for example in assembly languages there are often special instructions for
comparing to 0 (e.g. NEZ, not equals zero) which can save memory and also be faster. So as a programmer you may optimize your program by trying to use zeros if possible.
• In C and many other languages 0 represents the false value, a function returning 0 many times signifies an error during the execution of that function. However 0 also sometimes means success,
e.g. as a return value from the main function. 0 is also often used to signify infinity, no limit or lack of value (e.g. NULL pointer normally points to address 0 and means "pointing nowhere").
• Historically the concept of number zero seems to have appeared at least 3000 BC and is thought to signify an advanced abstract thinking, though it was first used only as a positional symbol for
writing numbers and only later on took the meaning of a number signifying "nothing".
It's common knowledge that dividing by zero is not defined (although zero itself can be divided), it is a forbidden operation mainly because it breaks equations (allowing dividing by zero would also
allow us to make basically any equation hold, even those that normally don't). In programming dividing by zero typically causes an error, crash of a program or an exception. In some programming
languages floating point division by zero results in infinity or NaN. When operating with limits, we can handle divisions by zero in a special way (find out what value an expression approaches if we
get infinitely close to dividing by 0).
See Also
Powered by nothing. All content available under CC0 1.0 (public domain). Send comments and corrections to drummyfish at disroot dot org.
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Survival Evaluation: Leveraging Deep Studying for Time-to-Occasion Forecasting | by Lina Faik | Apr, 2023
Illustration by the creator
Sensible Utility to Rehospitalization
Survival fashions are nice for predicting the time for an occasion to happen. These fashions can be utilized in all kinds of use instances together with predictive upkeep (forecasting when a machine
is more likely to break down), advertising analytics (anticipating buyer churn), affected person monitoring (predicting a affected person is more likely to be re-hospitalized), and way more.
By combining machine studying with survival fashions, the ensuing fashions can profit from the excessive predictive energy of the previous whereas retaining the framework and typical outputs of the
latter (such because the survival likelihood or hazard curve over time). For extra info, try the primary article of this collection here.
Nevertheless, in observe, ML-based survival fashions nonetheless require in depth characteristic engineering and thus prior enterprise data and instinct to result in satisfying outcomes. So, why not
use deep studying fashions as an alternative to bridge the hole?
This text focuses on how deep studying could be mixed with the survival evaluation framework to unravel use instances akin to predicting the probability of a affected person being (re)hospitalized.
After studying this text, you’ll perceive:
1. How can deep studying be leveraged for survival evaluation?
2. What are the widespread deep studying fashions in survival evaluation and the way do they work?
3. How can these fashions be utilized concretely to hospitalization forecasting?
This text is the second a part of the collection round survival evaluation. If you’re not acquainted with survival evaluation, it’s best to begin by studying the primary one here. The
experimentations described within the article had been carried out utilizing the libraries scikit-survival, pycox, and plotly. You will discover the code right here on GitHub.
1.1. Downside assertion
Let’s begin by describing the issue at hand.
We’re keen on predicting the probability {that a} given affected person can be rehospitalized given the accessible details about his well being standing. Extra particularly, we want to estimate this
likelihood at totally different time factors after the final go to. Such an estimate is important to observe affected person well being and mitigate their threat of relapse.
It is a typical survival evaluation drawback. The information consists of three components:
Affected person’s baseline information together with:
• Demographics: age, gender, locality (rural or city)
• Affected person historical past: smoking, alcohol, diabetes mellitus, hypertension, and so on.
• Laboratory outcomes: hemoglobin, whole lymphocyte depend, platelets, glucose, urea, creatinine, and so on.
• Extra details about the supply dataset here.
A time t and an occasion indicator δ∈{0;1}:
• If the occasion happens throughout the commentary length, t is the same as the time between the second the info had been collected and the second the occasion (i.e., rehospitalization) is
noticed, In that case, δ = 1.
• If not, t is the same as the time between the second the info had been collected and the final contact with the affected person (e.g. finish of research). In that case, δ = 0.
Determine 1 — Survival evaluation information, illustration by the creator. Observe: sufferers A, and C are censored.
⚠️ With this description, why use survival evaluation strategies when the issue is so much like a regression job? The preliminary paper provides a fairly good rationalization of the primary motive:
“If one chooses to make use of commonplace regression strategies, the right-censored information turns into a sort of lacking information. It’s often eliminated or imputed, which can introduce
bias into the mannequin. Subsequently, modeling right-censored information requires particular consideration, therefore the usage of a survival mannequin.” Supply [2]
1.2. DeepSurv
Let’s transfer on to the theoretical half with a bit of refresher on the hazard operate.
“The hazard operate is the likelihood a person won’t survive an additional infinitesimal period of time δ, given they’ve already survived as much as time t. Thus, a better hazard signifies a
better threat of dying.”
Supply [2]
Just like the Cox proportional hazards (CPH) mannequin, DeepSurv relies on the idea that the hazard operate is the product of the two features:
• the baseline hazard operate: λ_0(t)
• the chance rating, r(x)=exp(h(x)). It fashions how the hazard operate varies from the baseline for a given particular person given the noticed covariates.
Extra on CPH fashions within the first article of this collection.
The operate h(x) is often known as the log-risk operate. And that is exactly the operate that the Deep Surv mannequin goals at modeling.
In actual fact, CPH fashions assume that h(x) is a linear operate: h(x) = β . x. Becoming the mannequin consists thus in computing the weights β to optimize the target operate. Nevertheless, the
linear proportional hazards assumption doesn’t maintain in lots of functions. This justifies the necessity for a extra advanced non-linear mannequin that’s ideally able to dealing with giant volumes
of knowledge.
On this context, how can the DeepSurv mannequin present a greater various? Let’s begin by describing it. In response to the unique paper, it’s a “deep feed-forward neural community which predicts the
consequences of a affected person’s covariates on their hazard fee parameterized by the weights of the community θ.” [2]
How does it work?
‣ The enter to the community is the baseline information x.
‣ The community propagates the inputs via quite a few hidden layers with weights θ. The hidden layers include fully-connected nonlinear activation features adopted by dropout.
‣ The ultimate layer is a single node that performs a linear mixture of the hidden options. The output of the community is taken as the expected log-risk operate.
Supply [2]
Determine 2 — DeepSurv structure, illustration by the creator, impressed by supply [2]
Because of this structure, the mannequin may be very versatile. Hyperparametric search methods are sometimes used to find out the variety of hidden layers, the variety of nodes in every layer, the
dropout likelihood and different settings.
What concerning the goal operate to optimize?
• CPH fashions are educated to optimize the Cox partial probability. It consists of calculating for every affected person i at time Ti the likelihood that the occasion has occurred, contemplating
all of the people nonetheless in danger at time Ti, after which multiplying all these chances collectively. You will discover the precise mathematical system right here [2].
• Equally, the target operate of DeepSurv is the log-negative imply of the identical partial probability with an extra half that serves to regularize the community weights. [2]
Code pattern
Here’s a small code snippet to get an concept of how such a mannequin is carried out utilizing the pycox library. The entire code could be discovered within the pocket book examples of the library
here [6].
# Step 1: Neural web
# easy MLP with two hidden layers, ReLU activations, batch norm and dropout
in_features = x_train.form[1]
num_nodes = [32, 32]
out_features = 1
batch_norm = True
dropout = 0.1
output_bias = False
web = tt.sensible.MLPVanilla(in_features, num_nodes, out_features, batch_norm,
dropout, output_bias=output_bias)
mannequin = CoxPH(web, tt.optim.Adam)
# Step 2: Mannequin coaching
batch_size = 256
epochs = 512
callbacks = [tt.callbacks.EarlyStopping()]
verbose = True
log = mannequin.match(x_train, y_train, batch_size, epochs, callbacks, verbose,
val_data=val, val_batch_size=batch_size)
# Step 3: Prediction
_ = mannequin.compute_baseline_hazards()
surv = mannequin.predict_surv_df(x_test)
# Step 4: Analysis
ev = EvalSurv(surv, durations_test, events_test, censor_surv='km')
1.3. DeepHit
As a substitute of creating sturdy assumptions concerning the distribution of survival occasions, what if we may prepare a deep neural community that might be taught them immediately?
That is the case with the DeepHit mannequin. Particularly, it brings two vital enhancements over earlier approaches:
• It doesn’t depend on any assumptions concerning the underlying stochastic course of. Thus, the community learns to mannequin the evolution over time of the connection between the covariates and
the chance.
• It could actually deal with competing dangers (e.g., concurrently modeling the dangers of being rehospitalized and dying) via a multi-task studying structure.
As described right here [3], DeepHits follows the widespread structure of multi-task studying fashions. It consists of two principal components:
1. A shared subnetwork, the place the mannequin learns from the info a basic illustration helpful for all of the duties.
2. Job-specific subnetworks, the place the mannequin learns extra task-specific representations.
Nevertheless, the structure of the DeepHit mannequin differs from typical multi-task studying fashions in two features:
• It features a residual connection between the inital covariates and the enter of the task-specific sub-networks.
• It makes use of just one softmax output layer. Due to this, the mannequin doesn’t be taught the marginal distribution of competing occasions however the joint distribution.
The figures beneath present the case the place the mannequin is educated concurrently on two duties.
The output of the DeepHit mannequin is a vector y for each topic. It provides the likelihood that the topic will expertise the occasion okay ∈ [1, 2] for each timestamp t throughout the commentary
Determine 3 — DeepHit structure, illustration by the creator, impressed by supply [4]
2.1. Methodology
The information set was divided into three components: a coaching set (60% of the info), a validation set (20%), and a take a look at set (20%). The coaching and validation units had been used to
optimize the neural networks throughout coaching and the take a look at set for remaining analysis.
The efficiency of the deep studying fashions was in comparison with a benchmark of fashions together with CoxPH and ML-based survival fashions (Gradient Boosting and SVM). Extra info on these
fashions is accessible within the first article of the collection.
Two metrics had been used to judge the fashions:
• Concordance index (C-index): it measures the aptitude of the mannequin to supply a dependable rating of survival occasions primarily based on particular person threat scores. It’s computed
because the proportion of concordant pairs in a dataset.
• Brier rating: It’s a time-dependent extension of the imply squared error to proper censored information. In different phrases, it represents the common squared distance between the noticed
survival standing and the expected survival likelihood.
2.2. Outcomes
When it comes to C-index, the efficiency of the deep studying fashions is significantly higher than that of the ML-based survival evaluation fashions. Furthermore, there may be virtually no
distinction between the efficiency of Deep Surval and Deep Hit fashions.
Determine 4 — C-Index of fashions on the prepare and take a look at units
When it comes to Brier rating, the Deep Surv mannequin stands out from the others.
• When analyzing the curve of the Brier rating as a operate of time, the curve of the Deep Surv mannequin is decrease than the others, which displays a greater accuracy.
Determine 5— Brier rating on the take a look at set
• This commentary is confirmed when contemplating the combination of the rating over the identical time interval.
Determine 6 — Built-in Brier rating on the take a look at set
Observe that the Brier wasn’t computed for the SVM as this rating is just relevant for fashions which might be in a position to estimate a survival operate.
Determine 7— Survival curves of randomly chosen sufferers utilizing DeepSurv Mannequin
Lastly, deep studying fashions can be utilized for survival evaluation in addition to statistical fashions. Right here, as an example, we are able to see the survival curve of randomly chosen
sufferers. Such outputs can carry many advantages, particularly permitting a simpler follow-up of the sufferers which might be probably the most in danger.
✔️ Survival fashions are very helpful for predicting the time it takes for an occasion to happen.
✔️ They might help deal with many use instances by offering a studying framework and methods in addition to helpful outputs such because the likelihood of survival or the hazard curve over time.
✔️ They’re even indispensable in such a makes use of instances to use all the info together with the censored observations (when the occasion doesn’t happen throughout the commentary interval for
✔️ ML-based survival fashions are inclined to carry out higher than statistical fashions (extra info here). Nevertheless, they require high-quality characteristic engineering primarily based on
strong enterprise instinct to attain passable outcomes.
✔️ That is the place Deep Studying can bridge the hole. Deep learning-based survival fashions like DeepSurv or DeepHit have the potential to carry out higher with much less effort!
✔️ Nonetheless, these fashions are usually not with out drawbacks. They require a big database for coaching and require fine-tuning a number of hyperparameters.
[1] Bollepalli, S.C.; Sahani, A.Okay.; Aslam, N.; Mohan, B.; Kulkarni, Okay.; Goyal, A.; Singh, B.; Singh, G.; Mittal, A.; Tandon, R.; Chhabra, S.T.; Wander, G.S.; Armoundas, A.A. An Optimized
Machine Learning Model Accurately Predicts In-Hospital Outcomes at Admission to a Cardiac Unit. Diagnostics 2022, 12, 241.
[2] Katzman, J., Shaham, U., Bates, J., Cloninger, A., Jiang, T., & Kluger, Y. (2016). DeepSurv: Personalized Treatment Recommender System Using A Cox Proportional Hazards Deep Neural Network, ArXiv
[3] Laura Löschmann, Daria Smorodina, Deep Learning for Survival Analysis, Seminar info programs (WS19/20), February 6, 2020
[4] Lee, Changhee et al. DeepHit: A Deep Learning Approach to Survival Analysis With Competing Risks. AAAI Convention on Synthetic Intelligence (2018).
[5] Wikipedia, Proportional hazards model
[6] Pycox library
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How to Convert SCM to SCF
••• Jupiterimages/Brand X Pictures/Getty Images
SCM stands for standard cubic meter, also written m^3, and SCF stands for standard cubic foot, also written ft^3. Both units measure the volume of an object. Though the standard cubic meter is the
preferred measurement in most of the world, many in the United States still rely on the standard cubic foot. If you have a size of a package you want to ship in standard cubic meters but you want to
describe it in terms that more Americans would understand, you have to convert from SCM to SCF.
Multiply the volume in SCM by 35.3147 to convert to SCF because each SCM equals 35.3147 SCF. Therefore, for every 1 SCF you have, you have 35.3147 SCM. For example, if you have 40 SCM, multiply
40 by 35.3147 to get 1,412.59 SCF.
Divide the volume in SCM by 0.0283168 to convert to SCF. Each SCF equals 0.0283168 SCM. Therefore, you divide the volume in SCM by the number of SCF per SCM. In this example, divide 40 by
0.0283168 to get 1,412.59 SCF.
Check your conversion using an online cubic meters to cubic feet converter. Enter the number of cubic meters and press "Go."
About the Author
Mark Kennan is a writer based in the Kansas City area, specializing in personal finance and business topics. He has been writing since 2009 and has been published by "Quicken," "TurboTax," and "The
Motley Fool."
Photo Credits
Jupiterimages/Brand X Pictures/Getty Images
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10 Most Common ISEE Upper-Level Math Questions
Preparing for the ISEE Upper-Level Math test? Want a preview of the most common mathematics questions on the ISEE Upper-Level Math test? If so, then you are in the right place.
The mathematics section of ISEE Upper Level can be a challenging area for many test-takers, but with enough patience, it can be easy and even enjoyable!
Preparing for the ISEE Upper-Level Math test can be a nerve-wracking experience. Learning more about what you’re going to see when you take the ISEE Upper Level can help to reduce those pre-test
jitters. Here’s your chance to review the 10 most common ISEE Upper-Level Math questions to help you know what to expect and what to practice most. Try these 10 most common ISEE Upper-Level Math
questions to hone your mathematical skills and to see if your math skills are up to date on what’s being asked on the exam or if you still need more practice.
Make sure to follow some of the related links at the bottom of this post to get a better idea of what kind of mathematics questions you need to practice.
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10 Sample ISEE Upper Level Math Practice Questions
1- Emily and Daniel have taken the same number of photos on their school trip. Emily has taken 5 times as many photos as Claire and Daniel has taken 16 more photos than Claire. How many photos has
Claire taken?
A. 4
B. 6
C. 8
D. 10
2- Emily lives 5 \(\frac{1}{4}\) miles from where she works. When traveling to work, she walks to a bus stop \(\frac{1}{3}\) of the way to catch a bus. How many miles away from her house is the bus
A. \(4\frac{1}{3}\) miles
B. \(4\frac{3}{4}\) miles
C. \(2\frac{3}{4}\) miles
D. \(1\frac{3}{4}\) miles
3- Use the diagram below to answer the question.
Given the lengths of the base and diagonal of the rectangle above, what is the length of height h, in terms of s?
A. 2s \(\sqrt{6}\)
B. s \(\sqrt{7}\)
C. 5s
D. 5s\(^2\)
There are also purple marbles in the bag. Which of the following can NOT be the probability of randomly selecting a purple marble from the bag?
A. \(\frac{1}{10}\)
B. \(\frac{1}{4}\)
C. \(\frac{2}{5}\)
D. \(\frac{7}{15}\)
5- A square measures 6 inches on one side. By how much will the area be increased if its length is increased by 5 inches and its width decreased by 3 inches.
A. 1 sq decreased
B. 3 sq decreased
C. 6 sq decreased
D. 9 sq decreased
6- If a box contains red and blue balls in the ratio of \(2 : 3\) red to blue, how many red balls are there if 90 blue balls are in the box?
A. 40
B. 60
C. 80
D. 30
7- How many \(3 × 3\) squares can fit inside a rectangle with a height of 54 and width of 12?
A. 72
B. 52
C. 62
D. 42
8- David makes a weekly salary of $220 plus \(8\%\) commission on his sales. What will his income be for a week in which he makes sales totaling $1100?
A. $328
B. $318
C. $308
D. $298
9- \(4x^2y^3 + 5x^3y^5 – (5x^2y^3 – 2x^3y^5) = \)
A. \(–x^2y^3\)
B. \(6x^2y^3 – x^3y^5\)
C. \(7x^2y^3\)
D. \(7x^3y^5 – x^2y^3\)
10- If the area of trapezoid is 126 cm, what is the perimeter of the trapezoid?
Best ISEE Upper Level Math Prep Resource for 2024
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1- A
Emily = Daniel
Emily = 5 Claire
Daniel = \(16 +\) Claire
Emily = Daniel → Emily = \(16 +\) Claire
Emily = 5 Claire → 5 Claire = \(16 +\) Claire → 5 Claire \(–\) Claire = 16
4 Claire = 16
Claire = 4
2- D
\(\frac{1}{3}\) of the distance is \(5\frac{1}{4}\) miles. Then:
\(\frac{1}{3} × 5 \frac{1}{4} = \frac{1}{3} × \frac{21}{4}= \frac{21}{12}\)
Converting \(\frac{21}{12}\) to a mixed number gives:
\(\frac{21}{12}= 1\frac{9}{12}=1\frac{3}{4}\)
3- A
Use Pythagorean theorem:
Subtracting (s^2) from both sides gives: \( h^2=24s^2\)
Square roots of both sides: \(h=\sqrt{24s^2}=\sqrt{4×6×s^2 } =\sqrt 4 × \sqrt6 × \sqrt{s^2 }=2 × s × \sqrt6 = 2s\sqrt6\)
4- D
Let \(x\) be the number of purple marbles. Let’s review the choices provided:
A. \(\frac{1}{10}\), if the probability of choosing a purple marble is one out of ten, then:
\(Probability=\frac{number \space of \space desired \space outcomes}{number \space of \space total \space outcomes}=\frac{x}{20+30+40+x}=\frac{1}{10}\)
Use cross multiplication and solve for x. \(10x=90+x→9x=90→x=9\)
Since the number of purple marbles can be 9, then, the choice is the probability of randomly selecting a purple marble from the bag.
Use the same method for other choices.
B. \(\frac{1}{4}\)
C. \(\frac{2}{5}\)
D. \(\frac{7}{15}\)
The number of purple marbles cannot be a decimal.
5- B
The area of the square is 36 square inches.
\(Area of square=side×side=6×6=36\)
The length of the square is increased by 5 inches and its width decreased by 3 inches. Then, its area equals:
Area of \(rectangle=width×length=11×3=33\)
The area of the square will be decreased by 3 square inches.
6- B
Write a proportion and solve.
Use cross multiplication:
7- A
Number of squares equal to: \(\frac{54×12}{3×3} = 18 × 4 = 72\)
8- C
David’s weekly salary is $220 plus \(8\%\) of $1,100. Then:
\(8\% \space of \space 1,100=0.08 × 1,100 = 88\)
9- D
\(4x^2y^3 + 5x^3y^5 – (5x^2y^3 – 2x^3y^5) = 4x^2y^3 + 5x^3y^5 – 5x^2y^3 + 2x^3y^5 = – x^2y^3 + 7 x^3y^5\)
10- 13
The area of the trapezoid is:
\(Area=\frac{1}{2}h(b_1+b_2 )=\frac{1}{2}(x)(13+8)=126\)
\(y=\sqrt{5^2 +12^2 } = \sqrt{25+144} = \sqrt{169} = 13\)
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Thermal Expansion Calculator
Calculating Thermal Expansion
Thermal expansion refers to the tendency of a material to change its dimensions (length, area, or volume) in response to a change in temperature. This phenomenon is important in engineering and
construction because materials can expand or contract with temperature fluctuations, potentially causing stress or deformation in structures. Calculating thermal expansion allows engineers to
anticipate these changes and account for them in design.
The Thermal Expansion Formula
The linear thermal expansion of a material can be calculated using the formula:
\( \Delta L = \alpha L_0 \Delta T \)
• \( \Delta L \) is the change in length (m).
• \( \alpha \) is the coefficient of linear thermal expansion (1/°C or 1/K).
• \( L_0 \) is the original length of the material (m).
• \( \Delta T \) is the change in temperature (°C or K).
This formula calculates the change in length, but similar formulas exist for calculating changes in area and volume due to thermal expansion.
Step-by-Step Guide to Calculating Thermal Expansion
Follow these steps to calculate the thermal expansion of a material:
• Step 1: Determine the original length \( L_0 \) of the material before the temperature change.
• Step 2: Find the coefficient of linear thermal expansion \( \alpha \) for the material (this value is specific to each material).
• Step 3: Measure or estimate the temperature change \( \Delta T \) that the material will undergo.
• Step 4: Plug these values into the thermal expansion formula to calculate the change in length \( \Delta L \).
Example: Calculating Thermal Expansion of a Metal Rod
Suppose a steel rod has an original length of 5 meters and is exposed to a temperature increase of 50°C. The coefficient of linear thermal expansion for steel is approximately \( 12 \times 10^{-6} \,
\text{1/°C} \). Using the thermal expansion formula:
\( \Delta L = \alpha L_0 \Delta T \)
Substitute the values:
\( \Delta L = (12 \times 10^{-6}) \times 5 \times 50 = 0.003 \, \text{m} \)
The rod will expand by 0.003 meters (3 mm) when subjected to this temperature increase.
Types of Thermal Expansion
Thermal expansion can occur in three forms, depending on the dimensions of the object being measured:
• Linear expansion: Expansion along a single dimension (length).
• Area expansion: Expansion of a material’s surface area. The formula for area expansion is \( \Delta A = 2 \alpha A_0 \Delta T \).
• Volumetric expansion: Expansion of the material’s volume. The formula for volume expansion is \( \Delta V = 3 \alpha V_0 \Delta T \).
Factors That Affect Thermal Expansion
Several factors influence how much a material will expand when heated:
• Material type: Different materials have different coefficients of thermal expansion. Metals, for instance, generally expand more than ceramics or plastics.
• Temperature change: Larger temperature changes result in greater expansion or contraction.
• Initial dimensions: The larger the original dimensions of the material, the more it will expand for a given temperature change.
Practical Applications of Thermal Expansion
Understanding and calculating thermal expansion is essential in many engineering and construction applications, such as:
• Bridges and buildings: Expansion joints are included in the design to allow for thermal movement, preventing structural damage.
• Piping systems: Pipelines may expand or contract with temperature changes, so flexible sections or loops are used to accommodate this movement.
• Railway tracks: Gaps are left between sections of track to prevent buckling caused by thermal expansion during hot weather.
Example: Calculating Thermal Expansion of a Glass Window
Let’s calculate the thermal expansion of a glass window with an original area of 1.5 m². The coefficient of linear expansion for glass is \( 9 \times 10^{-6} \, \text{1/°C} \), and the temperature
increase is 30°C. First, we calculate the area expansion:
\( \Delta A = 2 \alpha A_0 \Delta T \)
Substitute the known values:
\( \Delta A = 2 \times (9 \times 10^{-6}) \times 1.5 \times 30 = 0.00081 \, \text{m}^2 \)
The glass window will experience an area expansion of 0.00081 m² (0.81 mm²).
Frequently Asked Questions (FAQ)
1. What materials expand the most with temperature changes?
Metals typically expand more than other materials due to their higher coefficients of thermal expansion. For example, aluminum and steel expand significantly with temperature changes, while materials
like glass and concrete expand less.
2. How do engineers account for thermal expansion?
Engineers design structures with expansion joints or flexible connections that allow for movement due to thermal expansion and contraction. These design elements help prevent damage or stress caused
by temperature fluctuations.
3. Can thermal expansion cause permanent deformation?
Thermal expansion usually causes temporary deformation. However, if a material is heated beyond its limits or experiences rapid temperature changes (thermal shock), it can lead to permanent
deformation or even cracking.
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How many Satellites and Channels require in your DGNSS?
In the realm of precision navigation and positioning, Differential GNSS (DGNSS) often emerges as a critical yet misunderstood technology. Despite its pivotal role in enhancing GPS accuracy, several
myths persist, clouding its true value. Today, let’s discuss some common misconception in the survey world related to DGNSS (or DGPS).
You may have heard people emphasize the importance of having 400, 800, or even 1200 channels in a DGNSS receiver. Have you ever thought deeper to understand what this actually means? Let’s clarify
this with some simple mathematical calculations.
There are a total of 120 operational satellites: GPS (31), Galileo (24), GLONASS (24), BeiDou (30), NavIC (7), and QZSS (4). These satellites are distributed across the orbital plane and are not all
visible from a particular point on Earth.
What is an Elevation Mask?
An elevation mask is a threshold angle that defines the minimum elevation above the horizon that a satellite must have to be considered in the positioning calculations of a DGNSS receiver. The
purpose of this mask is to filter out low-angle satellite signals that are more likely to be degraded by atmospheric interference, multipath effects, and obstructions.
Figure 1: Different Satellite Elevation Angles
What value to keep for Elevation Mask?
It is recommended to set the Elevation Mask to 15 degrees to use only high-quality satellites for positioning. However, even with an Elevation Mask set to 5 degrees, you wouldn’t be able to see even
half of the satellites simultaneously.
How many satellites with Elevation Mask as 5 degrees?
With an Elevation Mask of 5 degrees, you would only be able to track a maximum of 14 GPS, 6 Galileo, 6 GLONASS, 11 BeiDou, and 4 QZSS satellites. Out of the 7 NavIC satellites, only 5 are
operational, and many companies do not support NavIC.
So how many Channels?
Now, let’s talk about channels. The L1 and L2 bands require only one channel each, whereas the L5 band requires two channels. Interestingly, out of the 120 satellites, only 40–45 have L5 bands.
Here’s the mathematical calculation with a 5-degree Elevation Mask (in general, if you set it to 15 degrees, which is the industry standard, the number would be lower):
Total satellites being used: 14 (GPS) + 6 (Galileo) + 6 (GLONASS) + 11 (BeiDou) + 4 (QZSS) + 5 (NavIC) = 46 satellites. Not all of these would have L5 bands; let’s assume 20 do.
So, the total number of channels required would be: 46*1 (L1) + 46*1 (L2) + 20*2 (L5) = 132 channels. In reality, this is still an exaggerated number. In the real world, no more than 100 channels are
typically required. Claims of needing 400, 800, or even 1200 channels are overly futuristic and marketing/selling strategy. It will take more than 10–15 years to even approach the utilization of 300
Does All Satellites Used in DGNSS Provide the Same Level of Accuracy?
The accuracy of signals from GNSS satellites can vary due to several factors, including the satellite’s position, health, and atmospheric conditions affecting the signal’s travel to Earth.
Differential GNSS (DGNSS) enhances overall accuracy by correcting these variable errors. However, the effectiveness of DGNSS corrections is influenced by the quality of the correction data, which
depends on the proximity and condition of the reference station, as well as the quality of the communication link between the reference station and the DGNSS receiver. Therefore, it is crucial to
closely examine the quality of the satellites being used for positioning.
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Pythagorean Triple Problem in Sub-linear Time
The Pythagorean triple problem is as follows. Given an input integer \(n\), return integers \(a\), \(b\), \(c\) such that the two following conditions hold:
$$ a b c = n $$ $$ a^2 + b^2 = c^2 $$
I was interested in finding a solution to this problem that was both succint and had good asymptotic complexity. The solution I found runs in O(sqrt(n)) time by deconstructing the problem into the
well-known 3SUM problem.
Getting the Divisors
We know that the three numbers we generate must all multiply together to form \(n\). Therefore, each number must be a divisor of \(n\). There is a simple O(sqrt(n)) time algorithm that generates all
divisors of \(n\):
// @require n >= 1
export const divisors = (n: number) => {
const d = _.times(Math.sqrt(n) - 1)
.map(i => i + 1)
.filter(i => n % i === 0);
return _.uniq(d.concat([...d].reverse().map(i => n / i)));
The algorithm is expressed in TypeScript, in a functional form. The algorithm takes all numbers in the range of [1 ... sqrt(n)] and filters such numbers that \(n\) is divisble by. We are left with
all of the divisors up until \(\sqrt n\).
To then get the rest of the numbers, concatenate the current array with each divisor’s pair. This is because if \(i\) is a divisor, \(\frac{n}{i}\) is also guaranteed to be a divisor. All references
to _ are lodash.
Invoking the 3SUM Problem
We now have a list of numbers to search from to achieve the two conditions. The length of the list is on order of O(log(n)) because that is, up to a constant factor, how many divisors a given number
On inspection, we expect the second condition to be more “stringent” i.e. there exists fewer combinations which satisfy the condition. Luckily, there exists a body of knowledge on solving that sort
of problem.
The 3SUM Problem
The 3SUM problem is to, given a list of numbers \(A\), return a set of three numbers \(a\), \(b\), \(c\) such that the following conditions hold:
$$ a + b + c = 0 $$ $$ a, b, c \in A $$
There are many algorithms to solve this, including a relatively simple \(O(n^2)\) solution. However, this does not quite match our problem. However, if we squint our eyes a bit, we can see how it may
be applied. We may perform some simple algebra on our condition:
$$ a^2 + b^2 = c^2 $$ $$ a^2 + b^2 - c^2 = 0 $$
So we see if we include all negative numbers of our divisor into our search set \(A\), we’re much better off. As well, we square each number of our original divisor set. So, given a divisor set for
example, of 30:
$$ {1, 2, 3, 5, 6, 10, 15, 30} $$
We transform this set into the following:
$$ {-900, -225, -100, -36, -25, -9, -4, -1, 1, 4, 9, 25, 36, 100, 225, 900} $$
The 3SUM search is guaranteed to find a 3-set matching our original Pythagorean condition. However, it will also match false-positives constructed of more than one negative number. To filter these
out, we only consider solutions to the 3SUM problem which possess one negative number.
Putting it all Together
The following code implements the algorithm described above, taking the divisor set, transforming it, applying it to the 3SUM problem, and filtering the results. The overall complexity is \(O(\sqrt
{n})\) because the complexity of constructing the divisors is strictly more expensive than solving the 3SUM problem on the divisor set. The complexity could probably be improved via Pollard’s Rho
algorithm, at the cost of sacrificing simplicity.
// Returns [a, b, c] where a^2 + b^2 = c^2 and a * b * c = n
// If no such 3-tuple exists, returns [].
// Runs in O(sqrt(n)) time.
// @require n >= 1
export const pythagoreanTriplet = (n: number) => {
let d = divisors(n).map(x => x ** 2);
d = [...d]
.map(x => -x)
// O(log(n)^2)
const p = sum3(d)
.filter(x => _.countBy(x, y => y < 0).true === 1)
.map(x => x.map(y => Math.sqrt(Math.abs(y))).sort((a, b) => a - b))
.filter(x => x.reduce((a, y) => a * y) === n);
return p.length > 0 ? p[0] : [];
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Part 2: Evidence Based Investing is Dead. Long Live Evidence Based Investing!
Advisor Perspectives welcomes guest contributions. The views presented here do not necessarily represent those of Advisor Perspectives.
This is Part two of a two-part article series. Please see article one here.
Michael Edesess’ article, The Trend that is Ruining Finance Research, makes the case that financial research is flawed. In this two-part series, I examine the points that Edesess raised in some
detail. His arguments have some merit. Importantly however, his article fails to undermine the value of finance research in general. Rather, his points highlight that finance is a real profession
that requires skills, education and experience that differentiates professionals from laymen.
Edesess’ case against so-called evidence-based investing rests on three general assertions. First, there is the very real issue with using a static t-statistic threshold when the number of
independent tests becomes very large. Second, financial research is often conducted with a universe of securities that includes a large number of micro-cap and nano-cap stocks. These stocks often do
not trade regularly and exhibit large overnight jumps in prices. They are also illiquid and costly to trade. Third, the regression models used in most financial research are poorly calibrated to form
conclusions on non-stationary financial data with large outliers.
This article will tackle the first issued, often called “p-hacking,” and proposes a framework to help those who embrace evidence-based investing to make judicious decisions based on a more thoughtful
interpretation of finance research. Part one of this series addressed the other two issues.
P-hacking and scaling significance tests
When Fama, French, Jegadeesh, et al. published the first factor models in the early 1990s, it was reasonable to reject the null hypothesis (no effect) with an observed t-statistic of 2. After all,
the computational power and data at the time could not support data mining to the extent that it is now possible. Moreover, these early researchers were careful to derive their models very
thoughtfully from first principles, lending economic credence to their results.
However, as Cam Harvey has so assiduously noted, the relevant t-statistic to signal statistical significance must expand through time to reflect the number of independent tests. He suggests that,
based on several different approaches to the problem, current finance research should seek to exceed a t-statistic threshold of at least 3 to be considered significant. If the results are derived
explicitly through data mining, or through multivariate tests, the threshold should be closer to 4, while results derived from first principles based on economic or behavioral conjecture, and with a
properly structured hypothesis test, may be considered significant at thresholds somewhat below 3.
Harvey’s recommendations make tremendous sense. The empirical finance community – like so many other academic communities such as medicine and psychology – is guilty of propagating “magical thinking”
for the sake of selling associated investment products, journal subscriptions and advertising. With few exceptions, journals only publish papers with interesting and significant findings. As a
result, the true number of tests of significance in finance vastly exceeds the number of published journal articles.
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VLOOKUP NA Error – Causes and Solutions
How to Use Excel > Excel Formula > VLOOKUP NA Error – Causes and Solutions
VLOOKUP NA Error – Causes and Solutions
VLOOKUP NA Error - Causes and Solutions
VLOOKUP NA error is the most common error when using the VLOOKUP function. This error occurs because the lookup_value is not in the first column of table_array for several reasons.
There are eight possible errors when using the VLOOKUP function. Read the following article for a more detailed explanation of the other VLOOKUP errors
VLOOKUP NA error because of the wrong table_array range
There are two possibilities for the wrong table_array range. First, the table_array range shifts because of the process of copying a formula. This error has been explained in detail with the solution
in “The Wrong Cell Reference” article.
Second, the table_array range point to the wrong location. If this happens, you must fix the formula and point the table_array range to the correct place.
VLOOKUP NA error because the data is missing from the table_array
This error occurs because something is missing in table_array because it was deleted.
The picture above explains the formula in cell H5 is correct, the same as the formula in the other cell in the same column, but the word “Caffe Latte” disappears from the price table, this is what
causes #N/A error, even though there is a price in the right column. The VLOOKUP function cannot retrieve the price value if there is no data in the first column of table_array.
VLOOKUP NA error because the lookup_value looks for nonexisting value
This error occurs because lookup_value is not in the first column of table_array. The image below explains no “Hot Brewed Coffee” in the table prices, this is what causes #N/A error, although the
formula is correct.
Some people want this #N/A error not to occur, because it affects to the other formula. #N/A error appears in total bill (cell H10) due to #N/A error in cell H8.
You can use the solution below to avoid #N/A error from appearing because of nonexisting lookup_value in the first column of table_array.
The other alternatives, you can use the AGGREGATE function or array formula. The following article discusses how to prevent divide by zero errors. You can use the same solution to avoid #N/A error in
the total bill.
VLOOKUP NA error because there is a blank space
Blank space is hard to detect error. Blank space could be in the lookup_value or in the first column of table_array. If a blank space in the front, it is easier to find out, but if a blank space in
the back it will be not visible
Look at the image above; there are two #N/A errors in cell H5 and cell H7. #N/A error in cell H7 because of a blank space in front of the word “Salted Caramel Mocha”. Pay attention to cell F7, the
word “Salted Caramel Mocha” is a bit indented, that’s where the blank space is.
#N/A error in cell H5 because there is a blank space behind the word “Caffe Latte”. A cell containing blank space is cell A2, but the appearance is not visible, it must be checked by editing the
cell, of course, this will take time if there are many data.
VLOOKUP NA error because numbers are stored as text
The first column of table_array and the lookup_value must have the same data type. If it is text, then all must be text, if it is number, then all must be number.
The problem arises because numbers can be stored as text by adding single quotes in front of the number. An appearance is a number, but Excel acknowledges it as a text.
Look at the image above. The VLOOKUP function looks for the price based on the item code. As a result, all formulas return #N/A errors.
Look again, the lookup_value is a number (column G), and the first column of table_array (column A) is a number too. Then why the VLOOKUP function returns a #N/A error?
Look more closely. All item code in column A (table_array) is right aligned, while all item code in column G (lookup_value) is left aligned.
Numbers by default is right-aligned, and text by default is left aligned, this is the problem. lookup value is text while the first column of table_array is a number.
Another way to detect numbers stored as text is by the appearance of small green triangles on the top left of each cell. Excel will notify if there are numbers stored as text, whether to convert to
numbers or to be ignored.
The Solution
The solution for removing #N/A error is to convert all item code in the first column of table_array to text or to convert all item code in the lookup_value to the number.
For example, the chosen solution is to convert all item code in the lookup_value to numbers. See the image below for how to do it.
You got the prices after conversion completed.
Related Function
Related Article
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how to swap indices for partial derivatives
General questions (367)
I need Cadabra to understand that the indices of derivatives can be swapped. For example, part of my expression looks like this:
2 \chi w**(3) M**(\xi+1) \delta{z}
\partial_{a b}{z} \partial_{a c d}{z} v_{b} v_{c} v_{d} -
2 \chi w**(3) M**(\xi+1) \delta{z}
\partial_{a b}{z} \partial_{c d a}{z} v_{b} v_{c} v_{d}
In theory, these 2 terms should contract, but they do not.
The indices are given as:
{a,b,c,d,f,i,j,h,k,l,m,n,p,r,s,t,u,v,w,z}::Indices("flat",position = free).
{a,b,c,d,f,i,j,h,k,l,m,o,p,r,s,t,u,v,w,z}::Integer(1..N) .
Partial derivative as
Hi bin_go.
I'm noting there is plenty of space in your code to improve, but I'm not here for that (in particular because I don't know your workcase scenario).
So, let me illustrate the use of the algorithm indexsort with an example inspired in your code.
My example
First, I'll define the indices and the symbol for the partial derivative.
{a,b,c,d}::Indices("flat",position = free).
In this example, I'd define some functions of the "coordinates", but saying that they depends on the derivative
Define the expression
ex := 2 \chi w**(3) M**(\xi+1) \delta{z}
\partial_{a b}{z} \partial_{a c d}{z} v_{b} v_{c} v_{d} -
2 \chi w**(3) M**(\xi+1) \delta{z}
\partial_{a b}{z} \partial_{c d a}{z} v_{b} v_{c} v_{d};
Note that the result is not simplified, unless we sort the indices
Hope this would help.
Cheers, Dox.
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Cyclic vector
From Encyclopedia of Mathematics
2020 Mathematics Subject Classification: Primary: 15A Secondary: 47A1693B [MSN][ZBL]
Let $A$ be an endomorphism of a finite-dimensional vector space $V$. A cyclic vector for $A$ is a vector $v$ such that $v,Av,\dots,A^{n-1}v$ form a basis for $V$, i.e. such that the pair $(A,v)$ is
completely reachable (see also Pole assignment problem; Majorization ordering; System of subvarieties; Frobenius matrix).
A vector $v$ in an (infinite-dimensional) Banach space or Hilbert space with an operator $A$ on it is said to be cyclic if the linear combinations of the vectors $A^iv$, $i=0,1,\dots$, form a dense
subspace, [a1].
More generally, let $\mathcal A$ be a subalgebra of $\mathcal B(H)$, the algebra of bounded operators on a Hilbert space $H$. Then $v\in H$ is cyclic if $\mathcal Av$ is dense in $H$, [a2], [a5].
If $\phi$ is a unitary representation of a (locally compact) group $G$ in $H$, then $v\in H$ is called cyclic if the linear combinations of the $\phi(g)v$, $g\in G$, form a dense set, [a3], [a4]. For
the connection between positive-definite functions on $G$ and the cyclic representations (i.e., representations that admit a cyclic vector), see Positive-definite function on a group. An irreducible
representation is cyclic with respect to every non-zero vector.
[a1] M. Reed, B. Simon, "Methods of mathematical physics: Functional analysis" , 1 , Acad. Press (1972) pp. 226ff
[a2] R.V. Kadison, J.R. Ringrose, "Fundamentals of the theory of operator algebras" , 1 , Acad. Press (1983) pp. 276
[a3] S.A. Gaal, "Linear analysis and representation theory" , Springer (1973) pp. 156
[a4] A.A. Kirillov, "Elements of the theory of representations" , Springer (1976) pp. 53 (In Russian)
[a5] M.A. Naimark, "Normed rings" , Noordhoff (1964) pp. 239 (In Russian) Zbl 0137.31703
How to Cite This Entry:
Cyclic vector. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Cyclic_vector&oldid=52802
This article was adapted from an original article by M. Hazewinkel (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098.
See original article
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Degree of Polynomial: Definition, Types, Examples, How to Find Degree of Polynomial?
Polynomials play a fundamental role in mathematics, particularly in algebra. They are used to model various real-world phenomena, solve equations, and make predictions. One crucial aspect of
polynomials is their degree, which provides valuable information about their behavior and properties. In this article, we will explore the concept of the degree of a polynomial, its different types,
and how to find it in various scenarios.
Before diving into the degree of a polynomial, let’s first understand what a polynomial is. A polynomial is an algebraic expression consisting of variables, coefficients, and exponents. It is a sum
of terms, where each term is a product of a coefficient and one or more variables raised to non-negative integer exponents. Here’s an example of a polynomial:
f(x) = 3x^2 + 2x – 5
In this polynomial, 3x^2, 2x, and -5 are the terms, and 3, 2, and -5 are the coefficients. The variable x is raised to the exponents 2 and 1 in the respective terms.
What is the Degree of a Polynomial?
The degree of a polynomial is the highest exponent of the variable in the polynomial when it is written in standard form. It provides essential information about the polynomial’s behavior, including
the number of solutions it has and the shape of its graph.
To determine the degree of a polynomial, we look at the term with the highest exponent. Let’s take an example:
f(x) = 4x^3 + 2x^2 – 7x + 1
In this polynomial, the highest exponent is 3, which means the degree of the polynomial is 3. The degree indicates that the polynomial is a cubic polynomial.
Degree of Zero Polynomial
A zero polynomial is a special case where all the coefficients are zero. In this case, the polynomial has no terms with non-zero coefficients, and therefore, no term has an exponent. As a result, the
degree of the zero polynomial is undefined.
Degree of Constant Polynomial
A constant polynomial is a polynomial where all the terms have the same exponent, which is zero. In other words, it is a polynomial with no variables. For example, f(x) = 5 is a constant polynomial.
The degree of a constant polynomial is zero.
Degree of a Polynomial With More Than One Variable
In polynomials with more than one variable, the degree is determined by the sum of the exponents of the variables in each term. The highest sum of exponents in the polynomial gives us the degree.
For example, consider the polynomial f(x, y) = 3x^2y^3 + 2xy^4. The sum of exponents in the first term is 2 + 3 = 5, and in the second term, it is 1 + 4 = 5. Therefore, the degree of this polynomial
is 5.
Degree of Linear Polynomials
A linear polynomial is a polynomial of degree 1. It consists of only one term with a non-zero coefficient and a variable raised to the first power. For example, f(x) = 2x – 3 is a linear polynomial.
The degree of a linear polynomial is always 1.
Degree of Quadratic Polynomial
A quadratic polynomial is a polynomial of degree 2. It consists of a term with the variable raised to the second power, along with other terms with lower exponents. For example, f(x) = x^2 + 2x + 1
is a quadratic polynomial. The highest exponent is 2, so the degree of this polynomial is 2.
Degree of Cubic Polynomial
A cubic polynomial is a polynomial of degree 3. It consists of a term with the variable raised to the third power, along with other terms with lower exponents. For example, f(x) = 4x^3 – 2x^2 + x – 5
is a cubic polynomial. The highest exponent is 3, so the degree of this polynomial is 3.
Degree of Bi-quadratic Polynomial
A bi-quadratic polynomial is a polynomial of degree 4. It consists of a term with the variable raised to the fourth power, along with other terms with lower exponents. For example, f(x) = 3x^4 – 2x^2
+ x – 1 is a bi-quadratic polynomial. The highest exponent is 4, so the degree of this polynomial is 4.
How to Find the Degree of a Polynomial?
Finding the degree of a polynomial is a straightforward process. Here are the steps to follow:
Step 1: Write the polynomial in standard form, with the terms arranged in descending order of their exponents.
Step 2: Identify the term with the highest exponent.
Step 3: The highest exponent of the variable in that term is the degree of the polynomial.
Let’s illustrate this process with an example:
Example: Find the degree of the polynomial f(x) = 2x^3 – 4x^2 + 5x + 1.
Step 1: The polynomial is already in standard form.
Step 2: The term with the highest exponent is 2x^3.
Step 3: The highest exponent of the variable in that term is 3.
Therefore, the degree of the polynomial f(x) = 2x^3 – 4x^2 + 5x + 1 is 3.
Classification Based on Degree of Polynomial
Polynomials can be classified based on their degree. Here is a table that categorizes polynomials based on their degree and provides examples of each type:
Degree Polynomial Name
0 Constant/Zero
1 Linear
2 Quadratic
3 Cubic
4 Bi-quadratic
Let’s examine each type in more detail:
Constant/Zero Polynomial
A constant polynomial is a polynomial of degree zero. It consists of a single constant term with no variables. For example, f(x) = 3 is a constant polynomial. The degree of a constant polynomial is
Linear Polynomial
A linear polynomial is a polynomial of degree 1. It consists of a single term with the variable raised to the first power. For example, f(x) = 2x – 1 is a linear polynomial. The highest exponent is
1, so the degree of this polynomial is 1.
Quadratic Polynomial
A quadratic polynomial is a polynomial of degree 2. It consists of a term with the variable raised to the second power, along with other terms with lower exponents. For example, f(x) = x^2 + 3x + 2
is a quadratic polynomial. The highest exponent is 2, so the degree of this polynomial is 2.
Cubic Polynomial
A cubic polynomial is a polynomial of degree 3. It consists of a term with the variable raised to the third power, along with other terms with lower exponents. For example, f(x) = 4x^3 – 2x^2 + x – 3
is a cubic polynomial. The highest exponent is 3, so the degree of this polynomial is 3.
Bi-quadratic Polynomial
A bi-quadratic polynomial is a polynomial of degree 4. It consists of a term with the variable raised to the fourth power, along with other terms with lower exponents. For example, f(x) = 3x^4 – 2x^2
+ x – 5 is a bi-quadratic polynomial. The highest exponent is 4, so the degree of this polynomial is 4.
Degree of a Polynomial Applications
Understanding the degree of a polynomial is crucial in various mathematical applications. Here are a few examples of how the degree of a polynomial is used:
1. Determining the number of solutions: The degree of a polynomial provides information about the number of solutions it has. For example, a quadratic polynomial can have at most two solutions,
while a cubic polynomial can have up to three solutions.
2. Graphing polynomials: The degree of a polynomial helps in sketching its graph. Higher-degree polynomials tend to have more complex graphs with multiple turning points and behavior.
3. Solving equations: The degree of a polynomial helps in determining the number of solutions to polynomial equations. It provides a starting point for solving equations by factoring or using other
algebraic techniques.
4. Identifying patterns and properties: The degree of a polynomial is closely related to its algebraic properties. For example, the leading coefficient and degree of a polynomial determine its end
behavior and whether it has a positive or negative leading term.
Behavior under Polynomial Operations
Polynomials exhibit specific properties when subjected to various operations such as addition, multiplication, and composition.
When adding polynomials, the degree of the resulting polynomial is determined by the highest degree among the added polynomials. For example, if we add a quadratic polynomial to a cubic polynomial,
the resulting polynomial will have a degree of either 2 or 3, depending on the specific terms.
When multiplying polynomials, the degree of the resulting polynomial is the sum of the degrees of the multiplied polynomials. For example, if we multiply a quadratic polynomial by a linear
polynomial, the resulting polynomial will have a degree of either 2 + 1 = 3 or 2 + 0 = 2, depending on the specific terms.
When composing polynomials, the degree of the resulting polynomial is the product of the degrees of the composed polynomials. For example, if we compose a quadratic polynomial with a cubic
polynomial, the resulting polynomial will have a degree of 2 * 3 = 6.
Solved Examples on Degree of Polynomial
Now, let’s solve a few examples to solidify our understanding of the degree of a polynomial.
Example 1: Find the degree of the polynomial f(x) = x^4 + 3x^3 – 2x^2 + 5x – 1.
Solution: The term with the highest exponent is x^4. Therefore, the degree of the polynomial is 4.
Example 2: Find the degree of the polynomial f(x) = 2x^2 + 4x – 7.
Solution: The term with the highest exponent is x^2. Therefore, the degree of the polynomial is 2.
Example 3: Find the degree of the polynomial f(x, y) = 5x^3y^2 + 2x^2y^3 – 3xy^4.
Solution: The sum of exponents in the first term is 3 + 2 = 5, and in the second term, it is 2 + 3 = 5. Therefore, the degree of this polynomial is 5.
How Kunduz Can Help You Learn Degrees of Polynomial?
At Kunduz, we understand the importance of mastering the concept of the degree of a polynomial. That’s why we offer comprehensive courses and personalized tutoring to help you excel in mathematics.
Our expert tutors will guide you through the intricacies of polynomials, including determining their degrees, finding zeros, and classifying them based on their degree. With our step-by-step approach
and interactive learning methods, you’ll gain a solid understanding of the degree of a polynomial and its applications.
So, whether you’re preparing for exams or looking to enhance your mathematical skills, Kunduz is here to support you every step of the way. Enroll in our courses today and unlock your full potential
in mathematics!
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<QtMath> - Generic Math Functions
The <QtMath> header file provides various math functions. More...
Header: #include <QtMath>
qreal qAcos(qreal v)
qreal qAsin(qreal v)
qreal qAtan2(qreal y, qreal x)
qreal qAtan(qreal v)
int qCeil(qreal v)
qreal qCos(qreal v)
float qDegreesToRadians(float degrees)
double qDegreesToRadians(double degrees)
qreal qExp(qreal v)
qreal qFabs(qreal v)
int qFloor(qreal v)
qreal qLn(qreal v)
quint32 qNextPowerOfTwo(quint32 value)
quint64 qNextPowerOfTwo(quint64 value)
quint32 qNextPowerOfTwo(qint32 value)
quint64 qNextPowerOfTwo(qint64 value)
qreal qPow(qreal x, qreal y)
float qRadiansToDegrees(float radians)
double qRadiansToDegrees(double radians)
qreal qSin(qreal v)
qreal qSqrt(qreal v)
qreal qTan(qreal v)
Detailed Description
These functions are partly convenience definitions for basic math operations not available in the C or Standard Template Libraries.
The header also ensures some constants specified in POSIX, but not present in C++ standards (so absent from <math.h> on some platforms), are defined:
Constant Description
M_E The base of the natural logarithms, e = exp(1)
M_LOG2E The base-two logarithm of e
M_LOG10E The base-ten logarithm of e
M_LN2 The natural logarithm of two
M_LN10 The natural logarithm of ten
M_PI The ratio of a circle's circumference to diameter, π
M_PI_2 Half M_PI, π / 2
M_PI_4 Quarter M_PI, π / 4
M_1_PI The inverse of M_PI, 1 / π
M_2_PI Twice the inverse of M_PI, 2 / π
M_2_SQRTPI Two divided by the square root of pi, 2 / √π
M_SQRT2 The square root of two, √2
M_SQRT1_2 The square roof of half, 1 / √2
Function Documentation
Returns the arccosine of v as an angle in radians. Arccosine is the inverse operation of cosine.
See also qAtan(), qAsin(), and qCos().
Returns the arcsine of v as an angle in radians. Arcsine is the inverse operation of sine.
See also qSin(), qAtan(), and qAcos().
Returns the arctangent of a point specified by the coordinates y and x. This function will return the angle (argument) of that point.
See also qAtan().
Returns the arctangent of v as an angle in radians. Arctangent is the inverse operation of tangent.
See also qTan(), qAcos(), and qAsin().
int qCeil(qreal v)
Return the ceiling of the value v.
The ceiling is the smallest integer that is not less than v. For example, if v is 41.2, then the ceiling is 42.
See also qFloor().
Returns the cosine of an angle v in radians.
See also qSin() and qTan().
float qDegreesToRadians(float degrees)
This function converts the degrees in float to radians.
float degrees = 180.0f
float radians = qDegreesToRadians(degrees)
This function was introduced in Qt 5.1.
See also qRadiansToDegrees().
double qDegreesToRadians(double degrees)
This function converts the degrees in double to radians.
double degrees = 180.0
double radians = qDegreesToRadians(degrees)
This function was introduced in Qt 5.1.
See also qRadiansToDegrees().
Returns the exponential function of e to the power of v.
See also qLn().
Returns the absolute value of v as a qreal.
int qFloor(qreal v)
Return the floor of the value v.
The floor is the largest integer that is not greater than v. For example, if v is 41.2, then the floor is 41.
See also qCeil().
Returns the natural logarithm of v. Natural logarithm uses base e.
See also qExp().
This function returns the nearest power of two greater than value. For 0 it returns 1, and for values larger than or equal to 2^31 it returns 0.
This function was introduced in Qt 5.4.
This function returns the nearest power of two greater than value. For 0 it returns 1, and for values larger than or equal to 2^63 it returns 0.
This function was introduced in Qt 5.4.
This is an overloaded function.
This function returns the nearest power of two greater than value. For negative values it returns 0.
This function was introduced in Qt 5.4.
This is an overloaded function.
This function returns the nearest power of two greater than value. For negative values it returns 0.
This function was introduced in Qt 5.4.
Returns the value of x raised to the power of y. That is, x is the base and y is the exponent.
See also qSqrt().
float qRadiansToDegrees(float radians)
This function converts the radians in float to degrees.
float radians = float(M_PI)
float degrees = qRadiansToDegrees(radians)
This function was introduced in Qt 5.1.
See also qDegreesToRadians().
double qRadiansToDegrees(double radians)
This function converts the radians in double to degrees.
double radians = M_PI
double degrees = qRadiansToDegrees(radians)
This function was introduced in Qt 5.1.
See also qDegreesToRadians().
Returns the sine of the angle v in radians.
See also qCos() and qTan().
Returns the square root of v. This function returns a NaN if v is a negative number.
See also qPow().
Returns the tangent of an angle v in radians.
See also qSin() and qCos().
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2017 Newsletter Article 4, Textbooks and the SI Base Units. A Challenge for Authors and Editors.
The present system of physical units has developed over the last two centuries. Originally, the important units for science and commerce were weights and measures. Indeed, those remain in the names
of the relevant organizations that are responsible for units.
Prior to the French Revolution, there were thousands of different weights and measures in what we know as Western Europe. Never mind those in other parts of the world most of which were unknown to
the French revolutionists. As a part of that great upheaval, it was decreed that there would be a single system of weights and measures in the new republic. Science was highly developed in France and
the scientists were charged with the development of this “universal” system. From these beginnings came the present and widely-accepted SI (Système International).
For measures, the meter was defined as 10^-7 (or one ten-millionth) of a quadrant of the Earth’s circumference. Happily, such a quadrant (one-fourth of a great circle of the Earth) ran through France
from Dunkirk on the north coast to Barcelona just outside southern France. Measuring this distance and calculating that fraction of the measured quadrant was left to the surveyors who had at their
disposal a highly accurate method of measuring angles. Starting with a single linear measurement of the base of a single triangle two teams set out to measure hundreds of triangles across France on
the way between Dunkirk and Barcelona. They met near the town of Rodez. All that remained was to calculate the linear distance using simple trigonometry. Thus, the meter (or metre) was based on the
dimension of the Earth and was easily understood.
For weight the kilogram was defined as a cubic decimeter of water at 4°C that was known to be the temperature of greatest density of water. Interestingly, the kilogram depended on the meter as it was
necessary to have that linear measurement to determine the volume of water. How could this be done as the meter itself was not yet known? Each of the units, meter and kilogram, were determined as
“provisional” until the calculation of the meter was completed. Thus, the kilogram was based on a terrestrial phenomenon that was easily understood.
We now recognize the seven SI Base Units shown in figure 1.
Figure 1. SI (Système International) Base Units
In the late 1700’s prototypes of the meter and kilogram were rendered as a durable objects made of metal. They were deposited in the archives of the French Republic in December 1799. In May 1875,
seventeen countries signed the Metre Convention that established the International Bureau of Weights and Measures (BIPM)[1]. In 1921 the convention was extended to all physical measurements. The
United States of America was one of the original signers of the Metre Convention. Now, more than fifty countries are members of this convention.
Later, forty standard kilograms were produced using a platinum-iridium allow and each measuring 39 mm in diameter and 39 mm in height. They were cast in London by George Matthey and were “hammered,
polished and adjusted” to match the kilogram in the French archive. In 1889, 34 of these “witnesses“ were distributed while six remained in France. In 1890, K4 and K20 arrived in the US where K20 was
designated the primary standard kilogram for the US.
Following the distribution of the standard kilograms it was decided to return them periodically to France for comparison to the standard kilogram referred to as “le Grand K.” Results of these
comparisons are shown in Figure 2.
Figure 2. Comparison of standard kilograms to le Grand K.
Because le Grand K is the standard, it is always used as the reference; le Grand K is at the top of the hierarchy of standard kilograms. The results clearly show that the mass of the witnesses is
changing relative to le Grand K. There is the possibility that le Grand K itself is changing mass but it cannot be compared to any other mass as it is the standard. It is this consistent drift over
time that has led to the proposal to eliminate the “artefact” know as le Grand K.
Redefining the kilogram by eliminating le Grand K requires some other basis for the definition of the kilogram. Since 1968 the definition of the second has been based on the radiation of the ^133Cs
atom. Since 1983, the meter has been defined on the speed of light. In each case, the definition is a physical constant believed to be an invariant of nature. Prompted by the need to redefine the
kilogram, all SI Base Units will be defined on invariants of nature. Consensus values for the constants will be adopted as exact values with uncertainty.
While all seven SI Base Units are of importance in the practice of chemistry, the new definitions of the kilogram, kelvin and mol required particular attention as they are more complicated to
understand and teach.
Unit of amount of substance (mole)
In 1971 the following definition was adopted:
1. The mole is the amount of substance of a system which contains as many elementary entities as there are atoms in 0.012 kilogram of carbon 12; its symbol is “mol.”
2. When the mole is used, the elementary entities must be specified and may be atoms, molecules ions, electrons, other particles, or specified groups of such particles.
This is the current definition of the mole that will be replaced. The definition is based on the agreement that the mass of an atom of carbon-12 is exactly 12 and that agreement is enshrined in the
definition. All atomic and molecular weight measurements are referenced to this exact number by definition.
It’s a well-known chemistry fact that Avogadro’s number (or the Avogadro constant) is related to this definition. Accordingly, the proposed redefinition uses the Avogadro constant as an invariant of
• The mole is the unit of amount of substance of a specified elementary entity which may be an atom, molecule, ion, electron, other particle or specified groups of such particles; its magnitude is
set by fixing the numerical value of the Avogadro constant to be equal to exactly 6.022 141…x 10^23 when it is expressed in the unit mol^-1.
This might be an easier way to introduce the Avogadro constant as it will be the basis of the definition of the mole. What is lost is the fundamental concept of the mass of an atom of carbon-12 is
exactly. Most likely the agreement on the mass of carbon-12 will remain fundamental in chemistry even though it is lost in the proposed definition of the mole.
Much discussion has led to the belief that “amount of substance” is an ambiguous and inappropriate term for the mole concept. Most likely the term “chemical amount” or “amount of chemical substance”
will gain favor. This, too, may be clarifying and pedagogically more pleasing.
Unit of thermodynamic temperature (kelvin)
Presently, the definition is:
The kelvin, unit of thermodynamic temperature, is the fraction 1/273.16 of the thermodynamic temperature of the triple point of water.
Once again there is a terrestrial phenomenon used to define the SI Base Unit. It is easily understood and realized. For consistency however, there is a problem: Exactly what is water? It is known
that for most elements the distribution of the stable isotopes is not consistent throughout the world. Accordingly, in 2005 the definition was enriched:
This definition refers to water having the isotopic composition defined exactly by the following amount of substance ratios: 0.000 155 76 mole of ^2H per mole of ^1H, 0.000 379 9 mole of ^17O per
mole of ^16O, and 0.002 005 2 mole of ^18O per mole of ^16O.
The proposed definition is:
The kelvin, K, is the unit of thermodynamic temperature its magnitude is set by fixing the numerical value of the Boltzmann constant to be equal to exactly 1.380 65… 10^-23 when it is expressed in
the unit s^-2 m^2 kg K^-1, which is equal to J K^-1.
The proposed definition has the advantage of not requiring a further definition of water. It has the disadvantage of explaining the Boltzmann constant and visualizing just how that defines
thermodynamic temperature.
Comparison of the present and proposed definitions of thermodynamic temperature provides an opportunity to illustrate the concept of “mise en practique” so important to the SI Base Units. Roughly
translated it means put into practice. It is imperative that we convey the notion that mise en practique is entirely separate from the definition.
Note that the present definition provides the mise en practique; obtain the correct, isotopically balanced water and lower the temperature until ice appears. Under normal atmospheric pressure that’s
the triple point of water.
With the proposed definition, there is no such guidance. What is the experiment or procedure that will relate in a practical way the Boltzmann constant to thermodynamic temperature?
Author’s Note – Presented at the ACS National ACS Meeting in San Francisco April 3, 2017
[1] “BIPM is an intergovernmental organization under the authority of the General Conference of Weights and Measures (CGPM) and the supervision of the International Committee for Weights and Measures
(CIPM)” www.bipm.org
May 1, 2017 - 7:57am — muzyka
standard kilograms
Hi, Peter,
Can you tell us anything about how the standard kilograms are stored? Are they all stored in the same way? How are they handled? What kind of variation is there in the handling of the standards? I'm
trying to understand what might be causing these minute but systematic variations in their masses.
May 1, 2017 - 1:35pm — Peter Rusch
Storage of kilogram standards
As you may know, the IPK (international prototype of the kilogram) is stored at BIPM under vacuum insided triple bell-jars. I don't know for sure but believe that the other "copies" or "witnesses"
are stored similarly.
May 2, 2017 - 8:58am — Roy Jensen
There are copies of the kilogram ...
And every time they are compared, there is considerable drift. I can't find the article I recently read, but you have a picture in your post that illustrates this drift.
May 1, 2017 - 8:33am — Bob Belford
Invariant Constant
Hi Peter,
First, I would like to thank you for sharing this paper with us and giving us a chance to discuss such an important topic. I actually have two questions.
First, you state, "Prompted by the need to redefine the kilogram, all SI Base Units will be defined on invariants of nature. Consensus values for the constants will be adopted as exact values with
What is meant by "exact values with uncertainty". I thought uncertainty was a property of inexact values. For example, the number of letters in this sentence is an exact value, with no uncertainty.
But once printed, the weight of the ink has uncertainty, and is an inexact value. Do you mean, it is an exact value with an agreed upon number of significant digits?
Second, and please forgive me if I am missing the obvious, but I thought the new definition of the Kilogram was based on Planck's constant, but I am not seeing that in the article.
May 1, 2017 - 8:48am — mbishop
Practical difference for measurements?
I understand the interest in defining the SI base units in terms of “physical constants believed to be invariant of nature”, but I wonder if it will make a difference in real-world measurements. For
example, are there ways to determine temperature that would yield different values based on the current and proposed definitions of the kelvin? In other words, at least for temperature, is this an
abstract exercise with no practical effect?
I also find it curious that the definitions are based in part on defining nature’s constants as exact, even though we don’t know what the exact values are. For example, the kelvin would be described
as, “The kelvin, K, is the unit of thermodynamic temperature its magnitude is set by fixing the numerical value of the Boltzmann constant to be equal to exactly 1.380 65… 10-23 when it is expressed
in the unit s-2 m2 kg K-1, which is equal to J K-1.” Wikipedia says that the Boltzmann constant is 1.38064852(79)×10−23 J/K. If this value (or some other more accurate value) is used to calibrate
temperature measuring instruments, what happens to this calibration when we develop ways to determine the constant more accurately? Are we still stuck with the problem of a drifting definition?
Mark Bishop
May 1, 2017 - 1:51pm — Peter Rusch
In effect, yes, this is an
In effect, yes, this is an abstract exercise regarding the practice of chemistry as we know it. Keep in mind that there is to be a continuity; the kelvin will remain what it is today. Only the
definition will change. Also, yes, there is experimental error in measurements and the several values for the Boltzmann constant is exactly what one expects. Part of the new definitions is that a
single value will be selected by CODATA based on the best measurements. There will be no drifitng values unless and until someone somewhere raises the isse. The metrologists pont out that once the
speed of light was selected to define the meter (metre) no one was interested in measuring it any more.
May 1, 2017 - 4:56pm — mbishop
If the definition of kelvin
If the definition of kelvin changes, how can the "kelvin...remain what it is today", or do you mean the the "kelvin will stay essentially the same to the limits of our ability to measure
temperature"? Is the Boltzmann constant going to be defined to keep the kelvin the same??? If so, does this mean that the Boltzmann constant used in the definition (which may be the best value
available) will be defined as a value that is almost certainly not the exact value for the true Boltzmann constant? Doesn't that defeat the purpose of defining the base units in terms of constants
that are assumed to be invariant? If I understand the plan, wouldn't it be more true to define the kelvin as, "“the unit of thermodynamic temperature for which the magnitude is set by fixing the
numerical value of the Boltzmann constant to be equal to exactly 1.380 65 10-23 (which is close to the true value of the Boltzmann constant) when it is expressed in the unit s-2 m2 kg K-1, which is
equal to J K-1.” If the number associated with the Boltzmann constant is going to be fixed with an exact value, why are there dots in the number in the definition (1.380 65... 10-23)? Can we assume
that at some point the kelvin would be redefined using a more accurate value for the Boltzmann constant that may be determined in the future?
One of the things I do is edit scientific papers for Chinese scientists who want to publish in English-language journals, and the editor in me really doesn't like the "its" in the kelvin definition.
Note that I substituted "for which the" for "its".
Mark Bishop
May 2, 2017 - 7:11am — Peter Rusch
Notes on the kelvin
The new definitions do not change the SI Base Units; the kilogam is still a kilogram; one degree kelvin is still one degree kelvin. The exact values of the fundamental constants used in the
definitions will be consensus values that will be accepted as "true" values for the purpose of the defintions. As these values are not yet accepted by the International Conference on Weights and
Measures that will meet in 2018, the dots appear to indicate present uncertainty in the value.
May 1, 2017 - 1:41pm — Peter Rusch
Value with no uncertainty
The science of metrology is all about traceability and uncertainty. Usually the variety of measurements of any physical constant group around a central value; often these values are assumed to be in
a Gaussian distribution. Once the value of each of the defining physical constants is selected it will be assumed to have no uncertainty. Similar to the mass of carbon-12 that is 12.0000000000000...
Each value will have it's own number of significant digits. The currently proposed values are listed in the draft of NISTS's Brochure 9 for the SI. It's in draft as the values are fully accepted by
the International Conference on Weights and Measures that willmeet in 2018.
You are correct. I don't know how that could be missed in the paper.,
May 1, 2017 - 2:00pm — Peter Rusch
New definition of the kilogram
The kilogram, kg, is the unit of mass; its magnitude is set by fixing the numerical value of the Planck constant to be equal to exactly 6.626 068… x 10-34 when it is expressed in the unit s-1 m2 kg,
which is equal to J s.
May 1, 2017 - 4:36pm — SDWoodgate
From a teaching perspective
I think that the new definition of the mole will be easier to teach. It was always a bit of a stretch to get students to grasp that the number of particles in a mole of substance was the same as in
12 g of carbon-12. It is much better to tell them the number. Personally I like the term chemical amount because we use mol in a more general way than just for substances as in kJ per mol of reaction
for enthalpy changes.
May 2, 2017 - 7:03am — Peter Rusch
New definition of the mole.
Yes, it's useful to bring the value of the Avagadro constant into the definition of the mole. What is lost in doing so is the basis of the atomic mass scale of carbon-12 as exactly 12.00000... as it
is no longer part of the definition. It's a small matter as the uncertainty introduced in the mass of carbon-12 makes little difference in the practice of chemistry.
May 2, 2017 - 9:00am — Gustavo Avitabile
Unit Symbols
Hi Peter,
Thanks for sharing your paper.
I am afraid I found two items in this paper that require a correction. If I'm wrong, please let me know.
The first item is Fig. 1. I think capitalization of unit names does not agree with SI specifications. As far as I now, the 7 basic units names must be in lower case, except for electrical current A
and temperature T
The second item is statement "Under normal atmospheric pressure that’s the triple point of water". I think the triple point has nothing to do with atmospheric pressure; indeed, pressure is one of the
two parameters that make up the triple point (the other is temperature).
May 3, 2017 - 5:48pm — Peter Rusch
Present definition of the kelvin
This quote means that the present definition of thermodynamic temperature is a fraction of the triple point of water. That definition also provides the means of achieving the definition, the mise en
practique. The new definnition uses the Boltzmann constant and does not also contain the mise en practique. Indeed, all of the new definitions are independent of their respective mise en practique.
My concern about the redefinition of the kiogram is that the new definition presently relies on the use of a Watt balance that is a method of achieving the definition. In my view, that's not the
stated intention of the new definitions; they are promoted to be independent of the mise en practique.
May 2, 2017 - 9:06am — Roy Jensen
You've hit on something that is a pet peeve of mine -- the adherence to international standards by textbooks. Specifically the lack thereof. Sigh. Consider:
* SATP/STP were changed in 1982. Today, only about half of first-year textbooks present the correct information.
* inorganic nomenclature (IUPAC Red Book) was substantially revised in 2005. Basically zero textbooks present the correct information.
* states of matter: are the subscript? italicized? To the best of my knowledge, this has never changed. Why are first-year textbooks still all over the map with this?
Sorry about the rant. I enjoyed reading about the upcoming new standards, but I doubt textbooks will change anytime soon. But I will update my textbook when they become official.
May 3, 2017 - 5:50pm — Peter Rusch
My few discussions with textbook editors from major publishers have been about just doing what the users demand. If true, the CHED and others should lobby the textbook editors for the content they
May 3, 2017 - 5:58pm — DelmarLarsen
Or just make a viable alternative....
May 4, 2017 - 12:36am — Roy Jensen
Common errors
I disagree. There is no reason to make a "viable alternative", when there is logic to the established standard.
1. spaces separate words
2. subscripted and superscripted labels apply to the entity *immediately* preceeding or postceeding them.
3. variables are italicized; labels are not
WRONG: 35.4mL, 35.4 mL
CORRECT: 35.4 mL
WRONG: HCl(aq), HCl(aq)
CORRECT: HCl(aq)
and a myriad of others that are still used incorrectly.
May 4, 2017 - 12:59am — DelmarLarsen
My reference was short and
My reference was short and perhaps subject to confusion. I was referring to making an alternative to textbook publishers so you do not need to try to convince anyone.... that is the topic of the next
However, since we are on the topic, it is clearly important to establish and use standards as much as possible in both teaching and research. However, as with all ideologies, dogmatic adherence and
aggressive proselytizing of a specific standard (IUPAC approved or otherwise) can be detrimental to future advances (in the lab and otherwise); we must remember that science is dynamic. I personally,
could care less if a phase is subscripted in a chemical reaction as long as it is clear what it means and no one gets confused.
Try to remove the wavenumber unit from my research and my fellow spectroscopists and I would revolt. We should be teaching our students flexibility in their education (within reason), not rigid
May 3, 2017 - 7:57pm — Eric Nelson
Format rules
Permit me to speak in defense of the textbook publishers on the question of applying IUPAC rules.
I was involved a few years back in writing a textbook to help introduce students to chemistry. When it came time to print balanced equations, my proofreader applied what was cited as a relatively new
IUPAC standard deciding that no space should be permitted between coefficients and molecular formulas.
I argued that the space between the coefficients and formulas be put back in. My argument was reading comprehension. When learners of a language are initially taught to “decode text” (translate
alphabetic words into constituent sounds), generally instruction starts with short words that have spaces between them, rather than starting with cyclooctatetraene.
When students balancing an equation write Co(OH)[2 ].in handwriting, in my experience it takes a while for them to learn to make some letter o cases small and some large in a manner that lets them
distinguish these cases, especially when reading their own handwriting. while solving a problem Co(CH[3]CO[2])[2] can be especially troubling given how small some textbook fonts print subscripts on
print and on the screen. .
I personally think writing 2 Co(CH[3]CO[2])[2] instead of 2Co(CH[3]CO[2])[2] is a bit more “student centered” for beginners who are trying to learn to count atoms, and my co-author and publisher
graciously allowed the space.
Experts read the language fluently either way, but I don’t see the downside to putting the space after the coefficient as had long been the practice, and I think it helps students in breaking the
terms into their constituent parts.
In chem problems, should we adopt SI units -- and state all of our volumes in cubic meters?
-- rick nelson
May 3, 2017 - 11:04pm — mbishop
SI units and spaces in equations
Yes, I think we should model good behavior and use all SI units, except for examples of how to convert other units into SI units.
The liter is an accepted SI unit. The liter is a derived unit that is defined as exactly 0.001 m3.
I vote for no space between coefficients and formulas.
May 4, 2017 - 12:39am — Roy Jensen
I've run into that in my
I've run into that in my province, and actually convinced the Ministry of Education to adopt IUPAC standards. Now if I could convince my colleagues to do the same.
Specific to your statement: the knowledgable capitulate to the ignorant? Good Grief!
May 2, 2017 - 9:30am — rjlanc
Inorganic Nomenclature
I too have been dismayed by the slowness of uptake of Nomenclature rules and in some cases of authors making their own rules for use in their textbooks. I did a quick survey of the dozen or so texts
available in our library for a first year course back in 1987 and commented on it in Journal of Chemical Education, 1987, 64, 900-1.
For laboratory courses where names are expected, the more recent changeover from chloro to chlorido may take a generation ??!!
With respect to errors I can't forget when in high school the Physics teacher handed out a piece of paper for circulation to the class and told each student to mark 1 foot according to the ruler we
all carried around in those days. When the 30-40 of us had finished it was amazing to see that the difference between the shortest and longest foot was about half an inch!
May 3, 2017 - 5:51pm — Peter Rusch
Measuring the foot
Sounds to me like a re-play of Gauss' well-known experiment.
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Epsilon-N Proof of a Limit of a Sequence: Lim[2n/(3n+2)] = 2/3 – Article by G. Stolyarov II
Epsilon-N Proof of a Limit of a Sequence: Lim[2n/(3n+2)] = 2/3 – Article by G. Stolyarov II
G. Stolyarov II
July 12, 2014
Note from the Author: This proof was originally published on Associated Content (subsequently, Yahoo! Voices) in 2007. The article earned over 17,250 page views on Associated Content/Yahoo! Voices,
and I seek to preserve it as a valuable resource for readers, subsequent to the imminent closure of Yahoo! Voices. Therefore, this essay is being published directly on The Rational Argumentator for
the first time. ***
~ G. Stolyarov II, July 12, 2014
This is the first in a series of formal mathematical proofs I intend to present in order to assist anybody who has ventured into the challenging but fascinating world of advanced calculus and real
analysis. I start with a fairly basic proof: the limit of the nth term of a sequence as n becomes increasingly large. This is an epsilon-N proof, which uses the following definition: lim(n approaches
∞)x[n]= L iff for each real number ε>0, there exists a positive integer N(ε) such that if n ≥ N(ε), then │x[n]-L│< ε, i.e., L- ε < x[n]< L+ ε.
The epsilon-N proof has two component steps; first, we assume that our ε>0 is given and work backward to transform the inequality │x[n]-L│< ε in order to find an appropriate value of N(ε) that
corresponds to a given value of ε. Then, using the value of N(ε) we found, we work forward to show that if n ≥ N(ε), then │x[n]-L│< ε.
The proof also uses the Archimedean Property, which states that the set of positive integers is not bounded above. There are actually four equivalent conditions that are known as the Archimedean
1. If a,b are in R, a>0, b>0, then there is a positive integer n such that na>b
2. The set of positive integers is not bounded above.
3. For each real number x, there exists an integer n such that n ≤x < n+1
4. For each positive real number x, there exists a positive integer n such that 1/n ≤ x
Prove: lim(n approaches ∞)[2n/(3n+2)] = 2/3
Proof: Let ε>0 be given. Find N(ε)є Z^+ such that if N(ε) < n, then │2n/(3n+2)-2/3│< ε.
Working backward to transform the inequality: │2n/(3n+2)-2/3│< ε
│6n/[3(3n+2)]-2(3n+2)/[3(3n+2)]│< ε
│[6n-2(3n+2)]/[3(3n+2)]│< ε
│[6n-6n-4]/[3(3n+2)]│< ε
│-4/[3(3n+2)]│< ε
Since ε>0, (3n+2)>0, the above inequality is the same as
4/[3(3n+2)] < ε
4/(3ε) < (3n+2)
4/(3ε)- 2 < 3n
4/(9ε)- 2/3 < n
Now I work forward to prove the original result:
Let N(ε)є Z^+ э 4/(9ε)- 2/3 < N(ε).
Since 4/(9ε)- 2/3 is a real number, by the Archimedean Property it must be the case that some integer exists which is greater than that real number-since the set of positive integers is not bounded
above. Here, we call that integer N(ε).
For all n>N, if 4/(9ε)- 2/3 < N < n, then:
4/(3ε)- 2 < 3n
4/(3ε) < (3n+2)
4/[3(3n+2)] < ε
│-4/[3(3n+2)]│< ε
│6n/[3(3n+2)]-2(3n+2)/[3(3n+2)]│< ε
│2n/(3n+2)-2/3│< ε
I have demonstrated the above inequality, which is sufficient to demonstrate that lim(n approaches ∞)[2n/(3n+2)] = 2/3.
I have hence proved what was desired. Another way to express that the desired objective has been obtained (which I shall use in future proofs of this sort) is the abbreviation
“Q. E. D.” of the Latin “Quod Erat Demonstraterum,” which means “that which was to be demonstrated.”
One thought on “Epsilon-N Proof of a Limit of a Sequence: Lim[2n/(3n+2)] = 2/3 – Article by G. Stolyarov II”
1. Thanks a lot for your insight, can you help me work out this example Lim(as n approaches infinity) n+3/((n^2)-5)
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Simplifying Square Roots Worksheet Answers
Simplifying Square Roots Worksheet Answers. Worksheet additionally features a puzzle that enables college students to check their answers as they work. Sum of all three digit numbers divisible by 6.
In order to simplify the expression utilizing the calculator technique, we must spherical this answer at some point. This sheet focuses on Algebra 1 issues using real numbers.
These Free Simplifying Radical Worksheets workouts may have your youngsters engaged and entertained whereas they improve their abilities. These Free Simplify Radicals Answers Worksheets workout
routines will have your youngsters engaged and entertained whereas they improve their skills. Displaying all worksheets related to – Simplifying Square Rootss.
The following download contains three coloring pages. Students reduce aside triangles and then match the perimeters to kind a kite form. Students consider square roots of numbers and variables whose
components include excellent squares.
Simplifying Radicals Mazes Sq And Cube Roots
During your walk you stumble across an awesome discovery in nature, baby ducks. Will you be ready to help the ducks and make them happy, or will they runaway earlier than their mother returns?
Worksheet to teach and follow simplifying, multiplying and dividing square roots. Front of worksheet contains 9 examples to work with students as a way of teaching the material. Back of worksheet
contains 15 problems for impartial apply.
All Of Us Have Students That Make This Mistake!
Let a and b be real numbers and let m and n be integers. Key to Algebra provides a unique, confirmed way to introduce algebra to your students. New concepts are defined in simple language, and
examples are straightforward to observe.
So, take the additional time and do not assume they know or have mastered that the sq. root of an ideal square is a whole number. Add this one further step exhibiting them that any radical multiplied
by itself comes out to be simply the quantity underneath the novel. Then the students will understand that if you break down an ideal square the unconventional symbol is not current any more as a
result of the square root of an ideal square is an entire quantity.
Extra Numbers Interactive Worksheets
To get the worksheet in html format, push the button “View in browser” or “Make html worksheet”. This has the advantage that you can save the worksheet instantly from your browser (choose File →
Save) after which edit it in Word or different word processing program. Each worksheet is randomly generated and thus unique.
Square roots are some of the necessary concepts within the area of mathematics. Simplifying them to simpler types becomes handy for complicated calculations. This entails breaking a number into its
elements and then coupling the same numbers collectively, and leaving the numbers which are distinctive as they are, contained in the sq. root signal.
Interactive Resources You’ll Find A Way To Assign In Your Digital Classroom From Tpt
Sum of all three digit numbers divisible by 6. Sum of all three digit numbers divisible by 7.
Interactive resources you can assign in your digital classroom from TPT. A actually great exercise for permitting students to grasp the concept of Simplifying Roots. Members have exclusive facilities
to obtain a person worksheet, or a whole degree.
Note that we used the truth that the second property could be expanded out to as many terms as we have in the product under the unconventional. Also, don’t get excited that there are not any x’s
beneath the radical in the last reply. 4 is not the largest good square that components into 80.
We are going to be simplifying radicals shortly and so we ought to always subsequent outline simplified radical type. A radical is said to be in simplified radical type if each of the next are true.
Books 5-7 introduce rational numbers and expressions. Books 8-10 prolong coverage to the true number system.
Factorize the right squares in the numerator and denominator of each fraction, and consider the root. The digital model is a Google Form.You determine to take a walk after a tense day at college.
However, one widespread methodology of simplifying the unconventional values is by eliminating the novel signal itself. This may be carried out by factorizing the quantity that’s present under the
novel sign. Factorization of a quantity can be done in many ways, however the most typical of all is breaking the quantity into like pairs.
Now, return to the unconventional and then use the second and first property of radicals as we did in the first example. Now use the second property of radicals to break up the unconventional and
then use the first property of radicals on the primary time period.
Worksheet additionally includes a puzzle that permits college students to check their solutions as they work. Answer to puzzle is “Math is so much enjoyable, it is radical!”. Just tacky enough for
highschool college students to take pleasure in.
We will break the radicand up into good squares times phrases whose exponents are lower than 2 (i.e. 1). In order to seek out the precise value, we must really issue out the greatest perfect sq.
which is the factor of our base, and then take its square root. Some people call this factoring a sq. root.
NOT the square root of that entire number like within the image. There is more than one time period right here however every thing works in exactly the identical style.
25 scaffolded questions that start out comparatively straightforward and end with some real challenges. Assignment covering good squares and cubes and the means to break down perfect sq. roots and
perfect dice roots. I even have corresponding notes of the identical name.
Supply grade 6 college students with these pdf worksheets, so they turn into remarkably assured and fairly practiced at finding the sq. roots of good squares. They get to follow first fifty sq.
In this matching sq. puzzle, students will practice simplifying sq. roots. These are all basic problems with no variables.
Any exponents in the radicand can don’t have any components in widespread with the index. All exponents within the radicand have to be lower than the index. The quantity inside the unconventional
signal known as the radicand .
This is broadly observable within the instances of such numbers that have a proper square or dice roots. For example, in the case of number forty nine, if we’ve to calculate the square root of forty
nine, we can merely break 49 into a pair of 7’s. Index 2 can simply cancel out the novel sign, and we are able to acquire the answer of 7.
These math worksheets must be practiced regularly and are free to download in PDF codecs. One approach to make this lesson exponentially more difficult for your students is to assume they know more
than they do.
This maze is a great way for college kids to apply their skills with simplifying radicals using sq. roots and cube roots. This message decoder is a nice way for faculty kids to follow their skills
with simplifying radicals utilizing square roots and cube roots.
Click on the picture to view or download the image. Post navigation ← 5 Nbt 6 Worksheets Dividing Fractions On. This now satisfies the rules for simplification and so we’re accomplished.
Simply inputting the expression into the calculator will not do, as these are irrational, and rounding a solution makes it inexact. It is as a result of they don’t actually understand what makes a
perfect sq. come out to an entire quantity.
Do not assume they have a mastery of their radical guidelines . Don’t forget to look for perfect squares within the number as well. Our bother normally happens after we either can’t easily see the
reply or if the quantity underneath our radical sign is not an ideal sq. or an ideal dice.
Options embody the radicand range, limiting the square roots to excellent squares solely, font size, workspace, PDF or html formats, and more. This is a 20 query worksheet where students are
requested to circle all right statements simplifying square and cube roots, together with fractions.Common mistakes by students are addressed.
This sheet focuses on Algebra 1 problems using actual numbers. The worksheet has mannequin problems worked out, step-by-step.
If you need the answer to be an entire quantity, choose “good squares,” which makes the radicand to be a perfect sq. (1, 4, 9, sixteen, 25, and so on.). If you choose to permit non-perfect squares,
the answer is typically an endless decimal that is rounded to a sure number of digits.
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Generalized nevanlinna-pick problems on hardy space - Rare & Special e-Zone
THESIS 2016
Let m be a nonnegative integer, {z
} be a sequence of distinct points in open unit disk D of complex plane and {w
} be an arbitrary sequence of complex numbers. The generalized Nevanlinna-Pick problem on Hardy Space H
of D is to find a condition that determines whether there exists a function f in H
satisfying ❘❘f❘❘
≤ 1 and f
) = w
for each i = 1,,n. In this thesis, the necessary and sufficient conditions for this generalized Nevanlinna-Pick problem are obtained by giving a constructive proof. Also, a formula for the minimal
norm among all the solutions and a form for such function are obtained. With the utilization of minimal norm function, the analog of usual consequences in the classical Nevanlinna-Pick problem can be
deduced. This thesis also discu...[
Read more
Let m be a nonnegative integer, {z
} be a sequence of distinct points in open unit disk D of complex plane and {w
} be an arbitrary sequence of complex numbers. The generalized Nevanlinna-Pick problem on Hardy Space H
of D is to find a condition that determines whether there exists a function f in H
satisfying ❘❘f❘❘
≤ 1 and f
) = w
for each i = 1,...,n. In this thesis, the necessary and sufficient conditions for this generalized Nevanlinna-Pick problem are obtained by giving a constructive proof. Also, a formula for the minimal
norm among all the solutions and a form for such function are obtained. With the utilization of minimal norm function, the analog of usual consequences in the classical Nevanlinna-Pick problem can be
deduced. This thesis also discusses another generalized Nevanlinna-Pick problem, which would lead to a possibility of solving the Nevanlinna-Pick problem with boundary data on H
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A note on the hopes for Fully Homomorphic Signatures — Quick Math Intuitions
A note on the hopes for Fully Homomorphic Signatures
This is taken from my Master Thesis on Homomorphic Signatures over Lattices.
See also But WHY is the Lattices Bounded Distance Decoding Problem difficult?.
What are homomorphic signatures
Imagine that Alice owns a large data set, over which she would like to perform some computation. In a homomorphic signature scheme, Alice signs the data set with her secret key and uploads the signed
data to an untrusted server. The server then performs the computation modeled by the function
Alongside the result
The signature homomorphic signature, where homomorphic has the same meaning as the mathematical definition: ‘mapping of a mathematical structure into another one in such a way that the result
obtained by applying the operations to elements of the first structure is mapped onto the result obtained by applying the corresponding operations to their respective images in the second one‘. In
our case, the operations are represented by the function mapping is from the matrices
Notice how the very idea of homomorphic signatures challenges the basic security requirements of traditional digital signatures. In fact, for a traditional signatures scheme we require that it should
be computationally infeasible to generate a valid signature for a party without knowing that party’s private key. Here, we need to be able to generate a valid signature on some data (i.e. results of
computation, like without knowing the secret key. What we require, though, is that it must be computationally infeasible to forge a valid signature it must not be possible to cheat on the signature
of the result: if the provided result is validly signed, then it must be the correct result.
The next ideas stem from the analysis of the signature scheme devised by Gorbunov, Vaikuntanathan and Wichs. It relies on the Short Integer Solution hard problem on lattices. The scheme presents
several limitations and possible improvements, but it is also the first homomorphic signature scheme able to evaluate arbitrary arithmetic circuits over signed data.
Def. – A signature scheme is said to be leveled homomorphic if it can only evaluate circuits of fixed depth
Def. – A signature scheme is said to be fully homomorphic if it supports the evaluation of any arithmetic circuit (albeit possibly being of fixed size, i.e. leveled). In other words, there is no
limitation on the “richness” of the function to be evaluated, although there may be on its complexity.
Let us remark that, to date, no (non-leveled) fully homomorphic signature scheme has been devised yet. The state of the art still lies in leveled schemes. On the other hand, a great breakthrough was
the invention of a fully homomorphic encryption scheme by Craig Gentry.
On the hopes for homomorphic signatures
The main limitation of the current construction (GVW15) is that verifying the correctness of the computation takes Alice roughly as much time as the computation of
To us, this limitation makes intuitive sense, and it is worth comparing it with real life. In fact, if one wants to judge the work of someone else, they cannot just look at it without any preparatory
work. Instead, they have to have spent (at least) a comparable amount of time studying/learning the content to be able to evaluate the work.
For example, a good musician is required to evaluate the performance of Beethoven’s Ninth Symphony by some orchestra. Notice how anybody with some musical knowledge could evaluate whether what is
being played makes sense (for instance, whether it actually is the Ninth Symphony and not something else). On the other hand, evaluating the perfection of performance is something entirely different
and requires years of study in the music field and in-depth knowledge of the particular symphony itself.
That is why it looks like hoping to devise a homomorphic scheme in which the verification time is significantly shorter than the computation time would be against what is rightful to hope. It may be
easy to judge whether the result makes sense (for example, it is not a letter if we expected an integer), but is difficult if we want to evaluate perfect correctness.
However, there is one more caveat. If Alice has to verify the result of the same function amortized verification). Again, this makes sense: when one is skilled enough to evaluate the performance of
the Ninth Symphony by the Berlin Philharmonic, they are also skilled enough to evaluate the performance of the same piece by the Vienna Philharmonic, without having to undergo any significant further
work other than going and listening to the performance.
So, although it does not seem feasible to devise a scheme that guarantees the correctness of the result and in which the verification complexity is significantly less than the computation complexity,
not all hope for improvements is lost. In fact, it may be possible to obtain a scheme in which verification is faster, but the correctness is only probabilistically guaranteed.
Back to our music analogy, we can imagine the evaluator listening to a handful of minutes of the Symphony and evaluate the whole performance from the little he has heard. However, the orchestra has
no idea at what time the evaluator will show up, and for how long they will listen. Clearly, if the orchestra makes a mistake in those few minutes, the performance is not perfect; on the other hand,
if what they hear is flawless, then there is some probability that the whole play is perfect.
Similarly, the scheme may be tweaked to only partially check the signature result, thus assigning a probabilistic measure of correctness. As a rough example, we may think of not computing the
homomorphic transformations over the unlikely (and it quickly gets more so as the number of checked entries increases, of course) that the result is wrong. After all, to cheat, the third party would
need to guess several numbers in
Another idea would be for the music evaluator to delegate another person to check for the quality of the performance, by giving them some precise and detailed features to look for when hearing the
play. In the homomorphic scheme, this may translate in looking for some specific features in the result, some characteristics we know a priori that must be in the result. For example, we may know
that the result must be a prime number, or must satisfy some constraint, or a relation with something much easier to check. In other words, we may be able to reduce the correctness check to a few
fundamental traits that are very easy to check, but also provide some guarantee of correctness. This method seems much harder to model, though.
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Resize the number of tiles that this graph contains.
void Resize (
IntRect newTileBounds Rectangle of tiles that the graph should contain. Relative to the old bounds.
Resize the number of tiles that this graph contains.
This can be used both to make a graph larger, smaller or move the bounds of the graph around. The new bounds are relative to the existing bounds which are IntRect(0, 0, tileCountX-1, tileCountZ-1).
Any current tiles that fall outside the new bounds will be removed. Any new tiles that did not exist inside the previous bounds will be created as empty tiles. All other tiles will be preserved. They
will stay at their current world space positions.
This is intended to be used at runtime on an already scanned graph. If you want to change the bounding box of a graph like in the editor, use forcedBoundsSize and forcedBoundsCenter instead.
AstarPath.active.AddWorkItem(() => {
var graph = AstarPath.active.data.recastGraph;
var currentBounds = new IntRect(0, 0, graph.tileXCount-1, graph.tileZCount-1);
// Make the graph twice as large, but discard the first 3 columns.
// All other tiles will be kept and stay at the same position in the world.
// The new tiles will be empty.
graph.Resize(new IntRect(3, 0, currentBounds.xmax*2, currentBounds.ymax*2));
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Neural Networks and Clojure
Neural Networks
(ANNs) attempt to "learn" by modelling the behaviour of neurons. Although neural networks sound cool, there is no magic behind them!
Invented in 1957, by
Frank Rosenblatt
, the single layer perceptron network is the simplest type of neural network. The single layer perceptron network is able to act as a binary classifier for any linearly separable data set.
The SLP is nothing more than a collection of weights and an output value. The Clojure code below allows you to create a network (initially with zero weights) and get a result from the network given
some weights and an input. Not very interesting.
(defn create-network
(repeat in 0))
(defn run-network
[input weights]
(if (pos? (reduce + (map * input weights))) 1 0))
The clever bit is adapting the weights so that the neural network learns. This process is known as training and is based on a set of data with known expectations. The learning algorithm for SLPs is
shown below. Given an error (either 1 or -1 in this case), adjust the weights based on the size of the inputs. The
decides how much to vary the weights; too high and the algorithm won't converge, too low and it'll take forever to converge.
(def learning-rate 0.05)
(defn- update-weights
[weights inputs error]
(fn [weight input] (+ weight (* learning-rate error input)))
weights inputs))
Finally, we can put this all together with a simple training function. Given a series of samples and the expected values, repeatedly update the weights until the training set is empty.
(defn train
([samples expecteds] (train samples expecteds (create-network (count (first samples)))))
([samples expecteds weights]
(if (empty? samples)
(let [sample (first samples)
expected (first expecteds)
actual (run-network sample weights)
error (- expected actual)]
(recur (rest samples) (rest expecteds) (update-weights weights sample error))))))
So we have our network now. How can we use it? Firstly, let's define a couple of data sets both linearly separable and not.
adds some random noise to each sample. Note the cool # syntax for a short function definition (I hadn't seen it before).
(defn jiggle [data]
(map (fn [x] (+ x (- (rand 0.05) 0.025))) data))
(def linearly-separable-test-data
(take 100 (repeatedly #(jiggle [0 1 0])))
(take 100 (repeatedly #(jiggle [1 0 0]))))
(repeat 100 0)
(repeat 100 1))])
(def xor-test-data
(take 100 (repeatedly #(jiggle [0 1])))
(take 100 (repeatedly #(jiggle [1 0])))
(take 100 (repeatedly #(jiggle [0 0])))
(take 100 (repeatedly #(jiggle [1 1]))))
(repeat 100 1)
(repeat 100 1)
(repeat 100 0)
(repeat 100 0))])
If we run these in the REPL we can see that the results are perfect for the linearly separable data.
> (apply train ls-test-data)
(0.04982859491606148 -0.0011851610388172009 -4.431771581539448E-4)
> (run-network [0 1 0] (apply train ls-test-data))
> (run-network [1 0 0] (apply train ls-test-data))
However, for the non-linearly separable they are completely wrong:
> (apply train xor-test-data)
(-0.02626745010362212 -0.028550312499346104)
> (run-network [1 1] (apply train xor-test-data))
> (run-network [0 1] (apply train xor-test-data))
> (run-network [1 0] (apply train xor-test-data))
> (run-network [0 0] (apply train xor-test-data))
The neural network algorithm shown here is really just a
gradient descent
optimization that only works for linearly separable data. Instead of calculating the solution in an iterative manner, we could have just arrived at an optimal solution in one go.
More complicated networks, such as the multi-layer perceptron network have more classification power and can work for non linearly separable data. I'll look at them next time!
5 comments:
1. Nice post. I'm novice in FP. Now I'm trying to play with Clojure. It's seems for me this paradigma is much closer to my way of thinking than OO. I was looking for any implementations of ANN on
functional language. I'm planning to implement Restricted Boltzman Machine for humor and jokes extraction from text.
I'd like to know your opinion regarding using functional programming vs OO programming for ANN construction. Which one is better suited for this kind of tasks. As soon as I know with FP it is
much simpler to programm for multiprocessor hardware. But how about memory consumption. With BP algorithm we should create many matrix copy when adjusting wheights.
So anybody using FP languages for ANN?
2. I'm definitely not an expert in FP or ANN so take this with a pinch of salt!
I think your concerns about many matrix copies are (probably) unfounded. The Clojure data structures are designed to mutate efficiently; they share the structure that stays the same only the new
data changes. In addition, when you need to mutate you can do so in a safe way using transient.
Even if it is a little slower then you can probably win in the end with multi-processor support!
3. What is your opinion, the structure of typiclal NN can be best described from the point of view objects or functions?
And what structure would be better to use for NN structure, weights, inputs.
or maybe use Java's structure?
4. "train" function does look a lot like reduce so I thought it'd be nice if you made test data 'reducable' and refactor "train" into function that may be used by "reduce".
You can find source code at:
5. Thanks Paweł - that looks much cleaner!
Looking back now, I'm of the opinion anything that uses recur is almost certainly something you can do with a standard function. I guess recursion is a code smell in a functional language since
most likely you can express it in terms of a map / fold operation.
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Bioequivalence and Bioavailability Forum
Inflation type one error [RSABE / ABEL]
Hi Mikalai,
❝ I am not a statistician […]
So am I.
❝ We basically do the same things as in the usual average bioequivalence where we are able to preserve TIE at 5%, [... ]
No, we aren’t. In ABE we have fixed limits of the acceptance range,
a pre-specified Null Hypothesis. In ABEL the limits are random variables or in other words, the Null is generated ‘in face of the data’. That means that each study sets it own standards and if we
have a couple of HVDPs, each of them was approved according to different rules.
❝ What is behind this inflation, philosophically and mathematically?
this presentation
helps. In short: Reference-scaling is based on the true population parameters (hence the Greek letters \(\theta_s,\,\mu_T,\,\mu_R,\,\sigma_{wR}\)). The true standard deviation \(\sigma_{wR}\) of the
reference is unknown. We have only its estimate \(s_{wR}\) from the study. Imagine: The true within-subject CV of the reference is 27%. Hence, it is
an HVD(P) and we should use the conventional limits of 80.00-125.00%. However, by chance in our study we get an estimate of 35% and we expand the limits. Since the PE and the 90% are not affected it
means that the chance of passing BE increases. The chance to
not accepting the Null increases and this is the inflated type I error.
❝ I also bumped into this discussion where some ever argue about whether this concept even exists.
❝ https://daniellakens.blogspot.com/2016/12/why-type-1-errors-are-more-important.html
❝ I assume this is not related to multiple testing.
Nice one. Your assumption is correct.
❝ Also we have a statistical concept, but do we have any real proof of this concept? Does any❝ W know products that were initially registered and then withdrawn from the market because their initial
bioequivalence had been due to the inflation TIE?
No (twice). But these questions deserve a detailed discussion. More when I’ll be back from Athens.
Dif-tor heh smusma 🖖🏼 Довге життя Україна! []
Helmut Schütz
The quality of responses received is directly proportional to the quality of the question asked.
🚮 Science Quotes
Complete thread:
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An expert center specializing in macroeconomic modeling and developing models for forecasting the economy of Kazakhstan
Economic Modelling Development Center
Conducts research and provides analytical forecasts
To further the integration of Nazarbayev University into the community of advanced research institutes and improve cooperation in the international arena
An expert center specializing in macroeconomic modeling and developing models for forecasting the economy of Kazakhstan
Economic Modelling
Development Center
Conducts research and provides analytical forecasts
To further the integration of Nazarbayev University into the community of advanced research institutes and improve cooperation in the international arena
NAC Analytica conducts advanced research in applied macro- and microeconomics in collaboration with world-class scientists. The economy of Kazakhstan is still largely unexplored due to the lack of
research activities in these areas. The Economic Modeling Development Center at NAC Analytica was established to fill the gap.
Our focus is on modeling and applied research to develop recommendations for monetary policy, fiscal policy, the labor market, real sectors of the economy, and forecasting key macro variables.
The Center works in close cooperation with the Department of Economics of the School of Science and Humanities (Nazarbayev University). Students and faculty are encouraged to collaborate with us on
collaborative data collection, economic modeling, and econometric analysis to create a strong research environment in applied macro and microeconomics within the university. The Center also aims to
organize research seminars, symposiums and conferences aimed at disseminating research results and creating an international academic network in Kazakhstan.
NAC Analytica conducts advanced research in applied macro- and microeconomics in collaboration with world-class scientists. The economy of Kazakhstan is still largely unexplored due to the lack of
research activities in these areas. The Economic Modeling Development Center at NAC Analytica was established to fill the gap.
Our focus is on modeling and applied research to develop recommendations for monetary policy, fiscal policy, the labor market, real sectors of the economy, and forecasting key macro variables.
The Center works in close cooperation with the Department of Economics of the School of Science and Humanities (Nazarbayev University). Students and faculty are encouraged to collaborate with us on
collaborative data collection, economic modeling, and econometric analysis to create a strong research environment in applied macro and microeconomics within the university. The Center also aims to
organize research seminars, symposiums and conferences aimed at disseminating research results and creating an international academic network in Kazakhstan.
The paper outlines a structural macroeconometric model for the economy of Russia. The aim of the research is to analyze how the domestic economy functions, generate forecasts for important
macroeconomic indicators and evaluate the responses of main endogenous variables to various shocks. The model is estimated based on quarterly data starting from 2001 to 2019. The majority of the
equations are specified in error correction form due to the non-stationarity of variables. Stochastic simulation is used to solve the model for expost and ex-ante analysis. We compare forecasts of
the model with forecasts generated by the VAR model. The results indicate that the present model outperforms the VAR model in terms of forecasting GDP growth, inflation rate and unemployment rate. We
also evaluate the responses of main macroeconomic variables to VAT rate and world trade shocks via stochastic simulation. Finally, we generate ex-ante forecasts for the Russian economy under the
baseline assumptions.
A small scale open economy model is estimated for Kazakhstan via Bayesian methods. The model explicitly takes into account the dependence of the economy on commodity exports and also accounts for
risk premium shocks in the foreign exchange market. The main contribution of the research is that it is the first DSGE model in literature estimated via Bayesian methods for Kazakhstan. The results
of the model are used to determine the historical contribution of structural shocks to endogenous variables, forecast error variance decomposition of observed macroeconomic variables and impulse
responses of important endogenous variables to various shocks. It has been found that the output gap turned significantly negative during the Great Recession and the negative oil price shock. The
effect of contractionary monetary policy is found to be negative on output gap, but it negligibly affects the inflation rate in the economy. Risk premium shocks are found to account for almost 60% of
forecast error variance decomposition of nominal exchange rate of tenge over all horizons.
The paper builds a structural macroeconometric model for Kazakhstan to generate short-term and medium-term forecasts for main macroeconomic variables and conduct scenario analyses based on dynamic
simulation of the model. Due to the poor quality of quarterly data on GDP and its expenditure components, they have been adjusted using volume indexes. The model consists of aggregate supply,
aggregate demand, labor market, asset market, the central bank policy and government side equations. Most equations are estimated via econometric techniques and identities are explicitly introduced
in line with economic theory. We combine all the regression equations into a single model and solve for the baseline scenario from 2003 to 2017. The simulation results show that the structural
macroeconometric model approximates Kazakhstani economy reasonably well. Ex-ante forecasts under oil prices remaining around 50 and 60 US dollars per barrel are generated and compared with the
baseline forecast of the National Bank of the Republic of Kazakhstan.
The conventional wisdom assumes that terms of trade shocks are the main drivers of business cycle dynamics in emerging exporting economies. This paper studies the effect of terms of trade shocks on
key macroeconomic variables for the Kazakhstani economy. Empirical SVAR model estimates suggest that the terms of trade shocks account for 12 % of output variation for the economy. Further, three
sectoral DSGE model with estimated structural parameters predict the modest importance of the terms of trade shocks for small open economy.
Dynamic stochastic general equilibrium (DSGE) models are widely used by central banks, government agencies and financial organizations to conduct simulation and forecast relevant macroeconomic
indicators in the economy. The most important inputs into all DSGE models are structural parameters which are either calibrated from other sources or estimated via Bayesian methods. Using non-public
microlevel data, we estimate ten structural parameters for Kazakhstan: the elasticity of substitution between exports and imports, constant relative risk aversion, intertemporal elasticity of
substitution in consumption, Frisch elasticity of labor supply, the depreciation rate of physical capital, capital and labor shares, and the elasticity of substitution between tradable and
nontradable goods. Various econometric techniques such as fixed-effects, generalized method of moments (GMM), Arellano-Bond, and non-linear iterative maximum likelihood estimation are used to obtain
consistent estimates of the models' coefficients. The structural parameters can be used in calibrated DSGE models as fixed parameters or as prior information in Bayesian estimation of the models.
This paper presents a two-country macroeconometric model for the economies of Kazakhstan and Russia. The model can be used for interpreting the structural relationship between the two economies,
determining the degree of trade integration and implementing scenario analyses with various shocks. Single-country models are linked through bilateral trade and exchange rate equations. The baseline
simulation of the two-country model demonstrates a good accuracy in tracking the actual dynamics of macroeconomic indicators in both countries. Scenario analyses are conducted with a risk premium
shock in the bilateral exchange rate and a monetary policy shock in Russia to analyze the transmission mechanism of the shocks, and clarify on the kind of interdependency of the economies. The model
shows a larger influence of the risk premium shock on economic activity in Kazakhstan than in Russia. A two percentage point decline in the key rate does not impose significant inflationary pressure
while imports and the real exchange rate are the most affected variables in both countries.
The paper outlines a structural macroeconometric model for the economy of Russia. The aim of the research is to analyze how the domestic economy functions, generate forecasts for important
macroeconomic indicators and evaluate the responses of main endogenous variables to various shocks. The model is estimated based on quarterly data starting from 2001 to 2019. The majority of the
equations are specified in error correction form due to the non-stationarity of variables. Stochastic simulation is used to solve the model for expost and ex-ante analysis. We compare forecasts of
the model with forecasts generated by the VAR model. The results indicate that the present model outperforms the VAR model in terms of forecasting GDP growth, inflation rate and unemployment rate. We
also evaluate the responses of main macroeconomic variables to VAT rate and world trade shocks via stochastic simulation. Finally, we generate ex-ante forecasts for the Russian economy under the
baseline assumptions.
A small scale open economy model is estimated for Kazakhstan via Bayesian methods. The model explicitly takes into account the dependence of the economy on commodity exports and also accounts for
risk premium shocks in the foreign exchange market. The main contribution of the research is that it is the first DSGE model in literature estimated via Bayesian methods for Kazakhstan. The results
of the model are used to determine the historical contribution of structural shocks to endogenous variables, forecast error variance decomposition of observed macroeconomic variables and impulse
responses of important endogenous variables to various shocks. It has been found that the output gap turned significantly negative during the Great Recession and the negative oil price shock. The
effect of contractionary monetary policy is found to be negative on output gap, but it negligibly affects the inflation rate in the economy. Risk premium shocks are found to account for almost 60% of
forecast error variance decomposition of nominal exchange rate of tenge over all horizons.
The paper builds a structural macroeconometric model for Kazakhstan to generate short-term and medium-term forecasts for main macroeconomic variables and conduct scenario analyses based on dynamic
simulation of the model. Due to the poor quality of quarterly data on GDP and its expenditure components, they have been adjusted using volume indexes. The model consists of aggregate supply,
aggregate demand, labor market, asset market, the central bank policy and government side equations. Most equations are estimated via econometric techniques and identities are explicitly introduced
in line with economic theory. We combine all the regression equations into a single model and solve for the baseline scenario from 2003 to 2017. The simulation results show that the structural
macroeconometric model approximates Kazakhstani economy reasonably well. Ex-ante forecasts under oil prices remaining around 50 and 60 US dollars per barrel are generated and compared with the
baseline forecast of the National Bank of the Republic of Kazakhstan.
The conventional wisdom assumes that terms of trade shocks are the main drivers of business cycle dynamics in emerging exporting economies. This paper studies the effect of terms of trade shocks on
key macroeconomic variables for the Kazakhstani economy. Empirical SVAR model estimates suggest that the terms of trade shocks account for 12 % of output variation for the economy. Further, three
sectoral DSGE model with estimated structural parameters predict the modest importance of the terms of trade shocks for small open economy.
Dynamic stochastic general equilibrium (DSGE) models are widely used by central banks, government agencies and financial organizations to conduct simulation and forecast relevant macroeconomic
indicators in the economy. The most important inputs into all DSGE models are structural parameters which are either calibrated from other sources or estimated via Bayesian methods. Using non-public
microlevel data, we estimate ten structural parameters for Kazakhstan: the elasticity of substitution between exports and imports, constant relative risk aversion, intertemporal elasticity of
substitution in consumption, Frisch elasticity of labor supply, the depreciation rate of physical capital, capital and labor shares, and the elasticity of substitution between tradable and
nontradable goods. Various econometric techniques such as fixed-effects, generalized method of moments (GMM), Arellano-Bond, and non-linear iterative maximum likelihood estimation are used to obtain
consistent estimates of the models' coefficients. The structural parameters can be used in calibrated DSGE models as fixed parameters or as prior information in Bayesian estimation of the models.
This paper presents a two-country macroeconometric model for the economies of Kazakhstan and Russia. The model can be used for interpreting the structural relationship between the two economies,
determining the degree of trade integration and implementing scenario analyses with various shocks. Single-country models are linked through bilateral trade and exchange rate equations. The baseline
simulation of the two-country model demonstrates a good accuracy in tracking the actual dynamics of macroeconomic indicators in both countries. Scenario analyses are conducted with a risk premium
shock in the bilateral exchange rate and a monetary policy shock in Russia to analyze the transmission mechanism of the shocks, and clarify on the kind of interdependency of the economies. The model
shows a larger influence of the risk premium shock on economic activity in Kazakhstan than in Russia. A two percentage point decline in the key rate does not impose significant inflationary pressure
while imports and the real exchange rate are the most affected variables in both countries.
In this paper, we analyze fiscal rules in an oil-exporting economy using the DSGE model for Kazakhstan. The model is estimated using Bayesian methods, and we analyze the four fiscal rules of the
government of Kazakhstan in the face of a negative oil price shock. We find that fiscal policy in Kazakhstan has been pro-cyclical and some fiscal rules are redundant in the model. In addition, a
fiscal rule designed to limit the use of a country's sovereign wealth fund assets, contrary to conventional wisdom, results in a reduction in assets when there is a negative oil price shock. We also
find that two rules that cap interest payments on government debt and cap non-oil fiscal deficits lead to a more stable response of key endogenous variables to negative oil price shocks and do not
result in a significant reduction in sovereign wealth fund assets.
To study the role of CBDC in financial stability in Kazakhstan, we build a DSGE model with nominal and real rigidities, which includes the banking sector characterized by monopolistic competition.
The model is based on the work of Gerali et al. (2010), which examines the role of financial frictions and banking intermediation in business cycles. We complement the model with liquidity
preferences in the household utility function, which consists of cash and CBDC. We estimate the model via Bayesian methods using macroeconomic and financial data for Kazakhstan. The economy is
populated by a continuum of households (savers and borrowers) and entrepreneurs. Households consume, supply labor and accumulate housing capital. The utility function of entrepreneurs is determined
only by consumption, and they also produce homogeneous goods using the services of labor and physical capital. Banks in the model have market power in their intermediary activities. This allows banks
to adjust interest rates on loans and deposits in response to external shocks. In addition, banks must adhere to the identity of the balance sheet and follow the capital adequacy ratio. In line with
the literature on the CBDC and financial stability, we focus on the impact of the CBDC on bank intermediation in Kazakhstan (lending and borrowing). We analyze the impact of CDBC on financial
stability under four different scenarios. We choose the following variables to measure financial stability: banks' interest rate spread (difference between lending and deposit rates), banks'
equity-to-assets ratio, return on assets, and banks' return on equity.
The paper analyzes the level of productivity and competition in the domestic market of Kazakhstan. We show that the total factor productivity in many industries fell significantly from 2009 to 2017.
At the same time, from 3 to 10 of the largest firms in the industry occupy a significant market share in most industries, demonstrating the elements of oligopolistic competition in the market where
prices are set by a few large firms. An econometric analysis demonstrates that increased investments, profits, wages, subsidies, and the presence of employees aged under 30 or with higher education
have a significant positive effect on the level of enterprise productivity. Firms that are subsidized and carry out R&D experience a significant increase in productivity: with a 10% increase in
subsidies productivity increases by 2.2% on average. Subsidies are received annually by almost the same firms, which contributes to an increase in the market power of these firms. Statistics show
that 5 companies in the market receive up to about 80% of subsidies in manufacturing and agriculture. Such an uneven distribution of subsidies among firms contributes to the development of monopoly
in the market.
One of the aims of the current microeconometric study is to predict the potential demand for CBDC in Kazakhstan compared to its close alternatives using a framework provided by Li (2021) based on a
structural demand model. Assessing the potential demand for CBDC is important for understanding the impact of CBDC on banking products and the potential of CBDC usage in the country. CBDC, cash, and
deposits are considered product bundles of different attributes. Utility gains of households from holding each product depend on product attributes such as convenience, cost of use, security,
acceptance rate, anonymity, budgeting, household characteristics, and the unobserved households’ idiosyncratic preferences.
A structural macroeconometric model is built for the group of countries of EAEU and deterministic simulation of the model is conducted. Although the macroeconometric models are considered as outdated
in modern macroeconomic theory, it is still used by most research institutes due to their flexibility and data-oriented nature in contrast to general equilibrium models that are popular in academic
literature. The model consists of a system of equations estimated for each member country in the union and the economies are linked through bilateral trade equations and bilateral exchange rate
equations. As a result, we obtain a large structural model for EAEU. The present model allows one to generate forecasts for EAEU economies, the analyses of the degree of integration of EAEU member
countries since its formation, and the transmission of shocks from one member-state to another. Particularly, we analyze how monetary tightening and fiscal consolidation in each member country affect
other member countries, the degree of trade integration between the countries after the establishment of the EAEU and comparison of the dynamics of macroeconomic variables under the baseline model
and real world observations. In addition, other scenarios concerning positive/negative oil price shocks and sanctions on Russia are explored as part of the research.
In this paper we introduce a banking sector in a small open economy DSGE model in order to understand the role of banking intermediation in the transmission of monetary policy shocks and to analyze
how shocks that originate in credit matrkets are transmitted to the real economy. In doing so, we follow Geralli et al (2008) and assume that the banking sector is characterized as having features of
a monopolistic competition. Geralli et al (2008) support this assumption by referring to the empirical evidence which suggests that bank rates are heteregenous in the adjustment speed to changing
conditions in money market interest rates. Overall, the model features credit frictions and borrowing contraints, and a set of real and nominal rigidities. The imperfectly competitive banking sector
in the model collects deposits and/or borrows at the interbank market and lends to the private sector. These banks apply a time-varying and slowly adjusting mark-up over the policy rate in setting
different interest rates to households and firms.
The model is used to analyze two issues. First, we want to understand the role of banking sector in the transmission mechanism of monetary policy in Kazakhstan. Second, we would like to analyze the
effect of financial shocks, related to the bank rates and borrowing restrictions, on the real economy.
The main aim of the model is to be able to predict an economic recession in the next period. For that purpose, an econometric model, based on the Markov-switching, will be employed. The important
difference with other popular forecasting models is that the assumption of linearity of observed series is abandoned. Instead, this model uses the fact that most of the observed time series follow a
non-linear process. There is an abundant evidence of non-linearity of key macroeconomic variables.
Non-linearities in the present model arise if the observed series process is subject to discrete shifts in regime. Here, regime means states of the economy, either expansion or recession. The main
approach is to use Markov regime switching regression, presented in Hamilton (1989), to characterize changes in parameters of autoregressive process. To put differently, the model involves multiple
structures that characterize time series behavior in different regimes or states. Then, switching mechanism, which follows a first-order Markov chain, is used to capture complex dynamic patterns.
Further, statistical estimates of the states of the economy (expansion or recession) will be uncovered using a non-linear filter (smoother). This filter is similar to the Kalman filter but, unlike
the Kalman filter, it allows to provide non-linear inference about the unobserved state vector. Markov regime switching model is superior to logit linear models in its predictive power. In addition,
this model allows the policymaker to infer future probabilities of recession in real time.
The pension system of Kazakhstan has undergone through various legislative changes since its first reform in 1998. The current pension landscape of the country embodies a multi-pillar system with
three levels of pension components. We use the Overlapping Generations Model (OLG) to examine the existing pension system of Kazakhstan by highlighting the features of each pillar in the pension
block. OLG models are one the most advanced and widely used tools which can be employed to investigate the impact of fiscal policy changes and pension system reforms on the economy. In our model, the
economy is populated with active and retired generations. We introduce heterogeneity with professional categories (high, medium and low qualified specialists) differentiated by gender. The structural
parameters of the economy are estimated using micro-level data from 2009 to 2018. A number of scenario analyses can be conducted within the framework of the model to examine the effects of different
policy options and pension system reforms on macroeconomic dynamics. Specifically, we are interested in the consequences of replacement ratio shocks, increasing the retirement age for women and
raising the level of compulsory pension contributions made by employers. These topics are particularly relevant in the current economic context of Kazakhstan. Hence, the model can be an attractive
tool for government agencies to use in making policy related decisions.
The research is aimed to empirically analyze the impact of spatial and territorial agglomeration on the productivity of firms using panel data. The model allows us to answer the following questions:
1) What will be the economic benefits of clustering the economy? In particular, how much does firm productivity increase when other firms from the same or another sector decide to locate nearby?
2) How much do firms benefit from deciding where to locate?
3) Are there good reasons for government intervention in favor of industrial clusters?
One of the main changes in the trade policy of Kazakhstan, due to its admission to the CU, was the increase of tariff rates for non-ECU countries. Mkrtchyan and Gnutzmann (2012) and Jandosov and
Sabyrova (2011) concluded that there was a significant increase in Kazakhstan’s tariff protection level after its accession to the CU with the simple average tariff rate increasing from 6.45% to
12.02%. However, the accession of Russia and Kazakhstan to the WTO will decrease the tariff rates of the ECU and hence the trade diversion. Shepotylo and Tarr (2012) calculated that after Russia
implemented all commitments to the WTO, average weighted tariff rates of ECU would decrease from 13 in 2011 to 5.8% in 2020. There was no trade creation based on the removal of the tariffs within the
CU of Russia, Kazakhstan, Belarus, Armenia and Kyrgyzstan as, since 1994, these countries were in an FTA; hence, no extra tariff preference was given since the creation of the CU. Countries might
benefit from the decrease of non-tariff barriers (NTB), such as the abolition of the customs controls, adoption of the single system of phytosanitary norms, single system of customs regulation and
procedures. Research undertaken by the EBRD (2012) and ADB (2012) have shown that NTB have decreased since the establishment of the ECU. All of these papers used surveys to determine change in
non-tariff barriers between EEU countries. However, no one retrieve the tariff equivalent of non-tariff barriers for EEU countries before and after creation of the CU using econometric techniques. In
the first paper we will study non-tariff barriers between Kazakhstan and other EEU members on the industry level (2 or 4 HS digit numbers). Using gravity model we will retrieve the yearly tariff
equivalent of non-tariff barriers for all industries from 2000 to 2019. If data will show that non-tariff barriers are decreasing for a particular industry, then there might a possibility for trade
creating effect in this sector. We will quantify this effect in the second paper.
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Sampling Procedures for Inspection and Sampling Plans for Lot Inspection Using ISO 2859 Analysis
Published on
Sampling Procedures for Inspection and Sampling Plans for Lot Inspection Using ISO 2859 Analysis
Sampling is a technique in which samples are drawn at random (without any favor or bias). For this, suitable measures or procedures may be laid down and adopted according to the nature and
configuration of parts under inspection for ensuring complete randomness in sample selection. The term sample implies a subset chosen from the population.
Acceptance sampling is an important method used in sampling procedures. It is also an important field of standard quality control where a random sample is taken from a lot, and upon the results of
appraising the sample, the lot will be either rejected or accepted. It is also a procedure for inspecting quality of batches without doing 100% inspection. So, acceptance sampling is also known as
the “middle-of-the-road approach” between no inspection and 100% inspection. The cost associated with 100% inspection is often prohibitive, and the risks associated with lesser or 0% inspection are
very large.
Acceptance Sampling by Attributes. Evaluations using attributes are generally based on “go” and “no-go” gauge inspections. A sample is taken and if it contains too many non-conforming items then the
entire batch is rejected, otherwise, the batch is accepted.
Acceptance Sampling by Variables. Evaluations using variables may also be known as continuous measurement. The decision rule for selection and rejection of a lot in this type of sampling depends on a
statistical tool, such as mean and standard deviation.
A decision on whether to use Acceptance Sampling by Attributes or Acceptance Sampling by Variables will depend on the particular circumstances of each case. However, it is noteworthy to mention that
ISO 2859 deals with Acceptance Sampling by Attributes only.
Types of Sampling Plans
The most important element of acceptance sampling is choosing an appropriate sampling plan, which specifies the lot size, sampling size, number of samples, and acceptance/rejection criteria. Hence,
considering all these parameters, acceptance sampling is typically of three types: single, double, or multiple sampling plans.
Single Sampling Plan. In this type of sampling plan, the decision for acceptance or rejection of the lot is made on the basis of only one sample. The acceptance plan is known as a single sampling
plan. It generally follows the flowchart in Figure 1.
Figure 1: Let N = lot size, n = sample size, C = acceptance number, and d = the number of defective items in the sample. The standard rule would be: if d < C, accept the lot, but if d ≥ C, reject the
lot. Courtesy: Mahindra Teqo Pvt. Ltd.
Double Sampling Plan. In this type of sampling plan, the decision on acceptance or rejection of the lot is based on two samples. It generally follows the flowchart in Figure 2.
Figure 2: Let N = lot size, n1 = number of pieces in first sample, C1 = acceptance number for first sample, d1 = number of defective items in the first sample, n2 = number of pieces in second sample,
n1 + n2 = number of pieces in two samples combined, C2 = acceptance number for the two samples combined, and d2 = number of defective items in the two samples combined. The standard rule would be: if
d1 ≤ C1, accept the lot; if d1 > C2, reject the lot; but if C1 < d1 ≤ C2, then evaluate the two samples combined. The rule in that case would be: if d2 ≤ C2, accept the lot, but if d2 > C2, reject
the lot. Courtesy: Mahindra Teqo Pvt. Ltd.
Multiple Sampling Plan. In this plan, a decision to accept or reject the lot is taken after inspecting more than two samples. In this case, if the cumulative number of defective items in a sample is
more than the upper limit specified for that particular sample, the sampling procedure is terminated and the lot is rejected. On the other hand, if the total number of defective items in a sample is
less than or equal to the lower limit specified, then also sampling is terminated and the whole lot is passed.
Sampling Terms and Their Definitions
Several important terms apply to sampling procedures and plans. They include the following.
Acceptance Quality Limit (AQL). The AQL defines the percentage of defects at which consumers are willing to accept lots as “good.” The producers would like to design a sampling plan such that there
is high probability of accepting a lot that has a defect level less than or equal to the AQL. In other words, the AQL is the worst tolerable quality level.
Lot Tolerance Percent Defective (LTPD) or Rejectable Quality Level (RQL). These terms define the upper limit on the percentage of defects that a consumer is willing to accept. In other words, it is
the poorest level of quality that the consumer is willing to tolerate in an independent lot. The probability of accepting a lot of RQL represents a risk for the consumer.
Average Outgoing Quality (AOQ). AOQ represents the average percentage defectives in the outgoing products after inspection including all accepted and all rejected lots.
Producer’s Risk/Manufacturer’s Risk (α). α represents the probability that a lot containing AQL will be rejected.
Consumer’s Risk (β). β represents the probability that a lot containing defectives more than LTPD/RQL will be accepted.
Operating Characteristic Curve (OC Curve)
The OC curve is one of the major tools for representing and investigating the properties of a lot acceptance sampling plan. Figure 3 depicts an ideal curve. For the curve shown, all lots with less
than 3% defects have a probability of acceptance of 100% while all lots with more than 3% defectives have a probability of acceptance as 0%. However, the sample size needed for such a plan is
infinite. Hence, such a plan does not exist in reality. It may be noted that in this case, 3% is the worst tolerable quality level, that is, AQL.
Figure 3: An Ideal Operating Characteristic Curve. Courtesy: Mahindra Teqo Pvt. Ltd.
Let us consider a practical OC Curve (Figure 4). It shows probability of acceptance L(p) as a function of lot fraction defective (p). This curve is based on sampling plans. It aids in selection of
plans that are effective in reducing risks.
Here, the probability of acceptance can be found using the equation:
L(p) = 1 – L(r)
where, L(r) = probability of rejection.
Figure 4: A Practical Operating Characteristic Curve. Courtesy: Mahindra Teqo Pvt. Ltd.
Lot Inspection Using ISO 2859-2:2020
As per ISO 2859-2:2020, “Sampling Procedures For Inspection By Attributes – Part 2: Sampling Plans Indexed By Limiting Quality (LQ) For Isolated Lot Inspection,” the sampling plan is indexed by a
series of specified values of LQ, where β is less than 10%, except under some circumstances. This standard is valid both for inspection of non-conforming items as well as for inspection of
non-conformities per 100 items. It is intended to be used when both supplier and consumer regard the lot to be in isolation. This means that the lot is unique in the sense that only one of its types
is produced.
Tables 1 and 2 are useful in choosing the appropriate sample size corresponding to a fixed AQL for a particular lot inspection.
Table 1: Sample size code letter selection. Courtesy: Mahindra Teqo Pvt. Ltd.
Table 2: Single sampling plan for normal inspection. (Master table limited to AQL of 10%.) Courtesy: Mahindra Teqo Pvt. Ltd.
The steps to follow while selecting a sampling plan include:
1. Decide AQL and choice of sampling plan.
2. Based on lot size, choose sample size code letter.
3. Based on sample size code and AQL, choose sample size and accept/reject numbers.
Let us consider that we have received 1,000 solar PV modules from a supplier and we want to determine a sampling plan for inspection of microcracks using a go/no-go gauge. What will be the sample
size and acceptance number if AQL is 1% for General Inspection Level II?
Using Table 1, the sample size code letter corresponding to General Inspection Level II and Lot Size range of 501 to 1,200 (for lot size of 1,000 as stated) is “J.” Now, in Table 2, referring to
sample size code letter “J,” we get the prescribed number of samples, which is 80. From the same Table 2, for AQL of 1% and sample size as 80, we get the Acceptance number (Ac) = 2, and the Rejection
number (Re) = 3. This implies that if two or fewer defects are found in the sample, accept the lot. Similarly, if the number of defects is three or more, reject the lot.
A good sampling plan is necessary for inspection at the lowest cost that will provide the best representation for the entire lot. ISO 2859 has proved to be the most vital standard toward selecting a
good sampling plan. It is the most sought technical standard being used in the solar PV industry for sample size selection.
As evidence, we have three General Inspection Levels, that is, Level I, Level II, and Level III, and four Special Inspection Levels (S-1, S-2, S-3, and S-4). By default, General Inspection Level II
should be used while using normal inspection. Similarly, Special Inspection Levels should be used when inspection time required per unit is larger, or whenever some destructive testing needs to be
performed on sampled units.
The sampling methodologies and procedures described in this article are equally useful for power applications outside of solar PV. Sampling can be used on wind farm, gas turbine, and other power
plant components. However, to facilitate the ease for sample selection in the solar industry, this article illustrates a detailed approach based on expertise and comprehensive analysis in that
The sampling methodology, as described in this article, may be used in various applications in solar power plants. One of the applications is sampling of solar modules, as previously mentioned, to
ascertain plant performance. Moreover, the detailed approach discussed in this article may be used effectively in both construction and operational phases of a solar power plant.
The same methodology was adapted for module testing at various solar plants, and the observed results were found to be very closely matched with desired plant performance. However, individual plant
results may vary on a case-to-case basis.
Disclaimer: The information provided here is for general purposes and is offered in good faith. It is accurate to the best of our knowledge. Mahindra Teqo Pvt. Ltd. has not authenticated or validated
the information provided, and assumes no responsibility or liability to any party or any person for the consequences of the use of the information contained herein.
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Origin of Black-Box Operators
The question "where do black-box operators come from?" is really two different questions:
1. Where do nature create problems with exploitable structure?
2. Where do the black-box operators used in BBTools come from?
We answer the first question with a quotation from [20]:
Large matrices ... usually arise indirectly in the discretization of differential or integral equations. One might say, that if m is large, it is probably an approximation to ∞. It follows that most
large matrices of computational interest are simpler than their vast number of individual entries might suggest.
The operators used in BBTools generally have the following sources:
1. Matlab-matrices converted to black-box operators
2. Operators constructed from the library shipped with BBTools
3. Operators built by combining other operators
4. Custom operators based on algorithms constructed from scratch
A Matlab matrix can be converted to a black-box operator using either bbmatrix or blackbox. Operators, which are shipped with BBTools, can be found in the function reference.
The real power of BBTools comes from its ability to combine the other operators efficiently.
[20] Lloyd Nicholas Trefethen and David Bau III. Numerical Linear Algebra. SIAM, Philadelphia, PA, 1997. ISBN: 0-89871-487-7. (Book)
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Oxford Mathematics Public Lectures - Can Mathematics Understand the Brain?' - Alain Goriely
The human brain is the object of the ultimate intellectual egocentrism. It is also a source of endless scientific problems and an organ of such complexity that it is not clear that a mathematical
approach is even possible, despite many attempts.
In this talk Alain will use the brain to showcase how applied mathematics thrives on such challenges. Through mathematical modelling, we will see how we can gain insight into how the brain acquires
its convoluted shape and what happens during trauma. We will also consider the dramatic but fascinating progression of neuro-degenerative diseases, and, eventually, hope to learn a bit about who we
are before it is too late.
Alain Goriely is Professor of Mathematical Modelling, University of Oxford and author of 'Applied Mathematics: A Very Short Introduction.'
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Package com.luciad.multidimensional
package com.luciad.multidimensional
General interfaces and implementations to model multi-dimensional data, for example data that can change over time or can vary in height.
The main interface is ILcdMultiDimensional, which describes the axes over which a multi-dimensional object is defined. A TLcdDimensionAxis is defined by its type, unit and ranges in which data can be
A model can also implement ILcdMultiDimensionalModel to present its ILcdDimensions and become filterable.
• Class
This interface defines the possible values or intervals in which data is defined.
A multi-dimensional represents an object that can vary over multiple dimensions like time or height.
Models that support dimensional filtering, such as NetCDF, NVG and some LuciadFusion models, should implement this interface.
Builder for dimension axis values.
This class represents a dimension axis.
Builder for dimension axes.
An abstraction of a dimensional filter, to be applied to multi-dimensional models.
A builder for filters which reuses filter instances as much as possible.
This class represents an interval defined by a minimum and maximum value or a single value.
Denotes if a boundary of an interval is inclusive/exclusive.
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New Types of Matrix Models
Item 学位論文 / Thesis or Dissertation(1)
公開日 2010-02-22
タイト New Types of Matrix Models
タイト New Types of Matrix Models
言語 en
言語 eng
イプ識 http://purl.org/coar/resource_type/c_46ec
資源タ thesis
大鷲, 雄飛
× 大鷲, 雄飛
オオワシ, ユウヒ
ナ × オオワシ, ユウヒ
OHWASHI, Yuhi
× OHWASHI, Yuhi
授与 総合研究大学院大学
学位名 博士（理学）
述タイ Other
内容記 総研大甲第579号
値 数物科学研究科
値 14 素粒子原子核専攻
与年月 2002-03-22
値 2001
述タイ Other
Today particle physics except for gravity is well described by the standard model. However, gravity cannot be quantized in the same method because we cannot renormalize it. Therefore the main
problem of current particle physics is to establish a consistent quantum theory which contains both the standard model and gravity. Under these circumstances, the most hopeful and popular
candidate is the string theory.<br /><br /> The reason to favor the string theory is its wonderful nature. We can give as concrete examples that the theory has no ultraviolet divergence and
includes gravitational field as well as matter and gauge fields automatically. However, due to the infinite ground states, this theory has no capability to predict; therefore we cannot answer
why the standard model emerges. On the other hand it is possible to consider that this problem is the problem in the framework of perturbative formulation of the theory, because the completed
region of the string theory is only the perturbative region. So if the non-perturbative formulation of the theory is accomplished, it is quite likely that this problem is resolved. Of course,
it is pure speculation, but it seems quite probable that the non-perturbative effects turn infinite ground states into single one.<br /><br /> What must not be forgotten is that one theory
never finish before the non-perturbative formulation is completed. One of candidates for the non-perturbative formulation of the string theory at present is the string field theory. Although
a considerable number of studies have been conducted on these theories, the only successful string field theories so far are the ones formulated in the light-cone gauge. So it is not clear
whether we can extract some essential information of the non-perturbative effects. Another candidate is what is called the matrix model. With the advent of the BFSS model as a starter, many
proposals have been being made since. The common idea of these models is that they reproduce sting or membrane theory in the large-N limit. In a sense the matrix model is similar to the
lattice gauge theory, which is the non-perturbative formulation of the field theory, in that they can be analyzed using numerical simulation. Therefore it is reasonable to suppose that we
will develop current matrix models a little further and find the true model.<br /><br /> A virtue of the matrix model is that it has a possibility of putting an interpretation on the
space-time itself. However, some important questions such as "what would be the real mechanism to realize the 4-dimensional world from the 10(or 11)-dimensional universe" and "how is the
diffeomorphism introduced into the theory" remain unsettled. One of them is the question of background independence. Consider the IKKT model for example. This model has an SO(10) × SU(N)
内容記 symmetry, and this is just a symmetry like some theory was expanded around the flat background. Therefore we cannot deny the existence of different matrix model whose expansion around a
述 special background gets the IKKT model. On this point Smolin proposed a new type of matrix model in which the action is cubic in matrices. Matrices are built from the super Lie algebra osp( 1
|32; Ｒ), and one multiplet is pushed into a single supermatrix. Smolin's conjecture is that the expansions around different backgrounds of the osp(1|32; Ｒ) matrix model will reduce to the
BFSS or IKKT model. However, as far as the IKKT model is concerned, the theory made from Smolin's way dose not reproduce the supersymmetry of the IKKT model. That is, indeed the
10-dimensionality is realized, but the half of supersymmetry required by the IKKT model cannot be held. Anyway, the model described by a single matrix alone is very attractive, and Smolin's
courageous attempt demonstrated one concrete possibility.<br /><br /> Moreover, as Smolin's u(1|16, 16) model has demonstrated, the matrix models are not irrelevant to the loop quantum
gravity which is another approach to the Theory of Everything. Furthermore, it was pointed out that the matrix string theory has a connection with the matrix model based on the exceptional
Jordan algebra Ｊ, while B.Kim and A. Schwarz have discussed a tie-in between the IKKT model and the Jordan algebra ｊ with its spinor representation. For these reasons, doing research on
extended matrix model is very interesting and important. Over and above, we should not overlook the fact that several approaches which are very similar to the matrix model have been pursued
by other fields. It might be inferred from these circumstantial evidence that the attempt to renounce the space-time as a continuum holds one important key to the future progress of physics.
It seems at least that there is no need to relate the matrix model to the string theory alone.<br /><br /> For these purposes, we investigate new types of matrix models based on the complex
exceptional Jordan algebra and the super Lie algebras. In the former case, a matrix Chern-Simons theory is directly derived from the invariant on E<SUB>6</SUB>. It is stated that the same
argument as Smolin which derives an effective action similar to the matrix string theory can also be held in our model. The only difference is that our model has twice as many degrees of
freedom as Smolin's model has. One way to introduce the cosmological term is the compactification on directions. It is of great interest that the properties of the product space Ｊ<SUP>c</
SUP> × ｇ, in which the degrees of freedom of our model live, are very similar to those of the physical Hilbert space. In the latter case, we investigate three super Lie algebras, osp(1|32;
Ｒ), u(1|16,16), and gl(1|32;Ｒ). In paticular, we study the supersymmetry structures of these models and discuss possible reductions to the IKKT model. In addition to those, a different u(1|
16, 16) model from Smolin's, and some kind of topological effective action derived using Wigner-Inönü contraction are also discussed
値 有
述タイ Other
内容記 application/pdf
Other Formats
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graded algebra
Added the notion of strongly $\mathbb{N}$ -graded algebra and an associated reference
diff, v11, current
added pointer to:
• Gregory Karpilovsky, Chapter 2 of: The Algebraic Structure of Crossed Products, Mathematics Studies 142, North Holland 1987 (ISBN:9780080872537)
diff, v10, current
Regarding the sentence on the terminology “strongly graded”:
I have moved it from “Properties” to “Definition” (now here)
I have hyperlinked technical terms (notably epimorphism)
I have added link from the text to the reference (to give readers a chance to see that there is such a reference).
In the process, I added the actual definition number.
diff, v12, current
Thanks, I’ll try to do as good next time.
added algebra - contents to sidebar
diff, v13, current
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Recent journal publications
H. S. Dhindsa, V. J. Marton and G. W. F. Drake, "Search for light bosons with King and second-King plots optimized for lithium ions," Phys. Part. Nuclei, 53, 800 (2022), 5 pp.
B. M. Henson, J. A Ross, K. F. Thomas, C. N. Kuhn, D. K. Shin, S. S. Hodgman, Y. H. L. Y. Zhang, G. W. F. Drake, A. T. Bondy, A. G. Truscott, and K. G. H. Baldwin, "Measurement of a helium tune-out
frequency: an independent test of quantum electrodynamics," Science, 376, 199 (2022).
G.W.F. Drake, Harvir S. Dhindsa and Victor J. Marton, "King and second-King plots with optimized sensitivity for lithium ions," Phys. Rev. A, 104, L060801 (2021), 6 pages.
A.T. Bondy, D.C. Morton, and G.W.F. Drake, "Two-photon decay rates in heliumlike ions: Finite-nuclear-mass effects," Phys. Rev. A, 102, 052807 (2020), 10 pages.
G.W.F. Drake and E. Tiesinga, "Simplify Your Life," Nature Physics 16, 1242 (2020): https://rdcu.be/cbBQS
G.W.F. Drake, "Accuracy in Atomic and Molecular Data," J. Phys. B: At. Mol. Opt. Phys. 53, 223001 (2020).
A.T. Bondy, D.C. Morton, and G.W.F. Drake, "Two-photon decay rates in heliumlike ions: Finite-nuclear-mass effects," Phys. Rev. A 102, 052807 (2020)
X.-Q Qi, P.-P. Zhang, Z.-C. Yan, G.W.F. Drake, Z.-X. Zhong, T.-Y. Shi, S.-L. Chen, Y. Huang, H. Guan, and K.-L. Gao, "Precision Calculation of Hyperfine Structure and the Zemach Radii of Li-6.7(+)
Ions Phys. Rev. Lett. 125, 183002 (2020).
H. Guan, S.L. Chen, X.-Q Qi, S.Y. Liang, W. Sun, P.P. Zhou, Y. Huang, P.P. Zhang, Z.-X. Zhong, Z.-C. Yan, G.W.F. Drake, T.-Y. Shi, and K.L. Gao, "Probing atomic and nuclear properties with precision
spectroscopy of fine and hyperfine structures in the Li-7(+) ion," Phys. Rev. A 102, 030801 (2020).
G.W.F. Drake, J.G. Manalo, P.-P. Zhang, and K.G.H. Baldwin, "Helium tune-out wavelength: Gauge invariance and retardation corrections," Hyperfine inter. 240, 31 (2019), 8 pp.
Gordon W.F. Drake, Jung-Sik Yoon, Daiji Kato, and Grzegorz Karwasz,"Atomic and Molecular Data and their Applications," Euro. Phys. J. D 72, R49 (2018), 3 pp.
Donald C. Morton and G.W.F. Drake, "Oscillator strengths for spin-changing P–D transitions in He I including the effect of a finite nuclear mass and intermediate coupling," Can. J. Phys. 95, 828
(2017), 4 pp.
L.M. Wang, Chun Li, Z.-C. Yan, and G.W.F. Drake, "Isotope shifts and transition frequencies for the S and P states of lithium: Bethe logarithms and second-order relativistic recoil," Phys. Rev. A 95
R032504 (2017) 10 pp.
D. C. Morton and G. W. F. Drake, "Oscillator strengths for 1s^2 ^1S_0 - 1s2p ^3P_{1,2} transitions in helium-like carbon, nitrogen and oxygen including the effects of a finite nuclear mass," J. Phys.
B -- At. Mol. Opt. Phys. 49 234002 (2016), 7 pp.
H.-K. Chung, B. J. Braams, K. Bartschat, A. G. Csaszar, G. W. F. Drake, T. Kirchner, V. Kokoouline and J. Tennyson, "Uncertainty Estimates for Theoretical Atomic and Molecular Data”, J. Phys. D:
Applied Physics, 49 (2016); arXiv:1603.05923v2 [physics.atom-ph].
D.C. Morton, E. Schulhoff and G.W.F. Drake, "Oscillator strengths and radiative decay rates for spin-changing S-P transition in helium: finite nuclear mass effects," J. Phys. B. 48, 235001 (2015).
E. Schulhoff and G.W.F. Drake, "Electron emission and recoil effects following the beta-decay of helium-6 (6He)," Phys. Rev. A. 92, 050701 (2015).
L.M. Wang, C. Li, Z.-C. Yan, and G.W.F. Drake, "Fine structure and ionization energy of the 1s2s2p (4)P state of the helium negative ion He(-)," Phys. Rev. Lett. 113, 263007 (2014) (4 pages).
C. Estienne, M. Busuttil, A. Moini, and G.W.F. Drake, "Critical nuclear charge for two-electron atoms," Phys. Rev. Lett. 112, 173001 (2014) (4 pages).
Z.-T. Lu, P. Mueller, G.W.F. Drake et. al., "Colloquium: Laser probing of neutron-rich nuclei in light atoms," Rev. Mod. Phys. 85, 1383-1400 (2013).
L.M. Wang, Z.-C. Yan, H.X. Qiao et al., "Variational energies and the Fermi contact term for the low-lying states of lithium: Basis-set completeness," Phys. Rev. A 85, 052513 (2012).
A.S. Titi, and G.W.F. Drake, "Quantum theory of longitudinal momentum transfer in above-threshold ionization," Phys. Rev. A 85, 041404 (2012).
W. ElMaraghy, H. ElMaraghy, T. Tomiyama, L. Monostori, Complexity in engineering design and manufacturing, CIRP Annals - Manufacturing Technology, 61/2: 2012
A. Djuric, R. Al Saidi, W. ElMaraghy, Dynamics solution of n-DOF global machinery model, Robotics and Computer-Integrated Manufacturing, 28: 621-630 pp, 2012
M. Brodeur, T. Brunner, C. Champagne, et al., "First Direct Mass Measurement of the Two-Neutron Halo Nucleus He-6 and Improved Mass for the Four-Neutron Halo He-8," Phys. Rev. Lett. 108, 052504
D. Morton and G.W.F. Drake, 2011. "Spin-forbidden radiative decay rates from the 3 (3)P(1,2) and 3 (1)P(1) states of helium," Phys. Rev. A 83, 042503 (2011)
L. M. Wang, Z. -C. Yan, H.X. Qiao, et al., "Variational upper bounds for low-lying states of lithium," Phys. Rev. A 83, 034503 (2011) (4 pages)
W. Noertershaeuser, W., R. Sanchez, R. G. Ewald, G., et al. " Isotope-shift measurements of stable and short-lived lithium isotopes for nuclear-charge-radii determination," Phys. Rev. A, 83, 012516,
Donald C. Morton, Paul Moffatt, and G. W. F. Drake, "Relativistic corrections to He I transition rates," Can. J. Phys. 89, 1, (2011) (5 pages) Conference: International Conference on Precision
Physics of Simple Atomic Systems Location: Ecole de Physique, Les Houches, FRANCE Date: MAY 30-JUN 04, 2010
M. Zakova, Z. Andjelkovic, M.L. Bissell, K. Blaum, G.W.F. Drake, C Geppert, M. Kowalska, J. Kramer, A. Krieger, M. Lochmann, T. Neff, R. Neugart, W. N\"ortersh\"auser, R. Sanchez, F. Schmidt-Kaler,
D. Tiedemann, Z.-C. Yan, D.T. Yordanov, And C. Zimmermann, "Isotope shift measurements in the 2s(1/2) --> 2p(3/2) transition of Be+ and extraction of the nuclear charge radii for Be-7, Be-10, Be-11,"
J. Phys. G--Nucl. and Particle Phys., 37, 055107 (2010) (14 pages).
R. El-Wazni and G.W.F. Drake, "Energies for the high-L Rydberg states of helium: Asymptotic analysis," Phys. Rev. A 80, 064501 (2009) (4 pages).
R. Ringle, M. Brodeur, T. Brunner, S. Ettenauer, M. Smith, A. Lapierre, V.L. Ryjkov, P. Delheij, G.W.F. Drake, J. Lassen, D. Lunney, and J. Dilling, "High-Precision Penning-Trap Mass Measurements of
9,10Be and the One-Neutron Halo Nuclide 11Be," Phys. Lett.\ B 695, 170-174 (2009).
W. Nortershauser, D. Tiedemann, M. Zakova, Z. Andjelkovic, K. Blaum, M.L. Bissell, R. Cazan, G.W.F. Drake, C. Geppert, M. Kowalska, J. Kramer, A. Krieger, R. Neugart, R. Sanchez, F. Schmidt-Kaler,
Z-C. Yan, D.T. Yordanov, C. Zimmermann, "Nuclear Charge Radii of Be-7, Be-9, and One-Neutron Halo Nucleus Be-11", Phys. Rev. Lett., 102, 062503 (2009).
M. Smith, M. Brodeur. T. Brunner, S. Ettenauer, A. Lapierre, R. Ringle, V. L. Ryjkov, F. Ames, P. Bricault, G. W. F. Drake, P. Delheij, D. Lunney, F. Sarazin, and J. Dilling, First Penning-Trap Mass
Measurements of the Exotic Halo Nucleus, Phys. Rev. Lett., 101, 202501 (2008).
I.A. Sulai, Wu, Qixue, Bishof, M., Drake, G. W. F., Lu, Z. -T., Mueller, P., Santra, R., Hyperfine Suppression of 2(3)(S(1)-3 (3)P(J)Transitions in 3He, Phys. Rev. Lett. 101, 173001 (2008).
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Sorting is one of the important operations done on array. Here the array elements are arranged in ascending or descending order. In this part we will study three different sorting techniques
□ Bubble sort
□ Selection sort
□ Insertion sort
Bubble Sort:
Let us start with an example - 1, 4, -1, 0, -8, 10, 3, 7, -2, 9, is our array content and we want to arrange it in ascending order. First of all the first number (1) is considered (bubbled) first as
shown in the figure below. Then it is campared with the adjacent number (4), since we want to arrange them in ascending order so we need not change the position of the numbers (means the two 1, 4
numbers are already arranged). Next the bubble move to the next number (4) then we compare it with its adjacent number (-1) this time we have to change its position so 1 takes the position of 4 and 4
takes the position of 1 (1, 4). This process goes on till the bubble moves to the second last number. At the end of this process see that the largest number takes it position in the sorted array.
This is called pass one. In the second pass it bubbles the first number and same process is applied and it goes till the third number from the last, at the end of this pass second largest number
takes it position in the sorted array. In this way there will be
passes, where
is the number of numbers in the array. See the figures - the first figure shows how each pass is carried out, second figure shows all the passes.
Now let us write the function to sort an array having 10 integers-
#include <iostream.h>
void bubbleSort(int a[], int n){
int i, j, t;
for(i=1; i<=n-1; i++){
for(j=0; j<=(n-1)-i; j++){
void main(){
int a[10], i;
for(i=0; i<= 9; i++){
cout<<"Enter the element :"<<i+1<<":";
cout<<"The sorted array is as under:\n";
for(i=0; i<=9; i++){
cout<<a[i]<<" ";
In the "main()" function see that 10 is passed as number of element. In the function "bubbleSort()" the first loop counts the number of passes and the second loop bubbles the numbers, if necessary
changes the position of the numbers. See that the second loop ranges from 0 (first number) to the second last number (n-1 is equal to 9, 9-i, i.e 9-1 equal to 8 which is the position of the second
last number in the first pass); in the second pass it ranges from 0 to third last number and so on.
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Here are some selections from the OpenFL Developer's Guide to help you get started with the openfl.geom package:
The ColorTransform class lets you adjust the color values in a display object. The color adjustment or color transformation can be applied to all four channels: red, green,
ColorTransform blue, and alpha transparency.
The Matrix class represents a transformation matrix that determines how to map points from one coordinate space to another. You can perform various graphical transformations on
a display object by setting the properties of a Matrix object, applying that Matrix object to the matrix property of a Transform object, and then applying that Transform object
Matrix as the transform property of the display object. These transformation functions include translation (x and y repositioning), rotation, scaling, and skewing. Together these types
of transformations are known as affine transformations. Affine transformations preserve the straightness of lines while transforming, so that parallel lines stay parallel.
The Matrix3D class represents a transformation matrix that determines the position and orientation of a three-dimensional (3D) display object. The matrix can perform
Matrix3D transformation functions including translation (repositioning along the x, y, and z axes), rotation, and scaling (resizing). The Matrix3D class can also perform perspective
projection, which maps points from the 3D coordinate space to a two-dimensional (2D) view.
The Orientation3D class is an enumeration of constant values for representing the orientation style of a Matrix3D object. The three types of orientation are Euler angles, axis
Orientation3D angle, and quaternion. The decompose and recompose methods of the Matrix3D object take one of these enumerated types to identify the rotational components of the Matrix.
The PerspectiveProjection class provides an easy way to assign or modify the perspective transformations of a display object and all of its children. For more complex or custom
PerspectiveProjection perspective transformations, use the Matrix3D class. While the PerspectiveProjection class provides basic three-dimensional presentation properties, the Matrix3D class provides
more detailed control over the three-dimensional presentation of display objects.
Point The Point object represents a location in a two-dimensional coordinate system, where x represents the horizontal axis and y represents the vertical axis.
Rectangle A Rectangle object is an area defined by its position, as indicated by its top-left corner point(x, y) and by its width and its height.
The Transform class provides access to color adjustment properties and two- or three-dimensional transformation objects that can be applied to a display object. During the
transformation, the color or the orientation and position of a display object is adjusted (offset) from the current values or coordinates to new values or coordinates. The
Transform Transform class also collects data about color and two-dimensional matrix transformations that are applied to a display object and all of its parent objects. You can access
these combined transformations through the concatenatedColorTransform and concatenatedMatrix properties.
Utils3D The Utils3D class contains static methods that simplify the implementation of certain three-dimensional matrix operations.
The Vector3D class represents a point or a location in the three-dimensional space using the Cartesian coordinates x, y, and z. As in a two-dimensional space, the x property
represents the horizontal axis and the y property represents the vertical axis. In three-dimensional space, the z property represents depth. The value of the x property
Vector3D increases as the object moves to the right. The value of the y property increases as the object moves down. The z property increases as the object moves farther from the point
of view. Using perspective projection and scaling, the object is seen to be bigger when near and smaller when farther away from the screen. As in a right-handed
three-dimensional coordinate system, the positive z-axis points away from the viewer and the value of the z property increases as the object moves away from the viewer's eye.
The origin point (0,0,0) of the global space is the upper-left corner of the stage.
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What is Set Theory?
Set theory constitutes most of the foundation of modern mathematics, and was formalized in the late 1800s. Set theory describes some very fundamental and intuitive ideas about how things called
"elements" or "members" fit together into groups. Despite the apparent simplicity of the ideas, set theory is quite rigorous. In seeking to eliminate all arbitrariness in their theories,
mathematicians have fine-tuned set theory to an impressive degree over the years.
In set theory a set is any well-defined group of elements or members. Sets are usually symbolized by italicized capital letters like A or B. If two sets contain the same members, they can be shown as
equivalent with an equal sign.
The contents of a set can be described in simple English: A = all terrestrial mammals. Contents can also be listed within brackets: A = {bears, cows, pigs, etc.} For large sets, ellipsis may be
employed, where the pattern of the set is obvious. For example, A = {2, 4, 6, 8... 1000}. One type of set has zero members, the set known as the empty set. It is symbolized by a zero with a diagonal
line ascending left to right. Though seemingly trivial, it turns out to be quite important mathematically.
Some sets contain other sets, therefore being labeled supersets. The contained sets are subsets. In set theory, this relationship is referred to as "inclusion" or "containment," symbolized by a
notation that looks like the letter U rotated 90 degrees to the right. Graphically, this can be represented as a circle contained within another, larger circle.
Some common sets in set theory include N, the set of all natural numbers; Z, the set of all integers; Q, the set of all rational numbers; R, the set of all real numbers; and C, the set of all complex
When two sets overlap but neither is completely embedded within the other, the whole thing is called a union of sets. This is represented by a symbol similar to the letter U, but slightly wider. In
set notation, A U B means "the set of elements which are members of either A or B". Turn this symbol upside down, and you get the intersection of A and B, which refers to all elements which are
members of both sets. In set theory sets can also be "subtracted" from each other, resulting in complements. For example, B - A is equivalent to the set of elements that are members of B but not A.
From the above foundations, most of mathematics is derived. Nearly all mathematical systems contain properties that can be described fundamentally in terms of set theory.
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OpenSim 3.3 Documentation
The topics covered in this section include:
Laboratory Coordinates
Every set of (x, y, z) coordinates obtained from a motion capture system is given relative to some coordinate system. Typically, this coordinate system is called the laboratory coordinate system, or
simply laboratory coordinates. The laboratory coordinate system is generally an inertial frame fixed to the Earth. Before inputting any coordinates from motion capture into OpenSim, it is your
responsibility to ensure that all (x, y, z) coordinates have been transformed from the laboratory coordinate system to the model coordinate system used in OpenSim. Although you can define an
arbitrary model coordinate system, the standard convention used in OpenSim is as follows: Assume that the model is a full-body musculoskeletal model of the human body, standing in an upright position
on the ground. The origin of the model coordinate system is halfway between its feet. The x-axis of the model coordinate system points forward from the model, the y-axis points upward, and the z-axis
points to the right of the model.
If all positions and distances are converted to meters, then all (x, y, z) coordinates can be mapped from the laboratory coordinate system to the model coordinate system by an orthonormal
transformation. An orthonormal transformation can be represented by a 3 X 3 rotation matrix whose rows (and columns) are a set of orthogonal vectors of length one,. This matrix represents the
orientation of the laboratory coordinate frame in the model coordinate frame. So, to transform the coordinates of a point ^labP = (x, y, z) given in the laboratory coordinate frame to its
coordinates ^modelP = (x', y', z') in the model coordinate frame, you would employ the following transformation, where ^model[lab]R is the matrix whose columns are the vectors of the laboratory
coordinate frame specified in the model coordinate frame:
^modelP = ^model[lab]R * ^labP
External forces and moments are usually given in the coordinate system of a particular force sensor, such as a force plate, which may be different than the laboratory coordinate system. In this case,
the force and moment data must be transformed from the appropriate force sensor's coordinate system to the model coordinate system.
OpenSim is supported by the Mobilize Center , an NIH Biomedical Technology Resource Center (grant P41 EB027060); the Restore Center , an NIH-funded Medical Rehabilitation Research Resource Network
Center (grant P2C HD101913); and the Wu Tsai Human Performance Alliance through the Joe and Clara Tsai Foundation. See the People page for a list of the many people who have contributed to the
OpenSim project over the years. ©2010-2024 OpenSim. All rights reserved.
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Why Did the Music Teacher Need a Ladder for Math Worksheet Answers?
Why did the music teacher need a ladder for a math worksheet answers?
The music teacher needed a ladder for a math worksheet to demonstrate the concept of intervals or distances between musical notes. By using a ladder as a visual aid, the teacher could represent the
steps or jumps between notes in a tangible and relatable way, helping students grasp the mathematical aspect of music theory.
Explaining the Use of the Ladder for Math Worksheet Answers
Intervals and Distances in Music Theory: In music theory, intervals refer to the distances or gaps between two different musical notes. These intervals can be measured in terms of half steps or whole
steps. To help students understand this concept visually, the music teacher used a ladder as a metaphorical representation of the intervals. Visual Representation with the Ladder: By assigning each
rung of the ladder to a specific note and demonstrating the movement between the rungs, the teacher could illustrate the concept of intervals in a concrete manner. Students could physically see and
understand how the distance between notes corresponds to the steps on the ladder. This visual aid would enhance their comprehension of the mathematical relationships between musical notes and
contribute to their overall understanding of music theory.
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there are 2 portal frames ;- (3 member,same dimensions , same properties ).one has both fixed supports at column ends and the other has both hinged supports at column ends. From foundation point of vi
In the case of fixed bases, there will be bending moments at the column bases, which have to be accounted for in the foundation design. In the case of hinged bases, there will be no bending moments
transmitted to the foundations, whose design will be simpler (but higher moments will develop at the beam column junctions, compared to the fixed base case).
Whether it is appropriate to model the base as hinged or fixed depends on the soil conditions and type of foundations! It would be inappropriate, for example to assume hinged supports when the
foundation is resting on hard strata. But if the foundation is on relatively soft strata, even a small rotation at the base can release any assumed fixed end moment...
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sourCEntral -
complex − basics of complex mathematics
#include <complex.h>
Complex numbers are numbers of the form z = a+b*i, where a and b are real numbers and i = sqrt(−1), so that i*i = −1.
There are other ways to represent that number. The pair (a,b) of real numbers may be viewed as a point in the plane, given by X- and Y-coordinates. This same point may also be described by giving the
pair of real numbers (r,phi), where r is the distance to the origin O, and phi the angle between the X-axis and the line Oz. Now z = r*exp(i*phi) = r*(cos(phi)+i*sin(phi)).
The basic operations are defined on z = a+b*i and w = c+d*i as:
addition: z+w = (a+c) + (b+d)*i
multiplication: z*w = (a*c − b*d) + (a*d + b*c)*i
division: z/w = ((a*c + b*d)/(c*c + d*d)) + ((b*c − a*d)/(c*c + d*d))*i
Nearly all math function have a complex counterpart but there are some complex-only functions.
Your C-compiler can work with complex numbers if it supports the C99 standard. Link with −lm. The imaginary unit is represented by I.
/* check that exp(i * pi) == −1 */
#include <math.h> /* for atan */
#include <stdio.h>
#include <complex.h>
double pi = 4 * atan(1.0);
double complex z = cexp(I * pi);
printf("%f + %f * i\n", creal(z), cimag(z));
cabs(3), cacos(3), cacosh(3), carg(3), casin(3), casinh(3), catan(3), catanh(3), ccos(3), ccosh(3), cerf(3), cexp(3), cexp2(3), cimag(3), clog(3), clog10(3), clog2(3), conj(3), cpow(3), cproj(3),
creal(3), csin(3), csinh(3), csqrt(3), ctan(3), ctanh(3)
This page is part of release 4.09 of the Linux man-pages project. A description of the project, information about reporting bugs, and the latest version of this page, can be found at https://
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Grid Reference Utilities
In many countries, there are local grid reference systems used by local mapping, allowing easy grid references to be used instead of latitude/longitude. These script libraries provide ways to convert
these into their latitude/longitude equivalents. In addition, they provide extra functionality for grid reference systems used in the British Isles, offering ways to convert them into different
formats. They can also perform transforms between various different latitude/longitude Earth models, including those used outside the British Isles.
The mathematics used to convert between grid references and latitude/longitude (known as spherical or geodetic coordinates), and between different Earth models is extremely complicated. Minor
mistakes can easily become a source of major errors. A great deal of care must be taken to follow the conversion algorithms precisely, and that is what these scripts do for you. Using these scripts
allows you to simplify your own scripts, so they can concentrate on using the resulting values.
The scripts also offer methods devoted to geodesics; measuring distances between points, and calculating positions based on geodetic vectors. As well as map projections, there are methods to convert
between other types of geographic coordinate systems, such as between spherical and cartesian. There are also methods for geoids; adjusting altitudes based on local gravitational differences.
There are two companion scripts, one for PHP 5.0.3+, and one for JavaScript, allowing you to use them on both the server and client side. The APIs offered by the scripts are almost entirely
compatible with each other, receiving compatible inputs and producing compatible outputs. The scripts can be used either separately or together, to provide both client and server-side interaction
To download the script(s), and see the script license, use the links on the navigation panel at the top of this page.
Sample conversions
Take this Irish grid reference (pointing at approximately the middle of Lough Neagh in Northern Ireland):
That can also be expressed as (assuming truncation):
Which is equivalent to these Irish Transverse Mercator coordinates:
E 702528m, N 874441m
Which is roughly equivalent to this point on the Irish modified Airy 1830 Earth model:
54.6077240961°, -6.4119822285°
Which can also be expressed as:
54°36'27.806746"N, 6°24'43.136023"W
(or 54°36.46344576'N, 6°24.71893371'W)
Which is roughly equivalent to these GPS coordinates:
54.6078031260°, -6.4129391413°
(also expressed as 54°36'28.091254"N, 6°24'46.580909"W)
Which is equivalent to these UTM coordinates:
29U 667082 6054225
(also expressed as 667082mE, 6054225mN, Zone 29, Northern Hemisphere)
(Which cannot be represented in UPS, since the UPS grid does not extend this far.)
Which is roughly equivalent to this point on the UK Airy 1830 Earth Model:
54.6077391122°, -6.4119726199°
(also expressed as 54°36'27.860804"N, 6°24'43.101432"W)
Which is equivalent to these UK grid coordinates (rounded):
Which can also be expressed as this UK grid reference (truncated):
The JavaScript version can also be used to compute values without needing to reload the page. Test it here by modifying the contents of any of the boxes below:
Irish grid reference:
or with high precision
Irish grid coordinates:
or with 6 dp precision
ITM coordinates:
Point on the Irish modified Airy 1830 Earth model:
in DD format
in DD format with units
in DM format
in DMS format
GPS coordinates:
in DD format
in DD format with units
in DM format
in DMS format
UTM coordinates:
UPS coordinates:
with latitude bands
with hemisphere
Point on the UK Airy 1830 Earth model:
in DD format
in DD format with units
in DM format
in DMS format
UK grid coordinates:
or with 6 dp precision
UK grid reference:
or with high precision
If your browser supports geolocation, you could even feed the script your (a toolbar or dialog may appear, normally at the top of the page, asking for permission).
My High Accuracy WGS84 GPS To British OSGB36 Grid Reference Conversion tool performs similar conversions (based on Britain only) using the higher accuracy methods provided by these libraries.
These scripts were written by the author of the How To Create website; details at the bottom of the page. Script license is given on the sidebar at the top of the page. The algorithms and formulae
used are publicly published methods by various geodesy agencies and authors, and links are given throughout the code.
These scripts are normally configured to use polynomial transformation for conversion to Irish Grid, and can optionally use the OSTN15 transformation to British National Grid, and OSGM15 geoid data
for both British and Irish height datums. Coefficients and data for these transformations are available under BSD License. If you make use of these specific conversions within your software, then
like this page, your software must include the following copyright statement: "© Copyright and database rights Ordnance Survey Limited 2016, © Crown copyright and and database rights Land & Property
Services 2016 and/or © Ordnance Survey Ireland, 2016. All rights reserved.".
If you want to avoid having to include that copyright notice (you will still have to follow the How To Create license terms), you can choose to use Helmert transformations instead of OSTN15 and Irish
Grid polynomial transformation, and public geoid data instead of OSGM15. The results will not be as accurate.
A note on conversion accuracy
Various stages of the conversions operate on floating point numbers. There are limitations to how computers handle floating point numbers, and some accuracy and precision is lost at various stages of
the conversions. The overall effect of this is quite minimal (unless you start feeding the scripts coordinates that lie vast distances outside the expected grid areas), but do not expect the accuracy
to be completely perfect. There will also be occasional minor differences between the accuracies of the PHP and JavaScript versions of the script. You can choose to have the scripts return very high
precision results, but it is important to note that the precision can easily exceed the accuracy of the conversion methods themselves, and most definitely can exceed the measurement accuracy of most
GPS devices.
When converting between British grid references and GPS coordinates, or between different Earth models, errors will be introduced by the conversion algorithms. The most basic conversion is done using
Helmert transformations (one of the types of conversion suggested by Ordnance Survey), and these are known to be imperfect, due to the imperfect nature of Earth's own shape. The Helmert
transformations between the Ordnance Survey or Ordnance Survey Ireland and GPS Earth models have an error of up to ±5 metres horizontally and vertically. These errors are cumulative, so converting
from the OS to OSI model via the GPS model can produce an error of up to ±10 metres. Conversions between other models will depend on the accuracy of their ellipsoids, and the Earth's own shape at
that point.
These scripts can also do much more accurate conversions, including the polynomial transformation to the Irish Grid with an accuracy of ±0.4 m for 95% of coordinates, and the OSTN15 transformation to
British National Grid. OSTN15 is the method that defines the National Grid itself, when measurements are taken using the OS Net base stations, and is therefore considered to have the perfect accuracy
for grid transformation (traditional surveying accuracy is within ±0.1 m, when tested against OSTN15). Polynomial transformation is done by default when converting to Irish Grid.
Note that even when performing high accuracy conversions, the scripts cannot take continental drift into account. Though normally small enough to be below the accuracy for a GPS device, continental
drift does change the GPS coordinates for a specific point slowly over time; about 1 m per 40 years in Europe. In 2020, the error is a little under 1 metre. Differential GPS services in the British
Isles (and much of Europe) are almost always tied to ETRS89 rather than GPS's normal WGS84. Conversion from ETRS89 coordinates to British National Grid can be done without any substantial error
(floating point precision error is the only error worth mentioning) using OSTN15. Conversion between WGS84 and ETRS89 requires external data sources that keep track of the changes over time. While
these scripts can help prepare numbers for use with those external services, they cannot determine the continental drift themselves.
Supported grid formats
The scripts are designed to work with map grid formats used by normal maps and mapping software in the British Isles, while having an API to enable conversions to compatible map grids. In addition,
they support the most popular global grid reference formats. This means that by default, they support the British National Grid, the Irish Grid, the Irish Transverse Mercator, Universal Transverse
Mercator, Universal Polar Stereographic, GPS coordinates, coordinates for the Earth models used by the UK and Irish national grids, and geoids govering the British Isles and the World.
Example uses
Converting between grid references and latitude/longitude GPS coordinates
This is made extremely easy by the scripts, using just two method calls. The first converts the grid reference into numeric coordinates, and the second converts those coordinates into longitude/
$grutoolbox = Grid_Ref_Utils::toolbox();
$uk_grid_reference = 'NN 16667 71285';
//convert to a numeric reference
$uk_grid_numbers = $grutoolbox->get_UK_grid_nums($uk_grid_reference);
//convert to global latitude/longitude
$gps_coords = $grutoolbox->grid_to_lat_long($uk_grid_numbers,$grutoolbox->COORDS_GPS_UK,$grutoolbox->HTML);
var grutoolbox = gridRefUtilsToolbox();
var ukGridReference = 'NN 16667 71285';
//convert to a numeric reference
var ukGridNumbers = grutoolbox.getUKGridNums(ukGridReference);
//convert to global latitude/longitude
var gpsCoords = grutoolbox.gridToLatLong(ukGridNumbers,grutoolbox.COORDS_GPS_UK,grutoolbox.HTML);
With UTM or UPS, this is just a single method call. You could optionally also use dd_to_dms/ddToDms to convert it into the desired format.
To perform the reverse conversion, use the lat_long_to_grid/latLongToGrid then get_UK_grid_ref/getUKGridRef. To work with heights, this must be done in two steps, covered below in the section on
Helmert transformations.
Converting between UK and Irish grid references
The UK national grid reference system and the Irish national grid reference system do not line up conveniently at all, and although there are significant overlaps between them in various places,
there is normally no easy way to translate grid references from one to the other. Within Northern Ireland, for example, grid references are usually expressed using the Irish national grid reference
system. Using this library, it is easily possible to obtain the grid reference in the other system:
$grutoolbox = Grid_Ref_Utils::toolbox();
$irish_grid_reference = 'J0259874444';
//convert to a numeric reference
$irish_grid_numbers = $grutoolbox->get_Irish_grid_nums($irish_grid_reference);
//convert to global latitude/longitude
$gps_coords = $grutoolbox->grid_to_lat_long($irish_grid_numbers,$grutoolbox->COORDS_GPS_IRISH);
//convert to UK numeric reference
$uk_grid_numbers = $grutoolbox->lat_long_to_grid($gps_coords,$grutoolbox->COORDS_GPS_UK);
//convert to UK grid reference
print $grutoolbox->get_UK_grid_ref($uk_grid_numbers,5,$grutoolbox->HTML);
var grutoolbox = gridRefUtilsToolbox();
var irishGridReference = 'J0259874444';
//convert to a numeric reference
var irishGridNumbers = grutoolbox.getIrishGridNums(irishGridReference);
//convert to global latitude/longitude
var gpsCoords = grutoolbox.gridToLatLong(irishGridNumbers,grutoolbox.COORDS_GPS_IRISH);
//convert to UK numeric reference
var ukGridNumbers = grutoolbox.latLongToGrid(gpsCoords,grutoolbox.COORDS_GPS_UK);
//convert to UK grid reference
textNode.nodeValue = grutoolbox.getUKGridRef(ukGridNumbers,5,grutoolbox.TEXT);
A similar approach is used in reverse, or to convert between Irish Transverse Mercator and Irish national grid coordinates. The Irish Transverse Mercator coordinates can either be specified directly
as numeric coordinates (so they do not need the extra conversion step), or can be converted into an alternative format using add_grid_units/addGridUnits. Converting between most different grids is
done in this way, including UTM and UPS (for the small parts where they overlap), by converting the first reference to GPS coordinates, then converting from GPS coordinates to the other grid
reference system.
Converting between Earth models using Helmert transformation
Conversions between different Earth models are only a little harder, as you have to select the appropriate source and destination ellipsoids, and transformation set. You will only need to use this if
you are converting from another mapping system which does not use the WGS84/GRS80 Earth model (the scripts automatically apply it for you when converting between the UK/Irish grids and GPS
coordinates). It is possible to create your own transformation sets, but to make life easier, the scripts come with transformation sets for the following transforms:
Transformation sets available by default
Conversion From Via To
UK to GPS Airy_1830 OSGB36_to_WGS84 WGS84
GPS to UK WGS84 WGS84_to_OSGB36 Airy_1830
Irish to GPS Airy_1830_mod Ireland65_to_WGS84 WGS84
GPS to Irish WGS84 WGS84_to_Ireland65 Airy_1830_mod
Note that GPS software typically uses the WGS84 Earth model, which is a refined version of the GRS80 Earth model. The scripts use the values from WGS84, but you could also choose another ellipsoid,
such as the GRS80 Earth model used by Europe's ETRS89. In practice, this makes no significant difference to the resulting values, given the accuracy of the transforms (the practical difference
between the two ellipsoids is only about 0.15 mm). Some other ellipsoids produce substantially different coordinates, since they may be different sizes (to better represent the local shape of the
Earth), or have different meridians (0 degree longitude).
For example, to convert between the Airy 1830 Earth model used by Ordnance Survey to the modified Airy 1830 Earth model used by Ordnance Survey Ireland, you can use the following code:
$grutoolbox = Grid_Ref_Utils::toolbox();
$source_coords = Array(54.607720,-6.411990);
//get the ellipsoids that will be used
$Airy_1830 = $grutoolbox->get_ellipsoid('Airy_1830');
$WGS84 = $grutoolbox->get_ellipsoid('WGS84');
$Airy_1830_mod = $grutoolbox->get_ellipsoid('Airy_1830_mod');
//get the transform parameters that will be used
$UK_to_GPS = $grutoolbox->get_transformation('OSGB36_to_WGS84');
$GPS_to_Ireland = $grutoolbox->get_transformation('WGS84_to_Ireland65');
//convert to GPS coordinates
$gps_coords = $grutoolbox->Helmert_transform($source_coords,$Airy_1830,$UK_to_GPS,$WGS84);
//convert to destination coordinates
print $grutoolbox->Helmert_transform($source_coords,$WGS84,$GPS_to_Ireland,$Airy_1830_mod,$grutoolbox->HTML);
var grutoolbox = gridRefUtilsToolbox();
var sourceCoords = [54.607720,-6.411990];
//get the ellipsoids that will be used
var Airy1830 = grutoolbox.getEllipsoid('Airy_1830');
var WGS84 = grutoolbox.getEllipsoid('WGS84');
var Airy1830Mod = grutoolbox.getEllipsoid('Airy_1830_mod');
//get the transform parameters that will be used
var UKToGPS = grutoolbox.getTransformation('OSGB36_to_WGS84');
var GPSToIreland = grutoolbox.getTransformation('WGS84_to_Ireland65');
//convert to GPS coordinates
var gpsCoords = grutoolbox.HelmertTransform(sourceCoords,Airy1830,UKToGPS,WGS84);
//convert to destination coordinates
element.innerHTML = grutoolbox.HelmertTransform(sourceCoords,WGS84,GPSToIreland,Airy1830Mod,grutoolbox.HTML);
It is worth noting that when converting between grid references and latitude/longitude GPS coordinates, heights are ignored completely, since the conversion formulae do not use them. If you wanted to
convert heights at the same time, the conversions must be done in two steps, rather than using the COORDS_GPS_UK shortcut (which actually applies a Helmert transformation as well). The conversion
between grid coordinates and latitude/longitude must be done using the correct Earth model for the grid coordinates, and a Helmert transformation can then be used to convert between that Earth model
and the GPS Earth model, supplying the height during the Helmert transformation step. The conversion below is shown forwards in PHP, and in reverse in JavaScript:
$grutoolbox = Grid_Ref_Utils::toolbox();
$uk_grid_reference = 'NN 16667 71285';
$uk_height = 1345;
//convert to a numeric reference
$uk_grid_numbers = $grutoolbox->get_UK_grid_nums($uk_grid_reference);
//convert to global latitude/longitude
$uk_coords = $grutoolbox->grid_to_lat_long($uk_grid_numbers,$grutoolbox->COORDS_OS_UK);
//get the ellipsoids that will be used
$Airy_1830 = $grutoolbox->get_ellipsoid('Airy_1830');
$WGS84 = $grutoolbox->get_ellipsoid('WGS84');
//get the transform parameters that will be used
$UK_to_GPS = $grutoolbox->get_transformation('OSGB36_to_WGS84');
//convert to destination coordinates
$gps_coords = $grutoolbox->Helmert_transform($uk_coords,$uk_height,$Airy_1830,$UK_to_GPS,$WGS84);
//convert to degrees-minutes-seconds
print $grutoolbox->dd_to_dms($gps_coords,$grutoolbox->HTML) . ' height ' . $gps_coords[2];
//or it can just output the coordinates and height directly
print $grutoolbox->Helmert_transform($uk_coords,$uk_height,$Airy_1830,$UK_to_GPS,$WGS84,$grutoolbox->HTML);
var grutoolbox = gridRefUtilsToolbox();
var gpsString = '54°36\'27.792000"N, 6°24\'43.164000"W';
var gpsHeight = 1345;
//convert to decimal degrees
var gpsCoords = grutoolbox.dmsToDd(gpsString);
//get the ellipsoids that will be used
var WGS84 = grutoolbox.get_ellipsoid('WGS84');
var Airy1830 = grutoolbox.get_ellipsoid('Airy_1830');
//get the transform parameters that will be used
var GPSToUK = grutoolbox.get_transformation('WGS84_to_OSGB36');
//convert to destination coordinates
var ukCoords = grutoolbox.HelmertTransform(gpsCoords,gpsHeight,WGS84,GPSToUK,Airy1830);
var ukGridNumbers = grutoolbox.latLongToGrid(gpsCoords,grutoolbox.COORDS_OS_UK);
//convert to grid reference
textNode.nodeValue = grutoolbox.getUKGridRef(ukGridNumbers,5,grutoolbox.TEXT) + ' height ' + ukCoords[2];
Converting from other map systems
A large number of local mapping systems have been developed to serve particular countries or areas. In many cases these use the same approach as the mapping systems these scripts already deal with.
They begin with a model of the Earth that attempts to reflect its ellipsoidal shape at the target location. They then use a Transverse Mercator projection (or polar stereographic projection) of the
ellipsoid in order to produce the map. The map is aligned to one specific part of the Earth model (the true origin), and then usually offset to some other false grid origin, and scaled slightly using
a scale factor. This is the datum of that map's projection.
You would need to work out (if needed) the distance from any grid reference on that grid system to its false origin. So if the maps use anything like myriad letters, you need to convert from that
format to simple distances from the false origin - easting and northing coordinates. You then need to obtain the ellipsoid parameters used by the Earth model. Feed the ellipsoid parameters into the
create_ellipsoid/createEllipsoid method to get an ellipsoid parameter set (if you are lucky enough to find that the map uses one of the Earth models already provided by the scripts, you can just use
get_ellipsoid/getEllipsoid method to retrieve the appropriate ellipsoid parameter set). You then need to obtain the datum parameters used by the map projection (mentioned above). Feed all of those,
along with the ellipsoid parameter set, into the create_datum/createDatum method to get a datum parameter set.
Then feed your numbers into either grid_to_lat_long/gridToLatLong or polar_to_lat_long/polarToLatLong (depending on what type of projection the map system uses), along with your custom datum
parameter set, and it will return the geodetic coordinates of the point on that Earth model. If the mapping system uses the same Earth model as GPS (WGS84 or GRS80), then there is nothing more to do.
If it uses a different ellipsoid, you need to also get the Helmert transformation parameters for converting between that Earth model and the WGS84/GRS80 Earth model. Feed those into
create_transformation/createTransformation to get a custom Helmert transformation parameter set. Feed your geodetic coordinates, custom Earth model, Helmert transformation parameter set and the
script's existing Earth model for WGS84 as the destination Earth model, into Helmert_transform/HelmertTransform, and it will return GPS coordinates for the point.
A similar process works in reverse. That may all sound complicated, but the details of Earth models and datums for most mapping systems are easily available, as are the Helmert transformation
parameters, and these scripts take care of most of the rest for you. An example in JavaScript:
var grutoolbox = gridRefUtilsToolbox();
var sourceCoords = [12345,67890];
//get the ellipsoids that will be used
var earthModel = grutoolbox.createEllipsoid(6378206.4,6356583.8);
var datum = grutoolbox.createDatum(earthModel,0.99987,100000,200000,20,119);
var WGS84 = grutoolbox.getEllipsoid('WGS84');
//get the transform parameters that will be used
var modelToGPS = grutoolbox.createTransformation(-8,160,176,0,0,0,0);
//convert from grid coordinates to geodetic coordinates
var geoCoords = grutoolbox.gridToLatLong(sourceCoords,datum);
//convert to GPS coordinates
var gpsCoords = grutoolbox.HelmertTransform(geoCoords,earthModel,modelToGPS,WGS84);
And an example using a polar stereographic projection in PHP:
$grutoolbox = Grid_Ref_Utils::toolbox();
$source_coords = Array(12345,67890);
$earth_model = $grutoolbox->create_ellipsoid(6378206.4,6356583.8);
$datum = $grutoolbox->create_datum($earth_model,0.99987,100000,200000,20,119);
$WGS84 = $grutoolbox->get_ellipsoid('WGS84');
$model_to_gps = $grutoolbox->create_transformation(-8,160,176,0,0,0,0);
//convert from polar grid coordinates to geodetic coordinates
$geo_coords = $grutoolbox->polar_to_lat_long($source_coords,$datum);
$gps_coords = $grutoolbox->Helmert_transform($geo_coords,$earth_model,$model_to_gps,$WGS84);
Of course, it is possible that you need to convert from a mapping system that does not use a Transverse Mercator projection or basic polar stereographic projection. In these cases, you will need to
either find another script that supports that mapping system, or write one yourself. I can recommend the EPSG guides, in particular guide 7's "Coordinate Conversions and Transformations including
Formulas", as it has a very comprehensive listing of different mapping systems, and the algorithms needed to convert them to other coordinate systems.
Cartesian coordinates
GPS systems normally show their coordinates in latitude and longitude format. However, behind the scenes, they actually calculated the position using Cartesian X,Y,Z coordinates. These give the
location of a point in metres (normally) from the centre of the ellipsoid/planet. X is the distance towards the prime meridian of the ellipsoid's datum (towards 0°), Y is the distance towards the 90°
meridian, and Z is the distance towards the North Pole. This makes it very easy to perform calculations like the direct line of sight distance between two points, since the distances can be directly
related to each other.
At all points around the globe, physical distances are measures with the same resolution. This contrasts with latitude and longitude, where longitude bunches up at the poles such that moving a few cm
changes your longitude by 180°, while at the equator it would take 20000 km to change longitude by that much. However, it can seem very odd that moving horizontally across the Earth causes a shift in
not just X and Y, but Z too, and the coordinates therefore can be difficult to comprehend. Several tools may operate in X,Y,Z coordinates, so these scripts provide methods to easily convert between
them and latitude,longitude.
The conversion relies on knowing how big the ellipsoid is, in order to convert from coordinates based only on degrees, to coordinates that are given in metres. Therefore it needs to know which
ellipsoid your original coordinates relate to. For regular GPS, this is WGS84. For differential GPS in Europe, it is normally ETRS89. The resulting XYZ coordinates also relate to that same ellipsoid,
but only the datum part which says where the prime meridian is located, and therefore which direction X relates to.
$grutoolbox = Grid_Ref_Utils::toolbox();
$WGS84 = $grutoolbox->get_ellipsoid('WGS84');
$source_coords = Array(54.607720,-6.411990);
//height above the ellipsoid
$source_height = 66.341;
//convert to X,Y,Z
$xyz = $grutoolbox->lat_long_to_xyz($source_coords,$source_height,$WGS84);
//do something with the results
$some_other_xyz = ... etc. ...;
//convert back to latitude,longitude
$final_coords = $grutoolbox->xyz_to_lat_long($some_other_xyz,$WGS84);
var grutoolbox = gridRefUtilsToolbox();
var WGS84 = grutoolbox.getEllipsoid('WGS84');
var sourceCoords = [54.607720,-6.411990];
//height above the ellipsoid
var sourceHeight = 66.341;
//convert to X,Y,Z
var xyz = grutoolbox.latLongToXyz(sourceCoords,sourceHeight,WGS84);
//do something with the results
var someOtherXyz = ... etc. ...;
//convert back to latitude,longitude
finalCoords = grutoolbox.xyzToLatLong(someOtherXyz,WGS84);
Polynomial transformation
Polynomial transformation is a superior alternative to Helmert transformation. It can be used to transform latitude,longitude coordinates from one ellipsoid to another, but at the same time, it can
take into account the way that gravity varies over an area and the distortions that occur within it, which is something that a Helmert transformation cannot do. Distortions occur because of changes
in the density of the Earth's mantle over distances, and also because of the additional mass of nearby mountains. These can cause an area to to be level according to gravity, while a GPS might see it
as being a slope. The result is that horizontal distances, which is what mapping cares about, can be more or less than the distance around an ellipsoid. In reality, these differences produce a few
metres of error in various directions across in the British Isles.
The more coefficients that can be used, the more distortion fluctuations that can be represented. The scripts support third order polynomials, with 16 coefficient pairs. The primary use is to convert
to Irish Grid, where the coefficients represent 183 different fluctuations that have been measured across the island. The conversion does not need to know which ellipsoid the coordinates relate to,
but you need to make sure you are using them to convert in the correct direction from one ellipsoid to another.
The following example uses the built-in coefficients to translate from Irish grid references to ETRS89 coordinates using PHP. This is can be done much more simply by using the COORDS_GPS_IRISH type
with grid_to_lat_long/gridToLatLong, which uses polynomial transformation automatically. However, it is shown here expanded into a few steps so you can see how the process works in cases where you
might need to use the intermediate values. Of course, most GPS systems give results in WGS84 not ETRS89, so there will always be the continental drift error if those coordinates are converted
directly. However, a 40 cm + 79 cm error (in 2020) is still far better than the 5 m + 79 cm error when using Helmert transformation. Note that if your software uses the built-in 'OSiLPS'
coefficients, then you will need to check the copyright section above.
$grutoolbox = Grid_Ref_Utils::toolbox();
//convert Irish Grid reference to modified Airy 1830 coordinates
$irish_grid_reference = 'J0259874444';
$irish_grid_numbers = $grutoolbox->get_Irish_grid_nums($irish_grid_reference);
$ma1830_coords = $grutoolbox->grid_to_lat_long($irish_grid_numbers,$grutoolbox->COORDS_OSI);
//convert modified Airy 1830 coordinates to ETRS89 coordinates
$coefficients = $grutoolbox->get_polynomial_coefficients('OSiLPS');
$etrs89_coordinates = $grutoolbox->polynomial_transform($ma1830_coords,$coefficients);
This JavaScript example creates a custom set of third order polynomial coefficients, and converts in reverse using them. In this case, these coefficients might be used to apply or remove a
correctional shift without transforming it between ellipsoids:
var grutoolbox = gridRefUtilsToolbox();
var sourceCoords = [54.607720,-6.411990];
//define coefficients
var coefficients = grutoolbox.createPolynomialCoefficients(
//shift the coordinates
var shiftedCoords = grutoolbox.reversePolynomialTransform(sourceCoords,coefficients);
It is worth noting that polynomial transformation ignores heights completely. With the Irish Grid, accurate conversion of heights is done using geoids instead, or they can be approximated with
Helmert transformations.
Basic geodesics
Geodesics serve two purposes. The first is to work out where you would end up if you travelled in a given direction from a point for a certain distance around the curve of the Earth. The second is to
work out the distance and direction between two points around the curve of the Earth. They are based only on horizontal distance around an ellipsoidal Earth model, and ignore heights. The scripts
make this extremely easy.
The following example for PHP shows how to work out where you would end up after travelling from a point. If working with GPS coordinates, you will want the WGS84 ellipsoid.
$grutoolbox = Grid_Ref_Utils::toolbox();
$startpoint = array( 54.6078031260, -6.4129391413 );
print $grutoolbox->get_geodesic_destination( $startpoint, 1270.119, 137.24, $grutoolbox->get_ellipsoid('WGS84'), $grutoolbox->HTML );
The following example for JavaScript works out the distance and azimuth between two points:
var grutoolbox = gridRefUtilsToolbox();
var startpoint = [ 54.6078031260, -6.4129391413 ];
var endpoint = [ 51.884008219444446, -3.4364607166666667 ];
document.write( grutoolbox.getGeodesic( startpoint, endpoint, grutoolbox.getEllipsoid('WGS84'), grutoolbox.HTML ) );
Geodesics have certain limitations, and there are also ways they can be used to solve more complex problems. These are discussed in more detail in the more about geodesics section below.
Accepting input from users
Some of the methods provided by the scripts can accept string input, which can be used to interpret user-submitted input, which may or may not be in a valid format. In all cases, these methods accept
an extra parameter (called deny_bad_*), which can be used to determine if the script was able to recognise it as the given coordinate type, and if not, try another one.
Note that there are some cases where dms_to_dd/dmsToDd may return a result, when it may be more appropriate to interpret the string as generic numeric grid reference, such as where the string
contains '<number> ,<number> ' (with whitespace after the numbers) or '<number>m, <number>m'. As a result, the following example checks if it is a generic numeric grid reference first. Similarly, it
may also accept '1U, 123 456 ' (with a trailing space), even though that may be more appropriately interpreted as a UTM grid reference, so that check is also placed beforehand.
If you are expecting UPS grid references as input, you can also use ups_to_lat_long/upsToLatLong to detect those. However, the normal UPS format is identical to the Irish grid format, with the only
exception being that within the normal range, the numbers are usually two pairs of at least 6 digits, while with Irish grid references, they are normally only upto 5 digits. To detect this
difference, the ups_to_lat_long/upsToLatLong method accepts an additional min_length parameter, which causes the method to only accept a string as a UPS grid reference if the easting and northing are
at least 800000 (a number slightly below the minimum expected values within the UPS grids). This check must be made before the check for Irish grid references.
$coordsstring = $_GET['coords'];
if( function_exists('get_magic_quotes_gpc') && get_magic_quotes_gpc() ) { $coordsstring = stripslashes($coordsstring); }
$grutoolbox = Grid_Ref_Utils::toolbox();
if( $coords = $grutoolbox->parse_grid_nums($coordsstring,$grutoolbox->DATA_ARRAY,true) ) {
//it was interpreted as a generic numeric grid reference
//assume it was ITM and convert to lat/long
$coords = $grutoolbox->grid_to_lat_long($coords,$grutoolbox->COORDS_GPS_ITM);
} elseif( $coords = $grutoolbox->utm_to_lat_long($coordsstring,null,$grutoolbox->DATA_ARRAY,true) ) {
//it was interpreted as a UTM grid reference
//the method converts it to GPS coordinates; nothing else is needed
} elseif( $coords = $grutoolbox->ups_to_lat_long($coordsstring,$grutoolbox->DATA_ARRAY,true,true) ) {
//it was interpreted as a UPS grid reference
//the method converts it to GPS coordinates; nothing else is needed
} elseif( $coords = $grutoolbox->dms_to_dd($coordsstring,$grutoolbox->DATA_ARRAY,true) ) {
//it was interpreted as latitude/longitude coordinates
//assume it was GPS coordinates; nothing else is needed
} elseif( $coords = $grutoolbox->get_UK_grid_nums($coordsstring,$grutoolbox->DATA_ARRAY,true) ) {
//it was interpreted as a UK grid reference - convert to lat/long
$coords = $grutoolbox->grid_to_lat_long($coords,$grutoolbox->COORDS_GPS_UK);
} elseif( $coords = $grutoolbox->get_Irish_grid_nums($coordsstring,$grutoolbox->DATA_ARRAY,true) ) {
//it was interpreted as an Irish grid reference - convert to lat/long
$coords = $grutoolbox->grid_to_lat_long($coords,$grutoolbox->COORDS_GPS_IRISH);
} else {
//failed to recognise it as any format at all
var coordsString = document.getElementById('coordinateinput').value;
var coords, grutoolbox = gridRefUtilsToolbox();
if( coords = grutoolbox.parseGridNums(coordsString,grutoolbox.DATA_ARRAY,true) ) {
//it was interpreted as a generic numeric grid reference
//assume it was ITM and convert to lat/long
coords = grutoolbox.gridToLatLong(coords,grutoolbox.COORDS_GPS_ITM);
} else if( coords = grutoolbox.utmToLatLong(coordsString,null,grutoolbox.DATA_ARRAY,true) ) {
//it was interpreted as a UTM grid reference
//the method converts it to GPS coordinates; nothing else is needed
} else if( coords = grutoolbox.upsToLatLong(coordsString,grutoolbox.DATA_ARRAY,true,true) ) {
//it was interpreted as a UPS grid reference
//the method converts it to GPS coordinates; nothing else is needed
} else if( coords = grutoolbox.dmsToDd(coordsString,grutoolbox.DATA_ARRAY,true) ) {
//it was interpreted as latitude/longitude coordinates
//assume it was GPS coordinates; nothing else is needed
} else if( coords = grutoolbox.getUKGridNums(coordsString,grutoolbox.DATA_ARRAY,true) ) {
//it was interpreted as a UK grid reference - convert to lat/long
coords = grutoolbox.gridToLatLong(coords,grutoolbox.COORDS_GPS_UK);
} else if( coords = grutoolbox.getIrishGridNums(coordsString,grutoolbox.DATA_ARRAY,true) ) {
//it was interpreted as an Irish grid reference - convert to lat/long
coords = grutoolbox.gridToLatLong(coords,grutoolbox.COORDS_GPS_IRISH);
} else {
//failed to recognise it as any format at all
Try it here; input any one of the formats mentioned above:
You will need JavaScript support to test this script.
Note that deny_bad_* does not enforce a very strict restriction on the format to ensure that it is a perfectly valid sequence. There are cases where you could feed invalid data to the method, and it
will accept it, if it can interpret it as a potential match for the given type of data that it was expecting. For example, the string '796.345( 5632.222.g N, 999 E' is not a valid format for latitude
/longitude, but the scripts will be able to extract meaning from it - deny_bad_coords will therefore not cause the method to return false in that case. The purpose of deny_bad_* is to help your
script make sense out of the many potential types of data that users may feed the scripts, not to complain to them if they don't give a perfect string format.
The parse_grid_nums/parseGridNums method is an exception here, as it has a strict_nums parameter that allows you to force it to accept only a limited number of formats. This exists only for the
purpose of determining if a set of coordinates are given with units that make them most likely to be ITM grid coordinates, or if the coordinates are simply generic coordinates that could apply to any
of the numeric grid reference formats. You can choose to use this functionality at your own discretion.
Likewise, the dms_to_dd/dmsToDd method has the allow_unitless parameter that makes it accept simple number pairs, separated by a comma. These number pairs cannot be distinguished from grid
coordinates, so if you want to allow both, you will need to try parsing as grid coordinates first, and reject it if the grid coordinates are below 180, or negative. It is important to note, however
that '56.78,12.34' is valid for both grid coordinates and GPS coordinates, so you will need to be careful about which numbers will be considered valid for each of them. Note that dms_to_dd/dmsToDd
will accept any numbers, even those too large to be valid, and will wrap the longitude around to the right range. This means it cannot be used to detect whether the numbers are too large.
Users may still input values that extend far beyond the expected grids. While the scripts will force longitude to wrap-around when converting from Transverse Mercator or polar stereographic grid to
latitude/longitude, they do not attempt to prevent out-of-bounds latitudes, and will return whatever values the conversion algorithms produce. This can lead to unexpectedly high GPS coordinates, for
example, potentially enough to travel several quadrillion times around the World. You may wish to check the returned values are within the expected limits.
The exceptions here are shift_grid/shiftGrid, reverse_shift_grid/reverseShiftGrid and apply_geoid/applyGeoid, which also have a deny_bad_coordinates parameter. In these cases, it will refuse to
accept coordinates which are outside the grid, because there is no data covering that area. This allows the situation to be detected, so that warnings can be shown. The methods will also return false
if the data could not be loaded, for whatever reason.
Similarly, lat_long_to_utm/latLongToUtm and lat_long_to_ups/latLongToUps have a deny_bad_coordinates parameter to allow you to detect when the coordinates are outside the coordinate system region, so
you can switch to the other system instead. These coordinate systems are designed to work together. UTM can represent coordinates (badly) inside the polar regions using only a band letter, but UPS
cannot represent coordinates outside the polar regions at all. In this case, the parameter is not used for detecting invalid input strings.
The Earth's shape is not perfect, and gravity is influenced both by the mass of the crust, and the density of the mantle. Mountains have a significant affect on gravity and increase it noticeably,
but the mantle has the most substantial effect. Diverging tectonic plate boundaries increase gravity the most, with the strongest effects being near Iceland and Indonesia. The most significant low
points are in the World's oceans, with the Indian Ocean and Sri Lanka having the lowest gravity of any part of the planet. The effect of this is to alter the mean sea level at that point, with the
strongest gravity raising the sea level significantly. Ignoring tides, the sea level at Sri Lanka is 192 metres lower than nearby in Indonesia, compared with what would be expected on a perfectly
smooth spheroid. This is why in the UK, a GPS height is about 50 metres higher than the local mapping (orthometric) height; sea level in the UK is 50 metres higher than the World average. It also
means that almost all mountains are actually shorter compared with the surrounding terrain in orthometric height than they are in ellipsoidal height.
A geoid represents the vertical gravity effect over an area of the planet. A geoid can be thought of as the sea level, if the land were ignored, tides were removed, and ocean currents were stopped.
By subtracting the geoid undulation (the difference between the geoid height and the ellipsoid height), you can get an idea of the local terrain height. It still normally will not take the
fluctuations caused by tidal currents and ocean currents into account, so it is not perfect, but still better than nothing. It gives a better idea of whether you would feel like you were walking
uphill, and whether streams would flow in one direction or the other, while ellipsoid height can give the wrong impression (within the UK, the ellipsoid height would suggest that rivers cannot slowly
flow west, but the Rivers Severn and Clyde would disagree). The OSGM15 geoid is actually calibrated to use the local mapping sea level datums, so it does give real local mapping heights.
Geoids are calculated by precise gravimetric scans of the area, typically using satellites. These are then normally converted into mathematical formulas. These or the original data can be used to
export a grid of data points covering the area. A common file format for these data points is the .grd (grid) file format, and these scripts are designed to work with software that interprets the
.grd file format. These files can be very large. EGM96, a very popular global geoid, has a low density version called ww15mgh.grd, which is 9 MB. Higher density data covering just Ireland is just
over 1 MB.
The scripts expect data to be provided by an external database, with your own scripts communicating with the database and providing the data when needed. The data is expected to be provided in a
regular latitude-longitude grid of points, typically supplied by the .grd file format, and the scripts will ask for whichever data points they need for the current conversion. Firstly, the scripts
need to know the specification for the grid, which can all be found in a .grd file's header. When calling the apply_geoid/applyGeoid method, it must be given a record fetching callback function which
it can use to request data. If it needs data, it will pass the callback function an array of the data points it needs, giving the latitude, longitude, latitude index, longitude index, and record
index of the data point within the grid. This allows the record fetching callback function to choose whichever method it would like, to identify the data points to return. The callback must return
the data points that it got, in an array of geoid grd data point records; for that, it will need the data point value and the record index, for each point.
Processing the .grd file is fairly simple due to it being a very basic format. The first line contains 6 numbers which are used to make the geoid grd specification. Every line after that contains
several data points, which can be assigned an increasing record index starting from 0, beginning at the northwest corner of the grid, and progressing eastwards, one row at a time. If you wanted to
work with latitude and longitude indexes, or actual coordinates, the first 6 numbers are the minimum latitude, maximum latitude, minimum longitude, maximum longitude, latitude spacing and longitude
spacing. It seems like fairly simple mathematics to work out how many latitude rows there are, and how many longitude columns (watch out for the fencepost problem). However, note that files may use
truncated recurring numbers as the spacing, and so the numbers may not divide perfectly, and you need to work out the intent rather than the actual specified numbers. If it helps, I have also
released a .grd file parser for PHP, which can help to populate your database, and help you to visualise the data.
The following example shows how to get from ellipsoid height to geoid height, using the predefined EGM96_ww15mgh geoid:
function load_geoid_records( $name, $records ) {
global $grutoolbox;
$whereclause = array();
foreach( $records as $onerecord ) {
//this example database uses recordindex
$whereclause[] = "recordindex = '" . $onerecord['recordindex'] . "'";
$whereclause = implode( ' OR ', $whereclause );
$results = database_query_command( "SELECT * FROM `" . addslashes($name) . "` WHERE $whereclause" );
if( database_row_count($results) ) {
$resultarray = array();
while( $row = database_fetch_array($results) ) {
$resultarray[] = $grutoolbox->create_geoid_record( $row['recordindex'], $row['shift'] );
return $resultarray;
} else {
//database data did not match the expected grid
return array();
print $grutoolbox->apply_geoid( 54.6078, -6.4129, 66.341, true, $grutoolbox->get_geoid_grd('EGM96_ww15mgh'), 'load_geoid_records', $grutoolbox->HTML );
The scripts will cache every returned value automatically, so that subsequent requests that use the same data points will not need to load the data dynamically. It is also possible to load parts of
the data first, create geoid grd data point records for them, and use cache_geoid_records/cacheGeoidRecords to cache them. Once data has been cached, it continues to use up memory, which can become a
problem if large amounts of data gets loaded. If this is likely to be an issue, the delete_geoid_cache/deleteGeoidCache method may be used to delete the cached data for one geoid or all geoids.
Because the files are generally too large to transmit in full as Web page data, it is expected that JavaScript will need to use dynamic requests to a server, and as such, it will probably need to
operate asynchronously. The JavaScript version of the script therefore offers a return callback feature that can be used to run the code asynchronously if needed. If it is not provided, the scripts
will operate synchronously and return the values directly, just like the PHP version. Synchronous operation may be suitable for very small geoid grids, for JavaScript applications, for scripts that
cache the data first, or for scripts that run in a JavaScript (Web) worker using synchronous XMLHttpRequest to load the results. Note that while it is currently possible to use synchronous
XMLHttpRequest from a normal JavaScript thread, it is strongly discouraged as it locks up the page while waiting for a response, and browsers may stop supporting it in future.
If the return callback function is provided, then the method will call that function with the result instead, and asynchronous operation will be enabled. The record fetching callback function also
gets given a third parameter; a database callback function. When the results have loaded, the database callback function must be called with the results, instead of returning the results directly.
Make sure that you do not delete the geoid cache until the final result has been returned, since it will be operating asynchronously and may still need to use the cache. (You may note that this
relies on callbacks instead of promises, async functions or the convenient await syntax. This allows the script to work with popular legacy browsers such as Internet Explorer.)
Basically, for synchronous operation, call the method, and it returns the results. For asynchronous operation, call the method and pass it a callback function. In response, it will pass you a
callback function when it asks for data. You use that callback to pass back the data, and it will deliver the results to your callback.
The following example shows how to use both synchronous and asynchronous operation with JavaScript. The functions can also be implemented as anonymous functions, if needed:
function loadGeoidRecordsSynchronous( name, records ) {
//implement this dtabase connection yourself
... do something to get the data synchronously ...
... parse the response, such as JSON.parse ...
var newarray = [];
... for each record in the results ...
newarray.push( grutoolbox.createGeoidRecord( oneResult.index, oneResult.shift ); );
... then ...
//synchronous operation, return directly
return newarray;
var result = grutoolbox.applyGeoid(
54.6078, -6.4129, 66.341,
function loadGeoidRecordsAsynchronous( name, records, dataCallback ) {
//implement this dtabase connection yourself
...do something to get the data asynchronously, such as asynchronous XMLHttpRequest ...
dataFetcher.onload = function (responseText) {
... parse the response, such as JSON.parse ...
var newarray = [];
... for each record in the results ...
newarray.push( grutoolbox.createGeoidRecord( oneResult.index, oneResult.shift ); );
... then ...
//asynchronous operation, use the callback
function handleResults( result ) {
... data has been processed, use the result ...
... geoid cache can be deleted if needed ...
var anotherResult = grutoolbox.applyGeoid(
54.6078, -6.4129, 66.341,
Bilinear interpolated grid shifting
This is the highest accuracy method for shifting coordinates between earth models, taking into account all of the distortions caused by gravity fluctuations, and optionally applying geoid corrections
at the same time. Unlike polynomial transformation, it has no limits to how many different corrections it can apply, and it can be given as much or as little detail as needed, to cater to the
specific needs of a particular situation. Grid shifts can be used to convert between one idealised Earth model to another, or to correct inaccuracies in an Earth model, or do both at once.
The primary use for this is the simultaneous OSTN15 and OSGM15 transformation used by the British National Grid, to convert between ETRS89 GPS coordinates and OSGB36 National Grid references and
orthometric heights, bypassing the need to convert to Airy 1830 coordinates first. This transformation is so accurate that it is now used along with the OS Net GNSS base stations to define the
National Grid itself. When tested against land-based surveying techniques, there was found to be a 10 cm RMS distortion, most of which comes down to limitations in land based surveying techniques.
This means that when you start with ETRS89 coordinates, and convert to OSGB36 using OSTN15, no approximations are used in the transformation itself, and the transformation is 100% compatible with the
National Grid. Of course, your GPS readings can still have their own precision errors or accuracy errors. The grid corrections are given at a resolution of one data point per km. The data is
available as part of the Ordnance Survey OSTN15 developer pack, in a 40 MB file called OSTN15_OSGM15_DataFile.txt, which is in CSV format. It is perhaps best to use a proper database to store that
data, due to its size, and due to the speed at which a database can retrieve results without using too much memory. If it helps, I have also released an OSTN15 data file parser, which can help to
populate your database from the Ordnance Survey data file.
Just like when dealing with geoids, the scripts expect data to be provided by an external database, with your own scripts communicating with the database and providing the data when needed. The data
is expected to be provided in a regular easting,northing grid of points, and the scripts will ask for whichever data points they need for the current conversion. Firstly, the scripts need to know the
shift grid set; the specification for the grid. The OSTN15 shift grid set is provided by default, so you don't have to look up the details. When calling the shift_grid/shiftGrid and
reverse_shift_grid/reverse_shiftGrid methods, they must be given a record fetching callback function which it can use to request data. If they need data, they will pass the callback function an array
of the data points they need, giving the easting and northing for each. If the shift grid set specifies the number of columns, then the callback function will also be given the record index of the
data point within the grid, which the fetching callback can use to look up records. However, this is not mandatory, as a shift grid may have uneven numbers of columns, and cover an irregularly shaped
area. In all cases, the easting and northing will always be provided, and will be used internally to recognise data points. For this reason, shift grids should, if at all possible, use integers for
the coordinates and spacing - they are given to a resolution of 1 metre and it is very unlikely that a grid will need spacing that is not an exact number of metres. This is not mandatory though, and
it might work for some floating point numbers. However, it will almost certainly run into floating point precision problems recognising records, if you try to use grids with spacings like 0.333...
The callback must return the data points that it got, in an array of bilinear interpolation shift records; for that, it will need each data point's east, north and vertical shift values, as well as
their easting and northing.
When reversing the transformation, it is important to note that the positions of the data points relates to the unshifted coordinates, while the coordinates that you supply will be the shifted
coordinates. To reverse the transformation, the scripts have to try using the current position's shift, apply it, and then re-check the forward shift from that new position, until it finds the point
which shifts to the right position. During this process, it might need to load more and more records, if the shift takes it outside the original grid square. In the worst case, it might need to call
the record fetching callback function 4 times for different sets of coordinates (or more if the shifts are larger than the grid spacing), since it cannot predict this in advance. This will make it
slower than a forwards transformation.
To get from ETRS89 GPS coordinates to OSGB36 grid references, you need to firstly project the coordinates from latitude,longitude, into ETRS89 grid coordinates using the OSGB36 datum. This positions
them in the wrong place, since the two grids use different datums. Then you need to apply the OSTN15 shift to move the coordinates into the right location for the OSGB36 grid. To make this easier for
you, the scripts provide the ETRS89+OSGB36 datum as a datum called 'OSTN15'. The record fetching callback function can also return height datum information, which allows you to display a warning if
the shifting relied on more than one datum (in which case the height results can be nonsense). The following PHP example shows how to get from ETRS89 latitude and longitude to OSGB36 grid reference:
function load_shift_records( $name, $records ) {
global $grutoolbox;
$whereclause = array();
foreach( $records as $onerecord ) {
//this example database uses recordindex
//$whereclause[] = "easting = '" . $onerecord['easting'] . "' AND northing = '" . $onerecord['northing'] . "'";
$whereclause[] = "recordindex = '" . $onerecord['recordindex'] . "'";
$whereclause = implode( ' OR ', $whereclause );
$results = database_query_command( "SELECT * FROM `" . addslashes($name) . "` WHERE $whereclause" );
if( database_row_count($results) ) {
$resultarray = array();
while( $row = database_fetch_array($results) ) {
$resultarray[] = $grutoolbox->create_shift_record( $row['easting'], $row['northing'], $row['se'], $row['sn'], $row['sg'], $row['datum'] );
return $resultarray;
} else {
//database data did not match the expected grid
return array();
$etrs89_gps = array(51.8840136123,-3.4364538123);
$etrs89_height = 938.946;
$etrs89_grid = $grutoolbox->lat_long_to_grid( $etrs89_gps, $grutoolbox->get_datum('OSTN15'), $grutoolbox->DATA_ARRAY );
print $grutoolbox->shift_grid( $etrs89_grid, $etrs89_height, $grutoolbox->get_shift_set('OSTN15'), 'load_shift_records', $grutoolbox->HTML );
This would perform the reverse conversion, using the same load_shift_records function:
$osgb36_grid = array( 12345, 67890 );
$osgb36_height = 938.946;
$etrs89_grid = $grutoolbox->reverse_shift_grid( $os_grid, $osgb36_height, $grutoolbox->get_shift_set('OSTN15'), 'load_shift_records', $grutoolbox->DATA_ARRAY );
print $grutoolbox->grid_to_lat_long( $etrs89_grid, $grutoolbox->get_datum('OSTN15'), $grutoolbox->HTML );
print ', height ' . $etrs89_grid[2];
Just like with geoids, the scripts will cache every returned value automatically, so that subsequent requests that use the same data points will not need to load the data dynamically. It is also
possible to load parts of the data first, create bilinear interpolation shift records for them, and use cache_shift_records/cacheShiftRecords to cache them. Once data has been cached, it continues to
use up memory, which can become a problem if large amounts of data gets loaded. If this is likely to be an issue, the delete_shift_cache/deleteShiftCache method may be used to delete the cached data
for one shift grid or all shift grids.
Because the files are generally too large to transmit in full as Web page data, far larger than a typical geoid, JavaScript will almost certainly need to use dynamic requests to a server, and it will
probably need to operate asynchronously. The JavaScript version of the script therefore offers a return callback feature that can be used to run the code asynchronously if needed. If it is not
provided, the scripts will operate synchronously and return the values directly, just like the PHP version. Synchronous operation may be suitable for very small shift grids, for JavaScript
applications, for scripts that cache the data first, or for scripts that run in a JavaScript (Web) worker using synchronous XMLHttpRequest to load the results. Note that while it is currently
possible to use synchronous XMLHttpRequest from a normal JavaScript thread, it is strongly discouraged as it locks up the page while waiting for a response, and browsers may stop supporting it in
If the return callback function is provided, then the method will call that function with the result instead, and asynchronous operation will be enabled. The record fetching callback function also
gets given a third parameter; a database callback function. When the results have loaded, the database callback function must be called with the results, instead of returning the results directly.
Make sure that you do not delete the shift cache until the final result has been returned, since it will be operating asynchronously and may still need to use the cache. This is especially important
when reversing the shift transformation, as it will need to use the shift cache several times for each operation. (You may note that this relies on callbacks instead of promises, async functions or
the convenient await syntax. This allows the script to work with popular legacy browsers such as Internet Explorer.)
Basically, for synchronous operation, call the method, and it returns the results. For asynchronous operation, call the method and pass it a callback function. In response, it will pass you a
callback function when it asks for data. You use that callback to pass back the data, and it will deliver the results to your callback.
The following example shows how to use both synchronous and asynchronous operation with JavaScript. The functions can also be implemented as anonymous functions, if needed:
function loadShiftRecordsSynchronous( name, records ) {
//implement this dtabase connection yourself
... do something to get the data synchronously ...
... parse the response, such as JSON.parse ...
var newarray = [];
... for each record in the results ...
newarray.push( grutoolbox.createShiftRecord( oneResult.easting, oneResult.northing, oneResult.eastshift, oneResult.northshift, oneResult.heightshift, oneResult.datum ); );
... then ...
//synchronous operation, return directly
return newarray;
var etrs89Gps = [51.8840136123,-3.4364538123];
var etrs89Height = 938.946;
var etrsGrid = grutoolbox.latLongToGrid( etrs89Gps, grutoolbox.getDatum('OSTN15'), grutoolbox.DATA_ARRAY );
var result = grutoolbox.shiftGrid(
etrs89Gps, etrs89Height,
function loadShiftRecordsAsynchronous( name, records, dataCallback ) {
//implement this dtabase connection yourself
...do something to get the data asynchronously, such as asynchronous XMLHttpRequest ...
dataFetcher.onload = function (responseText) {
... parse the response, such as JSON.parse ...
var newarray = [];
... for each record in the results ...
newarray.push( grutoolbox.createShiftRecord( oneResult.easting, oneResult.northing, oneResult.eastshift, oneResult.northshift, oneResult.heightshift, oneResult.datum ); );
... then ...
//asynchronous operation, use the callback
function handleResults( result ) {
... data has been processed, use the result ...
var etrsLatLong = grutoolbox.gridToLatLong( etrs89Gps, grutoolbox.getDatum('OSTN15'), grutoolbox.DATA_ARRAY );
var etrsHeight = result[2];
... shift cache can be deleted if needed ...
var reversed = grutoolbox.reverseShiftGrid(
For those who are interested, the way that reverse iteration is possible both in synchronous and asynchronous operation, is to use a scoped function which calls itself recursively intead of using a
loop for iteration (this allows an asynchronous callback to run the scoped function after fetching data). This has the limitation that recursive functions can cause stack overflows. In this case, it
is limited to 100 iterations, and rarely uses more than 5. This is well within the stack sizes of even most legacy browsers - modern browsers have a stack size of several thousand.
More about geodesics
The scripts offer methods that can solve two problems: adding a geodetic vector to a point to calculate the resulting position, and finding the distance and bearing of the shortest route between two
points around the curve of the World - a geodesic. These are known as the Direct Problem and Inverse Problem respectively. There are several methods of doing each of these, some of which only work on
spheres, even though the World is better represented as a spheroid. The most complete approach (used by GeographicLib, if you wanted to try it) is a very complex and extremely lengthy one developed
by Charles Karney (2012) which uses several methods at once to obtain a nearly perfect result in all cases (it can also tell you where two vecors collide, and how far north a vector will travel).
However, there is a much more simple (relatively speaking) method that can work in the vast majority of cases, with minimal errors (typically less than 5 mm), designed to be used in places where
computation efficiency is important. That is what these scripts use.
Geodesics are calculated using a modified version of the Vincenty formulae, inspired by the works and modifications of Chris Veness, but reworked to cater to a few more situations. The modifications
are intended to cope with the less common situations where the Vincenty iteration does not converge, such as points on opposite sides of the World from each other, or points that are positioned
extremely close to each other. These are detected either by the the trigonometry functions returning values too small to be added to larger numbers (a limitation of floating point arithmetic), or by
the formulas requiring too many iterations.
The error handling for problematic coordinates tries to correct the returned distances and azimuths, with varying degrees of accuracy. For points on opposite longitudes from each other, or for points
where at least one is situated on a pole, it can calculate the correct azimuth. For points that are nearly opposite, it gives the most likely direction, but with a very tiny error (which normally
gets lost in rounding precision anyway). For points extremely close to each other, it returns a poor approximation of the azimuth between them (since at those tiny distances, trigonometry functions
cannot return anything useful), within 45°. However, this error gets worse closer to the poles, and is completely meaningless right beside the poles. This is really quite an unimportant correction
given that the distance between the points is so close to 0 that it gets rounded to 0 by the rounding precision anyway.
When error handling has been used, an extra parameter will be returned in any data arrays, saying how the problem was detected, the type of separation between the points, and the confidence in how
well it has approximated the distances and azimuths. The error handling approximates the returned distance, in many cases just giving the antipodal distance for points that are not quite antipodal,
or 0 for points that are very close to each other, so the mere presence of this return parameter indicates that the distance data is not entirely trustworthy. You can use this to display notices when
the distances may be inaccurate, or when the confidence in the azimuths is particularly bad. When returning text or HTML strings, this is signified with an approximation symbol in the returned
The azimuth - which is also known as a bearing in British English - is the direction of the geodetic vector as seen from the start or end points. At the poles, this may seem like nonsense because
every direction from the North Pole is southwards, and vice versa. However, with a geodesic, the azimuth will be the direction compared with the line of longitude which is given for a point. This
means that although 90°,0° and 90°,123° are the same point (both are the North Pole), they will produce different azimuths, as the first compares the direction with that of the prime meridian, and
the second compares the direction with that of the 123° meridian. For 90°,0°, an azimuth of 0° would continue "forwards" down the 180° meridian, while for 90°,123°, an azimuth of 123° would point
down the 180° meridian. For -90°,0°, an azimuth of 0° would point up the prime meridian. Polar longitude is relevant with geodesics.
The distance between two mountain tops
Geodetic distances are the distances around the ellipsoid being used, which is a rough approximation of the shape of the World, ignoring slopes and imperfections in the shape of the World. Since they
are being measured around a curve, the distances are further for points that are higher than that ellipsoid, either because they are high above sea level, or because they are on a part of the World
where sea level is higher than the ellipsoid being used. The distance of a geodesic must therefore be treated as only a rough approximation, and not the distance that you would need to travel in
order to follow it on the actual World's surface. Following the slopes and bumps of the surface would add a lot more distance, and a direct line of sight between points is shorter than the curved
distance between those same points. This should put that <= 5 mm error into perspective; no matter how perfect your formula is, the answer can never be more than a simple guide. Even for navigation
at sea where the surface is relatively flat and consistent, the ellipsoid is still only an approximation compared with a geoid (gravitational representation of the World), but there are no perfect
formulae for determining the shortest route between points on a geoid. Geodesics still have their uses - as long as you accept the limitations - such as approximating the length of a GPS track by
adding the geodesic distances between pairs of points, or approximating the shortest direction between distant islands.
Mountain tops are typically raised some distance above the WGS84 ellipsoid (some will be under it, depending on how poorly WGS84 represents the area). Assuming you know the height of your points
compared with the reference ellipsoid (such as a GPS altitude, which is the height above the WGS84 ellipsoid), you can use the height above the ellipsoid as a circle radius and a simple 2πr formula
to calculate how much further the distance should be over a full circle. For points 100 m above the ellipsoid, the difference over a full circle is 628 m. Assuming the World is a sphere, to get the
distance adjustment for your arc, you would need to multiply it by the length of your arc divided by the length of a full great circle. For two hills 10 km apart that rise 100 m above a sphere based
on the WGS84 major axis, the tops are over 15 cm further apart than the bases which sit perfectly on the sphere. This is how far an aircraft would have to fly to get between the tops if it maintained
a constant altitude above the sphere. The direct line of sight between the two hills is under a tenth of a millimetre longer than 10 km, yet another way to measure distances. To an aircraft, this
would feel like it was losing altitude for half the journey, then gaining altitude for the other half.
However, since the World is better represented as an oblate spheroid, the result will be slightly wrong, and the exact result should be different at different points and orientations on the
ellipsoid. It is, however, possible to get the distance by supplying a different ellipsoid. If the two points sit 100 m above the WGS84 ellipsoid, supply an ellipsoid with the major and minor axes
100 m larger than WGS84's. The resulting geodesic will be the correct length for the curved path between those points.
If the points are at different heights, then simple Pythagorean theorem (hypotenuse squared = adjacent squared + opposite squared) seems tempting to get the length of the diagonal between them, but
around a curve this needs an extra step. Work with the height half way between the points (rather than the higher or lower point's altitude) as the height of the new spheroid surface. The resulting
geodesic distance is the length of the adjacent, and the difference in height between the points is the opposite side.
It is also possible to work out the line of sight distance between the points, but not using geodesics. With each point, use lat_long_to_xyz/latLongToXYZ to get cartesian coordinates of the two
points. Then use three dimensional Pythagoras theorem to get the distance (square root of XDifference squared plus YDifference squared plus ZDifference squared). Of course, this will not actually
help to work out if the two points can actually be seen from each other around the curve of the World.
Great ellipse
It is easily possible to use the two geodesics methods to determine a great ellipse distance. If you are starting with two points, and you want a great ellipse that intersects both of them, use
get_geodesic/getGeodesic to get the initial azimuth. Once you have a start point and azimuth, use get_geodesic_destination/getGeodesicDestination to get the result of a geodetic vector in that
direction, of longer than half the distance around the ellipsoid. You can either specify the length 30000000 for most global ellipsoids, or for any oblate spheroid, a length of more than π times the
major axis radius (over half way around the equator) and less than 4 times the major axis radius (the shortest distance across an infinitely thin ellipsoid and back) will work perfectly. Then use
get_geodesic/getGeodesic to get from the resulting point back to the start of the ellipse. The initial length plus the final length add up to the length of the great ellipse.
$lat = 53;
$lon = -2;
$dir = 12;
$grutoolbox = Grid_Ref_Utils::toolbox();
$WGS84 = $grutoolbox->get_ellipsoid('WGS84');
$length1 = ( ( M_PI + 4 ) / 2 ) * max( $WGS84['a'], $WGS84['b'] );
$p2 = $grutoolbox->get_geodesic_destination( $lat, $lon, $length1, $dir, $WGS84, $grutoolbox->DATA_ARRAY );
$length2 = $grutoolbox->get_geodesic( $p2, $lat, $lon, $WGS84, $grutoolbox->DATA_ARRAY );
$greatellipselength = $length1 + $length2[0];
Changes in version 3.0, 23/12/2020
• Added support for cartesian ellipsoid coordinates
□ Added xyz_to_lat_long/xyzToLatLong and lat_long_to_xyz/latLongToXyz for converting between latitude,longitude and cartesian coordinates.
• Added support for OSTN15 and similar coordinate shifting systems
□ Added shift_grid/shiftGrid and reverse_shift_grid/reverseShiftGrid, for use with bilinear interpolation shifting.
□ Added get_shift_set/getShiftSet and create_shift_set/createShiftSet, for use with bilinear interpolation shifting.
□ Added create_shift_record/createShiftRecord, cache_shift_records/cacheShiftRecords and delete_shift_cache/deleteShiftCache, for use with bilinear interpolation shifting.
□ Added OSTN15 datum for transverse mercator projection.
□ Added OSTN15 shift set.
• Added support for .grd-based geoids such as Irish OSGM15 and global EGM96 ww15mgh.grd
□ Added apply_geoid/applyGeoid, for use with applying bilinear interpolated geoid heights.
□ Added get_geoid_grd/getGeoidGrd and create_geoid_grd/createGeoidGrd, for use with applying bilinear interpolated geoid heights.
□ Added create_geoid_record/createGeoidRecord, cache_geoid_records/cacheGeoidRecords and delete_geoid_cache/deleteGeoidCache, for use with bilinear interpolated geoid heights.
□ Added OSGM15_Belfast, OSGM15_Malin and EGM96_ww15mgh geoid grd sets.
• Added support for polynomial transformations
□ Added polynomial_transform/polynomialTransform and reverse_polynomial_transform/reversePolynomialTransform.
□ Added get_polynomial_coefficients/getPolynomialCoefficients and create_polynomial_coefficients/createPolynomialCoefficients.
□ Added OSiLPS polynomial coefficient set for converting Irish Grid to GPS.
□ grid_to_lat_long/gridToLatLong and lat_long_to_grid/latLongToGrid now use polynomial transformation when COORDS_GPS_IRISH is selected as the type parameter, with COORDS_GPS_IRISH_HELMERT
being available to use the less accurate Helmert transformation instead.
• Added support for geodesics - measuring distances between latitude,longitude points
□ Added get_geodesic/getGeodesic and get_geodesic_destination/getGeodesicDestination.
• Updated ellipsoids
□ Updated WGS84 ellipsoid to its most recent revision (0.1 mm difference at the poles, no effect on real usage).
□ Added GRS80 ellipsoid separately from the WGS84 ellipsoid
• Allowed floating point grid coordinates with HTML and text outputs
□ Added precise parameter to get_UK_grid_nums/getUKGridNums, get_Irish_grid_nums/getIrishGridNums, add_grid_units/addGridUnits, parse_grid_nums/parseGridNums, lat_long_to_grid/latLongToGrid,
lat_long_to_utm/latLongToUtm, lat_long_to_polar/latLongToPolar and lat_long_to_ups/latLongToUps to return floating point grid coordinates.
• Allowed more input formats
□ dms_to_dd/dmsToDd now recognises numbers with directions but not units.
□ Added allow_unitless parameter to dms_to_dd/dmsToDd, to accept number pairs.
□ dms_to_dd/dmsToDd now recognises multi-byte minute and second separators, to allow "smart quotes" (PHP only).
□ Helmert_transform/HelmertTransform now allows height to be specified separately when latitude and longitude are supplied as an array.
• Allowed UK and Irish grid references to use rounding or truncation
□ Added use_rounding parameter to get_UK_grid_ref/getUKGridRef, get_Irish_grid_ref/getIrishGridRef, get_UK_grid_nums/getUKGridNums and get_Irish_grid_nums/getIrishGridNums.
□ get_UK_grid_ref/getUKGridRef, get_Irish_grid_ref/getIrishGridRef, get_UK_grid_nums/getUKGridNums and get_Irish_grid_nums/getIrishGridNums now default to using truncation instead of rounding,
as officially specified.
• Made other outputs more useful
□ Increased precision of latitude and longitude strings to 10 decimal places for decimal degrees, and 8 decimal places for decimal minutes.
□ Added the $grutoolbox->TEXT_DEFAULT constant in PHP, which uses your default PHP encoding when outputting text.
• Bug fixes
□ Corrected handling of easting and northing with ups_to_lat_long/upsToLatLong.
□ parse_grid_nums now allows floats instead of integers when strict_nums is true, to match JavaScript version and documentation (PHP only).
□ utmToLatLong now returns false for invalid ellipsoids with invalid coordinates (JavaScript only).
□ Fixed a rounding issue that could cause dd_format/ddFormat to return 180°W instead of 180°E.
□ Fixed error handling with separate parameters with getUKGridNums (JavaScript only).
□ Corrected handling of invalid ellipsoid parameter with utm_to_lat_long/utmToLatLong and lat_long_to_utm/latLongToUtm.
□ Corrected handling of invalid coordinates and zones with upsToLatLong (JavaScript only).
□ Avoided a PHP decoding bug with TEXT_ASIAN output, so it should work on all installations now (PHP only).
□ Avoided a floating point limitation with very precise grid references in getUKGridNums and getIrishGridNums (JavaScript only).
□ Avoided -0 bug with Helmert_transform (PHP only).
□ add_grid_units/addGridUnits no longer rounds numbers when outputting data arrays (for consistency with other methods).
□ ITM projection now uses GRS80 ellipsoid instead of WGS84 ellipsoid (this makes no practical difference to any results except with really high precision ETRS89 coordinates), but is done for
the sake of being correct).
□ Corrected Airy 1830 ellipsoid and OSGB36 datum 'b' value to 6356256.909 (rounding error corrected in OS documentation).
• Other changes
□ get_ellipsoid/getEllipsoid, get_datum/getDatum, create_datum/createDatum and create_transformation/createTransformation now all return false instead of null in response to bad parameters, for
consistency with other methods.
□ 0.000000 no longer has its precision removed with latitude,longitude or height, for consistency with PHP and all other numbers (JavaScript only).
□ Helmert_transform/HelmertTransform now returns false for invalid ellipsoid or transformation parameter sets.
□ get_UK_grid_nums/getUKGridNums and get_Irish_grid_nums/getIrishGridNums can now deny additional letters when separate parameters are used.
□ parse_grid_nums/parseGridNums and dms_to_dd/dmsToDD can now detect and deny more error cases (such as numbers consisting only of decimal points).
□ Refactored common return values.
□ Conversion between radians and degrees is now done with inbuilt methods (PHP only).
□ Code style fixes.
□ Added a very extensive testsuite, to make sure everything works.
Changes in version 2.1.2 for PHP, 19/10/2020
• Corrected decimal degrees detection in dms_to_dd.
Changes in version 2.1.1 for PHP, 16/9/2020
• Corrected Helmert height handling.
Changes in version 2.1, 9/5/2010
• Added dd_format/ddFormat.
Changes in version 2.0 for PHP, 6/5/2010
• Released JavaScript companion script.
• Added support for Irish Transverse Mercator (ITM) grid references:
□ Added COORDS_GPS_ITM conversion type to lat_long_to_grid/grid_to_lat_long.
□ Added add_grid_units method.
□ parse_grid_nums now optionally recognises many more variations of numeric grid format, determined by the additional strict_nums parameter.
□ parse_grid_nums now optionally accepts a data array of coordinates, in the format produced by add_grid_units.
• Added support for Universal Transverse Mercator (UTM) grid references:
□ Added utm_to_lat_long and lat_long_to_utm methods.
• Added support for Universal Polar Stereographic (UPS) grid references:
□ Added ups_to_lat_long and lat_long_to_ups methods.
• Made it possible to convert from other local grid reference systems:
□ Added create_datum (and the related get_datum) to allow custom datums to be used.
□ Added polar_to_lat_long and lat_long_to_polar methods.
□ grid_to_lat_long and lat_long_to_grid now allow a custom datum.
□ grid_to_lat_long/lat_long_to_grid now reject invalid conversion types.
• Allowed grid references to be constructed and interpreted using floating point accuracy:
□ get_*_grid_ref, get_*_grid_nums, parse_grid_nums and grid_to_lat_long now accept floating point numbers as input.
□ get_*_grid_nums, parse_grid_nums and lat_long_to_grid now return floating point numbers for easting and northing when returning a data array, to allow you to construct more precise grid
□ get_*_grid_ref now accepts integers greater than 5 for figures, to make it return floating point numbers.
• Other enhancements:
□ Allowed methods to return text strings, in addition to the existing HTML strings and data arrays.
□ Added support for deny_bad_reference when passing an array to get_*_grid_nums.
□ Improved detection of invalid grid references in get_*_grid_nums.
• And lots of bug fixes:
□ Fixed a bug that caused grid_to_lat_long to produce big errors for some references.
□ Fixed a bug that caused grid_to_lat_long and lat_long_to_grid to produce wrapped-around coordinates when the grid overlaps the antimeridian.
□ Avoided a PHP bug that could cause -0 to be returned by some methods.
□ Corrected casing when passing an array to get_UK_grid_nums.
□ Corrected casing when passing a string to dms_to_dd.
□ Corrected grid numbers when passing negative out-of-grid coordinates to get_*_grid_ref.
• Moved API documentation onto its own dedicated Web page, and made it much more comprehensive.
Changes in version 1.1 and 1.2 for PHP
• Added only_dm parameter to dd_to_dms.
• Added deny_bad_reference parameter to get_*_grid_ref.
Changes in version 1.0 for PHP, 10/4/2010
API documentation
The API documentation has its own page.
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NEW FUNCTION FOR REPRESENTING IEC 61000-4-2 STANDARD ELECTROSTATIC DISCHARGE CURRENT
New function for representing electrostatic discharge (ESD) currents according to the IEC 61000-4-2 Standard current is proposed in this paper. Good agreement with the Standard defined parameters is
obtained. This function is compared to other functions from literature. Its first derivative needed for field calculations is analyzed in the paper. Main advantages are simplified choice of
parameters, possibility to obtain discontinuities in the decaying part, and zero value of the function first derivative at t=0+. Parametrs of the function are obtained by using Least-squares method
C. Paul, "Introduction to Electromagnetic Compatibility", ed. 2, John Wiley & Sons, 2006.
EMC – Part 4-2: Testing and Measurement Techniques – Electrostatic Discharge Immunity Test. IEC International Standard 61000-4-2, basic EMC publication, 1995+A1:1998+A2:2000.
EMC – Part 4-2: Testing and Measurement Techniques – Electrostatic Discharge Immunity Test. IEC International Standard 61000-4-2, Ed. 2, 2009.
EMC – Part 4-3: Testing and Measurement Techniques - Radiated Radio-Frequency Immunity Test. IEC International Standard 61000-4-3, Ed. 2, 2002.
EMC – Part 4-3: Testing and Measurement Techniques - Radiated Radio-Frequency Immunity Test. IEC International Standard 61000-4-3 (77B/339/FDIS), Ed. 3, 2006+A1:2007.
T. Ishida, G. Hedderich, "Recent Status of IEC 61000-4-2 and IEC 61000-4-3", EMC’09 Kyoto, 2009, pp. 821-824.
T. C. Moyer, R. Gensel, "Update on ESD testing according to IEC 61000-4-2", EM Test.
D. Pommerenke, M. Aidam, "ESD: waveform calculation, field and current of human and simulator ESD", J. Electrostat., No.38, 1996, pp. 33-51.
O. Fijuwara, H. Tanaka, Y. Yamanaka, "Equivalent circuit modeling of discharge current injected in contact with an ESD gun", Electr. Eng. Japan, No.149, 2004, pp. 8-14.
N. Murota, "Determination of characteristics of the discharge current by the human charge model ESD", Simulator Electron. Commun. Japan, No.80, 1997, pp. 49-57.
K. Wang, D. Pommerenke, R. Chundru, T. Van Doren, J. L. Drewniak, A. Shashindranath, "Numerical modeling of electrostatic discharge generators", IEEE Transactions on EMC, Vol.45, No.2, May 2003, pp.
S. V. Berghe, D. Zutter, "Study of ESD signal entry through coaxial cable shields", J. Electrostat., No.44, 1998, pp. 135-148.
Z. Yuan, T. Li, J. He, S. Chen, R. Zeng, "New mathematical descriptions of ESD current waveform based on the polynomial of pulse function", IEEE Transactions on EMC, Vol.48, No.3, 2006, pp. 589-591.
V. Javor, "Multi-Peaked Functions for Representation of Lightning Channel-Base Currents," 31st Int. Conf. on Lightning Protection ICLP 2012, Proc. of papers, DOI: 10.1109/ICLP.2012.6344384 ,Vienna,
V. Javor, "Approximation of a double-peaked lightning channel-base current", COMPEL: The Int. Journal for Comp. and Mathematics in Electrical and Electronic Engineering, Vol.31, No.3, 2012, pp.
G. P. Fotis, I. F. Ganos and I. A. Stathopulos "Determination of discharge current equation parameters of ESD using genetic algorithms," Electronic Letters, Vol.42, No.14, July 2006.
G. P. Fotis, F. E. Asimakopoulou, I. F. Ganos and I A. Stathopulos, "Applying genetic algorithms for the determination of the parameters of the electrostatic discharge current equation", Measurement
Science and Technology, Vol.17 (2006), pp. 2819-2827, 2006.
K. Lundengard, M. Rančić, V. Javor, S. Silvestrov, "Estimation of pulse function parameters for approximating measured lightning currents using the Marquard least-squares method", EMC Europe 2014,
2014, (accepted for publication).
R. K. Keenan, L. K. A. Rosi, "Some fundamental aspects of ESD testing", in Proc. IEEE Int. Symp. on Electromagnetic Compatibility, Aug. 12-16, 1991, pp. 236-241.
F. Heidler, “Travelling current source model for LEMP calcula¬tion,” in Proc. 6th Int. Zurich Symp. EMC, Zurich, Switzerland, pp. 157-162, Mar. 1985.
K. Wang, J. Wang, X. Wang, "Four order electrostatic discharge circuit model and its simulation", TELKOMNIKA, Vol.10, No.8, pp. 2006-2012, Dec. 2012.
G. P. Fotis, L. Ekonomou, "Parameters’ optimization of the electrostatic discharge current equation", Int. Journal on Power System Optimization., Vol.3, No.2, pp. 75-80, 2011.
S. Shenglin, B. Zengjun, L. Shange, "A new analytical expression of standard current waveform", High Power Laser and Particle Beams, Vol.15, No.5, pp. 464-466, 2003.
R. Chundru, D. Pommerenke, K. Wang, T. Van Doren, F. P. Centola, J. S. Huang "Characterization of human metal ESD reference discharge event and correlation of generator parameters to failure levels –
Part I: Reference event", IEEE Transactions on EMC, Vol.46, No.4, pp. 498-504, Nov. 2004.
V. Javor, "Modeling of lightning strokes using two-peaked channel-base currents", Int. Journal of Antennas and Propagation, Vol. 2012, Article ID 318417, doi: 10.1155/2012/318417, 2012.
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ISSN: 2217-5997 (Online)
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Cirq and A Simple Quantum Circuit
After discovering the world of quantum computers a few months ago I have been trying to take in as much information on them as possible. With the many free courses, Wikipedia pages, and dwave's leap
program I can now say that I have a basic understanding of quantum computers.
However, this article is geared more towards using Cirq. In the future, I may compile all of my notes and write a series on quantum computers and circuits, but for now, this is it.
What is Cirq
So you may be asking what is cirq? Well, cirq is a software library used to create quantum circuits that you can then use to solve complex problems(I have not gotten this far yet but will have
something on this in the near future).
Set Up
So first make sure you have python installed on your computer. If not then go to https://www.python.org/downloads/ and set it up for your system. Then once this is done go to your terminal and enter
I tried this with anaconda and pip3, but could only get it to work with pip. I did not research this issue, so if anyone was an idea as to why I am having this issue, please enlighten me.
Things You Should Know
Before we go any further let me explain a few things we will be using.
In a quantum circuit, there are various gates that increase or decrease the likelihood of the input(1 or 0) changing upon reaching its output. We are only going to use two in this article; the
Hadamar Gate or H gate and Pauli's X Gate or just the X gate.
The H gate increases the probability of the input changing to fifty percent. So if a qubit's (don't worry I will get to what this is next) initial state is 1 then there is a fifty percent chance it
will become a 0 once it reaches its output. The X gate just flips the state of the qubit(so 1 to 0, or 0 to 1).
At this point, you are probably wondering what a qubit is. Well to put it simply a qubit is short for a quantum bit, so like classical computer bits, it can have two states 1 or 0. However, since it
is at the quantum level qubits have an advantage over classical bits; superposition. This means that unlike classical bits which must switch between 1 and 0 a qubit can be 1, 0, or both. If you want
to read some more on this click here
Circuit Setup
Ok so with your new knowledge of qubits, and the two gates explained above lets set up a simple quantum circuit.
So first you need to import cirq, decide on the number of qubits, and create a new circuit.
import cirq
length = 3 #will produce length**2 qubits
qubits = [
cirq.GridQubit(i, j)
for i in range(length)
for j in range(length)
#This creates 9 grid qubits and gives
#them positions (i, j) in the circuit
circuit = cirq.Circuit() #creates a blank circuit
Then determine the gates that will be present with each qubit, and append them to the circuit
circuit.append(cirq.H(q) for q in qubits if (q.row + q.col) % 2 == 0)
#append an H gate at qubit q if even position
circuit.append(cirq.X(q) for q in qubits if (q.row + q.col) % 2 != 0)
#append a X gate at qubit q if odd position
(0, 0): ───H───
(0, 1): ───X───
(0, 2): ───H───
(1, 0): ───X───
(1, 1): ───H───
(1, 2): ───X───
(2, 0): ───H───
(2, 1): ───X───
(2, 2): ───H───
There you go, the output above is a basic quantum circuit. As you can imagine this can be extended to much larger circuits. I recently created something like this and will go through its creation and
purpose in a few weeks:
I hope you enjoyed this and I look forward to continuing writing on this subject in the future.
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Newtonian Quantum Gravity
1. Introduction
The quantum treatment of gravity has been researched for more than fifty years. The focus of the research efforts was the investigation into the concept of “gravitational quantum field”. This article
presents a new method that can solve this puzzle. Gravity is a phenomenon generated by the undulating nature of the universe. To support this claim, in this paper, I deduce a more generalized
expression than Newton’s law of gravity from simple considerations related to wave function, probability density function, and a hypothesis of how all particles in the universe are related. I also
consider space to be a three-dimensional surface of a sphere with radius equal to the age of the universe, so that the universe can be represented as a sphere of radius equal to the distance from the
observer to the current particle horizon, which can be approximately expressed as ${R}_{h}=\text{π}R$ , where R is the Hubble radius. I have used the basic concept of the probability function of a
particle and considered that the energy of a particle is distributed in space according to this function of probability density. The objective of this theory of “quantum gravity” is the
identification of the quantum substrate from which macroscopic gravity emerges [1] [2] and to pave the way for the complete quantum analysis of the gravitational phenomenon. Likewise, both the
problem of dark matter and dark energy may, in principle, be solved by extended theories of gravity [3] ; an example of this can be the results obtained.
The hypothesis of this work is that the wave function of each particle deforms the wave functions of all particles of the universe, thereby generating the gravitational phenomenon.
2. Methods
We define the probability $\Psi {\left(r,t\right)}^{2}=\frac{{\sum }_{i=1}^{N}\text{ }\text{ }{\phi }_{i}{\left(r,t\right)}^{2}}{N}$ , where N is the number of
particles in the entire universe and ${\phi }_{i}$ is the wave function of each particle. Let us now consider a closed spherical surface S such that the value of $\Psi {\left(r,t\right)}^{2}$ at all
points on that surface is the same, ${\phi }_{s}$ ; S is an equiprobable surface. The interior volume of that closed surface is ${V}_{s}$ . If we define a reference point 0, the value of ${V}_{s}$
will depend on ${\phi }_{s}$ and given that the particles in the universe are in perpetual motion, there will be a probability density flow entering and exiting through the surface S because of which
${V}_{s}$ and S will be time-dependent. Consider the situation prior to the detection of the particle, when the parameter t has the value ${t}_{0}$ . The particles do not always have a definite
energy, but irrespective of their energy and for the specific case that I discuss, I consider that the energy is distributed throughout the universe according to its probability density function. The
hypothesis of this work is that the probability density functions of the remaining particles affect the probability distribution of each of them so that the probability density function of a particle
includes the probability distribution $\Psi {\left(r,{t}_{0}\right)}^{2}$ defined above. We can call $\phi \left(r,{t}_{0}\right)$ as the wave function of the particle already deformed by $\Psi {\
left(r,{t}_{0}\right)}^{2}$ and ${\phi }_{0}\left(r,{t}_{0}\right)$ to the intrinsic part of the particle. Therefore, the probability density function of a particle can be given as.
$\phi {\left(r,{t}_{0}\right)}^{2}=\frac{1}{2}{\phi }_{0}{\left(r,{t}_{0}\right)}^{2}+\frac{1}{2}\text{Ψ}{\left(r,{t}_{0}\right)}^{2}.$ (1)
It is not easy to calculate this probability density function, $\text{Ψ}{\left(r,{t}_{0}\right)}^{2}$ , but by doing some simplification, it is feasible to calculate the total probability of finding
a particle inside ${V}_{s}$ ; this probability is denoted by ${P}_{s}$ . Because no particle in the universe is excluded, it can be calculated as the quotient obtained by dividing the energy
contained within the volume ${V}_{s}$ by the total energy of the universe; ${V}_{u}$ is the volume of the universe. This probability would also be the result of applying the volume integral of $\Psi
{\left(r,{t}_{0}\right)}^{2}$ throughout the volume ${V}_{s}$ . With $\rho \left(r\right)$ as the average energy density in ${V}_{s}$ , we obtain:
$\underset{{V}_{s}}{\overset{}{\int }}\text{ }\text{ }\Psi {\left(r,{t}_{0}\right)}^{2}\text{d}V=\frac{\rho \left(r\right){V}_{s}\left(r\right)}{\rho \left({R}_{h}\right){V}_{u}\left({R}_{h}\
right)}.$ (2)
${R}_{h}$ is the radius of the universe and $\rho \left({R}_{h}\right)$ is the average density of the universe. The Equation (2) is an expression of the probability of finding some particle inside $
{V}_{s}$ .
For a spherical surface of radius r and with centre at the centre of mass, which
is the point O, we have ${V}_{s}=\frac{4}{3}\text{π}{r}^{3}$ . Moreover, because the universe is consi-
dered a sphere of radius ${R}_{h}$ , (2) can be expressed as
$\underset{{V}_{s}}{\overset{}{\int }}\Psi {\left(r,{t}_{0}\right)}^{2}\text{d}V=\frac{\rho \left(r\right){r}^{3}}{\rho \left({R}_{h}\right){R}_{h}^{3}}.$ (3)
Applying the same reasoning to the area from the outer system to ${V}_{s}$ , we can arrive at a similar expression
$\underset{{V}_{os}}{\overset{}{\int }}\Psi {\left(r,{t}_{0}\right)}^{2}\text{d}V=\frac{\rho \left({R}_{h}-r\right)\left({R}_{h}^{3}-{r}^{3}\right)}{\rho \left({R}_{h}\right){R}_{h}^{3}}.$ (4)
where $\rho \left({R}_{h}-r\right)$ is the average density outside of S. This would be the probability of finding a particle outside S. If we assume that $r\ll {R}_{h}$ , then we can approximate $\
rho \left({R}_{h}-r\right)$ to $\rho \left({R}_{h}\right)$ and the expression for the whole universe can be given as
$\int \Psi {\left(r,{t}_{0}\right)}^{2}\text{d}V=\frac{\rho \left(r\right){r}^{3}}{\rho \left({R}_{h}\right){R}_{h}^{3}}+\frac{\left({R}_{h}^{3}-{r}^{3}\right)}{{R}_{h}^{3}}=1.$ (5)
This function, as defined above, is a sum of probabilities. Now, consider a particle that is free and at rest, i.e. the indeterminacy of r is of the order of ${R}_{h}$ ; its wave function is evenly
distributed throughout the universe. As mentioned previously, the hypothesis of this work is that the probability density functions of the particles affect the probability distribution of each of
them so that the probability density function of a particle includes the previously defined density function. All particles in the universe are related. Substituting in (1) the values of $\Psi {\left
(r,{t}_{0}\right)}^{2}$ from above, the expression for the probability density of the particle is obtained as
$\phi {\left(r,{t}_{0}\right)}^{2}=\frac{1}{2}\left({\phi }_{0}{\left(r,{t}_{0}\right)}^{2}+\frac{\rho \left(r\right){r}^{3}}{\rho \left({R}_{h}\right){R}_{h}^{3}}+1-\frac{{r}^{3}}{{R}_{h}^{3}}\
right).$ (6)
This would be valid for a free particle before being detected on surface S. The
normalizing constant of the complete wave function is $\frac{1}{2}$ . In the expression $\frac{\rho \left(r\right){r}^{3}}{\rho \left({R}_{h}\right){R}_{h}^{3}}$ , we can see a product of four
factors corresponding to four degrees of freedom of the system. The expression $\frac{{r}^{3}}{{R}_{h}^{3}}$ corresponds to the three degrees of freedom of a three-dimensional space. The factor $\
frac{\rho \left(r\right)}{\rho \left({R}_{h}\right)}$ is independent of the
previous degrees of freedom and depends on the history of the system; the density can vary over time. To analyze this phenomenon, we can study the collapse of the wave function only in that spatial
degree of freedom. The particle is detected at the distance r, e.g. on the surface of the sphere indicated above, with an
indeterminacy of $\Delta r$ ; the corresponding statistical factor, $\frac{r}{{R}_{h}}$ , is reduced to the factor $\frac{\Delta r}{\Delta {R}_{h}}$ . This new factor must meet two conditions to be
able to have a
defined measure. First, its value should be 1; given that there is certainty, this condition is fulfilled if $\Delta r=\Delta {R}_{h}$ , that is, the indeterminacy does not depend on the observer.
Second, both $\Delta r$ and $\Delta {R}_{h}$ should be greater than 0. Both conditions are assured by the principle of uncertainty. In this case,
$\Delta r{p}_{g}\ge \frac{\hslash }{2}.$ (7)
The momentum acquired by the particle is ${p}_{g}$ . After the collapse of the wave function, the probability distribution of the particle, with ${\phi }^{\prime }$ and ${{\phi }^{\prime }}_{0}$ as
the wave functions of the particle after detection, is as follows:
${\phi }^{\prime }{\left(r,t\right)}^{2}=\frac{1}{2}\left({{\phi }^{\prime }}_{0}{\left(r,t\right)}^{2}+\frac{\rho \left(r\right){r}^{2}}{\rho \left({R}_{h}\right){R}_{h}^{2}}+1-\frac{{r}^{2}}{{R}_
{h}^{2}}\right).$ (8)
This expression remains normalized. The $\frac{\rho \left(r\right){r}^{2}}{\rho \left({R}_{h}\right){R}_{h}^{2}}$ factor of the probability
density function maintains the degree of freedom due to the density factor and two additional spatial degrees of freedom. That is, neither the system knows the location of the particle (i.e. it
ignores the $\phi$ and $\theta$ coordinates) nor the particle knows how energy is distributed inside $V$ . From this and by applying the formalism of quantum mechanics to the left side of the
previous equality, we can obtain the total average energy of the particle, ${E}_{1}$ , which includes the average mechanical energy as the average kinetic energy due to the gravitational phenomenon $
{K}_{g}$ , that is, ${E}_{1}=m{c}^{2}+{K}_{g}$ . If we apply the same to the first term on
the right, we would obtain $\frac{1}{2}m{c}^{2}$ , where $m{c}^{2}$ is the average mechanical
energy of the particle. The second, third, and fourth terms in the second part of the equality are probability distributions. If, as we indicated at the beginning, the energy of the particle can be
considered distributed according to its probability density function, we would obtain:
${E}_{1}=\frac{1}{2}m{c}^{2}+\frac{1}{2}m{c}^{2}\frac{\rho \left(r\right){r}^{2}}{\rho \left({R}_{h}\right){R}_{h}^{2}}+\frac{1}{2}m{c}^{2}-\frac{1}{2}m{c}^{2}\frac{{r}^{2}}{{R}_{h}^{2}}.$ (9)
If the average mechanical energy $\left(\frac{1}{2}m{c}^{2}+\frac{1}{2}m{c}^{2}\right)$ is denoted by ${E}_{0}$ , we
obtain the following result:
${E}_{1}={E}_{0}+\frac{1}{2}m{c}^{2}\frac{\rho \left(r\right){r}^{2}}{\rho \left({R}_{h}\right){R}_{h}^{2}}-\frac{1}{2}m{c}^{2}\frac{{r}^{2}}{{R}_{h}^{2}},$ (10)
$m{c}^{2}=m{c}^{2}+{K}_{g}-\frac{1}{2}m{c}^{2}\frac{\rho \left(r\right){r}^{2}}{\rho \left({R}_{h}\right){R}_{h}^{2}}+\frac{1}{2}m{c}^{2}\frac{{r}^{2}}{{R}_{h}^{2}}.$ (11)
Finally, we obtain the energy balance for a particle linked to a gravitational system as
$0={K}_{g}-\frac{1}{2}m{c}^{2}\frac{\rho \left(r\right){r}^{2}}{\rho \left({R}_{h}\right){R}_{h}^{2}}+\frac{1}{2}m{c}^{2}\frac{{r}^{2}}{{R}_{h}^{2}},$
${K}_{g}=\frac{1}{2}m{c}^{2}\frac{{r}^{2}}{{R}_{h}^{2}}\left(\frac{\rho \left(r\right)}{\rho \left({R}_{h}\right)}-1\right).$ (12)
If we calculate ${K}_{g}$ for a sphere of radius $r$ and $M+{m}_{0}$ is its total internal energy, where ${m}_{0}$ is its component that contributes to $\rho \left({R}_{h}\right)$ , i.e.
${m}_{0}=\rho \left({R}_{h}\right)\frac{4}{3}\text{π}{r}^{3}$ , and assuming that $\rho \left({R}_{h}\right)$ is equal to the critical density ( $\frac{3{c}^{2}}{8\text{π}G{R}_{h}^{2}}$ ), then we
find that
${K}_{g}=\frac{GMm}{r}.$ (13)
Expression (12) is the kinetic energy of a free-falling particle in a gravitational system when it falls with initial velocity 0 from an infinite distance or ${R}_{h}$ . When it is at rest on the
surface of Earth, i.e. ${K}_{g}=0$ , the energy indicated in (13) would be as follows:
${E}_{p}=-\frac{GMm}{r}.$ (14)
If the particle has a kinetic energy not due to gravity, its energy can be expressed as
$E=K-\frac{GMm}{r}.$ (15)
As we shall see below, it is important to note that the contribution of the energy ${m}_{0}$ is not included in (13). In systems with high densities, this contribution is irrelevant; however, in
systems with small densities, the expression of gravity given in (12) does not coincide with Newton’s law. The force of gravity calculated with Newton’s law will always be greater than that obtained
with (13). For example, observations of the large-scale structure of the universe have given rise to the need to postulate the concept of dark energy because they are very low-density systems; the
application of formulation (13) instead of Newton’s law eliminates the need to postulate this concept of dark energy.
The expression $-\frac{1}{2}m{c}^{2}\frac{\rho \left(r\right){r}^{2}}{\rho \left({R}_{h}\right){R}_{h}^{2}}$ can be called the “gravitational term”; the expression $\frac{1}{2}m{c}^{2}\frac{{r}^{2}}
{{R}_{h}^{2}}$ can be called the “expansive term”. It must be considered
that the temporal evolution of the wave function indicated in (1) depends on the whole universe; therefore, the observed energy of the particle depends on how the whole universe evolves, both inside
S and outside of S. This study considers the Mach’s principle from this point of view. The acceleration of a particle “moves” the entire universe by modifying the probability density function of all
particles in the universe (see Section 3.1).
3. Results and Discussion
3.1. Force of Inertia and Mach’s Principle
The expansive term introduced above is equivalent to the energy of a simple
harmonic oscillator with an elastic constant of $\frac{m{c}^{2}}{{R}_{h}^{2}}$ . It does not depend on the
mass of the system, but only on the distance to the origin of coordinates and the mass of the particle. It is valid for any other frame of reference. This leads us to think that the particle is
attached to the rest of the universe by multitude of springs with the same elastic constant and different elongations. The particle is
confined at the centre of a scalar field of the form $V=\frac{1}{2}m{c}^{2}\frac{{r}^{2}}{{R}_{h}^{2}}$ . If the particle
is at rest or in uniform rectilinear motion, the force resulting from these springs is 0. Thus, just as the force represented by the weight is the reaction to the force due to the gravitational term,
the force of inertia can be identified as the reaction from the scalar field that binds the particle to the universe through any force that is applied on the particle. The centrifugal force must be
considered to be of the same nature as the gravitational force.
If a particle is accelerated from rest with an acceleration $a$ , it will have tra-
velled a distance $dr$ in time $dt$ , where $dr=\frac{1}{2}ad{t}^{2}$ . The force exerted by the
oscillators in the direction of the movement of the particle will be
$F=K\left(r-dr\right)$ , the force in the opposite direction is $F=-K\left(r+dr\right)$ , and the resultant of both will be $F=-2Kdr$ . If we calculate the sum of all the oscillators of both
hemispheres, we obtain the following result:
$F=-m4\text{π}\frac{{c}^{2}}{{R}_{h}}.$ (16)
If the expression $4\text{π}\frac{{c}^{2}}{{R}_{h}}$ is called ${a}_{0}$ , then for another acceleration, the displacement in the same time $dt$ will be $\Delta r$ so that $\frac{\Delta r}{dr}=\frac
{a}{{a}_{0}}$ ; consequently, we
can deduce that for any acceleration,
$F=-m4\text{π}\frac{{c}^{2}}{{R}_{h}}\frac{a}{{a}_{0}}.$ (17)
This expression is equivalent to Newton’s 2nd law. The force of inertia has the same nature as the gravitational force.
3.2. Gravitational Scalar Field Model
Consider the scalar field of the previous point $V\left(r\right)=\frac{1}{2}m\frac{{c}^{2}}{{R}_{h}^{2}}{r}^{2}$ . Now consider the fundamental oscillator with energy $\frac{1}{2}\frac{{c}^{2}}{{R}_
{h}^{2}}{r}^{2}$ , $m=1$ . The natural frequency of this elementary oscillator will be ${\omega }_{0}=\frac{c}{{R}_{h}}$ , its energy in the ground state will be ${E}_{0}=\frac{1}{2}\hslash {\omega }
_{0}$ , and the mass corresponding to that energy will be ${m}_{0}=\frac{1}{2}\frac{\hslash }{c{R}_{h}}$ ;
we could say that based on the known values of the constants, the value of the mass is approximately $4.3×{10}^{-70}\text{\hspace{0.17em}}\text{kg}$ . Its wavelength according to the expression,
${E}_{0}=\frac{1}{2}\hslash {\omega }_{0}$ , would be $2\text{π}{R}_{h}$ , its spatial indeterminacy is of the order of the par-
ticle horizon, that is to say, is at rest with respect to the universe. We can imagine that the universe is filled with these particles (henceforth called as gravitons) and I consider that they are
characterized by an elementary charge of mass at rest of value 1, which we can call the elementary charge of mass at rest. Their wavelength indicates that they would be standing waves whose main
nodes could be identified with the gravitons indicated. The interactions between them are given by the previous scalar field. Each graviton is connected to the rest by springs with the same constant
but with different elongations. The uniformity of the universe implies that the force of the springs located in one hemisphere exerting force at one point will be compensated for by the force exerted
by the springs of the other hemisphere so that, in principle, these springs would tend to be at rest. We have seen in the previous discussion that an external force exerted on some of them would
trigger a reaction from the rest, which we have called inertia.
These springs would tend to completely fill the universe due to the initial inequalities between the hemispheres, eliminating these inequalities. Let ${r}_{0}$ be called the average distance between
them at equilibrium. The force exerted be-
tween two of them separated by ${r}_{0}$ will be ${F}_{0}=\frac{{c}^{2}}{{R}_{h}^{2}}{r}_{0}$ . However, this force is
what a frame of reference anchored in one of them would observe; if we chose the mass centre as the origin of the reference frame (an inertial frame of refer-
ence), the force applied to each of them will be ${F}_{0}=\frac{1}{2}\frac{{c}^{2}}{{R}_{h}^{2}}{r}_{0}$ . From the above, it
can be said that the force of attraction between the gravitons, with respect to an
inertial frame of reference, is of the form $F=\frac{1}{2}\frac{{c}^{2}}{{R}_{h}^{2}}r$ . The average separation
distance between the adjacent gravitons will be ${r}_{0}$ and will have an average density of ${\rho }_{0}=1$ ; therefore, there would be a graviton in each cube of edge ${r}_{0}$ . If we identify
the universe as a 3-sphere, the gravitons will tend to cover the entire universe uniformly.
Let us now imagine a sphere with radius $n{r}_{0}$ . Within that sphere there is an excess of N free gravitons with centres of mass at the centre of the sphere. Symmetrical to this sphere and to its
right is another identical sphere but with gravitons corresponding to the average density ${\rho }_{0}$ ; in each of these spheres, there
will be $\frac{4}{3}\text{π}{n}^{3}{r}_{0}^{3}$ gravitons corresponding to this density. The density of the left sphere will be $\frac{\frac{4}{3}\text{π}{n}^{3}{r}_{0}^{3}+N}{\frac{4}{3}\text{π}{n}^
{3}{r}_{0}^{3}}$ . At the point of contact of both spheres, there is
another graviton called the test graviton. By choosing the mass centre of the complete system of the free N gravitons, the test graviton and the gravitons of both spheres corresponding to the density
${\rho }_{0}$ as the origin of the frame of reference (an inertial frame of reference), the acceleration induced by the gravitons of both spheres on the test graviton will be as follows:
$g=\frac{1}{2}\frac{{c}^{2}}{{R}_{h}^{2}}n{r}_{0}\left(\frac{\frac{\frac{4}{3}\text{π}{n}^{3}{r}_{0}^{3}+N}{\frac{4}{3}\text{π}{n}^{3}{r}_{0}^{3}}}{{\rho }_{0}}-1\right),$
$g=\frac{3{c}^{2}}{8\text{π}{R}_{h}^{2}{\rho }_{0}}\frac{N}{{n}^{2}{r}_{0}^{2}}.$ (18)
If $\frac{3{c}^{2}}{8\text{π}{R}_{h}^{2}{\rho }_{0}}$ is termed as G, then the above expression is similar to that for
Newton’s law of gravity except that in (18), the mass corresponding to the density-
${\rho }_{0}$ ( $\frac{4}{3}\text{π}{n}^{3}{r}_{0}^{3}$ ) is not considered, which can lead to errors for very low-density
systems. A particle model could be postulated such that a particle of mass at rest m could be identified with a stable set of N coordinated gravitons that move at the same time. As an example, it can
be stated that an electron would be a coordinated set of $2×{10}^{39}$ gravitons, which is approximately the same ratio as that of a hurricane to an air molecule.
It is important to say here that expression (12) has been applied to the system described here because, as indicated previously, neither the system of N gravitons knows where the test graviton is (it
ignores the $\phi$ and $\theta$ coordinates), nor does the test graviton know how the N gravitons are distributed within the sphere; that is, the gravitons have not been detected, and the
gravitational force is the effect of the interaction between the gravitons caused by their asymmetric
distribution. The expression $\frac{M}{{m}_{0}}$ is the only information we can have of the in-
ternal distribution of gravitons. If the anterior sphere is large enough so that its inner density is ${\rho }_{0}$ , the acceleration described in (18) would be null. This force between the free
system of N gravitons and the rest of the gravitons deforms the sphere, making it smaller until the force is balanced by that exerted by the rest of the gravitons in the universe. This deformation
can be identified as the deformation considered for space by the General Theory of Relativity (GTR); in fact, it would be possible to identify this network of gravitons with the space concept of the
GTR. The distance from which it seems that gravity does not cause a curvature in space is the distance from which it seems that dark energy accelerates the expansion. However, the reality is that
from that distance, the curvature is always null because the gravitons are in equilibrium. The photons can be identified by the excitations transmitted throughout this network of gravitons. I propose
that the true gravitational field is the scalar field indicated above,
$V\left(r\right)=\frac{1}{2}m\frac{{c}^{2}}{{R}_{h}^{2}}{r}^{2}$ , and the gravitational field which was considered until now is
a side effect of the unequal distribution of energy.
The fact that the gravitational potential between the gravitons depends on ${r}^{2}$
and not $\frac{1}{r}$ can allow the quantification of this field model without the prob-
lems encountered in previous attempts to find the gravitational quantum field. Additionally, a quantum theory of gravity based on this model would be compatible with GTR.
3.3. Anomalous Velocities in the Halos of Spiral Galaxies and Dark Matter
The anomalous speeds of rotation in the halos of the spiral galaxies have forced researchers to postulate the existence of some type of invisible matter that can justify those speeds. The nature of
this dark matter is a mystery. However, the model presented in the previous section can illuminate this enigma. I have indicated before that the material particles can be considered as coordinated
sets of gravitons travelling at the same time. Therefore, there may also be large accumulations of gravitons that are not yet in equilibrium, which travel freely through the universe and have not yet
coordinated to form material particles. These gravitons would have the same effect as the so-called dark matter.
Additionally, the eventual coordinated union of a sufficient number of gravitons to form material particles may justify the enigmatic presence of gas and dust in the spiral galaxies. Considering the
rate that stars are created in these galaxies, the proportion of gas in them must have reduced to zero millions of years ago.
3.4. Gravitational Constant
The constant G was discovered by Newton. It is an empirical constant that does not focus on the nature of the gravitational phenomenon. The expression
$\frac{GMm}{{r}^{2}}$ is an empirical formula deduced from experience, and not from the
knowledge of the nature of the gravitational phenomenon. However, a correct formulation of the gravitational phenomenon derived from the knowledge of its nature can add a practical structure to G as
opposed to a constant proportionality that is adjusted according to experience. The expression (18) allows us to ob-
tain a response to this problem. If we substitute ${\rho }_{0}$ for $\frac{3{M}_{h}}{4\text{π}{R}_{h}^{3}}$ , where ${M}_{h}$ is
the total energy of the universe and ${R}_{h}$ is the radius of the particle horizon, and then compare the results with a system with spherical symmetry, we obtain the following value for G:
$G=\frac{{c}^{2}{R}_{h}}{2{M}_{h}}.$ (19)
G gives information about the relationship between the radii of particles in the universe and the total energy of the universe.
3.5. Average Density of the Universe
Heretofore, we have been postulating that ${\rho }_{0}$ is equal to the critical density. However, the gravitational term of the expression (18) implies that the average density of the universe is
intrinsically equal to the critical density. Let us imagine that the average density of the universe is not ${\rho }_{0}$ but ${{\rho }^{\prime }}_{0}$ . In that case, Newton would have measured
another gravitational constant ${G}^{\prime }$ . Considering a system with spherical symmetry, we have:
$\frac{1}{2}m{c}^{2}\frac{{r}^{2}}{{R}_{h}^{2}}\left(\frac{\rho }{{{\rho }^{\prime }}_{0}}-1\right)=\frac{{G}^{\prime }Mm}{r}.$ (20)
By substituting $\rho$ for $\frac{3M}{4\text{π}{r}^{3}}$ , we obtain
${\rho }^{\prime }\left({R}_{h}\right)=\frac{3{c}^{2}}{8\text{π}{G}^{\prime }{R}_{h}^{2}}.$ (21)
The same expression of critical density again. The universe would be inherently flat, and its average density would always coincide with the critical density. It is not necessary to postulate the
existence of an inflationary period to justify the flatness of the universe.
3.6. Cosmological Implications
The expression (18) involves changing the hitherto accepted conception of the origin of the universe. Measurements of the brightness of distant stars indicate that Newton’s constant G has been the
same for all such observations. However, expression (18), which is equivalent to Newton’s law of gravity for denser systems such as stars, implies that the relationship between the mass of the
and the radius of the universe, $\frac{{R}_{h}}{M}$ , has been constant throughout the life of the
universe, as is apparent from the result 3.4. There are two options that fulfill this relationship, a stationary universe or the black hole proposed by Schwarzschild.
In a black hole, it is true that ${c}^{2}=\frac{2GM}{r}$ , where M is the mass of the black hole
and r is its radius. If we discard the stationary universe option, then the universe would be a black hole contained in another outer universe [4] [5] . Therefore, our universe cannot be considered
an isolated system. The expansion of the universe is comparable to the expansion of the 3-sphere that would form the event horizon of a black hole as it grows.
4. Conclusion
Although the results of this work allow a macroscopic view of the gravitational phenomenon in the field of quantum mechanics, it is necessary to achieve a complete formulation of gravity that
includes elementary particles at small distances or with great energies. I believe that a complete gravitational theory of the quantum field that includes the range of elementary particles is
possible if we consider that the graviton is not directly affected by the gravitational interaction, but that the real interaction between them is long range and directly proportional to distance.
The gravitational theory of the quantum field could explain the true nature, birth, and evolution of our universe.
To my wife for her patience and sincere collaboration in the correction of the English language used in this paper. In addition, my thanks to Google whose translation service has been vital in the
correct presentation of this study. My thanks also to Editage Author Services for their invaluable help in the correct presentation of this work.
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What are betting units? Do you want to read the predictions of our experts? - cst2021.org
How to control the profit from the units invested?
To control the profit or loss from the units wagered, we must again resort to the yield, although in this case we are going to learn how it is calculated and how it is interpreted.
If our yield was -8%, this would mean that we are losing $8 per month for every $100 invested and therefore we would have to modify our betting strategy. This is how profit or possible losses are
controlled by the units invested.
What are betting units?
In short, a betting unit is the amount that we decide to invest in a bet and that, under normal conditions, is usually equivalent to 1% of our bankroll (money we have available to invest). This term,
very often used by tipsters, will vary depending on the bankroll we have. Let’s see an example.
• If we have a bankroll of $100. A betting unit can be 1$.
• If we have a bankroll of $500. One betting unit can be $5.
So, the value of a betting unit will change depending on the money we have to bet (bankroll).
How to calculate betting units?
As we mentioned before, one betting unit is equivalent to 1% of our bankroll. Therefore, it is convenient to handle bankrolls such as $100, $200 or $300, as in these cases one betting unit would be
equivalent to $1, $2 or $3, respectively. However, no matter how much bankroll you have, it is very easy to calculate how much a betting unit is worth.
Is there a formula for calculating betting units?
To calculate betting units we simply follow a simple rule of 3, taking into account that each betting unit is equivalent to 1% of the bankroll.
• 357 (bankroll) = X$ (betting unit)
• 100 (bankroll) = $1 (betting unit)
By subtracting we have that:
In this way, we can calculate how much a betting unit is worth, regardless of the bankroll we have.
Is a unit the same as a stake?
Although they are similar concepts, they really need to be differentiated, as they talk about similar but really different things. Let’s see what each one means:
Betting unit: It represents a % of our bankroll. One unit usually has an equivalence of 1%, so for bankrolls of $100 it would be equivalent to $1.
Stake: Represents the amount of units we put on a bet. In this case, the stake is directly related to the confidence you have in a certain event. Therefore, it is normal to use a Stake scale from 1
to 10, so we can bet from 1% of our bankroll up to 10% depending on how confident we are that we will be successful.
What are the units won in sports betting?
Units won in sports betting is what is known as “yield”, a concept commonly used among bettors and which represents the % we win or lose for every $100 wagered.
It is possible to calculate this yield using a simple formula, so we can learn how to manage these winning units in a simple way.
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Quantitative touchdown localization for the MEMS problem with variable dielectric permittivity
C. Esteve, Ph Souplet. Quantitative touchdown localization for the MEMS problem with variable dielectric permittivity, NONLINEARITY, Vol. 31, No. 11 (2018). DOI: 10.1088/1361-6544
Abstract. We consider a well-known model for micro-electromechanical systems (MEMS) with variable dielectric permittivity, based on a parabolic equation with singular nonlinearity. We study the
touchdown or quenching phenomenon. Recently, the question whether or not touchdown can occur at zero points of the permittivity profile f, which had long remained open, was answered negatively in Guo
and Souplet (2015 SIAM J. Math. Anal. No. 47, pp. 614–25) for the case of interior points, and we then showed in Esteve and Souplet (2017 arXiv:1706.04375) that touchdown can actually be ruled out in
subregions of Ω where f is positive but suitably small.
The goal of this paper is to further investigate the touchdown localization problem and to show that, in one space dimension, one can obtain quite quantitative conditions. Namely, for large classes
of typical, one-bump and two-bump permittivity profiles, we find good lower estimates of the ratio ρ between ρ is rigorously obtained as the solution of a suitable finite-dimensional optimization
problem (with either three or four parameters), which is then numerically estimated. Rather surprisingly, it turns out that the values of the ratio ρ are not ‘small’ but actually up to the
order~0.3, which could hence be quite appropriate for robust use in practical MEMS design.
The main tool for the reduction to the finite-dimensional optimization problem is a quantitative type I, temporal touchdown estimate. The latter is proved by maximum principle arguments, applied to a
multi-parameter family of refined, nonlinear auxiliary functions with cut-off.
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Samacheer Kalvi 6th Maths Solutions Term 2 Chapter 5 Information Processing Intext Questions
You can Download Samacheer Kalvi 6th Maths Book Solutions Guide Pdf, Tamilnadu State Board help you to revise the complete Syllabus and score more marks in your examinations.
Tamilnadu Samacheer Kalvi 6th Maths Solutions Term 2 Chapter 5 Information Processing Intext Questions
Question 1.
Check whether the Tree diagrams are equal or not.
(i) Their algebraic expressions are a × (b – c) and (a × b) – (a × c)
∴ distributive property of multiplication over subtraction
∴ They are equal
(ii) Their algebraic expressions are a × (b – c) and (a × b) – c
Both are not equal [By BODMAS rule]
Question 2.
Check whether the following algebraic expressions are equal or not by using Tree diagrams.
(i) (x + y) + z and x + (y + z)
(ii) (p × q) × r and p × (q × r)
(iii) a – (b – c) and (a – b) – c
(i) (x + y) + z and x + (y + z)
The tree diagrams are
Subtraction is not associative and the expressions are not equal.
(ii) (p × q) × r and p × (q × r)
The tree diagram is
Multiplication is associative
∴ Both expressions are equal.
(iii) a – (b – c) and (a – b) – c
Their tree diagrams are
Subtraction is not associative
∴ Both expressions are not equal.
Leave a Comment
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Price and Value of Swaps - AnalystPrep | CFA® Exam Study Notes
Remember that a swap contract involves a series of periodic settlements with a final settlement at maturity. Swap price (or par swap rate) is a periodic fixed rate that equates the present value (PV)
of all future expected floating cash flows to the PV of fixed cash flows.
The swap rate is equivalent to the forward rate, \(F_0(T)\); it satisfies no-arbitrage conditions. On the other hand, the current market reference rate (MRR) is the “spot” price. Therefore, from the
fixed-rate payer perspective, the periodic value is given by:
$$\text{Periodic settlement value}=(\text{MRR}-\text{S}_{\text{N}})\times\text{Notional amount}\times\text{Period}$$
The swap value on any settlement date is calculated as the current settlement value using the above formula plus the present value of all the remaining future swap settlements.
Like all other forward commitments, the value of a swap contract at initiation is zero.
Note that it’s our assumption that MRR is set at the beginning of each interest period and has the same periodicity and day count as the swap rate. In addition, the net of fixed and floating
differences is exchanged at the end of each period.
Examples: Calculating the Swap Value and Effect of Varying MRRs
FinnLay LTD has entered a 4-year interest rate swap with a financial institution with a notional amount of USD 100 million. The contract states that FinnLay signed to receive a semiannual USD fixed
rate of 2.5% and, in turn, pay a semiannual market reference rate (MRR).
The MRR is expected to equal the respective implied forward rates (IFRs).
Scenario 1
If at the beginning of the sixth month, the MRR is 0.85%, the first swap settlement value from Finnlay’s perspective is closest to:
$$\begin{align*}\text{Periodic settlement value}&=(\text{MRR}-\text{S}_{\text{N}})\times\text{Notional Amount}\times\text{Period}\\&=(2.5\%-0.85\%)\times\text{USD 100m}\times0.5\\&=\text{USD 0.825m}\
Scenario 2
If implied forward rates remain constant as set at trade inception, how will this affect the MTM value from Finnlay’s perspective immediately after the first settlement?
The swap price (or fixed swap rate) of 2.5% is set at the initiation of the trade, which equates to the PV of fixed versus floating payments.
If there is no change in interest rate expectations, the PV of remaining floating payments rises above the PV of fixed payments.
As such, Finnlay, as a fixed receiver, realizes an MTM loss on the swap because:
$$\sum\text{PV}(\text{Floating payments paid})>\sum\text{PV}(\text{Fixed payments received})$$
Scenario 2
If implied forward rates decline just after initiation, how will this affect the MTM value from Finnlay’s perspective ?
A decrease in expected forward rates just after initiation will reduce the PV of floating payments while the fixed swap rate will remain constant.
Since FinnLay is the fixed-rate receiver, it will realize an MTM gain because:
$$\sum\text{PV}(\text{Floating payments paid})<\sum\text{PV}(\text{Fixed payments received})$$
Invest Capital Inc has signed a three-year swap contract to receive a fixed interest rate of 2.5% on a semiannual basis and pay a 6-month USD MRR. The notional amount of the swap contract is USD
Assume that the initial 6-month MRR sets at 0.56%, and MRR is expected to be upward sloping. The first settlement value in six months from Invest Capital is closest to:
A. $970.
B. $1,940.
C. $2,500.
The correct answer is A.
From the fixed-rate payer perspective, the periodic value is given by:
$$\begin{align*}\text{Periodic settlement value}&=(\text{S}_{\text{N}}-\text{MRR})\times\text{Notional Amount}\times\text{Period}\\&=(2.5\%-0.56\%)\times\text{USD 100,000}\times0.5\\&=$970\end
B is incorrect. It is calculated as \(=2.5\%-0.56\%\times\text{USD 100,000}\). It omits the period in the formula.
C is incorrect. It is the amount of the fixed interest amount after six months.
Sep 14, 2019
Forward Commitments versus Contingent ...
Derivatives typically fall into one of two classifications, either forward commitments or contingent... Read More
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Marco Valtorta
Marco Valtorta: Talks
Marco Valtorta Talk given in the "Last Lecture" Series organized by the student group Scholars United at the University of South Carolina, 2019-11-07
Mohammad Ali Javidian and Marco Valtorta. An Overview of the Back Door and Front Door Criteria (pdf), a presentatoin based on Sections 3.3 and 3.4 in: Judea Pearl. Causality, 2nd ed. Cambridge
University Press, 2009. The do-calculus rules are included.
Mohammad Ali Javidian and Marco Valtorta. An Illustrated Proof of the Front-Door Adjustment Theorem (pdf), a proof of Theorem 3.3.4 (Front-Door Adjustment) in: Judea Pearl. Causality, 2nd ed.
Cambridge University Press, 2009.
Discrete Graphical Models with One Hidden Variable (pdf), presentation (of work with Elizabeth Allman, John Rhodes, and Elena Stanghellini) at the 2011 SIAM Conference on Applied Algebraic Geometry
(AG-11), Raleigh, North Carolina, October 7, 2011.
Causality in Communication: The Agent-Encapsulated Bayesian Network Model (pdf), presentation (of work with Scott Langevin) at the special session "Recent Developments in Graphical Models" of the
14th Internation Conference on Applied Stochastic Models and Data Analysis (ASMDA-11), June 10, 2011.
Instantiation to Support the Integration of Logical and Probabilistic Knowledge (pdf), presentation (of work with Jingsong Wang) at the First Workshop on Grounding and Transformations for Theories
with Variables (GTTV 2011), May 16, 2011.
How Does Watson Work? (ppt), (pptx version here) presentation given on April 4, 2011 at Benedict College.
An Introduction to Pearl's Do-Calculus of Intervention (ppt), presentation given on October 6, 2010 at the workshop "Parameter Identification in Graphical Models," the American Institute of
Mathematics, Palo Alto, California, October 4-8, 2010.
Colloquia on Soft Evidential Update
USC seven-minute madness, September 10, 2004 (in postscript format, with speaker notes)
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