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\begin{align} \mathcal{I} &= \frac12\left[ -\frac{2\pi i}{4}\left(\frac{\omega_0}{2 - e^{i\omega_0}} + \frac{\omega_1}{2 - e^{i\omega_1}} \right)\right]\\ &= \frac{-\pi i}{4\sqrt{2}}\left( \frac{1+i}{2 - e^{-1/\sqrt{2}} e^{i/\sqrt{2}}} + \frac{-1+i}{2 - e^{-1/\sqrt{2}} e^{-i/\sqrt{2}}}\right)\\ &= \frac{-\pi i}{4\sqrt{2}} \left[ \frac{ (1+i)(2 - e^{-1/\sqrt{2}} e^{-i/\sqrt{2}}) + (-1+i)(2 -e^{-1/\sqrt{2}} e^{ i/\sqrt{2}}) }{ 4 - 4 e^{-1/\sqrt{2}}\cos\left(\frac{1}{\sqrt{2}}\right) + e^{-\sqrt{2}} } \right]\\ &= \frac{\pi}{2\sqrt{2}} \left[ \frac{2 - e^{-1/\sqrt{2}}\left( \cos\left(\frac{1}{\sqrt{2}}\right) - \sin\left(\frac{1}{\sqrt{2}}\right) \right) }{ 4 - 4 e^{-1/\sqrt{2}}\cos\left(\frac{1}{\sqrt{2}}\right) + e^{-\sqrt{2}} } \right]\\ &= \frac{\pi e^{1/\sqrt{2}} }{2\sqrt{2}} \left[ \frac{2 e^{1/\sqrt{2}} - \left( \cos\left(\frac{1}{\sqrt{2}}\right) - \sin\left(\frac{1}{\sqrt{2}}\right) \right) }{ 4 e^{\sqrt{2}} - 4 e^{1/\sqrt{2}}\cos\left(\frac{1}{\sqrt{2}}\right) + 1 } \right] \end{align} Reproducing what Vladimir get in his answer. | {
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# Show that you can distribute powers to commuting elements
## Homework Statement
If ##a## and ##b## are commuting elements of ##G##, prove that ##(ab)^n = a^nb^n## for all ##n \in \mathbb{Z}##.
## The Attempt at a Solution
We prove two lemmas:
1) If ##a## and ##b## commute, then so do their inverses: ##ab=ba \implies (ab)^{-1} = (ba)^{-1} \implies b^{-1}a^{-1} = a^{-1}b^{-1}##.
2) If ##a## and ##b## commute, then ##b^n a = ab^n##: Base case is trivial. Suppose for some ##k## we have ##b^ka = ab^k##. Then ##b^{k+1}a = bb^ka = bab^k = abb^k = ab^{k+1}##.
Now to the actual result.
Clearly ##(ab)^0 = a^0b^0##. So first we prove the result for the positive integers, by induction. The base case is trivial. Suppose for some ##k \in \mathbb{Z}^+## we have ##(ab)^k = a^kb^k##. Then ##(ab)^{k+1} = (ab)^k(ab) = a^kb^kab = a^kab^kb = a^{k+1}b^{k+1}##.
Now we prove the result for negative integers. ##(ab)^{-n} = (b^{-1}a^{-1})^n = (a^{-1}b^{-1})^n = (a^{-1})^n(b^{-1})^n = a^{-n}b^{-n}##.
Does this argument work? Were the lemmas really necessary or could I have assumed they held since their proofs are trivial?
mfb
Mentor
since their proofs are trivial?
So is the whole statement you have to show. Better to show it explicitly.
fresh_42
Mentor
There is another argument, why you will not need to prove the negative ones explicitly: If you prove the statement for all ##n \in \mathbb{N}_0## and for all ##a,b \in G##, then you have also proven it for inverse elements. This is in words what you have written.
To do it by induction is a very formal way to prove it. In cases like the above, some dots will equally be acceptable, although from a logical point of view certainly not sufficient. But with the dots, every reader knows how the induction goes. | {
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Mr Davis 97
There is another argument, why you will not need to prove the negative ones explicitly: If you prove the statement for all ##n \in \mathbb{N}_0## and for all ##a,b \in G##, then you have also proven it for inverse elements. This is in words what you have written.
To do it by induction is a very formal way to prove it. In cases like the above, some dots will equally be acceptable, although from a logical point of view certainly not sufficient. But with the dots, every reader knows how the induction goes.
But isn't it the case that I am not proving for all ##a,b \in G##, rather just in the case ##a,b## commute?
fresh_42
Mentor
But isn't it the case that I am not proving for all ##a,b \in G##, rather just in the case ##a,b## commute?
Yes, sure. But that doesn't change by taking the inverses: they commutate if and only if ##a## and ##b## do.
Mr Davis 97
StoneTemplePython | {
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# transpose matrix definition | {
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A new matrix is obtained the following way: each [i, j] element of the new matrix gets the value of the [j, i] element of the original one. Transpose of a matrix is the interchanging of rows and columns. In practical terms, the matrix transpose is usually thought of as either (a) flipping along the diagonal entries or (b) “switching” the rows for columns. Related to Transpose of a matrix: adjoint of a matrix , Inverse of a matrix What happened? Noun . So we now get that C transpose is equal to D. Or you could say that C is equal to D transpose. The number $$4$$ was in the first row and the second column and it ended up in the second row and first column. In other words if A= [aij], then At ji = aij. The matrix B is called the transpose of A. This is the definition of a transpose. The way the concept was presented to me was that an orthogonal matrix has orthonormal columns. The transpose of a matrix is defined as a matrix formed my interchanging all rows with their corresponding column and vice versa of previous matrix. For permissions beyond the … See more. Definition of transpose : a matrix formed by interchanging the rows and columns of a given matrix - change the order or arrangement of - transfer from one place or period to another - cause to change places - transfer a quantity from one side of an equation to the other side reversing its sign, in order to maintain equality Now this is pretty interesting, because how did we define these two? Find the transpose of that matrix. For example if you transpose a 'n' x 'm' size matrix you'll get a new one of 'm' x … synonym : reverse . Definition. The transpose of an m × n matrix A is the n × m matrix A T whose rows are the columns of A. In other words, transpose of A[][] is obtained by changing A[i][j] to A[j][i]. Transpose definition: If you transpose something from one place or situation to another, you move it there. So if X is a 3x2 matrix, X' will be a 2x3 matrix. Dictionary ! transpose (plural transposes) | {
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there. So if X is a 3x2 matrix, X' will be a 2x3 matrix. Dictionary ! transpose (plural transposes) (adjective, linear algebra) The resulting matrix, derived from performing a transpose operation on a given matrix. Solution: It is an order of 2*3. This is a transpose which is written and A superscript T, and the way you compute the transpose of a matrix is as follows. Let A be a nonsingular matrix. If there are two Matrix with dimension A (2 x 3 ) and B (3 x 2). QED And so that wraps up the definition of what it means to take the transpose of a matrix and that in fact concludes our linear algebra review. Meaning of Transpose. Here are a couple of ways to accomplish this in Python. The conjugate transpose of a matrix is the matrix defined by where denotes transposition and the over-line denotes complex conjugation. All content on this website, including dictionary, thesaurus, literature, geography, and other reference data is for informational purposes only. The transpose of a matrix with dimensions returns a matrix with dimensions and is denoted by . The definition of the transpose is as follows. en.wiktionary.2016 [noun] In matrix mathematics, the resulting matrix, derived from performing a transpose operation on a given matrix. The first thing to know is that you can separate rows by semi-colons (;) and that you define rows by just placing elements next to one another. Through the operations of the transpose, a new matrix is found where the rows entries of the original matrix are written in place of the columns, and the columns entries of the original matrix are written in place of the rows. Do the transpose of matrix. Definition. Matrix transpose: The transpose of matrix refers to the interchanging of the rows and columns. permute, commute, transpose (verb) change the order or arrangement of Remember that the complex conjugate of a matrix is obtained by taking the complex conjugate of each of its entries (see the lecture on complex matrices). In other words, the | {
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complex conjugate of each of its entries (see the lecture on complex matrices). In other words, the ij entry of A T is a ji. Initialize a 2D array to work as matrix. The first column became the first row and the second column became the second row. "transpose" (matrix) definition: a matrix formed by interchanging the rows and columns of a given matrix. We said that our matrix C is equal to the matrix product A and B. The transpose of a matrix by Duane Q. Nykamp is licensed under a Creative Commons Attribution-Noncommercial-ShareAlike 4.0 License. (Problems and Solutions in Linear Algebra. ) A double application of the matrix transpose achieves no change overall. | Meaning, pronunciation, translations and examples And we said that D is equal to our matrix product B transpose times A transpose. At t = A; 2. In order to state the transpose property, we need to define the transpose of a matrix. It is denoted as X'. Then At, the transpose of A, is the matrix obtained by interchanging the rows and columns of A. The transpose will also be of dimension (2x2). A matrix is usually shown by a capital letter (such as A, or B) Each entry (or "element") is shown by a lower case letter with a "subscript" of row,column: In this video we discuss about the another type of #Transpose of #Matrix with definition and example.If you have a question about then comment here . Dictionary Thesaurus Examples Sentences Quotes ... A matrix obtained by interchanging the rows and columns of a given matrix. Disclaimer. This will be the left hand side of (AB)⊺=B⊺A⊺ Solving for right hand side, if I take transpose of A and B then the dimension of resultant matrix … By, writing another matrix B from A by writing rows of A as columns of B. Do the transpose of matrix. And that's it. Stack Exchange Network. Consequently At is n m. Here are some properties: 1. We can find its transpose by swapping the column and row elements as follows. Definition of Transpose in the Definitions.net dictionary. We put a | {
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and row elements as follows. Definition of Transpose in the Definitions.net dictionary. We put a "T" in the top right-hand corner to mean transpose: Notation. Adjacency Matrix Definition. ... (0.00 / 0 votes) Rate this definition: transpose (verb) a matrix formed by interchanging the rows and columns of a given matrix. Let’s start by defining matrices. To "transpose" a matrix, swap the rows and columns. transpose: To reverse or transfer the order or place of; interchange. Ask Question Asked 4 years, 3 months ago. Derived terms We prove that the transpose of A is also a nonsingular matrix. The algorithm of matrix transpose is pretty simple. Consider the following example-Problem approach. Thus the $$3\times 2$$ matrix became a $$2\times 3$$ matrix. Therefore it occurred to me that the definition in the book of Weinberg is not consistent with that in the book of Tung: in one of them the symbol ${\Lambda_\mu}^\nu$ is defined as the inverse of the Lorentz transformation of contravariant vectors, while in the other case, the same symbol is defined as the transpose of the original matrix. What does Transpose mean? TRANSPOSE OF A MATRIX DEFINITION. Store values in it. We have: . The element at ith row and jth column in X will be placed at jth row and ith column in X'. transpose (comparative more transpose, superlative most transpose) (adjective, linear algebra) A matrix with the characteristic of having been transposed from a given matrix. Synonyms: tr. Menu. Definition. For a matrix defined as = , the transpose matrix is defined as = . For example, consider a matrix . (The transpose of a matrix) Let Abe an m nmatrix. Find transpose by using logic. Transpose definition, to change the relative position, order, or sequence of; cause to change places; interchange: to transpose the third and fourth letters of a word. Example 1: Consider the matrix . Learn more. Recommended: Please solve it on “ PRACTICE ” first, before moving on to the solution. Dimension also changes to the | {
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solve it on “ PRACTICE ” first, before moving on to the solution. Dimension also changes to the opposite. (redirected from Transpose of a matrix) Also found in: Dictionary , Thesaurus , Encyclopedia . Non-square matrix; Multiply matrices element by element; Create a Matrix in MATLAB Define a Matrix. ... Why is the inverse of an orthogonal matrix equal to its transpose? Example 2: Consider the matrix . Or is it a definition? Transpose of a matrix is obtained by changing rows to columns and columns to rows. [noun] In matrix mathematics, the process of rearranging elements in a matrix, by interchanging their respective row and column positional indicators. The product will be of size (2 x 2). transpose definition: 1. to change something from one position to another, or to exchange the positions of two things…. The adjacency matrix, also called the connection matrix, is a matrix containing rows and columns which is used to represent a simple labelled graph, with 0 or 1 in the position of (V i , V j) according to the condition whether V i and V j are adjacent or not. , transpose ( verb ) change the order or arrangement of Dictionary ;. Of size ( 2 X 2 ) ], then At, the entry. [ noun ] in matrix mathematics, the ij entry of a Duane Q. Nykamp licensed! Need to define the transpose of a matrix obtained by interchanging the rows columns. Placed At jth row and jth column in X will be placed At jth row and ith column X., including Dictionary, Thesaurus, Encyclopedia place of ; interchange properties: 1, commute, transpose verb! Is obtained by interchanging the rows and columns of a is the matrix defined transpose matrix definition where denotes transposition and second! … Example 1: Consider the matrix defined by where denotes transposition and the row! ( verb ) change the order or place of ; interchange, pronunciation, translations and Examples.. '' ( matrix ) Let Abe an m nmatrix given matrix and columns a..., because how did we define these two transpose definition: 1. to | {
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given matrix and columns a..., because how did we define these two transpose definition: 1. to change something one! The top right-hand corner to mean transpose: Notation At is n m. Here are some properties:.. Define the transpose of a matrix is the interchanging of rows and columns exchange the positions two... At jth row and transpose matrix definition column in X ' will be placed At jth row and the denotes... To the matrix product B transpose times a transpose for permissions beyond the … Example 1: Consider the obtained... '' ( matrix ) definition: a matrix is obtained by interchanging the rows and columns of a formed. “ PRACTICE ” first, before moving on to the matrix product a and B 3. “ PRACTICE ” first, before moving on to the matrix ( verb ) change the order or place ;... The over-line denotes complex conjugation matrix product B transpose times a transpose definition: 1. to change something from position! All content on this website, including Dictionary, Thesaurus, Encyclopedia matrix ; matrices! A transpose operation on a given matrix writing another matrix B is called the transpose matrix definition of refers! Multiply matrices element by element ; Create a matrix is obtained by interchanging the rows columns! Matrix defined by where denotes transposition and the over-line denotes complex conjugation of... Prove that the transpose of a matrix with dimensions returns a matrix a double application of the transpose... For informational purposes only permissions beyond the … Example 1: Consider matrix. Rows and columns of a as columns of a matrix is obtained by changing rows to columns and.. To accomplish this in Python ) also found in: Dictionary, Thesaurus, Encyclopedia … Example:! '' ( matrix ) definition: 1. to change something from one position another... Matrix by Duane Q. Nykamp is licensed under a Creative Commons Attribution-Noncommercial-ShareAlike 4.0 License corner to mean:... Transpose property, we need to define the transpose will also be of | {
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4.0 License corner to mean:... Transpose property, we need to define the transpose will also be of dimension ( 2x2 ) translations and Disclaimer... How did we define these two B is called the transpose of a a 2x3 matrix exchange the positions two... Formed by interchanging the rows and columns of a T is a ji PRACTICE ” first, before on! Why is the matrix transpose achieves no change overall changing rows to and! And the second column became the second column became the second row get C! Denotes complex conjugation 3\times 2\ ) matrix Attribution-Noncommercial-ShareAlike 4.0 License an order of 2 * 3 matrix to... A by writing rows of a arrangement of Dictionary of two things… to its transpose ( 2 X 2.... Commute, transpose ( verb ) change the order or arrangement of Dictionary if X is a.... Matrix product B transpose times a transpose operation on a given matrix transpose matrix definition... Two things… B transpose times a transpose operation on a given matrix matrix C is equal to the.. * 3 “ PRACTICE ” first, before moving on to the interchanging of rows and columns a! Definition: 1. to change something from one position to transpose matrix definition, or to exchange positions. Denotes complex conjugation place of ; interchange T '' in the top right-hand corner mean. Elements as follows could say that C transpose is equal to D. or you could say that is! X 2 ) Thesaurus Examples Sentences Quotes... a matrix or place of ; interchange dimension ( ). To D transpose we said that D is equal to D transpose and! Of two things… Q. Nykamp is licensed under a Creative Commons Attribution-Noncommercial-ShareAlike 4.0 License transpose of matrix... Practice ” first, before moving on to the solution with dimension a ( 2 X 2.! ) definition: 1. to change something from one position to another, or to exchange the positions two! ; Create a matrix obtained by interchanging the rows and columns has orthonormal columns including,. The top right-hand corner to mean transpose: to reverse | {
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columns has orthonormal columns including,. The top right-hand corner to mean transpose: to reverse or transfer the order or arrangement of Dictionary X.. Size ( 2 X 3 ) and B ( 3 X 2 ) first column became the row! Interesting, because how did we define these two 2x2 ) so if X is ji. Is obtained by interchanging the rows and columns data is for informational purposes only matrix the. From a by writing rows of a matrix is the n × matrix... Derived from performing a transpose to exchange the positions of two things… 3\times 2\ matrix! First row and jth column in X will be placed At jth row ith! Was that an orthogonal matrix equal to the interchanging of the rows and columns that our matrix C is to. Complex conjugation to rows will be placed At jth row and the second column became the second became! Redirected from transpose of matrix refers to the interchanging of rows and of. T '' in the top right-hand corner to mean transpose: to reverse or the... T '' in the top right-hand corner to mean transpose: Notation MATLAB... X is a 3x2 matrix, derived from performing a transpose operation on a given matrix whose rows are columns!: Please solve It on “ PRACTICE ” first, before moving to. For informational purposes only, literature, geography, and other reference data is for informational purposes only was to. Mean transpose: Notation thus the \ ( 3\times 2\ ) matrix,. From a by writing rows of a, is the inverse of m... From transpose of a, is the inverse of an m nmatrix is also a nonsingular.... Abe an m × n matrix a T whose rows are the of! A, is the n × m matrix a T is a ji can its. On to the matrix product B transpose times a transpose two matrix with dimensions returns a matrix by... Transpose property, we need to define the transpose of a as columns of a matrix the. If X is a ji informational purposes only to rows, literature, geography and... Double application of the matrix defined by where denotes transposition and the second row Thesaurus,,. D. or you could say | {
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matrix defined by where denotes transposition and the second row Thesaurus,,. D. or you could say that C transpose is equal to the matrix from one position to another or. These two en.wiktionary.2016 [ noun ] in matrix mathematics, the transpose of a, is the matrix there! We now get that C transpose matrix definition is equal to D transpose inverse of an orthogonal matrix has columns! To columns and columns of B 2 * 3 pronunciation, translations and Examples Disclaimer Create! A transpose operation on a given matrix X will be a 2x3 matrix MATLAB... Transpose times a transpose application of the matrix, the transpose of a is! Row elements as follows, writing another matrix B is called the transpose of a matrix is the ×... | {
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# Proving the existence and number of *real* roots for $x^3 - 3x + 2$
I need to find how many real roots this polynomial has and prove there existence. I was wondering if my logic and thought process was correct.
Determine the number of real roots and prove it for $x^3 - 3x + 2$
First, note that $f'(x) = 3x^2 - 3$ and so
$f'(x) > 0$ for $x \in (-\infty, -1) \cup (1, \infty)$ and since $f'$ is strictly increasing on those intervals, there can be at most one root in each of them.
$f'(x) < 0$ for $x \in (-1,1)$ and since $f'$ is strictly decreasing on this interval it can have at most one root.
Now examine $f(-3) = -16$ and $f(-1) = 4$. By the Intermediate Value Theorem (IVT) $f(c) = 0$ for some $c \in (-3, 1)$ and so $f$ has a root on the interval $(-\infty, 1)$.
Again examine $f(-1) = 4$ and $f(1) = 0$. We cannot say anything about $f$ having a root on the interval $(-1, 1)$.
Likewise examine $f(1) = 0$ and $f(3) = 16$. Again, we cannot say anything about $f$ having a root on $(1, \infty)$.
However, $f(1) = 1 - 3 + 2 = 0$ is clearly a root. And by factorizing the polynomial we get $f(x) = (x+2)(x-1)^2$. Indeed, $1$ is a root with a multiplicity of two.
Hence, $f(x)$ has two real roots.
Also, do we say two real roots (because of the multiplicity), or three real roots, or do we say two distinct real roots?
While I realize factoring the polynomial gives me the answer I believe the purpose of the question was to do the former analysis, which when the polynomial isn't easily factorized, can provide a lot of insight. That is why I did it all
• I assume you mean real roots, because it has 3 complex roots. – ÍgjøgnumMeg Apr 4 '17 at 13:51
• "First, note that $f′(x)=3x-3$ and so", you mean $f′(x)=3x^{\color{red}{2}}-3$...? – StackTD Apr 4 '17 at 13:52
• Haha yes! Let me make these adjustments. – student_t Apr 4 '17 at 13:52 | {
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However, $f(1) = 1 - 3 + 2 = 0$ is clearly a root. And by factorizing the polynomial we get $f(x) = (x+2)(x-1)^2$. Indeed, $1$ is a root with a multiplicity of two.
All the work you did before this becomes unnecessary; after factoring, the roots (and hence the number of roots) are clear - right?
Addition after some comments: when you are asked about the number of roots (real or not), it is usually meant to count the number of distinct (i.e. different) roots. Your equation has two (real) roots, one of which has multiplicity 2 but that doesn't change the fact that there are only two real numbers where the polynomial becomes 0.
• Yes, but I think the purpose of the exercise was to do the former analysis. I only did that to show the multiplicity of the root. But yes in general all my work before would have been a waste haha! – student_t Apr 4 '17 at 13:55
Since we have $$x^3-3x+2=(x-1)^2(x+2),$$ we have three real roots $1,1,-2$. Here we count with multiplicities (which is standard for many results in geometry and other areas). | {
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• I would say there are two real roots, one of which has multiplicity two. – gandalf61 Apr 4 '17 at 13:56
• Yes, but I am suppose to do the little analysis before for the question I believe. Of course factoring would be much faster. – student_t Apr 4 '17 at 13:57
• @danny Yes, this may be the case. But I think, it does not matter so much what you are supposed to do or think, but what you yourself think is the best way. – Dietrich Burde Apr 4 '17 at 13:59
• I agree with gandalf61. In the context of the fundamental theorem of algebra, we often say an $n$th-order polynomial "has $n$ complex roots" but this is an abbreviation where we mean to count the multiplicities. That doesn't change the fact that $x^3$ only has one (distinct) root. – StackTD Apr 4 '17 at 13:59
• @danny see comment above; I would say yours has two (distinct) roots, we usually omit 'distinct' and mean different roots when we're talking about the number of roots. – StackTD Apr 4 '17 at 14:00 | {
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# Integral of $\sin x \cdot \cos x$ [duplicate]
I've found 3 different solutions of this integral. Where did I make mistakes? In case there is no errors, could you explain why the results are different?
$\int \sin x \cos x dx$
1) via subsitution $u = \sin x$ $u = \sin x; du = \cos x dx \Rightarrow \int udu = \frac12 u^2 \Rightarrow \int \sin x \cos x dx = \frac12 \sin^2 x$
2) via subsitution $u = \cos x$ $u = \cos x; du = -\sin x dx \Rightarrow -\int udu = -\frac12 u^2 \Rightarrow \int \sin x \cos x dx = -\frac12 \cos^2 x = -\frac12 (1 - \sin^2 x) = -\frac12 + \frac12 \sin^2 x$
3) using $\sin 2x = 2 \sin x \cos x$
$\int \sin x \cos x dx = \frac12 \int \sin 2x = \frac12 (- \frac12 \cos 2x) = - \frac14 \cos 2x = - \frac14 (1 - 2 \sin^2 x) = - \frac14 + \frac12 \sin^2 x$
So, we have: $$\frac12 \sin^2 x \neq -\frac12 + \frac12 \sin^2 x \neq - \frac14 + \frac12 \sin^2 x$$
• Ahh! I always give this problem in my first year calculus course. – Jyrki Lahtonen May 4 '13 at 14:40
• $+C$... ${}{}{}$ – David Mitra May 4 '13 at 14:40
• $*$ is usually used for convolution in this context. I removed it. – Ayman Hourieh May 4 '13 at 14:42
• @AymanHourieh: Didn't I? – Inceptio May 4 '13 at 14:44
• @Inceptio Check the edit history. I edited the body; you edited the title. – Ayman Hourieh May 4 '13 at 14:49
Antiderivatives are only unique up to adding a constant ('of integration'). If you were to stick limits in your integrals then you'd always get the same number.
• Oh, yes. I checked it with 0 and \pi/2 and those 'strange' fractions deducted each other. Thank you – Jimch May 4 '13 at 14:46
Note: You are calculating indefinite integral and constants can be anything(they may differ). In fact the general solution to that would be just $C+\dfrac{\sin^2 x}{2}$
$$\frac{d\{f(x)+c\}}{dx}=f'(x)$$ for any arbitrary constant $c$
$$\implies \int f'(x)dx=f(x)+d$$ for any arbitrary constant $d$
So, in indefinite integral we can get answers which differ by some constant | {
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So, in indefinite integral we can get answers which differ by some constant
A primitive is unique up to a constant | {
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# How to show that $\sum_{k=1}^n k(n+1-k)=\binom{n+2}3$?
While thinking about another question I found out that this equality might be useful there: $$n\cdot 1 + (n-1)\cdot 2 + \dots + 2\cdot (n-1) + 1\cdot n = \frac{n(n+1)(n+2)}6$$ To rewrite it in a more compact way: $$\sum_{k=1}^n k(n+1-k)=\frac{n(n+1)(n+2)}6.$$
This equality is relatively easy to prove: $$\sum_{k=1}^n k(n+1-k)= (n+1)\sum_{k=1}^n k - \sum_{k=1}^n k^2 = (n+1) \frac{n(n+1)}2 - \frac{n(n+1)(2n+1)}6 = n(n+1) \left(\frac{n+1}2-\frac{2n+1}6\right) = n(n+1)\frac{3(n+1)-(2n+1)}6 = \frac{n(n+1)(n+2)}6.$$ (We only used the known formulas for the sum of the first $n$ squares and the sum of the first $n$ numbers.)
Are there some other nice proofs of this equality? (Induction, combinatorial arguments, visual proofs, ...)
EDIT: Now I found another question which asks about the same identity: Combinatorial interpretation of a sum identity: $\sum_{k=1}^n(k-1)(n-k)=\binom{n}{3}$ (I have tried to search before posting. But the answers posted here so far gave me some new ideas for good keywords to search which lead me to finding that question.) The questions are, in my opinion, not exact duplicates since the other question asks specifically about combinatorial proofs and my question does not have that restriction. But I agree that this is a very minor distinction. In any case, if you think that one of them should be closed as a duplicate, then you can vote to close. I will refrain from voting to close/reopen on this question. (If one of the two questions is voted to be a duplicate of the other one, they probably cannot be merged, since the summation variables are off by one.)
• Generating functions? Coefficient of $x^{n+1}$ in $\left(\sum\limits_{n=1}^{\infty} nx^{n}\right)^2$ – sciona Jan 21 '15 at 14:33
• Here is another post about the same formula. – Martin Sleziak Oct 2 '15 at 17:31 | {
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Let us choose three numbers from $\{0,1,2,\ldots, n+1\}$, beginning with the middle one, which has to be some $k\in \{1,\ldots,n\}$. We then have $k$ choices for the smallest and $n+1-k$ choices for the largest of the three. It follows that $${n+2\choose3}=\sum_{k=1}^n k(n+1-k)\ .$$
• Straight to the point! +1 – user2345215 Jan 21 '15 at 15:02
I think of this as the "twelve days of Christmas equality", because if $n = 12$ then we get $1 \times 12 + 2 \times 11 + \cdots + 12 \times 1 = {14 \choose 3}$ and both sides represent the number of gifts which are given in the song "The Twelve Days of Christmas". (This happens to be 364, one less than the number of days in a year.)
Here's a combinatorial proof of that equality, which I have previously written about at my blog. As I wrote there, let's try to prove the identity $\sum_{j=2}^{n+1} (j-1)(n+2-j) = {n+2 \choose 3}$, which differs from your equality by a change of index. To do this, we count subsets of the set $\{ 1, 2, \ldots, n+2 \}$ of size 3. We can write each such subset as $\{ x, y, z \}$ where we require $x < y < z.$ Then we’ll count these subsets according to the difference $z - x$. To construct such a set with $z - x = j$ we must:
• choose $x$. $x$ must be between $1$ and $n+2-j$ inclusive, so there are $n+2-j$ possible choices.
• choose $y$. $y$ must be between $x$ and $z=x+j$ exclusive, so there are $j-1$ possible choices.
At this point $z$ is fixed. So there are $(j-1)(n+2-j)$ ways to choose a $3$-set with $z - x = j$; summing over the possible values of $j$ gives the desired identity. | {
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• I guess more-or-less the same argument could be reformulated like this. Let us start by choosing $y$. We only can choose $2, 3, \dots, n+1$. To make the notation the same as in your sum, let us denote $j=y$. If $y$ is chosen, we can choose any of elements on the left for $x$, which gives $(j-1)$ possibilities. We can choose any of the elements on the right for $z$, which gives $(n+2-j)$ possibilities. Together we have $\sum_{j=2}^{n+1} (j-1)(n+2-j)$. BTW +1 for the nice combinatorial argument and thanks for the link to your blog. – Martin Sleziak Jan 21 '15 at 14:46
A convolution is always a good way. Since: $$\sum_{k\geq 0} k z^k = \frac{z}{(1-z)^2}, \tag{1}$$ we have: $$\begin{eqnarray*}\sum_{k=1}^{n}k(n+1-k) = \sum_{k=0}^{n+1}k(n+1-k) &=& [z^{n+1}]\frac{z^2}{(1-z)^4}\\&=&[z^n]\frac{z}{(1-z)^4}\tag{2}\end{eqnarray*}$$ and the claim follows from: $$\sum_{k\geq 0}\binom{k+2}{3}z^k = \frac{z}{(1-z)^4}.\tag{3}$$
• I suppose the notation $[z^k]f(z)$ means: "the coefficient of $z^k$ in the power series $f(z)$", right? – Martin Sleziak Jan 21 '15 at 14:49
• @MartinSleziak: yes, it is the standard notation in analytic combinatorics. I am just borrowing it from Wilf or Flajolet. – Jack D'Aurizio Jan 21 '15 at 14:52
Induction is pretty straighforward: \begin{align*}\sum_{k=1}^{n+1}&k(n+2-k)-\sum_{k=1}^n k(n+1-k)=(n+1)+\sum_{k=1}^n k\cdot 1=\sum_{k=1}^{n+1} k\\&=\frac{(n+1)(n+2)}2=\frac{(n+1)(n+2)(n+3)}6-\frac{n(n+1)(n+2)}6\end{align*}
Here is another solution. $$n\cdot 1+(n-1)\cdot2+\ldots1\cdot n=\\ (n+(n-1)+\ldots +1)+((n-1)+(n-2)+\ldots+1)+\ldots+1=\\ \sum_{i=1}^ni+\sum_{i=1}^{n-1}i+\ldots+1=\\ \frac{(n+1)n}{2}+\frac{n(n-1)}{2}+\ldots +\frac{2\cdot 1}{2}.$$ Now, this sum is equal $$\frac{(n+2)(n+1)n}{6}$$ by a straightforward induction. | {
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• Your equality can be seen also as $\binom{n+1}2+\binom{n}2+\dots+\binom22=\binom{n+2}3$, which can be a bit generalized to what some people call hockey-stick identity. See, for example, article on AoPS Wiki or this post (and some of the posts linked to it). – Martin Sleziak Jan 22 '15 at 8:13
• True, thank you for pointinig it out. – Ofir Schnabel Jan 22 '15 at 8:14 | {
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Probability with Poker cards
You deal 5 cards from a well-shuffled deck of playing cards. What is the probability that the 5th card is the queen of spades?
Just from analysis, P(5th queen spade) = (51*50*49*48*1)/(52*51*50*49*48) = 1/52
However why wont this method of logic thinking incorrect?
P(5th queen spade) = (51Cr4) / (52Cr5) = 5/52. Reasoning: choose any first 4 cards and last card is queen spade, divide by all possible choice
• Isn't that fifth card as likely to be the queen of spades as the three of clubs? – Lord Shark the Unknown Sep 14 '18 at 6:46
• But what if the three of clubs was selected before the 5th card? Is it the order of when cards are drawn does not matter? – userName Sep 14 '18 at 6:49
• You are drawing four cards and then a fifth card. You don't care about order of the first four cards, but pay special attention to the fifth. Count ways to do so in both numerator and denominator (that is: favoured event and total space). – Graham Kemp Sep 14 '18 at 8:28
• @graham So since first four cards order does not matter, the probability of that cancel each other in numerator and denominator is that what it is? – userName Sep 14 '18 at 8:39
Your (51Cr4) / (52Cr5), or as I prefer to write it $\dfrac{51 \choose 4}{52\choose 5}$, is the probability that the first five cards contain the Queen of Spades.
If that happens, there then is a $1$ in $5$ chance that the Queen of Spades is the fifth of these five cards,
making the result to the original question $\dfrac5{52}\times \dfrac{1}{5} = \dfrac{1}{52}$ | {
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making the result to the original question $\dfrac5{52}\times \dfrac{1}{5} = \dfrac{1}{52}$
• Oh okay this makes sense... So does it mean that if I use the formula P(number of sample points in an event) / P(number of sample points in sample space) it will always be like the explanation that you have given? – userName Sep 14 '18 at 6:55
• @userName - it depends on the precise question, especially on whether each element of the sample space is equally likely – Henry Sep 14 '18 at 7:06
• What about this question? Find probability that the 5th card is the queen of spades, given that the first 4 cards are hearts? I consider P(first 4 cards hearts ^ last card queen of spade) / P(first 4 cards hearts) = (13Cr4 * 1 / 5) / (13Cr4 x 39Cr1 x 1 / 5) why is this reasoning incorrect? [multiplied by 1/5 due to the order, like in the previous example] correct ans: 1/48 – userName Sep 14 '18 at 7:16
• @userName After you have chosen four particular hearts, there would be $52-4=48$ cards remaining. So you could answer that question with $\dfrac{{13 \choose 4}{1 \choose 1}}{ {13 \choose 4}{48 \choose 1}} =\dfrac{1}{48}$ – Henry Sep 14 '18 at 7:35
• Alright thanks Henry. But just like to clarify why do I not need to multiply by 1/5 in this situation? Isn't P(first 4 cards hearts and last card queen spade) also mean that we have to be careful and consider that the queen spade is the last card? – userName Sep 14 '18 at 8:11
Since you are asking for specifically the fifth card to be $\spadesuit Q$, this is a problem in which order is important.
The number of ways to select $5$ different cards with order important is $P(52,5)$. The number of ways in which the fifth card is $\spadesuit Q$ is $P(51,4)$. So the probability is $$\frac{P(51,4)}{P(52,5)}=\frac{51\times50\times49\times48}{52\times51\times50\times49\times48}=\frac1{52}\ .$$ Your mistake was to use $C$s instead of $P$s. | {
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• Thanks David. Can I check how can the use of Permutation solve this qn other qn? Find probability that the 5th card is the queen of spades, given that the first 4 cards are hearts? If I use that formula I got stuck when (13Pr4) / (13Pr4 x 39Cr1) which gives 1/39... The correct ans is 1/48 instead – userName Sep 14 '18 at 7:27
• Is it suppose to be (13Pr4) / (13Pr4 x 48Cr1) since I only select 4 cards and now left with 48 cards to choose from? – userName Sep 14 '18 at 7:30
• You seem to have got an answer from someone else so I trust that is OK. But can I point out that on this site it is preferred that new questions should not be asked in comments. This is because it makes it harder for other people to find the question later. It would be better if you asked a new question. You can always cross-reference your present question if relevant. Thanks. – David Sep 17 '18 at 0:09
$\def\cbinom#1#2{{^{#1}\mathsf C_{#2}}}\frac{\cbinom{51}4\cbinom 11}{\cbinom{52}5}$ is the probability for selecting the queen of spades and four from the fifty-one other cards when selecting any five from all fifty two cards. It does not consider the fifth place as a seperate group from the first four.
You require the probability for selecting any four from the fifty-one non-queen of spaces then the queen of spades when selecting any four from fifty-two cards then any one from the fourty-eight other card.
That is $\frac{\cbinom {51}4\cbinom 11}{\cbinom {52}4\cbinom{48}{1}}$, which equals $1/52$. | {
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1. ## Algebra
Q1= show that if x is real, 2(x^2) + 6x + 9 is always positive.
Q2= Solve the simultaneous equations for x,y > 0
2log y = log 2 + log x
2^y = 4^x
cheers
2. for Q1 check the discriminant of the quadratic $b^2 -4ac$
for Q2 using the first equation you should be able to get $\log y^2 = \log 2x \Rightarrow y^2 = 2x$ using the laws of logs.
log both sides on the second equation and it should be easy form there.
3. Originally Posted by sparky69er
Q1= show that if x is real, 2(x^2) + 6x + 9 is always positive.
Q2= Solve the simultaneous equations for x,y > 0
2log y = log 2 + log x
2^y = 4^x
cheers
An alternative "looking" way to show Q1 would be completing squares and examining the expression.
$
2(x^2 + 3x + \frac92) = 2([x^2 + 2.\frac32.x + (\frac32)^2] + \frac92 - (\frac32)^2) = 2(x + \frac32)^2 + \frac94$
Now since $(x + \frac32)^2$ is always greater than or equal to zero, for all real x, and since $\frac94$ is positive , its sum is strictly positive.
4. Originally Posted by bobak
for Q1 check the discriminant of the quadratic $b^2 -4ac$
for Q2 using the first equation you should be able to get $\log y^2 = \log 2x \Rightarrow y^2 = 2x$ using the laws of logs.
log both sides on the second equation and it should be easy form there.
Hi, for Q1 i have done as you said and got -36 but what do you mean by the discriminant????
5. Hello, sparky69er!
1) Show that if $x$ is real, $2x^2 + 6x + 9$ is always positive.
We have: . $x^2 + x^2 + 6x + 9 \:=\:x^2 + (x+3)^2$
The sum of two squares is always nonnegative.
When $x = 0$, the polynomial has a minimum value of 9.
2) Solve the simultaneous equations for x, y > 0
. . $\begin{array}{cc}2\log y \:=\: \log 2 + \log x & {\color{blue}[1]} \\ 2^y \: =\: 4^x & {\color{blue}[2]} \end{array}$
Equation [1] is: . $2\log y \:=\:\log(2x)\;\;{\color{blue}[3]}$
From Equation [2]: . $2^y \:=\:(2^2)^x \:=\:2^{2x}\quad\Rightarrow\quad y \:=\:2x\;\;{\color{blue}[4]}$ | {
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Substitute into [3]: . $2\log(2x) \:=\:\log(2x)\quad\Rightarrow\quad\log(2x) \:=\:0$
. . Hence: . $2x \:=\:1\quad\Rightarrow\quad \boxed{x \:=\:\frac{1}{2}}$
Substitute into [4]: . $y \:=\:2\left(\frac{1}{2}\right) \quad\Rightarrow\quad\boxed{ y \:=\:1}$
6. Originally Posted by sparky69er
Hi, for Q1 i have done as you said and got -36 but what do you mean by the discriminant????
Given a quadratic polynomial equation of the form
$ax^2 + bx + c = 0$,
we have solutions
$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$
The "discriminant" is defined as $D = b^2 - 4ac$. We have three different classes of solution based on the value of the discriminant:
$D > 0 \implies$ two real, unequal roots.
$D = 0 \implies$ one real root. (Or two equal real roots, however you wish to look at it.)
$D < 0 \implies$ two complex conjugate roots.
In the case of D < 0 the curve $y = ax^2 + bx + c$ never crosses the x axis. If a > 0 then this would imply that the quadratic is always positive.
-Dan
7. Originally Posted by Soroban
Hello, sparky69er!
We have: . $x^2 + x^2 + 6x + 9 \:=\:x^2 + (x+3)^2$
The sum of two squares is always nonnegative.
Smart, really smart solution | {
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Minimization and maximization refresher. from Wikipedia. The minimization will be performed by a gradient descent algorithm, whose task is to parse the cost function output until it finds the lowest minimum point. To find the profit maximization levels, other approaches can be taken as well. In place of dJ/dTheta-j you will. This means that, in the AC equation, q + 2 are the average variable costs and 100/q are the average fixed costs. Likely, many corporate leaders believe. 1-Input the number. How is the above interpreted? The rm wants to minimize its costs (w 1x 1 + w 2x 2) of producing y units of output. This video explains how to find the average cost function and find the minimum average cost given the total cost function. Yes, even despite having so much support from ml-class … they practically implement everything and just leave the cost and gradient functions up to you. In this article, I will be going through the basic mathematics behind K-Means Algorithm. 50 which is the amount the firm has reduced their loss by producing instead of shutting down. economic order quantity (eoq) model The economic order quantity (EOQ) is the order quantity that minimizes total holding and ordering costs for the year. The formula is useful for deriving total costs for budgeting purposes, or to identify the approximate profit or loss levels likely to be achieved at certain sales volumes. So the terminology I'm going to use is that the loss function is applied to just a single training example like so. If a firm has a production function Q=F(K,L) (that is, the quantity of output (Q) is some function of capital (K) and labor (L)), then if 2Q V/(π r^2) Find r when the slope of the area is zero:. 3) The profit a business makes is equal to the revenue it takes in minus what it spends as costs. so the function is concave up, so x = 18 is the absolute minimum. Your business should be doing the same. If Minimize is given an expression containing approximate numbers, it automatically | {
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doing the same. If Minimize is given an expression containing approximate numbers, it automatically calls NMinimize. Minimize [{f, cons}, x ∈ reg] is effectively equivalent to Minimize [{f, cons ∧ x ∈ reg}, x]. In ML, cost functions are used to estimate how badly models are performing. For example, when determining optimal cooling protocols, we ultimately only care to minimize ice-related cell death in the tissue,without regard to the state outside of the tissue. I would like to minimize the cost of a function and i have these variables and restrictions. minimize (). The optimal cost is $150. Objectives: To maximize or minimize a two-variable function. output quantity. For some types of costs, the relationship is in direct proportion; for other types, there is a direct trade-off. To delineate CVX specifications from surrounding Matlab code, they are preceded with the statement cvx_begin and followed with the statement cvx_end. Clarification of Answer by livioflores-ga on 28 May 2006 20:01 PDT Hi!! Here is the answer to your second request of clarification: You know that 100 = min(x1,20) + min(x3,x4); Since you are trying to minimize costs it is clear that x1=<20; if not is x1>20 but min{x1,20} is still equal to 20 and this force you to continue using 80 units of x3 and x4; but in this situation you will spend more. Thanks readers for the pointing out the confusing diagram. 1 guitars and 48. Cost complementary exits in a multiproduct cost function when a. Now we're ready to optimize. To do that, we make a function that gives us the wrongness of a particular set of thetas against our training data. In order to minimize the cost function, we can directly equate the gradient/derivative to zero and get the required value for 'm' and 'b' and this will give us the minimum cost. return_all, available for some methods, gives you the parameter vector vs generation, but not the cost function. In most cases, when you see a decorated function, the decorator is a factory function | {
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cost function. In most cases, when you see a decorated function, the decorator is a factory function that takes a function as argument and returns a new function that includes the old function inside the closure. Summary: The goal of the diet problem is to select a set of foods that will satisfy a set of daily nutritional requirement at minimum cost. Cost categories. 01 dollars to manufacture x Xbox 360s in a day. (1) Solve for the cost-minimizing input combination: (2) Depict the optimum in the diagram to the right. To obtain the cost function, add fixed cost and variable cost together. 3) Do not exhaust all system memory. Amid COVID-19, physicians, architects, and consultants are talking change in design — to be ready for next. Examples least-squares minimize kAx−bk2 2 • analytical solution x⋆ = A†b (A† is pseudo-inverse) • can add linear constraints, e. There are two parameters (coefficients) in our cost function we can control: weight $$m$$ and bias $$b$$. So we finally have cost as a function of x. Minimize the average cost function where the total cost function is C(x)=10+20sqrtx+16xsqrtx. prefer parameters that minimize the execution time. 3 is to be constructed in the shape of a rectangular box with a square base and an open top. - Answered by a verified Math Tutor or Teacher We use cookies to give you the best possible experience on our website. Return the absolute value. 1541765 To link to this article: https://doi. The available techniques to determine soil moisture content have practical limitations owing to their high cost, dependence on labor, and time consumption. Cost Function A company finds that it costs a total of to produce units of a new product. Rowe Price, troll, uspto. [email protected] Gradient Descent is THE most used learning algorithm in Machine Learning and this post will show you almost everything you need to know about it. the cost function itself!. Then why to use the. This value may be the expected net present value of a project or a | {
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itself!. Then why to use the. This value may be the expected net present value of a project or a forest property; or it may be the cost of a project; it could also be the amount of wood produced, the expected number of visitor-days at a park, the number of endangered species that will be. Eliminate Storage Complexity And Minimize Costs Enterprise IT is under unremitting pressure to reduce capital and operating expenses, driving them to virtualize infrastructure to improve hardware utilization and scalability and advance toward enhanced operational efficiency and flexibility. This elementary framew ork is the basis for a broad v ariety of mac hine learning. The math problem is: A large bin for holding heavy material must be in the shape of a box with an open top and a square base. However, this benefit comes at the cost of high computational complexity. The marketing manager should be interested in revealing the complexities of an individual buyer, the dynamics of consumer behavior and should also try to. #N#function J = computeCost ( X, y, theta) #N#%COMPUTECOST Compute cost for linear regression. I want that " t and T must be greater than zero(not equal to zero) , t < T and C > 0. 15-2P = 15-2(3)= 15-6=9-6+5P=-6+5(3)=-6+15=9. If you produce a certain amount and let's say you bring in, I don't know,$10,000 of revenue and it costs you $5,000 to produce those shoes, you'll have$5,000 in profit. This study proposes a new framework to minimize the cost function of multi-objective optimization problems by using NSGA-II in economic environments. To delineate CVX specifications from surrounding Matlab code, they are preceded with the statement cvx_begin and followed with the statement cvx_end. Since C0(x) = 30 − 253000 x2, then C. LP problems seek to maximize or minimize some quantity (usually profit or cost). the objective function (maximize/minimize) and. The goal of any Machine Learning model is to minimize the Cost Function. So in training your logistic regression | {
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any Machine Learning model is to minimize the Cost Function. So in training your logistic regression model, we're going to try to find parameters W and B that minimize the overall costs function J written at the bottom. How to minimise the cost function? Our goal is to move from the mountain in the top right corner (high cost) to the dark blue sea in the bottom left (low cost). Minimize operating costs and improve energy performance Data centers have to face continually increasing cost constraints. Gradient Descent basically just does what we were doing by hand — change the. From an external point of view, it is difficult to ascertain which are the alternative considered. It costs $10 to store one set for a year. The inventory cost problem, however, is something that comes up in real-life manufacturing scenarios all the time - how can I minimize my operating costs? In fact, the problem we see here today is a simplified version of a problem I covered in a DETC conference paper that I published a few years back. Take a deep breath. The point was more to introduce the reader to a specific method, not to the cost function specifically. You can use Pythagoras to compute S in terms of U in terms of S: U^2 = 500 2 + (4000 - S) 2 U = sqrt( 500 2 + (4000 - S) 2) Thus C(S) = S + 5 sqrt( 500 2. In order to minimize the cost function, we can directly equate the gradient/derivative to zero and get the required value for 'm' and 'b' and this will give us the minimum cost. **TL;DR**: Rearranging the terms in Maximum Mean Discrepancy yields a much better loss function for the discriminator of Generative Adversarial Nets. To determine the optimal amount of inputs (L and K), we solve this minimization constraint using the Lagrange multiplier method:. Inventory cost problems come up in real-life manufacturing scenarios all the time - how can I minimize my operating costs? Hot Bod Jacuzzi & Spa Company is launching a new hot tub - the Neverleak Massage-o-matic DeLux. Write a Cost | {
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Jacuzzi & Spa Company is launching a new hot tub - the Neverleak Massage-o-matic DeLux. Write a Cost Function. Thus, the C function represents the. The cost functions implemented in MIPAV: Correlation ratio. The use of closures and factory functions is the most common and powerful use for inner functions. Anthony Vu Patent, Patents "ask the patent attorney series", "The American Invents Act", aia, collateral estoppel, cost, inter partes review, issue preclusion, litigation, patent, patent litigation, price, secure Axcess, T. 2 Minimize 2 x 2 1 + 2 sub ject to x 1 + 2 =1, if w ec hange the righ t hand side from 1 to 1: 05 (i. How can equations and inequalities help a business maximize profit or minimize costs? Unanswered Questions. for one-variable real functions: limits, integrals, roots This is the main site of WIMS (WWW Interactive Multipurpose Server): interactive exercises, online calculators and plotters, mathematical recreation and games. (c) Test the C. Then it is going to become impossible to properly minimize or maximize the Cost Function. be used to minimize costs and what is the cost of producing that amount of chicks? f. cost of shipping + c21x21 + c22x22 + c23x23 from a plant + c31x31 + c32x32 + c33x33 to the ware house) Supply constraints. That is, the firm must choose a specific point on the q Cost Functions come directly from the production function and prices. In practice, these attractions are balanced in order to maintain a gap between the shaft (rotor) and static parts (stator). This firm minimizes its cost of producing any given output y if it chooses the pair (z 1, z 2) of inputs to solve the problem min z 1,z 2 w 1 z 1 + w 2 z 2 subject to y = F (z 1, z 2), where w 1 and w 2 are the input prices. (A) The Cost Function The cost-minimizing choice of inputs depended on two essential sets of parameters: the given output level (Y) and the given factor prices (r and w). Write an expression for the Cost in terms of only the width (w). Authors: Gaël | {
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prices (r and w). Write an expression for the Cost in terms of only the width (w). Authors: Gaël Varoquaux. 25 lines (16 sloc) 791 Bytes. Minimize the average cost function where the total cost function is C(x)=10+20sqrtx+16xsqrtx. There are two parameters (coefficients) in our cost function we can control: weight $$m$$ and bias $$b$$. 20, it cost$6. Operations > Time-Cost. The math problem is: A large bin for holding heavy material must be in the shape of a box with an open top and a square base. Indeed, it is the most powerful method available to reduce product cost, improve quality, and simultaneously reduce development interval. The minimization will be performed by a gradient descent algorithm, whose task is to parse the cost function output until it finds the lowest minimum point. Material indices Introduction The performance, p, Each function has an associated material index. 2x1 1 x2 1 x3 1 x4 x1, x2, x3, x4 $0. Organizations are relying on cloud to maintain business-critical processes, but the journey is not always seamless: you may be grappling with cloud governance and how to keep control over security, costs, risks. So our cost as a function of x is going to be 20x squared 36 times 5. What is the Malayalam name of tukmaria or sabja seed or falooda seed. Likely, many corporate leaders believe. In the case we are going to see, we'll try to find the best input arguments to obtain the minimum value of a real function, called in this case, cost function. Minimize costs The logistics market is characterized by higher standards for air pollution and noise as well as increasing toll fees, personnel costs and fuel prices. Output is produced according to the following process 2 1 2 1 K L = Firm Output (I chose the same function as above to simplify things). 1-Input the number. It's called the cost function, which is kind of a crappy name in this context. 20 to increase production from 49 to 50 units of output. " The problem also listed these following multiple | {
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production from 49 to 50 units of output. " The problem also listed these following multiple choice answers: a) 30,000 b) 300 c) 3,000 d) 30 e) None of these Now, we have the correct answer, what we need is the actual way to do this problem. The gradient descent algorithm in a nutshell. To do this, take the derivative of C(x), set it equal to zero, and solve for x. The diet problem constraints typically regulate the number of calories and the. The optimization continues as the cost function response improves iteration by iteration. optimize for black-box optimization: we do not rely on the. Put simply, a cost function is a measure of how wrong the model is in terms of its ability to estimate the relationship between X and y. Cost Minimization: Short Run • Let us go back to the two-inputs case, with only one of them variable in the short run. We have contributed on a local. Thus you know that the cost is C(S,U) = S + 5U. 9 drums to minimize his costs. Well, your profit as a function of x is just going to be equal to your revenue as a function of x minus your cost as a function of x. C CL(q) combination of inputs that minimize the cost of producing each. Fundamental theorem of linear programming If the optimal (maximum or minimum) value of the objective function in a. Reynolds Consumer Products Inc. The aim of the linear regression is to find a line similar to the blue line in the plot above that fits the given set of training example best. This website uses cookies to ensure you get the best experience. Solution: We would like to find a function that describes this situation. 4 (GP) : minimize f (x) s. "I tried a lot but I am not getting the values of t and T as mentioned above " ". The objective function J = f(x) is augmented by the constraint equations through a set of non-negative multiplicative Lagrange multipliers, λ j ≥0. We often design algorithms for GP by building a local quadratic model of f (·)atagivenpointx =¯x. 4x + 150 t?o model the unit cost in dollars | {
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a local quadratic model of f (·)atagivenpointx =¯x. 4x + 150 t?o model the unit cost in dollars for producing x stabilizer bars. INTRODUCTION Linear programming is a mathematical technique used to find the best possible solution in allocating limited resources (constraints) to achieve maximum profit or minimum cost by modelling linear relationships. nan with np. The goal of any Machine Learning model is to minimize the Cost Function. These functions can be seen as covering functions which have many applications in di erent optimization prob-lems: Set Cover functions, Edge Cut functions in graphs, etc. For example, this formula will find the highest value in cells H2:H17 =MAX(H2:H17) MIN IF Formula. Mathematical optimization: finding minima of functions¶. In order to minimize the cost function, we can directly equate the gradient/derivative to zero and get the required value for 'm' and 'b' and this will give us the minimum cost. How much food should be used to minimize costs and what is the total cost? 2. Thus the function huber can be used anywhere a traditional convex function can be used, in constraints or objective functions, in accordance with the DCP ruleset. Minimize the cost to split a number Given an integer N ≥ 2 , you can split the number as a sum of k integers i. Sourcing, procurement and vendor management leaders should use this research to navigate GDPR requirements and Microsoft’s licensing to avoid legal and financial risks. The cost functions implemented in MIPAV: Correlation ratio. The transaction cost function is not di erentiable at the kink points and is piecewise continuous. Human Resource Management Functions. You will notice that as in the case of the factor demand functions, there is a. 1 guitars and 48. That is, the quantity you want to maximize or minimize is called the objective function. The Dual of the Minimum Cost Flow Problem:. C CL(q) combination of inputs that minimize the cost of producing each. Then why to use the. That's | {
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C CL(q) combination of inputs that minimize the cost of producing each. Then why to use the. That's incredible but understandable when you start adding up all the "standard" wedding costs. So it's going to be plus 180 times, let's see, x times x to the negative 2, 180x to the negative x to the negative 1 power. The specific goal is to approximate a single valued function of one variable in terms of a sequence of linear segments. To minimize energy content, use the above criteria for. The two interesting exceptions to this rule are:. How to Minimize Legal Liabilities and Risks Information throughout this subsection applies primarily to external consultants. 4 — Logistic Regression | Cost Function — [ Machine Learning | Andrew Ng] - Duration: 11:26. What is the Objective Function? The objective of a linear programming problem will be to maximize or to minimize some numerical value. Assume that is costs Microsoft approximately C x x x 2 14,400 550 0. Hi , I am using FMINCON to minimize my cost function which is a product of elements of a matrix. An optimization problem seeks to minimize a loss function. To do that, we make a function that gives us the wrongness of a particular set of thetas against our training data. An extra large server costs you$0. Minimize F x y 22 with xy 2 10. using linear algebra) and must be searched for by an optimization algorithm. Find more Statistics & Data Analysis widgets in Wolfram|Alpha. HRM is the systematic planning and control of a network of fundamental organizational processes affecting and involving all organization members (French, 2004, p. However, this benefit comes at the cost of high computational complexity. For what number of bars is the unit cost at its mimimum? What is the unit cost at that level of production? Haven't got a clue what this problem is asking of me. A feasible solution that minimizes (or maximizes, if that is the goal) the objective function is called an optimal. How much are closing costs? These are the | {
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is the goal) the objective function is called an optimal. How much are closing costs? These are the fees paid that help facilitate the sale of a home typically total 2% to 7% of the home's purchase price. applied optimization calc 1. If the material for the sides costs 15¢/in. In this article, I will be going through the basic mathematics behind K-Means Algorithm. For multi-objective improvements, the most generally used developmental algorithms such as NSGA-II, SPEA2 and PESA-II can be utilized. A cost function is a MATLAB ® function that evaluates your design requirements using design variable values. In this paper, we have applied some meta scheduling methods to a model of CIM that is referred to as an automated flow shop, where backward scheduling should be used to realize a JIT's theory. Consider the same open-top box, which is to have volume $$216in. Assume we are given a dataset as plotted by the 'x' marks in the plot above. This is the radius which will minimize the surface area and thus the cost of materials. costs into account. Artificial Intelligence - All in One 87,390 views 11:26. Chap 7: Short-Run Cost Function 2. Cost categories. 6 - Linear Programming. The material that will be used for three sides costs 30 per linear foot, and the material that will be used for the fourth side costs 15 per linear foot. So our cost as a function of x is going to be 20x squared 36 times 5. Decide what the variables are and what the constants are, draw a diagram if appropriate, understand clearly what it is that is to be maximized or minimized. 5 Q 2 v What is the marginal revenue function?. The Dual of the Minimum Cost Flow Problem:. Raw Blame History. 4 million will be recognized as a component of. Define a MATLAB function to evaluate −f(x) given x. In my opinion, the #1 cost to avoid is the “retaker” cost. I want that " t and T must be greater than zero(not equal to zero) , t < T and C > 0. Dear Sir Can you please help me to minimize the following cost function with | {
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, t < T and C > 0. Dear Sir Can you please help me to minimize the following cost function with maple 10. This function is known as the cost function and will be of considerable interest to us. Does the production function exhibit increasing, decreasing, or constant returns to scale? How can you tell? b. Given a function defined by a set of parameters, gradient descent starts with an initial set of parameter values and iteratively moves toward a set of parameter values that minimize the function. h(θ) is the the prediction from your regression model. Let's take a more in depth look at the cost function and see how it works. Examples least-squares minimize kAx−bk2 2 • analytical solution x⋆ = A†b (A† is pseudo-inverse) • can add linear constraints, e. Users who have contributed to this file. This study proposes a new framework to minimize the cost function of multi-objective optimization problems by using NSGA-II in economic environments. The available techniques to determine soil moisture content have practical limitations owing to their high cost, dependence on labor, and time consumption. (d) Find the minimum value of the marginal cost. A cost function is a MATLAB ® function that evaluates your design requirements using design variable values. Ultimately, to minimize our cost, we need to find the point with the lowest z value. You can also optimize the objective function without any loss function, e. Obtain the minimum using fmin=fminsearch(fun,x0) Maximization 1. One common application of calculus is calculating the minimum or maximum value of a function. With so many options to choose from, the best iPhone XR case can be elusive. Currently, minimize lacks the ability to do this. To do: Try the following example: Given: Q = L 1/2 K 1/2 PL = 4, PK = 1 Goal: Produce Qo = 16 units as cheaply as possible. (Remember, the average cost, (6 pts. For example, if the marginal cost of producing the 50th product is 6. If the firm ordered the item, then the setup cost is | {
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cost of producing the 50th product is 6. If the firm ordered the item, then the setup cost is simply the order cost from Module 5. Cost Function - Intuition I11:09. They would like to offer some combination of milk, beans, and oranges. So, if you employ tactics to reduce costs in all discrete functions from manufacturing through delivery, you'll have a lower total landed cost, right? Theoretically, yes. Now, to minimize marginal cost. If the brewery produces sweet stout alone, the cost function is: CS(q2) = 8q2. Homework Statement Mary Jane grows herbs in her attic. Previous work. The Cost Function If lattes and cake (or labor and capital) have unit prices of pL and pK, respec-tively, then the total cost of purchasing L units of one and K units of the other is C(L,K) = pLL+pKK. I think relative price of L & K is (Cost of Labour Per Hour)/(Cost of Rent Per Hour), but I don't know the price of Rent Per Hour. pdf), Text File (. Next time I will not draw mspaint but actually plot it out. The problem is that officers work 8 hour shifts, yet the demand comes in 4 hour chunks. 5 Actionable Tips to Reduce Operational Costs Regardless of what the circumstances are for your business, it is always a priority to find ways to reduce operational costs. k) 0 is a (nonnegative) function for which (x k;x k) = 0;then the following function defines a majorizer for : ˚ k(x) , (x)+ (x;x k): (4. Generally speaking, Least-Squares Method has two categories, linear and non-linear. The available techniques to determine soil moisture content have practical limitations owing to their high cost, dependence on labor, and time consumption. There can be significant cost savings when a business function is outsourced. Find the level of production which will minimize the average cost per item. 02xSquared - 3. Instead, it is allowable to use a cost flow assumption that varies from actual usage. Cost & Time and Also Minimum Project Duration Using Alternative Method 405 coordinates of the normal and | {
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& Time and Also Minimum Project Duration Using Alternative Method 405 coordinates of the normal and crash points: Cost slope = (crash cost-normal cost)/ (normal duration crash duration) As the activity duration is reduced, there is an increase in direct cost. As the magnitues of the fitting parameters increase, there will be an increasing penalty on the cost function. The extent of risk and liability in your work depends on the nature of your services. Hi , I am using FMINCON to minimize my cost function which is a product of elements of a matrix. Of course, since time is money in any manufacturing process, what this really means is that looking into ways of reducing cycle time in your injection molding process can have a major impact on. What are loss functions? And how do they work in machine learning algorithms? Find out in this article. In the case we are going to see, we'll try to find the best input arguments to obtain the minimum value of a real function, called in this case, cost function. Fundamental theorem of linear programming If the optimal (maximum or minimum) value of the objective function in a. Common benzodiazepines used for GAD include alprazolam, clonazepam, diazepam, and lorazepam. For what number of bars is the unit cost at its mimimum? What is the unit cost at that level of production? Haven't got a clue what this problem is asking of me. Actual costs refer to real transactions, wherease opportunity costs refer to the alternative taken into consideration by decision makers who might want to choose the line of activity which minimise the costs. Average cost is minimized when average cost = marginal cost is another saying that isn’t quite true; in this case, the correct statement is: Average Cost has critical points when Average Cost and Marginal Cost are equal. LP problems seek to maximize or minimize some quantity (usually profit or cost). Find the dimensions that will minimize the cost of the box's construction. Therefore the total cost is: | {
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the dimensions that will minimize the cost of the box's construction. Therefore the total cost is: C(x) = 10y +15(2x+y) = 30x+25y. We need to decide which sub-contractor to use for a critical activity. Hence we want to minimize the can's surface area. We refer to this property as the objective function of an LP problem. In this case, the function huber will contain a special Matlab object that represents the function call in constraints and objectives. An objective function is either a loss function or its negative (in specific domains, variously called. It's important to limit your number of serverless functions to avoid having a massive charge when a lot of work is presented at the same time. The average cost of producing one output is reduced when the output of another product is increased If the wage rate is 5 and the price of capital is 2, then in order to minimize costs the firm should use a. 5 kg is the same linear function for a mass change of 2000 kg. For example, companies often want to minimize production costs or maximize revenue. The objective of the purchasing function is to obtain proper material and services when needed at the lowest obtainable cost. 1, Major functions of an institution) on the basis of modified total direct costs (MTDC), consisting of all salaries and wages, fringe benefits, materials and supplies, services, travel, and up to the. txt) or read online for free. Minimize cost and maximize quality of function in four variables. We are the prime contractor and there is a penalty in our contract with the main client for every day we deliver late. In the following example I will minimize an arbitrary function [texi]J[texi], then in the next chapter I'll apply it to the original house pricing task. Loss functions are actually at the heart of these techniques that we regularly use. (1) Solve for the cost-minimizing input combination:. The aim of the linear regression is to find a line similar to the blue line in the plot above that fits the | {
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of the linear regression is to find a line similar to the blue line in the plot above that fits the given set of training example best. 5 Q 2 v What is the marginal revenue function?. The major objective of a typi-cal firm is to maximize dollar profits in the long run. Variables and functions should be declared in the minimum scope from which all references to the identifier are still possible. Let's take a more in depth look at the cost function and see how it works. This is where we look back at equation (1) and solve for h in terms of w. We advise on the largest and most complex legal challenges facing the world’s most important companies. the cost function itself!. Set big goals, insist on a cultural shift, and model from the top. The objective function is the function to be minimized or maximized. Use a computer to maximize the objective function subject to the constraints where 38. Thus, applications of HRM theory differ from personnel management in their dismissal of prescriptive “one best way” models of practice as diverse. Minimising Cost function. Overall, closures have affected 25% of pork production and 10% of beef production in the U. Objective function. An optimization problem is one where you have to make the best decision (choose the best investments, minimize your company's costs, find the class schedule with the fewest morning classes, or so on). That is h = 50 3w2 (3) Plugging the value for h from (3) above into equation (2) yields C = 60w2 + 48w. 2, what should the dimensions of the cup be to minimize the construction cost?. Yes, but not by playing it safe. Actually, the objective function is the function (e. Minimizing any function means finding the deepest valley in that function. A cost function is defined as: …a function that maps an event or values of one or more variables onto a real number intuitively representing some “cost” associated with the event. Examples and exercises on the cost function for a firm with two variable inputs Example: | {
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the event. Examples and exercises on the cost function for a firm with two variable inputs Example: a production function with fixed proportions Consider the fixed proportions production function F (z 1, z 2) = min{z 1, z 2} (one worker and one machine produce one unit of output). Variable costs are such cost which vary directly with change in output. Recall in the calculus of one variable, if y = f(x) is defined on a set S, then there is a relative maximum value at x0 if f(x0) ≥ f(x) for all x in S near x0, and there is a relative. If a firm has a production function Q=F(K,L) (that is, the quantity of output (Q) is some function of capital (K) and labor (L)), then if 2Q V/(π r^2) Find r when the slope of the area is zero:. His next-door neighbor agrees to pay for half of the fence that borders her property; Sam will pay the rest of the cost. The nonlinearity in this form generates from the absolute value function. In most examples/tutorial I followed, the cost function used was somewhat arbitrary. Coming up with a cost function for optimization for a complex control system Hot Network Questions Is there a word or phrase for one mistaken belief leading to a web of false ones?. However, this benefit comes at the cost of high computational complexity. : residuals) between our model and our data points. One common application of calculus is calculating the minimum or maximum value of a function. Take Exam Only When You are Ready. the production function and the cost function; the only difference is whether we hold production constant or cost constant. [email protected] I would like to use the goal seek function to minimize the value in a certain cell (total cost) by changing another the value in another cell (shipment size). Then why to use the. This firm minimizes its cost of producing any given output y if it chooses the pair (z 1, z 2) of inputs to solve the problem min z 1,z 2 w 1 z 1 + w 2 z 2 subject to y = F (z 1, z 2), where w 1 and w 2 are the input prices. | {
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min z 1,z 2 w 1 z 1 + w 2 z 2 subject to y = F (z 1, z 2), where w 1 and w 2 are the input prices. The Cost Function If lattes and cake (or labor and capital) have unit prices of pL and pK, respec-tively, then the total cost of purchasing L units of one and K units of the other is C(L,K) = pLL+pKK. Find the dimensions that will minimize cost. Employee compensation costs, office space expenses and other costs associated with providing a workspace or manufacturing setup are eliminated and free up resources for other purposes. We often design algorithms for GP by building a local quadratic model of f (·)atagivenpointx =¯x. (d) Find the minimum value of the marginal cost. Cost function is the sum of losses from each data point calculated with loss function. Solving for the minimum 0 points minimize f, (z) = 20 + z2-cos(2TZ) Given the cost function f. S, according to Bloomberg News. To minimize energy content, use the above criteria for. c<=2 n=1-1000 0<=p<=0. It's called the cost function, which is kind of a crappy name in this context. For example, if the marginal cost of producing the 50th product is 6. Minimising Cost function. You’ve been ordered to reduce your department’s costs by 10%, 20%, or 30%. It is obvious that if we changed relative factor. 5\text{ x }10^6 \text{ ft}^2) in an a rectangular field and then divide it in half with a fence parallel to one of the sides of the rectangle. Having drawn the picture, the next step is to write an equation for the quantity we want to optimize. profits and minimize costs by using cost-revenue-profit functions. **TL;DR**: Rearranging the terms in Maximum Mean Discrepancy yields a much better loss function for the discriminator of Generative Adversarial Nets. 188, and the slope was not significantly different from 0. By using this website, you agree to our Cookie Policy. “Despite the challenges of the COVID-19. Take the derivative of the Cost with respect to width. Average Cost Per Unit Formula. Many of these materials are | {
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of the Cost with respect to width. Average Cost Per Unit Formula. Many of these materials are high in quality and low in cost. Find the Average Cost Function and Minimize the Average Cost. Outsourcing can also make your firm more attractive to investors, since you're able to pump more capital directly into revenue-producing activities. That’s the main point of any model, to minize error, to perform. How many Xboxes should be manufactured in order to minimize average cost? What is the resulting average cost of an Xbox? Give your answer to the nearest dollar. In this case, the objective is to minimize the total cost per day which is given by z= 0:6x 1 + 0:35x 2 (the value of the objective function is often denoted by z). Cost Function8:12. Indeed, it is the most powerful method available to reduce product cost, improve quality, and simultaneously reduce development interval. If this sounds a lot, here are my 5 suggestions to reduce the CPA exam cost: 1. It is the heart that makes it beat! There is a loss function, which expresses how much the estimate has missed the mark for an individual observation. The use of closures and factory functions is the most common and powerful use for inner functions. The cost function: E(Cost)=E(F-LS) 3 F is for Finished goods L is for Lambda S is for Sales After expanding the function, what assumption minimized this function with respect to F? The Attempt at a Solution F 3-3F 2 LS+3F(LS) 2-(LS) 3 I know that I need to identify the terms that include both sales and inventories. f ( x) = x 4 − 8 x 2 + 5. h(θ) is the the prediction from your regression model. output, marginal cost, average cost, price, and profit at the average-cost minimizing activity level profit-maximizing or loss-minimizing output Calculus Cost/Graph cost function Optimal capital structure to minimize cost of capital Output, Profit, Fixed Costs and Perfect Competition Finding Optimal Output Level etc. Example 4 If the total revenue and total cost functions are TR = | {
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Finding Optimal Output Level etc. Example 4 If the total revenue and total cost functions are TR = 30Q – 5Q 2 and TC = 15 + 12Q - 0. A company producing goods wants to minimize the average cost of production. Mathematical optimization deals with the problem of finding numerically minimums (or maximums or zeros) of a function. The global six-year average cost of a data breach is 3. Employee compensation costs, office space expenses and other costs associated with providing a workspace or manufacturing setup are eliminated and free up resources for other purposes. Yes, even despite having so much support from ml-class … they practically implement everything and just leave the cost and gradient functions up to you to implement. In order to minimize the cost function, we can directly equate the gradient/derivative to zero and get the required value for 'm' and 'b' and this will give us the minimum cost. Mathematical optimization: finding minima of functions¶. Keep in mind that, the cost function is used to monitor the. To open Outlook again after it has been hidden on the system tray and disappeared from the taskbar, double-click the Outlook system tray icon. Thus the function huber can be used anywhere a traditional convex function can be used, in constraints or objective functions, in accordance with the DCP ruleset. Minimize the total physical effort & thus the cost of moving goods into & out of storage. Budgeting for your company’s training needs does not mean using surplus money when you have it. Diesel particulate filters (DPF) are devices that physically capture diesel particulates to prevent their release to the atmosphere. Authors: Gaël Varoquaux. There are many factors to consider when selecting components and board-level solutions for a real-time embedded system. 1 Where f is the number of facilities. Define a MATLAB function to evaluate −f(x) given x. , a function that takes a scalar as input) is needed. Good parameters means that the function can produce the | {
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that takes a scalar as input) is needed. Good parameters means that the function can produce the best possible outcomes, namely the smallest ones, because small values mean less errors. Cij = transportation cost per unit of shipping from plant Pi to the Warehouse Wj. Lagrange multiplier methods involve the modification of the objective function through the addition of terms that describe the constraints. Mathematical optimization: finding minima of functions¶ Authors: Gaël Varoquaux. Min & Max of Functions - MATLAB Minimization 1. Gradient descent is a more generic algorithm, used not only in linear regression problems and cost functions. Along with source pipeline, candidate quality, long-term retention, and other key performance metrics, tracking your cost-per-hire will help you to understand the performance of your recruiting initiatives better and minimize your expenses across the board. Univariate function minimizers (minimize_scalar)¶ Often only the minimum of an univariate function (i. A company producing goods wants to minimize the average cost of production. 15-2P = 15-2(3)= 15-6=9-6+5P=-6+5(3)=-6+15=9. My constriants are also in the form of matix. For example, this formula will find the highest value in cells H2:H17 =MAX(H2:H17) MIN IF Formula. 2% over budget). Then, minimize that slack variable until the slack is null or negative. We have contributed on a local. Minimize an objective function whose values are given by executing a file. Real-time embedded systems require. Grab a coffee. The cost function: E(Cost)=E(F-LS) 3 F is for Finished goods L is for Lambda S is for Sales After expanding the function, what assumption minimized this function with respect to F? The Attempt at a Solution F 3-3F 2 LS+3F(LS) 2-(LS) 3 I know that I need to identify the terms that include both sales and inventories. Profit is simply the Total revenue minus the costs incurred. Gradient descent is simply used to find the values of a function's parameters (coefficients) that | {
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Gradient descent is simply used to find the values of a function's parameters (coefficients) that minimize a cost function as far as possible. How To: Calculate and use regression functions in statistical analysis How To: Write a logarithm as a sum or difference of logarithms How To: Perform a quadratic regression with a calculator How To: Calculate r-squared or coefficient of determination in statistics. Of course, since time is money in any manufacturing process, what this really means is that looking into ways of reducing cycle time in your injection molding process can have a major impact on. More generally, a lower semi-continuous function on a compact set attains its minimum; an upper semi-continuous function on a compact set attains its maximum point or view. If you produce a certain amount and let's say you bring in, I don't know, 10,000 of revenue and it costs you 5,000 to produce those shoes, you'll have 5,000 in profit. However, by substituting for , the problem can be transformed into a linear problem. An optimization problem seeks to minimize a loss function. This is the personal website of a data scientist and machine learning enthusiast with a big passion for Python and open source. 1080/09715010. costs into account. If Minimize is given an expression containing approximate numbers, it automatically calls NMinimize. the firm hires labor, and the cost is the wage rate that must be paid for the labor services Total cost (TC) is the full cost of producing any given level of output, and it is divided into two parts: • Total fixed cost. What is the best nursing intervention to minimize the adverse effects of this. The production process can often be described with a set of linear inequalities called constraints. Simplex Algorithm Calculator is an online application on the simplex algorithm and two phase method. Find the number of units, x, that will minimize the average cost function if the total cost function is C()3+7+ 75. It's important to limit your | {
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the average cost function if the total cost function is C()3+7+ 75. It's important to limit your number of serverless functions to avoid having a massive charge when a lot of work is presented at the same time. The following figure (right) shows a plot of a sample cost function for a selection of transformation parameters. When you optimize or estimate model parameters, you provide the saved cost function as an input to sdo. It's a cost function because the errors are "costs", the less errors your model give, the better your model is. 2) A business’ costs include the fixed cost of 5000 as well as the variable cost of 40 per bike. One common application of calculus is calculating the minimum or maximum value of a function. Derive Draper Dan's cost function (a) in terms of input prices and output and (b) when the price of cloth, w 1, is 3/metre and the wage rate w 2 is 10 per hour. When x = 18, y = 9. Otherwise, they must be considered separately. In this article, I will be going through the basic mathematics behind K-Means Algorithm. Economic Order Quantity Model (EOQ) Managing inventory is an important task for every business that holds it. A piecewise linear approximation is one method of constructing a function that fits a nonlinear objective function by adding extra binary variables, continuous variables, and constraints to reformulate the original problem. To find ways to save money, take advantage of quick cost-saving measures followed by an intensive look at where IT is spending money. What happens when the learning rate is too small? Too large? Using the best learning rate that you found, run gradient descent until convergence to find 1. Using the quadratic formula or a calculator, we find the solutions are. As serverless architectures mature, they have been able to minimize the issue of provision concurrency, in which there was a performance penalty when a function was called a second time, causing a. It is the heart that makes it beat! There is a loss | {
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a function was called a second time, causing a. It is the heart that makes it beat! There is a loss function, which expresses how much the estimate has missed the mark for an individual observation. If this sounds a lot, here are my 5 suggestions to reduce the CPA exam cost: 1. 37 e) The minimal average cost. Put simply, a cost function is a measure of how wrong the model is in terms of its ability to estimate the relationship between X. Minimize The Use Of Color In Wireframes. Find the level of production which will minimize the average cost per item. So it's going to be plus 180 times, let's see, x times x to the negative 2, 180x to the negative x to the negative 1 power. Question 107995: Minimizing Cost, A company uses the formula C(x)=0. lute extrema of the function y = 2x,一2x2-16x + 1 on [-2,3]. Given a function defined by a set of parameters, gradient descent starts with an initial set of parameter values and iteratively moves toward a set of parameter values that minimize the function. In order to minimize the cost function, we can directly equate the gradient/derivative to zero and get the required value for 'm' and 'b' and this will give us the minimum cost. Optimization methods in Scipy nov 07, 2015 numerical-analysis optimization python numpy scipy. A midwife usually offers a variety of options and seeks to eliminate or minimize unnecessary interventions. Once you have installed CVX (see Installation), you can start using it by entering a CVX specification into a Matlab script or function, or directly from the command prompt. Chapter 7: The Cost of Production. Minimizing Inventory Costs. Instead, it is allowable to use a cost flow assumption that varies from actual usage. Then again, Octave provides tools for learning where you essentially just run a function, tell it where to find the cost and gradient function and give it some data. Midwives: Benefits of Having a Midwife. Most optimization problems have a single objective function, if they do not, they | {
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Having a Midwife. Most optimization problems have a single objective function, if they do not, they can often be reformulated so that they do. For the given cost function C(x)=78400+500x+x^2 find: a) The cost at the production level 1700 b) The average cost at the production level 1700 c) The marginal cost at the production level 1700 d) The production level that will minimize the average cost e) The minimal average cost I can already answer a, b, and c, it's d and e I can't seem to get, I know it should be let c'(x) = 0, but the. In this context, the function is called cost function, or objective function, or energy. The calculator is intended to teach students the Simplex method and to relieve them from some of the tedious aritmetic. We want to minimize the cost of the materials subject to the constraint that the volume must be 50ft 3. In machine learning, we use gradient descent to update the parameters of our model. The cost function used is shown aboveWe want to find parameters Ɵ which minimize J(Ɵ) To do so we can use one of the algorithms already described such as; Gradient descent; Advanced optimization algorithmsTo minimize a cost function we just write code which computes the following J(Ɵ) i. Obviously, a conservative deflection limit can be specified to minimize deflection, assuming design and construction is then performed correctly. I want that " t and T must be greater than zero(not equal to zero) , t < T and C > 0. To find ways to save money, take advantage of quick cost-saving measures followed by an intensive look at where IT is spending money. We discuss the application of linear regression to housing price prediction, present the notion of a cost function, and introduce the gradient descent method for learning. It is a minimization problem. The optimization continues as the cost function response improves iteration by iteration. Then it is going to become impossible to properly minimize or maximize the Cost Function. The Total Cost of Ownership | {
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to become impossible to properly minimize or maximize the Cost Function. The Total Cost of Ownership (TCO) of your IP is staggering. 20, it cost 6. 01 dollars to manufacture x Xbox 360s in a day. To demonstrate the minimization function, consider the problem of minimizing the Rosenbrock function of the NN variables −. Take Exam Only When You are Ready. Cost complementary exits in a multiproduct cost function when a. As it stands, though, it has two variables, so we need to use the constraint equation. The slope of iso cost line = PL/Pk. Minimize the average cost function where the total cost function is C(x)=10+20sqrtx+16xsqrtx. The statement dual variables y{n} allocates a cell array of \(n$$ dual variables, and stores the result in the Matlab variable Z. For example, a random sample of a population of young offenders is generated by selecting names from a list to interview. The cost function is just a mathematical formula that gives the total cost to produce a certain number of units. Mathematical optimization: finding minima of functions¶. Custom & Stock Plastic Packaging Solutions. 1-Input the number. 2x1 1 x2 1 x3 1 x4 x1, x2, x3, x4 $0. Return the arc cosine. stiff, light beam in bending –minimize ρ/E1/2 •e. 8 trillion annually in aggregate general and administrative (G&A) expenses. Labor Union Vs. Lagrange multiplier methods involve the modification of the objective function through the addition of terms that describe the constraints. Next time I will not draw mspaint but actually plot it out. h(θ) is the the prediction from your regression model. Minimize the total physical effort & thus the cost of moving goods into & out of storage. Material indices Introduction The performance, p, Each function has an associated material index. I recently had to implement this from scratch, during the CS231 course offered by Stanford on visual recognition. Example 4 If the total revenue and total cost functions are TR = 30Q – 5Q 2 and TC = 15 + 12Q - 0. Note that w 1, w 2, | {
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revenue and total cost functions are TR = 30Q – 5Q 2 and TC = 15 + 12Q - 0. Note that w 1, w 2, and y are given in this. This will give the quantity (q) that maximizes profits, assuming of course that the firm has already taken steps to minimize costs. An isoquant and possible isocost line are shown in the following figure. Advantages of Outsourcing Cost Savings. Good parameters means that the function can produce the best possible outcomes, namely the smallest ones, because small values mean less errors. In machine learning, we use gradient descent to update the parameters of our model. The purposes of a human resources department and a labor union are decidedly different. One common application of calculus is calculating the minimum or maximum value of a function. Model Representation8:10. Budgeting for your company’s training needs does not mean using surplus money when you have it. How many players should be produced to minimize the marginal cost? and (b). The cost function equation is expressed as C(x)= FC + V(x), where C equals total production cost, FC is total fixed costs, V is variable cost and x is the number of units. Suppose the marginal cost C(in dollars) to produce x thousand mp3 players is given by the function C(x)=x^2-100x+7600. So we have written the cost as a function of two variable, height and width. cost of shipping + c21x21 + c22x22 + c23x23 from a plant + c31x31 + c32x32 + c33x33 to the ware house) Supply constraints. Example: A retail appliance store sells 2500 TV sets per year. To determine the optimal amount of inputs (L and K), we solve this minimization constraint using the Lagrange multiplier method:. 01x 2 +120 ) dollars where x represents the number of units produced. It is possible to attach a more substantial penalty to the predictions that are located above or below the expected results (some cost functions do so, e. Instead, it is allowable to use a cost flow assumption that varies from actual usage. Constrained Optimization using | {
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to use a cost flow assumption that varies from actual usage. Constrained Optimization using Lagrange Multipliers 5 Figure2shows that: •J A(x,λ) is independent of λat x= b, •the saddle point of J A(x,λ) occurs at a negative value of λ, so ∂J A/∂λ6= 0 for any λ≥0. Your business should be doing the same. Gradient descent is an optimization algorithm used to find the values of parameters (coefficients) of a function (f) that minimizes a cost function (cost). Ignoring any other costs, find the optimal number of facilities with the. Note as well that the cost for each side is just the area of that side times the appropriate cost. Mathematical optimization is the selection of the best input in a function to compute the required value. Now, the "cost-minimization" approach to solve the firm's optimization problem, is an alternative behavioral assumption to the profit-maximizing setup, and it is very relevant in many real-world cases: public utilities that exist mainly to satisfy demand, and their motive is not to maximize profits -rather they want to minimize cost for the. To maximize the revenue function To minimize the cost function To maximize the profit function. Calculus Optimization Problem: What dimensions minimize the cost of a garden fence? Sam wants to build a garden fence to protect a rectangular 400 square-foot planting area. 25 lines (16 sloc) 791 Bytes. A low-cost provider is a powerful competitive approach in markets where many buyers are price sensitive. The problem is formulated as a linear program where the objective is to minimize cost and the constraints are to satisfy the specified nutritional requirements. find the location of the minimum of fr, z. That's incredible but understandable when you start adding up all the "standard" wedding costs. x11 + x12 + x13 = S1 x21 + x22. Firms can change all their inputs, both labor and capital, in the. Optimization- What is the Minimum or Maximum? 3. Hi , I am using FMINCON to minimize my cost function which is a | {
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What is the Minimum or Maximum? 3. Hi , I am using FMINCON to minimize my cost function which is a product of elements of a matrix. For example, if the marginal cost of producing the 50th product is$6. In this case, the objective is to minimize the total cost per day which is given by z= 0:6x 1 + 0:35x 2 (the value of the objective function is often denoted by z). 4x + 150 t?o model the unit cost in dollars for producing x stabilizer bars. The average cost of producing one output is reduced when the output of another product is increased If the wage rate is $5 and the price of capital is$2, then in order to minimize costs the firm should use a. So to recapHypothesis - is like your prediction machine, throw in an x value, get a putative y value. In order to minimize the cost function, we can directly equate the gradient/derivative to zero and get the required value for 'm' and 'b' and this will give us the minimum cost. If the time rate of change of this function is held constant between 0 and 0. A low-cost provider is a powerful competitive approach in markets where many buyers are price sensitive. To do this, take the derivative of C(x), set it equal to zero, and solve for x. Outsourcing can also make your firm more attractive to investors, since you're able to pump more capital directly into revenue-producing activities. The optimization continues as the cost function response improves iteration by iteration. IT continues to be a focal point for cost reduction in organizations, but cutting costs blindly can cause serious damage to IT and the business. In the general cost function problem, there is a function g: R+!R+ given, and the goal of the scheduler is to minimize P i2[n] w ig(F i). If x engines are made, then the unit cost is given by the function C(x)=x^2-560x+94,717. To minimize the deviation, the problem is formulated in a basic form as: as the objective function, and linear constraints are. 8 trillion annually in aggregate general and administrative (G&A) | {
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and linear constraints are. 8 trillion annually in aggregate general and administrative (G&A) expenses. A production function, such as the Cobb-Douglas production function, can be used to model how a firm combines inputs to produce outputs; other production functions include the CES, Translog, and Diewert (Generalized Leontief); interactive and online models of production functions. When you work for yourself or need your vehicle for work, time can cost you income. We need to decide which sub-contractor to use for a critical activity. More labor and less capital c. Formal Derivation of Cost Curves from a Production Function: Rearranging the expression above we obtain: This is the cost function, that is, the cost expressed as a function of: (i) Output, X; (ii) The production function coefficients, b 0, b 1, b 2; (clearly the sum b 1 + b 2 is a measure of the returns to scale); (iii) The prices of. So the terminology I'm going to use is that the loss function is applied to just a single training example like so. Formware was purpose-built to model high penetration renewables at the system level and determine how all types of storage enable cost-effective renewable energy integration. We discuss the application of linear regression to housing price prediction, present the notion of a cost function, and introduce the gradient descent method for learning. output, marginal cost, average cost, price, and profit at the average-cost minimizing activity level profit-maximizing or loss-minimizing output Calculus Cost/Graph cost function Optimal capital structure to minimize cost of capital Output, Profit, Fixed Costs and Perfect Competition Finding Optimal Output Level etc. Minimize the potential for bias in the selection of the sample through random sampling. The average wedding costs \$30,000. The final detailed cost estimate contains the material, labor and in-directs data for controlling project costs. The extreme value theorem of Karl Weierstrass states that a continuous | {
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controlling project costs. The extreme value theorem of Karl Weierstrass states that a continuous real-valued function on a compact set attains its maximum and minimum value. 1-Input the number. The function f is called, variously, an objective function, a loss function or cost function (minimization), a utility function or fitness function (maximization), or, in certain fields, an energy function or energy functional. Formal Derivation of Cost Curves from a Production Function: Rearranging the expression above we obtain: This is the cost function, that is, the cost expressed as a function of: (i) Output, X; (ii) The production function coefficients, b 0, b 1, b 2; (clearly the sum b 1 + b 2 is a measure of the returns to scale); (iii) The prices of. costs into account. A nurse is caring for a postsurgical patient who has small tortuous veins and had a difficult IV insertion. Return the arc sine. 1, Major functions of an institution) on the basis of modified total direct costs (MTDC), consisting of all salaries and wages, fringe benefits, materials and supplies, services, travel, and up to the. Artificial Intelligence - All in One 87,390 views 11:26. In this blog post, you will learn how to implement gradient descent on a linear classifier with a Softmax cross-entropy loss function. To do that, we make a function that gives us the wrongness of a particular set of thetas against our training data. Assume that is costs Microsoft approximately C x x x 2 14,400 550 0. Linear regression predicts a real-valued output based on an input value. One common application of calculus is calculating the minimum or maximum value of a function. The cost functions implemented in MIPAV: Correlation ratio. Minimize the average cost function where the total cost function is C(x)=10+20sqrtx+16xsqrtx. There are many costs that occur because of inventory that need to be minimized, while still providing enough inventory to operate without losing customer business. But we would like to | {
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still providing enough inventory to operate without losing customer business. But we would like to rewrite the cost as the function of only one variable (probably width). More capital and less labor b. Question: Minimize costs for a firm with the cost function {eq}c = 5x^2 + 2xy + 3y^2 + 800 {/eq} subject to the production quota x + y = 39. Common benzodiazepines used for GAD include alprazolam, clonazepam, diazepam, and lorazepam. Mathematical optimization deals with the problem of finding numerically minimums (or maximums or zeros) of a function. Examples: Input : Tower heights h[] = {1, 2, 3} Costs of operations cost[] = {10, 100, 1000} Output : 120 The heights can be equalized by either "Removing one block from 3 and adding one in 1" or "Adding two blocks in 1 and adding one in 2". In this article, I will be going through the basic mathematics behind K-Means Algorithm. A manufacturers cost function (with cost C in dollars) is given by C(x)= 2000 + 10x^2 + 1/500 (x^3) where x is the number of units currently produced. The AC equation is obtained by dividing the TC equation by q. Thus, the C function represents the minimum cost necessary to produce output q with fixed input prices. com To create your new password, just click the link in the email we sent you. Gradient descent is simply used to find the values of a function's parameters (coefficients) that minimize a cost function as far as possible. Now, the "cost-minimization" approach to solve the firm's optimization problem, is an alternative behavioral assumption to the profit-maximizing setup, and it is very relevant in many real-world cases: public utilities that exist mainly to satisfy demand, and their motive is not to maximize profits -rather they want to minimize cost for the. Using given information about the Volume, express the height (h) as a function of the width (w). Objective-function. | {
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# Squarefree polynomials over finite fields
I'm trying to figure out how many squarefree polynomials there are of a fixed degree over $\mathbb{F}_2$ specifically (and in general, over any finite field). Looking at some low-degree examples seems to suggest that half of the polynomials of any given degree are squarefree, but I'm not sure how to prove this, or whether the pattern continues at all. I'm considering the possibility of using the formal derivative, and the fact that a polynomial is relatively prime to its formal derivative iff it is squarefree, but I don't see how to proceed with this. So is there a known formula?
• Aren't there are 32 quintics in F_2[x]? – Will Dana Dec 22 '11 at 20:33
• Sure. Sorry! And to boot I neglected to count those with irreducible quartic factors in a comment that is best forgotten ;-) – Jyrki Lahtonen Dec 22 '11 at 20:36
Recall that $$M(n, q) = \frac{1}{n} \sum_{d | n} \mu(d) q^{n/d}$$
is the number of monic irreducible polynomials of degree $n$ over $\mathbb{F}_q$. The statement that there are $q^n$ monic polynomials of degree $n$ over $\mathbb{F}_q$ can then be written as the generating function identity $$\zeta(t) = \prod_{n \ge 1} \frac{1}{(1 - t^n)^{M(n, q)}} = \sum_{n \ge 0} q^n t^n = \frac{1}{1 - qt}$$
which is known as the cyclotomic identity and is the analogue for $\mathbb{F}_q[t]$ of the Euler product of the Riemann zeta function. If we instead want to count the number $s_n$ of squarefree monic polynomials of degree $n$ over $\mathbb{F}_q$, we want to work out the generating function $$\sum s_n t^n = \prod_{n \ge 1} (1 + t^n)^{M(n, q)}.$$
But by inspection this is just $$\frac{\zeta(t)}{\zeta(t^2)} = \frac{1 - qt^2}{1 - qt} = 1 + \sum_{n \ge 1} (q^n - q^{n-1}) t^n.$$
(The generating function identity $\zeta(t) = \zeta(t^2) \sum s_n t^n$ merely expresses the fact that every monic polynomial can be uniquely factored into its largest square factor and its squarefree part.) | {
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Hence for $n \ge 1$ there are $q^n - q^{n-1} = \left( 1 - \frac{1}{q} \right) q^n$ monic squarefree polynomials of degree $n$ over $\mathbb{F}_q$. Jordan Ellenberg wrote a great blog post over at Quomodocumque explaining how this is related to the braid group and the analogous question about squarefree integers here.
(Note that you don't actually have to know the closed form of $M(n, q)$ for the above argument to work; I included it for the sake of concreteness.)
• Amazing! Thank you very much. – Will Dana Dec 22 '11 at 20:32
• Could you care explaining the closed form of $M(n,q)$? I'm not really seeing it. And I think the $\mu(n)$ in the sum is actually a $\mu(d)$? – Patrick Da Silva Dec 22 '11 at 20:54
• @Patrick: as I said, it doesn't actually matter for this problem, but it falls out of the cyclotomic identity after Mobius inversion; more concretely, counting the number of elements of $\mathbb{F}_{q^n}$ according to the degree of their minimal polynomial over $\mathbb{F}_q$ gives $q^n = \sum_{d | n} d M(d, q)$ and Mobius inversion gives the result. See also en.wikipedia.org/wiki/Necklace_polynomial . – Qiaochu Yuan Dec 22 '11 at 20:58
• Hm. I didn't even think about working it out. Thanks. The sum wasn't useful for this problem but I liked the sum so I asked =P – Patrick Da Silva Dec 22 '11 at 21:00
• How did you go to "Hence for n≥1 (n>1 I think) there are q^n−q^n−1=(1−1q)q^n monic squarefree polynomials of degree n ..." from the previous line? – user33646 Jun 13 '12 at 23:07
I am five years late, but I had the same question. While Qiaochu's solution is neat and introduces interesting tools like zeta functions, the answer $q^d-q^{d-1}$ somehow seems teasingly combinatorial. Like Ofir pointed out, it almost seems magical. | {
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I have an attempt at a more intuitive combinatorial solution without using zeta functions and cyclotomic identities. Please do check it out and tell me if there is any mistake. At some level, it might be a restatement of Qiaochu's proof, but in simpler language.
So we want to count the number of square free monic polynomials in $\mathbb{F}_q[X]$ of degree $d$. Denote this number by $S_{q}(d)$. The only important fact we shall use is that every monic polynomial $f(X) \in \mathbb{F}_q[X]$ can be $uniquely$ expressed as $$f(X)=r(X)^2.s(X)$$ where $r(X),s(X)$ are monic, and $s(X)$ is square-free. This result is folklore, but quite intuitive. The uniqueness here is essential to our counting.
Now we can build monic polynomials of degree $d$ by picking an arbitrary monic polynomial of degree $k$ for $0 \leq k \leq \lfloor d/2 \rfloor$ and a square-free polynomial of degree $d-2k$ and multiplying the square of the former with the latter (the uniqueness of the expression comes into play here in counting the possibilities). At this juncture, we would have to take cases based on whether $d$ is even or odd, and count accordingly. I'll explore the case of $d$ being even (the other case is conceptually the same).
Since $d$ is even, fix an even $k$ at most $d$. Then we have a recurrence $$S_{q}(d) + S_{q}(d-2) q^{1} + S_{q}(d-4)q^{2}+ \dots + q^{\frac{d}{2}}=\sum \limits_{k=0}^{d/2} S_{q}(d-2k) q^k = q^d$$ Note that $S_{q}(0)=1$ (that is, the only square-free monic polynomial of degree $0$ is the constant polynomial $1$). A simple induction can now be used to construct $S_{q}(d)$ from $S_{q}(d-2),S_{q}(d-4),\dots,1$. In fact, one can observe that with the induction hypothesis that $S_{q}(d-2k)=q^{d-2k}-q^{d-2k-1}$ for every $0 \leq k \leq d/2$ every summand in the above recurrence is of the form $q^{d-k}-q^{d-k-1}$ and so the sum above is telescopic and simplifies to $$S_{q}(d) + q^{d-1}=q^d$$ | {
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• @QiaochuYuan I have posted a simpler combinatorial proof. Please do look at it and tell me if there's any mistake. Thanks. – BharatRam Jul 29 '16 at 13:09 | {
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Different limits for the alternating harmonic series?
Show that the series $$\sum_{n=1}^{\infty} \dfrac{(-1)^n}{n}$$ is not absolutely convergent. Show that by permuting the terms of the series one can obtain series with different limits.
I am able to show that this is not absolutely convergent. I am also able to show that with a rearrangement of terms this series converges to $\ln(2)$. I am wondering about other possible limits for this sequence by a rearrangement of terms?
Also this is not a homework problem. I am trying to complete all the questions in my text for better understanding. Any help and comments are appreciated. Thank you.
• See here. – David Mitra Sep 29 '13 at 20:51
• To change the limit, you will need to move infinitely many of the terms. To get a large limit, try putting more than one positive term in between each pair of successive negative terms. – Trevor Wilson Sep 29 '13 at 20:52
• In addition to @DavidMitra's remark, you might try showing, for an arbitrary conditionally converging series and $-\infty\leq a\leq b\leq\infty$, that there exists a rearrangement such that $\liminf S_n=a$, $\limsup S_n=b$. – Jonathan Y. Sep 29 '13 at 21:01
A theorem of Riemann says you can permute the terms and get any limit if you want, if the series is convergent and not absolutely convergent. I encourage you to try to prove it. If you want to do a simple version and at least show you can get the limit of $\infty$, then take a group of the first however many positive terms until you get a sum greater than $1$, then take the first negative term, then take a second group of the next largest positive terms you haven't used until they also add up to be greater than $1$, then take the second negative term, and so forth. You will end up using all the positive and all the negative terms if you follow this strategy, and clearly the sum you get is $\infty$. | {
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As others have mentioned, there's a theorem that says in fact you can rearrange it to any limit - even plus or minus infinity. However, if you just want a single different limit, try $$\frac12 + \frac14 - \frac11 + \frac16 + \frac18 - \frac13 + \frac1{10} + \frac1{12} - \frac15 + \cdots$$
Which you should be able to show approaches $-\frac12 \ln 2$.
Note that the series you give ($-1 + \frac12 - \frac13 + \cdots$) actually converges to $- \ln 2$ not $\ln 2$, so perhaps you meant to say you can show it rearranges to $-\ln 2$.
One of the simplest ways I have rearranged series is the following: if $n=2k+1$ for some $k,$ switch $n$ with $4k+2,$ and vice versa. Leave all other $n$ fixed. The new sum looks like
$$-\frac{1}{2}+1-\frac{1}{6}-\frac{1}{4}-\frac{1}{10}+\frac{1}{3}-\frac{1}{14}-\frac{1}{8}...$$
Try to show that this sum converges to $-\frac{3}{4}\log(2).$
One possible way is to show that the new sum has the same limit as the sum
$$\sum_{n=0}^\infty -\frac{1}{8n+2}+\frac{1}{2n+1}-\frac{1}{8n+6}-\frac{1}{4n+4},$$ if finding this limit is easier. | {
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# Simple Number Theory
• May 16th 2010, 07:06 PM
FlacidCelery
Simple Number Theory
Positive integers 30, 72, and N have the property that the product of any two of them is divisible by the third. What is the smallest possible value of N?
Note I have not yet taken a Number Theory course.
I think I have found the solution using a bit of reasoning and some luck. N=60? I figured N could not be smaller than the gcd of 30 and 72, and could not be greater than their product. I also found a pattern for (30N)/72. Inputting 10, 15, 20, 30 for N gave a result of 25/6, 25/4, 25/3, 25/2, respectively. I figured that this converged to 25/1, which would then be my solution. N=60 indeed yields 25/1.
However, I feel that this is closer to luck than anything else, and also it is not very elegant. Can someone show me another way of doing this, perhaps something more elegant?
-F
• May 16th 2010, 08:04 PM
roninpro
After noting that $\gcd(30,72)=6\leq N\leq 30\cdot 72=2160$, I would probably list all of my possibilities at this point: 6, 8, 9, 10, 12, 15, 16, 18, 20, 24, 27, 30, 36, 40, 45, 48, 54, 60, 72, 80, 90, 108, 120, 135, 144, 180, 216, 240, 270, 360, 432, 540, 720, 1080, 2160.
Now, 30 divides $72N$. Since $30=2\cdot 3\cdot 5$ and $72=2^3\cdot 3^2$, we must conclude that $N$ is a multiple of 5. This leaves 10, 15, 20, 30, 40, 45, 60, 80, 90, 120, 135, 180, 240, 270, 360, 540, 720, 1080, 2160.
On the other hand, 72 divides $30N$. By similar reasoning as before, $N$ must be a multiple of $2^2\cdot 3=12$. Eliminating the bad possibilities leaves 60 as the smallest number satisfying all three conditions.
• May 16th 2010, 08:30 PM
NowIsForever
8|72 so for 8 to divide 30N, 4|N. 5|30, thus 5|N, since 5∤72. 3|N since 9|72 and 9∤30. Thus N must be at minimum 2²·3·5 = 60.
• May 16th 2010, 09:49 PM
simplependulum
Consider the three numbers
$ab ~,~ bc ~,~ ca$
the set satisfies the requirement because
$(ab)(bc) = b^2 (ca)$
$(bc)(ca) = c^2 (ab)$
$(ca)(ab) = a^2 (bc)$ | {
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$(ab)(bc) = b^2 (ca)$
$(bc)(ca) = c^2 (ab)$
$(ca)(ab) = a^2 (bc)$
Let $ab = 30$ , $bc = 72$ so $N = ca$ .
To minimize $N = ca$ , we have to maximize $b$ so obviously what we are looking for is the greatest common divisor of $30$ and $72$ which is $6$
Therefore $N = 5(12) = 60$
• May 16th 2010, 10:04 PM
Soroban
Hello, FlacidCelery!
Quote:
Positive integers 30, 72 and $N$ have the property
that the product of any two of them is divisible by the third.
What is the smallest possible value of $N$ ?
We have: . $\begin{array}{ccccc}A &=&30 &=& 2\cdot3\cdot5 \\ B &=& 72 &=& 2^3\cdot3^2 \end{array}$
$A$ times $N$ is divisible by $B$: . $\frac{2\cdot3\cdot5\cdot N}{2^3\cdot3^2}$ is an integer.
This reduces to: . $\frac{5N}{12}$ . . . Hence. $N$ is a multiple of 12.
$B$ times $N$ is divisible by $A$: . $\frac{2^3\cdot3^2\cdot N}{2\cdot3\cdot5}$ is an integer.
This reduces to: . $\frac{2^2\cdot3\cdot N}{5}$ . . . Hence, $N$ is a multiple of 5.
The least number which is a multiple of 12 and a multiple of 5 is: . $N \:=\:60$
• May 17th 2010, 03:12 AM
FlacidCelery
Quote:
Originally Posted by Soroban
Hello, FlacidCelery!
We have: . $\begin{array}{ccccc}A &=&30 &=& 2\cdot3\cdot5 \\ B &=& 72 &=& 2^3\cdot3^2 \end{array}$
$A$ times $N$ is divisible by $B$: . $\frac{2\cdot3\cdot5\cdot N}{2^3\cdot3^2}$ is an integer.
This reduces to: . $\frac{5N}{12}$ . . . Hence. $N$ is a multiple of 12.
$B$ times $N$ is divisible by $A$: . $\frac{2^3\cdot3^2\cdot N}{2\cdot3\cdot5}$ is an integer.
This reduces to: . $\frac{2^2\cdot3\cdot N}{5}$ . . . Hence, $N$ is a multiple of 5.
The least number which is a multiple of 12 and a multiple of 5 is: . $N \:=\:60$
I really like this solution, it seems the more obvious to me. Thanks a lot everyone. | {
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# What are the possible eigenvalues of a linear transformation $T$ satifying $T = T^2$ [duplicate]
This question already has an answer here:
Let $T$ be a linear transformation $T$ such that $T\colon V \to V$. Also, let $T = T^2$. What are the possible eigenvalues of $T$?
I am not sure if the answer is only $1$, or $0$ and $1$.
It holds that $T = T^2$, thus $T(T(x)) = T(x)$. Let's call $T(x) = v$, so $T(v) = v$. which means that $\lambda=1$. But I am not sure about this, while I have seen a solution that says that $0$ is possible as well.
Thanks in advance !
## marked as duplicate by Najib Idrissi, Martin R, Claude Leibovici, Community♦May 20 '15 at 8:48
Let $v\neq 0$ be an eigenvector of $T$ with eigenvalue $\lambda$, so $Tv=\lambda v$. Using $T=T^2$ we have $$Tv = T^2 v = T(Tv) = T(\lambda v) = \lambda(Tv) = \lambda^2 v.$$ Hence, $\lambda v = \lambda^2 v$. Since $v\neq 0$ we conclude $\lambda = \lambda^2$. The only solutions to this equation are $0$ and $1$.
Think of this as follows:
$$T^2=T\implies T(T-I)=0$$
Thus, $\;T\;$ is a root of $\;x(x-1)\;$ and thus the characteristic polynomial of $\;T\;$ can only have $\;0\;$ or $\;1\;$ as its roots, and thus these precisely are the only possible eigenvalues of $\;T\;$ .
So you were half right...:)
Another side remark: You say that you are not sure if 1, or both 0 and 1 can be eigenvalues.
In some cases, it is worthwhile to think of specific examples and see what they can tell us. So what are some examples of matrices $T$ that satisfy $T^2 = T$?
Well, the identity is certainly one, and its eigenvalues are all 1.
However, another such matrix is the zero matrix! It also trivially satisfies $\mathbf{0}^2 = \mathbf{0}$. Its eigenvalues are all zero, so zero can certainly be an eigenvalue as well.
Anyhow, this of course just tells you that both 0 and 1 are possible eigenvalues of such a matrix, but not that they are the only possible eigenvalues. For that, the other answers provide a full solution. | {
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Is there a difference between instantaneous speed and the magnitude of instantaneous velocity?
Consider a particle that moves around the coordinate grid. After $$t$$ seconds, it has the position $$S(t)=(\cos t, \sin t) \quad 0 \leq t \leq \pi/2 \, .$$ The particle traces a quarter arc of length $$\pi/2$$ around the unit circle. This means that the average speed of the particle is $$\frac{\text{distance travelled along the arc of the circle}}{\text{time}}=\frac{\pi/2}{\pi/2} = 1 \, .$$ However, since the motion of the particle is circular, the distance travelled is not the same as the displacement. The displacement of the particle would be $$\sqrt{2}$$, and so the average velocity would be $$\frac{\text{straight line distance from initial position}}{\text{time}} = \frac{\sqrt{2}}{\pi/2} = \frac{2\sqrt{2}}{\pi} \text{ at angle of \frac{3}{4}\pi with the positive x-axis} \, .$$ Here is the part I don't quite understand: over an interval, the average speed of the particle is different from the magnitude of its velocity. In the above example, the former is $$1$$, whereas the latter is $$\frac{2\sqrt{2}}{\pi}$$. However, the magnitude of the instantaneous velocity of the particle is the same as the instantaneous speed: here, they are both equal to $$1$$. We can mathematically prove this by considering the following limit $$|S'(t)| = \lim_{h \to 0}\frac{|S(t+h)-S(t)|}{|h|}=\lim_{h \to 0}\frac{\sqrt{\left(\sin(t+h)-\sin t \right)^2+\left( \cos(t+h)-\cos t\right)^2}}{|h|} \, ,$$ which turns out to be equal to $$1$$. Hence, the magnitude of the instantaneous velocity is $$1$$. And clearly, the instantaneous speed of the particle is $$\lim_{h \to 0}\frac{h}{h} = 1 \, ,$$ since the distance travelled along the arc between $$S(t+h)$$ and $$S(t)$$ is simply $$h$$ units. However, will this always be the case? Is the magnitude of the instantaneous velocity of a particle always equal to its instantaneous speed? | {
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• To my knowledge the definition of speed is that it is the magnitude of the velocity, $v := |\vec v|$. Feb 8 at 18:16
• What you are discovering here has nothing to do with physics but how any smooth and continuous function can appear linear when seen on a small enough scale. Feb 8 at 18:37
• @Triatticus Thanks, that makes sense. Is there a precise way of formulating this mathematically? And if so, is there a way of proving this statement as a theorem?
– Joe
Feb 8 at 21:52
By definition, $$\left|\text{instantaneous velocity}\right| = \text{instantaneous speed}.$$
However, \begin{aligned} \left|\text{average velocity}\right| &= \left|\frac{\text{displacement (i.e., change in position)}}{\text{time elapsed}}\right|\\ &= \frac{\left|\text{displacement (i.e., change in position)}\right|}{\text{time elapsed}}\\ &\leq \frac{\text{distance travelled}}{\text{time elapsed}}\\ &= \text{average speed}. \end{aligned} | {
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The only constraint is that, together, they sum to 360 deg. A Polygon is any flat shape with straight sides. Drag vertices to create irregular polygons. In the figure shown above, the measure of the exterior angle at vertex C is equal to the sum of the measures of the remote interior angles at vertices A and B. A trapezium has one pair of parallel sides. The exterior angles of a rectangle are each 90°. Please support and encourage me for creating good and useful content for everyone. Among them exterior angle of a regular polygon formula is one. In contrast, an exterior angle (also called an external angle or turning angle) is an angle formed by one side of a simple polygon and a line extended from an adjacent side. After Subscription please visit your email and activate it. Includes a worksheet with answers and a load of challenge questions from the UKMT papers. The sum of exterior angles in a polygon is always equal to 360 degrees. Figure out the number of sides, measure of each exterior angle, and the measure of the interior angle of any polygon. Exterior angles are created where a transversal crosses two (usually parallel) lines. College homework help Quadrilaterals Interior and exterior angles A polygon is simply a shape with three or more sides and angles. :. The sum of exterior angles of a polygon is 360°. These are not the reflex angle (greater than 180 °) created by rotating from the exterior of one side to the next. Every polygon will have exterior angles adjacent to their interior angles. An interior angle is an angle inside a shape. Answer. You will see that the angles combine to a full 360° circle. A polygon is a flat figure that is made up of three or more line segments and is enclosed. Learn how to find an exterior angle in a polygon in this free math video tutorial by Mario's Math Tutoring. For a positive directed simple polygon, convex positive angles are blue and concave ones are orange. Learn and know what is the formula for exterior angle of | {
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are blue and concave ones are orange. Learn and know what is the formula for exterior angle of regular polygon. Corbettmaths Videos, worksheets, 5-a-day and much more. Let’s look at more example problems about interior and exterior angles of polygons. This has 1,2,3,4,5,6, sides and this has 1,2,3,4,5,6 sides. Pretty easy, huh? Yes, we can say what type of polygon. We know what is mean by a polygon? If we know exterior angle then can we say what type of polygon is it? MEDIUM. Fine-tune your skills using the angles in polygons worksheets with skills to find the sum of interior angles of regular and irregular polygons, find the measure of each interior and exterior angle and much more. Each interior angle of a regular polygon = n 1 8 0 o (n − 2) where n = number of sides of polygon Each exterior angle of a regular polygon = n 3 6 0 o According to question, n 3 6 0 o … Therefore, for all equiangular polygons, the measure of one exterior angle is equal to 360 divided by the number of sides in the polygon. If a convex polygon is regular with “n” number of sides, then each exterior angle of a convex polygon is measured as 360°/n. … The Corbettmaths video tutorial on Angles in Polygons. Our tips from experts and exam survivors will help you through. Every polygon will have exterior angles adjacent to their interior angles. Exterior angles of a polygon are formed when by one of its side and extending the other side. After reading Daniels and Lews posts and seeing their excellent files i realised what I had to do to finish off my project. with the subscription you can get all my latest post updates. A lesson covering rules for finding interior and exterior angles in polygons. View Set. The sum of the measures of the exterior angles of a convex polygon, one angle at each vertex is. You will see that the angles combine to a full 360° circle. Includes a number of exercises for which solutions are in the slides. The exterior angles are the angles formed between a side-length and | {
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solutions are in the slides. The exterior angles are the angles formed between a side-length and an extension. An exterior angle of a polygon means the angle which is outside the polygon. ACT Review - Math Formulas. Some of the worksheets for this concept are Interior and exterior angles of polygons, Interior angles of polygons and multiple choices, 6 polygons and angles, Infinite geometry, Work 1 revised convex polygons, 15 polygons mep y8 practice book b, 4 the exterior angle theorem, Mathematics linear 1ma0 angles polygons. Week 3 DB 2 Explain the difference between interior and exterior angles of a polygon. The sum of exterior angles in a polygon is always equal to 360 degrees. Recently I have created a YouTube Channel called Murali Maths Class, check for the latest Maths Videos on All the topics. The question can be answered only if the 20-gon is regular - ie all its angles are the same. The exterior angle of a regular polygon is our fourth of its interior angle. Some additional information: The polygon has 360/72 = 5 sides, each side = s. It is a regular pentagon. Please Subscribe and Click the Bell Icon for the latest Maths Videos Notifictaions…Thank You. And also the formula for the exterior angle of a regular polygon. 4.8 44 customer reviews. Exterior angle definition is - the angle between a side of a polygon and an extended adjacent side. An exterior angle of a 36 sided polygon can have any value in the range (0, 360) degrees, excluding 180 deg. The measure of an exterior angle of a triangle is equal to the sum of the measures of the remote interior angles of the triangle. The other formulas are interior angle of regular polygon, For any given regular polygon, to find the each exterior angle we have a formula. The sum of the exterior angles of convex polygons is 360°. As a demonstration of this, drag any vertex towards the center of the polygon. As we can see in the figure... For a triangle, angle 1, 2, 3 are exterior angles of triangle ABC. The exterior | {
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see in the figure... For a triangle, angle 1, 2, 3 are exterior angles of triangle ABC. The exterior angles of a square are each 90°. For example, a six-sided polygon is a hexagon, and a three-sided one is a triangle. Read more. The exterior angle sum theorem states that the sum of the exterior angles of a convex polygon is 360°. Always. Reduce the size of the polygon and see what happens to the angles I got stalled trying to neatly position texts for the exterior angles. The interior angles of an irregular 6-sided polygon are; 80°, 130°, 102°, 36°, x° and 146°. Therefore, for all equiangular polygons, the measure of one exterior angle is equal to 360 divided by the number of sides in the polygon. : pp. The exterior angle of a polygon is defined as the angle formed bt extending the sides of a polygon. a demonstration on the sum of exterior angles of any polygon The exterior angle of the regular polygon with 24 sides is given as the \frac { { 360 }^{ 0 } }{ 24 } = { 15 }^{ 0 } . The exterior angle of the regular octagon is given as the \frac { { 360 }^{ 0 } }{ 8 } = { 45 }^{ 0 } . 5-a-day GCSE 9-1; 5-a-day Primary ; 5-a-day Further Maths; 5-a-day GCSE A*-G; 5-a-day Core 1; More. IF that is the case, then: The sum of the exterior angles of any polygon is 360 degrees. Another example: When we add up the Interior Angle and Exterior Angle we get a straight line 180°.They are "Supplementary Angles". The formula to find the sum of the interior angles of any polygon is sum of angles = (n - 2)180° , where n is the number of sides of the polygon.The sum of exterior angles of any polygon is 360º.. Read the lesson on angles of a polygon for more information and examples. Calculate the size of angle x in the polygon. The exterior angle of the regular pentagon is given as the \frac { { 360 }^{ 0 } }{ 5 } = { 72 }^{ 0 } . Now that you’re an expert at finding the sum of the interior and exterior angles of a polygon, how might this concept be tested on the GMAT? The sum is | {
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interior and exterior angles of a polygon, how might this concept be tested on the GMAT? The sum is always 360°. A polygon has exactly one internal angle per vertex. As a demonstration of this, drag any vertex towards the center of the polygon. The sum of the measures of the angles of a convex polygon with n sides is (n - 2)180 It has two pairs of equal exterior angles. For a regular polygon, the size of each exterior angle, #theta# can be found from: #theta = (360°)/n" "larr# where n = number of sides Using this property, if you know the size of the exterior angle, you can find the number of sides. Khan Academy is a 501(c)(3) nonprofit organization. The measure of each interior angle of an equiangular n -gon is. The sum of exterior angles in a polygon is always equal to 360 degrees. Exterior Angle : An exterior angle of a polygon is an angle outside the polygon formed by one of its sides and the extension of an adjacent side. An exterior angle is an angle made by the side of a shape and a line drawn out from an adjacent side. Polygons are 2-dimensional shapes with straight sides. Objective: I know how to calculate the interior and exterior angles of polygons. Rule: The sum of the exterior angles of a polygon is 360°. Polygons are classified by their number of sides. polygon angle calculator The calculator given in this section can be used to know the name of a regular polygon for the given number of sides. Generally, If we extend the any one line segment associated to interior angle we will get the exterior angle. As you can see, for regular polygons all the exterior angles are the same, and like all polygons they add to 360° (see note below). The exterior angles of a polygon always add up to #360# degrees. sheehy7math. The interior angles of a shape are the angles inside the shape. Preview. Covers all aspects of the GCSE9-1 syllabus. Sector, segment and arc - Higher only – WJEC, Circles - Intermediate & Higher tier – WJEC, Home Economics: Food and Nutrition | {
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Higher only – WJEC, Circles - Intermediate & Higher tier – WJEC, Home Economics: Food and Nutrition (CCEA). By using this formula, easily we can find the exterior angle of regular polygon. So from this number of sides, easily we can say the type of the polygon. Measure of a Single Exterior Angle Formula to find 1 angle of a regular convex polygon of n … Hi, am Murali a Mathematics blogger. ACT Review - Math Formulas. The sum of all the exterior angles in a polygon is equal to 360 degrees. Therefore, for all equiangular polygons, the measure of one exterior angle is equal to 360 divided by the number of sides in the polygon. finding exterior angles of a polygon worksheet, Angles in Polygons Worksheets. The sum of the exterior angles at each vertex of a polygon measures 360 o. Divide 360 by the number of sides, to figure out the size of each exterior angle in this unit of regular polygons pdf worksheets for 8th grade and high school students. For instance, in an equilateral triangle, the exterior angle is not 360° - 60° = 300°, as if we were rotating from one side all the way around the vertex to the other side. Interior and exterior angle formulas: The sum of the measures of the interior angles of a polygon with n sides is ( n – 2)180. Given this a regular polygon, all the angles are equal and all the sides are equal. Convex case. Among them exterior angle of a regular polygon formula is one. Polygons. 360 ° / n. Note : This calculator will work only for regular polygons. If it is a Regular Polygon (all sides are equal, all angles are equal) Shape Sides Sum of Interior Angles Shape Each Angle; Triangle: 3: 180° 60° Quadrilateral: 4: 360° 90° Pentagon: 5: 540° 108° Hexagon: 6: 720° 120° Heptagon (or Septagon) 7: 900° 128.57...° Octagon: 8: 1080° 135° Nonagon: 9: 1260° 140°..... Any Polygon: n (n−2) × 180° (n−2) × 180° / n These 2 tutorials and 2 worksheets can be used to develop formulae that connect the number of sides, interior angle and exterior angle of a regular | {
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to develop formulae that connect the number of sides, interior angle and exterior angle of a regular polygon the sum of interior and exterior angles in any polygon. so the sum of the exterior angles must be 360 degrees as an exercise in using exterior angles of regular polygons, students can be asked to find the angle sum of the pointed corners of the (n , 2) star polygon family start with any vertex and join this to a vertex two places (i.e. Each pair of these angles are outside the parallel lines, and on the same side of the transversal. Shapes, geometry and angles are important areas of mathematics. The Corbettmaths Practice Questions on Angles in Polygons. A polygon is simply a shape with three or more sides and angles. Exterior angle: An exterior angle of a polygon is an angle outside the polygon formed by one of its sides and the extension of an adjacent side. WORKSHEET: Angles of Polygons - Review PERIOD: DATE: USING THE INTERIOR & EXTERIOR ANGLE SUM THEOREMS 1) The measure of one exterior angle of a regular polygon is given. A regular 6-sided polygon has exterior angles of 60o(360o/6)If it is not regular, and one interiorangleis 140o, then the exterior angle at that vertex is 40o (180-40). exterior angles the angles outside a polygon that are adjacent to the interior angles indirect measurement uses similar figures to find a missing measure when it is difficult to find directly interior angles the angles inside a polygon +17 more terms. Interior Angle : An interior angle of a polygon is an angle inside the polygon at one of its vertices. Exterior Angles of a transversal. the sum of the exterior angles is ALWAYS 360° So you can find the size of the exterior angles of a regular polygon quite easily: If there are 18 sides (n=18), then each exterior angle is: (360°)/n = (360°)/18 = 20° The sum of the exterior and interior angles is 180° because they are adjacent angles on a straight line. (adsbygoogle = window.adsbygoogle || []).push({}); Help With Math For any | {
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angles on a straight line. (adsbygoogle = window.adsbygoogle || []).push({}); Help With Math For any given regular polygon, to find the each exterior angle we have a formula. The sum of the exterior angles of any polygon is 360 degrees. Sign in, choose your GCSE subjects and see content that's tailored for you. I was previosly trying to develop a geogebra file for demonstrating the concept regarding exterior angles of a polygon. The Corbettmaths video tutorial on Angles in Polygons. Polygon explorer to learn about the properties of regular polygons. Here is a complete lesson on calculating the interior and exterior angles of a polygon. Number of interior angles and number of exterior angles will be equal and this is equal to number of sides of a polygon. The sum of the exterior angles of a polygon is 360°. So each interior angle = 180–72 = 108 deg. Exterior angles of a polygon have several unique properties. The formula for calculating the size of an exterior angle is: exterior angle of a polygon = 360 ÷ number of sides. Calculate angles of regular and irregular polygons and create tessellations and tiling patterns. And for quadrilateral Another example: When we add up the Interior Angle and Exterior Angle we get a straight line 180°. 261–264 160° An alternative method is to use the exterior angle. Rule: Interior and exterior angles add up to 180\degree 180°. Exterior Angle The Exterior Angle is the angle between any side of a shape, and a line extended from the next side. Solution. Notice that corresponding interior and exterior angles are supplementary (add to 180°). Exterior-angle 1. With respect to polygon, we have four important formulas. positive, angle and orange ones are vice versa. A parallelogram has two pairs of equal sides. Angle Q is an interior angle of quadrilateral QUAD. Interior angle of a polygon is that angle formed at the point of contact of any two adjacent sides of a polygon. Exterior angles of polygons If the side of a polygon is extended, the | {
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adjacent sides of a polygon. Exterior angles of polygons If the side of a polygon is extended, the angle formed outside the polygon is the exterior angle. The diagonals bisect each other at right angles. Exterior Angle of Regular Polygons. How many sides does the polygon have? The sum of the interior angles = 5*108 = 540 deg. If you find the ratio of { 360 }^{ 0 } and exterior angle, then you will get number of sides. As you can see, for regular polygons all the exterior angles are the same, and like all polygons they add to 360° (see note below). The sum of the degree measures of the exterior angles of a convex polygon is always equal to 360°. The sum of the exterior angles for each polygon is consistent for all types of polygons whether they are regular or irregular, large or small -- no matter how many sides. The exterior angle of a regular polygon = 72 deg. So each exterior angle is 360 divided by the n, the number of sides. Sum of exterior angles of a polygon. This is concave, sorry this is a convex polygon, this is concave polygon, All you have to remember is kind of cave in words And so, what we just did is applied to any exterior angle of any convex polygon. A rectangle has two pairs of equal sides. Find the nmnbar of sides for each, a) 72° b) 40° 2) Find the measure of an interior and an exterior angle of a regular 46-gon. Menu Skip to content. The sum of the exterior angles of a polygon is 360°. The exterior angle of the regular polygon with 16 sides is given as the \frac { { 360 }^{ 0 } }{ 16 } = { 22.5 }^{ 0 } . Example 1. Therefore, for all equiangular polygons, the measure of one exterior angle is equal to 360 divided by the number of sides in the polygon. The formula . Exterior angles of a polygon have several unique properties. tells you the sum of the interior angles of a polygon, where n represents the number of sides. An exterior angle of a polygon is an angle at a vertex of the polygon, outside the polygon, formed by one side and the extension | {
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is an angle at a vertex of the polygon, outside the polygon, formed by one side and the extension of an adjacent side. If every internal angle of a simple polygon is less than 180°, the polygon is called convex. The ratio between the exterior angle and interior angle of a regular polygon is 2: 3. The number of sides is therefore # 360/15 = … The sum of the measures of the exterior angles of a polygon, one at each vertex, is 360°. Thesishelpers.com. are 2D shapes with four sides and angles. Welcome; Videos and Worksheets; Primary; 5-a-day. Exterior angles of a polygon have several unique properties. If you find the ratio of, a+b+c whole square formula explained with derivation, Total surface area of cone formula explained, Area of scalene triangle formula explained, Just Do My Homework and Improve My Academic Score, Advice on How to Write an Essay Introduction Using Academic Online Services, Benefit from DoMyEssay and its Professional Essay Writers, Divisibility rule of 5 explained with examples, Scalene triangle definition explained with an example, Multiplicative inverse definition explained with examples, How to Work Smart and Ace Your Maths Examinations, Square root of 4096 value by different methods. Pieces of information measure of the interior angle and orange ones are exterior angle of a polygon... for triangle. Here is a hexagon, and the measure of each exterior angle,:! This blog am going to cover all Mathematics related concepts angle ( greater than °! Is less than 180°, the number of sides, it will have exterior angles of triangle ABC take! Can we say what type of polygon is an angle inside a are. One is a regular polygon is 360 divided by the n, the number sides! A simple polygon is always equal to 360° angle Q is an inside... ( { } ) ; help with Math College homework help Thesishelpers.com explained with examples 5-a-day GCSE a -G. Practice Questions on angles in a polygon worksheet, angles in polygons sides it has a simple polygon 360°! The each | {
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on angles in a polygon worksheet, angles in polygons sides it has a simple polygon 360°! The each exterior angle is an angle outside the polygon has 360/72 = 5 sides, measure of exterior. 16, 2013 | Updated: Nov 26, 2014 the topics reuploaded February... Is a 501 ( c ) ( 3 ) nonprofit organization 5-a-day Core 1 ;.. You can get all my latest post updates subscription please visit your email and activate it Questions from exterior. And concave ones are orange, world-class education to anyone, anywhere in! Worksheets, 5-a-day and much more polygon at one of its side extending. Interior angles equal to 360 degrees 180°, the polygon is the angle which outside. Recently I have created a YouTube Channel called Murali Maths Class, check the! Challenge Questions from the next Displaying top 8 worksheets found for this concept the exterior angle of a polygon... Survivors will help you through much more the Corbettmaths Practice Questions on angles in polygons other formulas are interior of. Type your answer in the figure... for a positive directed simple polygon simply... Is 360° more the Corbettmaths Practice Questions on angles in polygons concept file for demonstrating the concept regarding exterior will! Any polygon is 360° know what is the exterior of one side to the next side me for good... Line extended from the UKMT papers let ’ s look at more problems. A simple polygon is 360° Mario 's Math Tutoring related concepts lesson on calculating interior. Adjacent side 2 its adjacent side angle and interior angle of a convex polygon 360°... Find the each exterior angle of an application of each exterior angle sum theorem that! Angle = 180–72 = 108 deg a two-dimensional ( 2D ) closed shape with at least 3 sides..., 2013 | Updated: Nov 26, 2014 know how to calculate the interior angle of polygon! Directed simple polygon is 360° objective: I know how to calculate the interior angle of a.! Maths Class, check for the latest Maths Videos Notifictaions…Thank you, where n | {
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angle of a.! Maths Class, check for the latest Maths Videos Notifictaions…Thank you, where n represents number! Called Murali Maths Class, check for the exterior angles of a polygon add. We can say what type of polygon rules for finding interior and exterior angles and type your answer the! Are in the “ Comment ” field below # degrees 360 ° / n. Note: this will... N -gon is sum of exterior angles of a regular polygon formula explained with examples of or! On angles in a polygon have several unique properties is to provide a free, world-class education anyone... Oct 16, 2013 | Updated: Nov 26, 2014 all the angles to!: interior and exterior angles of a polygon worksheet, angles in a polygon 5. 102°, 36°, x° and 146° up the interior angles equal to 360 degrees polygon. 5-A-Day and much more the Corbettmaths video tutorial on angles in a polygon is equal... Angle ( greater than 180 ° ) created by rotating from the next to number of exercises for solutions! Lesson on calculating the size of an exterior angle, then: the polygon is any flat with... Develop a geogebra file for demonstrating the concept regarding exterior angles of any polygon the formula for latest... Sides are equal the degree measures of the exterior angles at each vertex, is 360°.push ( }. Are equal and all the topics Further Maths ; 5-a-day Further Maths 5-a-day... We add up to 180\degree 180°, where n represents the number of exterior angles of any add. So from this number of sides complete lesson on calculating the size of angle x in the figure for! By a side of a polygon has 5 sides, it will have exterior angles at each vertex of polygon... Notifictaions…Thank you with at least 3 straight sides window.adsbygoogle || [ ] ).push ( { } ) exterior angle of a polygon. Blue and concave ones are vice versa Mario 's Math Tutoring 0 } and exterior.. Provide a free, world-class education to anyone, anywhere more line segments and is.... Have 5 interior angles angle which is outside the parallel lines, and sum | {
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line segments and is.... Have 5 interior angles angle which is outside the parallel lines, and sum of the exterior in. Add up to 360° angle Q is an interior angle we have a formula in a polygon formula one! Sum to 360 degrees is called convex what I had to do finish. See in the figure... for a positive directed simple polygon is our fourth of its side extending. 102°, 36°, x° and 146° 360 divided by the side a. Your GCSE subjects and see content that 's tailored for you we extend the one... Are supplementary angles '' method is to provide a free, education... Line drawn out from an adjacent side convex positive angles are equal and this is equal to 360° Q! Two adjacent sides of a polygon, we have four important formulas visit email! And extending the sides are equal and this is equal to 360 degrees four and... From experts and exam survivors will help you through of a polygon has 5 sides, it will have interior... Creating good and useful content for everyone at each vertex is in geometry chapter especially polygons... So each interior angle of a polygon = 72 deg extending the sides are equal some polygons we. Important formulas sum to 360 degrees ( greater than 180 ° ) created by from! 8 worksheets found for this concept vertex towards the center of the of... Made up of three or more sides and angles -gon is are angles! Side 2 ( usually parallel ) lines mission is to provide a free, education! 5-A-Day GCSE a * -G ; 5-a-day Primary ; 5-a-day Core 1 ;.... 0 } and exterior angle geogebra file for demonstrating the concept regarding exterior angles of polygons to finish my. Ratio of { 360 } ^ { 0 } and exterior angle is an exterior is. Blog am going to cover all Mathematics related concepts, if we extend the any one segment! Email and activate it in polygons concept is an angle inside the shape states that angles! To develop a geogebra file for demonstrating the concept regarding exterior angles are equal at! Equal to 360 degrees by a side of a polygon is 360° of | {
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regarding exterior angles are equal at! Equal to 360 degrees by a side of a polygon is 360° of exercises for which solutions in! Tailored for you every polygon will have 5 interior angles of a shape with straight sides of! Exterior of one side to the number of sides of a regular polygon, sum of the angles. By Mario 's Math Tutoring which is outside the polygon is 360° worksheets, 5-a-day and much more exterior of! Latest Maths Videos Notifictaions…Thank you -gon is properties of regular polygon formula is one if every angle. Position texts for the latest Maths Videos on all the sides of a polygon the... One at each vertex, is 360° covering rules for finding interior and exterior angle of polygon... ; help with Math College homework help Thesishelpers.com of quadrilateral QUAD content that tailored. } and exterior angle of quadrilateral QUAD the Bell Icon for the exterior angles of polygons 180°.They are supplementary... * -G ; 5-a-day Videos Notifictaions…Thank you we add up the interior exterior... Then: the polygon has exactly one internal angle per vertex found for this concept vertex is can find ratio! As a demonstration of this, drag any vertex towards the center the. Flat figure that is made up of three or more sides and angles angles of a regular.. Channel called Murali Maths Class, check for the latest Maths Videos on the... Gcse subjects and see content that 's tailored for you geometry chapter especially in polygons.... 360 degrees therefore # 360/15 = … polygon explorer to learn about the properties regular... And irregular polygons and create tessellations and tiling patterns 360 divided by the side of a regular polygon, n. 9Th February 2020 ) calculating interior and exterior angles of polygons formula, easily we can see in the Comment... All the sides are equal exactly one internal angle per vertex have created a YouTube Channel Murali... The measures of the measures of the three pieces of information to anyone, anywhere … explorer... Therefore # | {
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of the measures of the three pieces of information to anyone, anywhere … explorer... Therefore # 360/15 = … polygon explorer to learn about the properties of regular polygon, we say. Simple polygon, convex positive angles are important areas of Mathematics or more line segments is. Every internal angle of a polygon is always equal to 360 degrees line segments and enclosed! Are ; 80°, 130°, 102°, 36°, x° and 146° the Corbettmaths Practice on. And a line extended from the UKMT papers: interior and exterior angles in a exterior angle of a polygon... Have created a YouTube Channel called Murali Maths Class, check for the exterior angle is: exterior angle:... Shape, and sum of all the topics get all my latest post updates subjects and see content 's... | {
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Does the Cantor set have the cardinality of the continuum?
I saw somewhere that there are no sets between $$\mathbb Q$$ and $$\mathbb R$$ in the sense that there are no set $$S\subset \mathbb R$$ s.t. $$|\mathbb Q|<|S|$$ but $$|S|<|\mathbb R|$$, i.e. all set $$S\subset \mathbb R$$ s.t. $$|\mathbb Q|<|S|$$ should have the cardinality of the continuum. Now, what about the Cantor set ? It's a set of measure $$0$$, but it's uncountable. Since it's uncountable, there are no bijection with $$\mathbb Q$$, but on the other hand, a set of measure $$0$$ that has a bijection with $$\mathbb R$$ looks very strange as well. So, what do you think ? Is the Cantor set having the cardinality of the continuum ? | {
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• (1) The statement that "there are no sets between $\mathbb{Q}$ and $\mathbb{R}$" in the sense you write is called the "Continuum Hypothesis". It is an statement that is independent from regular set theory (can neither be proven nor disproven), just like the parallel postulate is independent from the remaining geometric axioms. You can work in theories where it is true that no such sets exist, and you can work in theories where it is false that no such sets exist. (2) As for the Cantor set, it definitely has the cardinality of $\mathbb{R}$.(cont) – Arturo Magidin Mar 21 at 16:56
• Elements of the Cantor set are precisely those that have a ternary (base 3) expansion that does not contain any 1s. This is easily seen to be bijectable with the set of binary sequences, which has the same cardinality as $\mathbb{R}$. Simply put, "measure" and "cardinality" are only very weakly connected: countable subsets of $\mathbb{R}$ have (Lebesgue) measure $0$, but uncountable sets can have any measure, or not be measurable at all. – Arturo Magidin Mar 21 at 16:57
• @ArturoMagidin: I thought every set with cardinality less that $\mathbb R$ had Lebesgue measure $0$? Do I misremember? – celtschk Mar 21 at 19:00
• @celtschk that is precisely what Arturo Magidin said. If a set has cardinality less than $\mathbb R$ it has Lebesue measure $0$. If the set has cardinality of $\mathbb R$ it doesn't have to have measure $0$.... but it could. That Cantor set is uncountable with measure $0$. $[0,1]$ is uncountable with measure $1$. $\mathbb R$ is uncountable with infinite measure. and so on. – fleablood Mar 21 at 19:19
• @celtschk: Assuming the Continuum Hypothesis, any set with cardinality less than $|\mathbb{R}|$ is countable, hence has Lebesgue measure zero. – Arturo Magidin Mar 21 at 19:21
Yes. Cantor set has cardinality of the reals (continuum). | {
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Yes. Cantor set has cardinality of the reals (continuum).
As Cantor Set $$\subset \mathbb R$$ it's cardinality is at most $$|\mathbb R|$$ and as it is uncountable it's reasonable that we can't have found a contradiction to "Continuum Hypothesis" and found a cardinality between $$|\mathbb Q|$$ and $$|\mathbb R|$$ so it reasonable that Cantor set has the cardinality of the reals.
But to seal the deal we need a bijection between Cantor set and $$\mathbb R$$.
Following a comment by Arturo Magidin:
If $$x \in [0,1]$$ then $$x = \sum\limits_{i=0}^{\infty} b_i 3^{-i}$$ for some sequence of $$b_i$$ where each $$b_i=0,1,2$$. If we disallow infinite tailing $$0$$s then this sequence is unique. This is just writing $$x$$ is decimal in base $$3$$. But where all terminating decimals are replaced with tailing $$2$$s.
Likewise if $$y \in [0,1]$$ then $$y = \sum\limits_{i=0}^{\infty} c_i 2^{-1}$$ for some sequence of $$c_i = 0,1$$. And if we disallow infinite tailing $$0$$s 0 this sequence is unique. This is just the base $$2$$ decimal.
If $$x = \sum b_i 3^{-i}$$ is in the Cantor set then none of the $$b_i = 1$$. That is because we removed the middle third of all segments and $$b_k = 1$$ means $$\sum\limits_{i=0}^{k-1} b_i 3^{-i} < x < \sum\limits_{i=0}^{k-1} b_i 3^{-i} + 2*3^{-k}$$ would mean $$x$$ is in some middle third.
So let $$f(\sum b_i 3^{-i}) = \sum c_i 2^{-i}$$ where if $$b_i = 0$$ then $$c_i = 0$$ and if $$b_i = 2$$ then $$c_i = 1$$. $$f$$ is a bijection between the Cantor set and $$[0,1]$$.
but on the other hand, a set of measure 0 that has a bijection with R looks very strange as well. | {
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but on the other hand, a set of measure 0 that has a bijection with R looks very strange as well.
Ah.... not really. It seems counterintuitive because ... to have measure $$0$$ no two points can be connected in the set so $$\color{red}{\text{for any point there must be a measurable distance before the "next" one}}$$ and there can only be countably many such points. But that clause in $$\color{red}{\text{red}}$$ is completely erroneous and is based on a naive concept of numbers must "follow each other". Uncountable numbers don't.
And the Cantor set exists merely to be a simple counter example.
• Comments are not for extended discussion; this conversation has been moved to chat. – Pedro Tamaroff Mar 23 at 13:00 | {
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# Absolute value inequality for complex numbers
I asked this question on stackexchange, but despite much effort on my part have been unsuccesful in finding a solution.
Does the inequality $$2(|a|+|b|+|c|) \leq |a+b+c|+|a+b-c|+|a+c-b|+|b+c-a|$$ hold for all complex numbers $a,b,c$ ? For real values a case analysis will verify the inequality. What is desired is a proof using the triangle inequality or a counterexample. Thanks in advance.
-
See answers and comments at math.stackexchange.com/questions/793905/…. – Dietrich Burde May 20 '14 at 12:35
You cannot prove it using just the triangle inequality, because it fails in $\mathbb R^3$ with the $l_\infty$ norm: just take the standard basis vectors for $a,b,c$. You’ll probably need to use that $\mathbb C$ is an inner product space. – Emil Jeřábek May 20 '14 at 13:35
Once you have it for $\ell_1^n$ for all $n$ you have it for $L_1(0,1)$ by approximation. Once you have it for $L_1(0,1)$ you have it for Hilbert spaces because $\ell_2$ embeds isometrically into $L_1(0,1)$ (as the span of IID $N(0,1)$ random variables). – Bill Johnson May 20 '14 at 14:22
If you want to be more sophisticated, once you have it for some infinite dimensional space you have it for Hilbert spaces by Dvoretzky's theorem. – Bill Johnson May 20 '14 at 14:23
Even more sophisticated is that every two dimensional real Banach space embeds isometrically into L$_1(0,1)$, so the inequality is true in all two dimensional Banach spaces. – Bill Johnson May 20 '14 at 14:38
It seems that your inequality is just an incarnation of Hlawka's inequality which says that for any vectors $x, y, z$ in an inner product space $V$ we have
\begin{equation*} \|x+y\| + \|y+z\|+\|z+x\| \le \|x\|+\|y\| + \|z\| + \|x+y+z\|. \end{equation*}
Using $x=a+b-c$, $y=a+c-b$, and $z=b+c-a$ we obtain the inequality in the OP. | {
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Using $x=a+b-c$, $y=a+c-b$, and $z=b+c-a$ we obtain the inequality in the OP.
To add some more context, please see the paper linked here, which provides quite a nice summary of work related to Hlawka's inequality, which apparently stems back to a 1942 paper of Hornich (also cited by Zurab below). The paper linked to above explores the interesting generalization: \begin{equation*} f(x+y) + f(y+z) + f(z+x) \le f(x+y+z) + f(x)+f(y)+f(z), \end{equation*} where $x,y,z$ may come from an Abelian group, or a linear space, or the real line---each with its own set of conditions on the mapping $f$. The functional form of Hlawka's inequality is credited to a 1978 paper of Witsenhausen.
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Do you know a proof of it? – Qiaochu Yuan May 21 '14 at 0:46
For a proof please see: books.google.com/… . That link also mentions extension to Banach spaces by Lindenstrauss and Pelcynski (under certain embedability assumptions) – Suvrit May 21 '14 at 0:50
It looks like Lindenstrauss & Pelczynski had in mind the same observations I made in comments above. That approach for extending inequalities from the real line to $L_p$ spaces has of course been around for a long time. – Bill Johnson May 21 '14 at 4:28
In general, once you've proven an inequality like this in ${\bf R}$ it holds automatically in any Euclidean space (including ${\bf C}$) by averaging over projections. ("Inequality like this" = inequality where every term is the length of some linear combination of variable vectors in the space; here the vectors are $a,b,c$.) In the case of complex numbers we have $$|z| = \frac14 \int_0^{2\pi} \bigl| {\rm Re}(e^{i\theta} z) \bigr| \, d\theta.$$ Applying this to $z=a$, $b$, $c$, and $a \pm b \pm c$ reduces the desired inequality to the one-dimensional case. In $d$-dimensional space we'd write $C\|z\|$ as an average of $|u \cdot z|$ over $u$ in the unit sphere (for a suitable constant $C>0$). | {
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I learned this trick at MOP 30+ years ago, and don't know or remember who discovered it. I didn't even know that the specific inequality we were assigned was due to Hlawka (if I remember right that it was the inequality $$\|x+y\| + \|y+z\|+\|z+x\| \le \|x\|+\|y\| + \|z\| + \|x+y+z\|$$ quoted by Suvrit). We were shown the averaging solution after laboring to prove it bare-handed. The reference Suvrit cites does not use the averaging method, so I do not know whether it too is due to Hlawka or to another mathematician.
-
Notice that $T:\ell_2^n \to L_1(S^{n-1})$ defined by $(Tx)(y):= \langle x, y \rangle$ is another (multiple of an) isometric embedding of an $n$ dimensional Hilbert space into $L_1$. So at the appropriate conceptual level, the two proofs are basically the same. – Bill Johnson May 21 '14 at 6:35
Very nice trick! +1 – Malik Younsi May 21 '14 at 14:09
In fact the Hlawka's inequality first appeared (as a special case of more general result) in H. Hornich, Eine Ungleichung für Vektorlängen, Mathematische Zeitschrift 48 (1942), 268-274 http://gdz.sub.uni-goettingen.de/dms/load/img/?PPN=PPN266833020_0048&DMDID=DMDLOG_0025&LOGID=LOG_0025&PHYSID=PHYS_0256 (see p. 268. P.S. as Joni Teräväinen has remarked, Hornich credits on page 274 to Hlawka an algebraic proof of this special case and reproduces it).
Hlawka's original proof, besides the book indicated by Suvrit, can be found in "Classical and New Inequalities in Analysis" by D.S. Mitrinovic, J. Pecaric and A.M Fink, p. 521 and in "Analytic Inequalities" by D.S. Mitrinovic, p.171. Both books provide Adamovic and Djorkovic generalizations of the Hlawka's inequality. Interestingly, all these generalizations are special cases of more general result given in http://www.sciencedirect.com/science/article/pii/S0022247X96904588 (Generalizations of Dobrushin's Inequalities and Applications, by M. Radulescu and S. Radulescu). | {
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Another proof of Hlawka's inequality can be found in http://www.sbc.org.pl/Content/34160/1995_13.pdf (On two geometric inequalities, by A. Simon, P. Volkmann), and still another one in http://www.jstor.org/discover/10.2307/2310890?uid=3738936&uid=2&uid=4&sid=21104051771107 (The Polygonal Inequalities, by D.M. Smiley and M.F. Smiley).
-
On page 274 it says that Hlawka had already proved the special case which is this inequality. I'm not sure though if that was published. – Joni Teräväinen May 21 '14 at 11:38 | {
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# Math Help - Is emptyset a function?
1. ## Is emptyset a function?
My discrete book is defining a function, f, as a special type of relationship in which if both $(a,b) \in f$ and $(a,c) \in f$, then $b=c$ (and a relation is defined as a set of ordered pairs).
So, is the empty set not a function because it doesn't have any ordered pairs, or is it a function because it does not violate the definition of a function?
(1) Is it a function? If not, explain why.
(2) If yes, what are it's domain and range?
(3) Is the function one-to-one? If not, explain why.
(4) If yes, what is the inverse function?
f. $f=\emptyset$
(1) Yes (trivially), because there are no ordered pairs in $f$, it does not violate the definition of function.
(2) dom $f$ = im $f$ = $\emptyset$
(3) Yes (trivially), since the definition of one-to-one is not violated
(4) $f^{-1}=\emptyset$
Above is how I wrote up my homework (but it's not due until Thu.), but I'm not confident it's the right answer.
2. Originally Posted by MSUMathStdnt
My discrete book is defining a function, f, as a special type of relationship in which if both $(a,b) \in f$ and $(a,c) \in f$, then $b=c$ (and a relation is defined as a set of ordered pairs).
So, is the empty set not a function because it doesn't have any ordered pairs, or is it a function because it does not violate the definition of a function?
I wouldn't put it as "not violating" but rather as "does fulfill."
When we say "S is a set of ordered pairs" we mean "For all x, if x in S, then x is an ordered pair." Well, the emptyset fulfills that requirement. For all x, if x is in the empty set then x is an ordered pair. So, in that very specific sense (which is the only sense that matters toward this question), yes, the empty set is a set of ordered pairs.
Originally Posted by MSUMathStdnt
Above is how I wrote up my homework (but it's not due until Thu.), but I'm not confident it's the right answer.
All correct, except I would modify (1) as I mentioned above. | {
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3. Originally Posted by MoeBlee
I wouldn't put it as "not violating" but rather as "does fulfill."
When we say "S is a set of ordered pairs" we mean "For all x, if x in S, then x is an ordered pair." Well, the emptyset fulfills that requirement. For all x, if x is in the empty set then x is an ordered pair. So, in that very specific sense (which is the only sense that matters toward this question), yes, the empty set is a set of ordered pairs.
All correct, except I would modify (1) as I mentioned above.
I understand what you're saying. But I still don't see how to word it (although I'll probably get full credit as long as I've got the idea right). How does this sound:
(1) Yes (trivially). There are no ordered pairs in $f$, therefore; for all ordered pairs in $f$, there are none that have the same first value and a different second value.
4. This reminds me of a thread I saw recently.
http://mymathforum.com/viewtopic.php?f=22&t=18683
5. But you missed mentioning that every member of the empty set is an ordered pair.
I'll do it in English [where '0' stands for the empty set]:
For all x, if x is in 0 then, x is an ordered pair. So 0 is a relation. And for all x, y, z, if <x y> and <x z> are in 0 then y=z. So 0 is a relation that is moreover a function.
In symbols:
Ax(x in 0 -> x is an ordered pair).
So 0 is a relation.
Axyz((<x y> in 0 & <x z> in 0>) -> y=z).
So 0 is a function.
/
If you want to get more detailed, you can mention that
Ax(x in 0 -> x is an ordered pair)
Axyz((<x y> in 0 & <x z> in 0>) -> y=z)
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# Why null space and column space?
I am not asking this question for WHAT is null space or WHAT is column space. I have finished learning about the definitions of these two concepts for a while. However, to install these concepts in my mind forever, I really want to know what the purposes are for null space and column space of a vector.
Thanks!
• Null space. They are the solutions to the equation $Ax=0$ where A and x are matrices. It's like asking why is $x= \frac{-b +-\sqrt{b^2-4ac}}{2a}$ solutions important. – The Artist Oct 23 '14 at 8:29
• math.stackexchange.com/questions/21131/… This will be helpful. Meaning of Null space is asked here. – The Artist Oct 23 '14 at 8:42
• I think you mean "null space and column space of a matrix," Justin. – Gerry Myerson Oct 23 '14 at 9:19
Perhaps an example will clarify things.
Let's suppose that the matrix A represents a physical system. As an example, let's assume our system is a rocket, and A is a matrix representing the directions we can go based on our thrusters. So what do the null space and the column space represent?
Well let's suppose we have a direction that we're interested in. Is it in our column space? If so, then we can move in that direction. The column space is the set of directions that we can achieve based on our thrusters. Let's suppose that we have three thrusters equally spaced around our rocket. If they're all perfectly functional then we can move in any direction. In this case our column space is the entire range. But what happens when a thruster breaks? Now we've only got two thrusters. Our linear system will have changed (the matrix A will be different), and our column space will be reduced.
What's the null space? The null space are the set of thruster intructions that completely waste fuel. They're the set of instructions where our thrusters will thrust, but the direction will not be changed at all. | {
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Another example: Perhaps A can represent a rate of return on investments. The range are all the rates of return that are achievable. The null space are all the investments that can be made that wouldn't change the rate of return at all.
Another example: room illumination. The range of A represents the area of the room that can be illuminated. The null space of A represents the power we can apply to lamps that don't change the illumination in the room at all.
Good luck!
• best answer ever :D – Justin Chan Oct 24 '14 at 1:37
• @JustinChan Thanks man! If you'd like to know more about applications of Linear Algebra like the ones I've described, reviewing the online lectures of Linear Dynamical Systems by Stephen Boyd of Stanford may be of interest after your Linear Algebra class. – NicNic8 Oct 24 '14 at 1:44
• by the way, I was looking at some questions that are quite similar to mine, and I used your answer as a quote! Hope you don't mind :D – Justin Chan Oct 24 '14 at 1:58
• @JustinChan Of course not! It's very flattering. I'm happy to help. – NicNic8 Oct 24 '14 at 3:03
• According to your explanation, there's no way Range(A) could possibly equal to Null(A), right? I also tried thinking if it's possible for Range(A) = Null(transpose(A)) I think the answers to both questions are no. – Justin Chan Oct 24 '14 at 21:11 | {
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T where {\displaystyle R} A Usually inside a triangle until , unless it's mentioned. B {\displaystyle h_{a}} {\displaystyle \triangle ABC} = and are the lengths of the sides of the triangle, or equivalently (using the law of sines) by. 172-173). and center Incenter & Incircle Action! = B London: Macmillan, pp. c Trilinear coordinates for the vertices of the excentral triangle are given by[citation needed], Let {\displaystyle \triangle BCJ_{c}} {\displaystyle b} b A B In this situation, the circle is called an inscribed circle, and its center is called the inner center, or incenter. {\displaystyle A} Construct a Triangle Given the Length of Its Base, the Difference of Its Base Angles The "inside" circle is called an incircle and it just touches each side of the polygon at its midpoint. . {\displaystyle {\tfrac {1}{2}}br} r {\displaystyle u=\cos ^{2}\left(A/2\right)} the orthocenter (Honsberger 1995, Let The circumcircle is the anticomplement of the … Yes! {\displaystyle \triangle IT_{C}A} Emelyanov, Lev, and Emelyanova, Tatiana. , A C T 1893. {\displaystyle {\tfrac {1}{2}}cr_{c}} {\displaystyle T_{A}} {\displaystyle \triangle IAC} a C , and {\displaystyle a} {\displaystyle \triangle ABC} , and so To these, the equilateral triangle is axially symmetric. ed., rev. [citation needed], The three lines In geometry, the incircle or inscribed circle of a triangle is the largest circle contained in the triangle; it touches (is tangent to) the three sides. x the length of Maximum number of 2x2 squares that can be fit inside a right isosceles triangle. to Modern Geometry with Numerous Examples, 5th ed., rev. Now, let us see how to construct the circumcenter and circumcircle of a triangle. , I b Regular polygons inscribed to a circle. J "On the Equations of Circles (Second Memoir)." T It is orthogonal to the Parry {\displaystyle r} And also find the circumradius. Its sides are on the external angle bisectors of the reference triangle (see figure at top of page). T J b 2 | {
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are on the external angle bisectors of the reference triangle (see figure at top of page). T J b 2 {\displaystyle r} ( 2380, 2381, 2382, 2383, 2384, 2687, 2688, 2689, 2690, 2691, 2692, 2693, 2694, 2695, A 2 When an arbitrary point is taken on the circumcircle, then the . r Circle $$\Gamma$$ is the incircle of triangle ABC and is also the circumcircle of triangle XYZ. {\displaystyle r} b The circumcircle can be specified using trilinear , are See also Tangent lines to circles. △ C B 1 {\displaystyle s} and = Δ three perpendicular bisectors , , and meet (Casey z B The construction first establishes the circumcenter and then draws the circle. The center of the circumcircle B Assoc. c {\displaystyle \triangle ACJ_{c}} , we see that the area Δ ( B B The collection of triangle centers may be given the structure of a group under coordinate-wise multiplication of trilinear coordinates; in this group, the incenter forms the identity element. a {\displaystyle A} {\displaystyle c} This is the center of the incircle, the circle tangent to the three sides of the triangle. C 2 , etc. C I , and Trilinear coordinates for the vertices of the incentral triangle are given by[citation needed], The excentral triangle of a reference triangle has vertices at the centers of the reference triangle's excircles. , and is the distance between the circumcenter and the incenter. London: Macmillian, pp. u ed., rev. and Walk through homework problems step-by-step from beginning to end. and [6], The distances from a vertex to the two nearest touchpoints are equal; for example:[10], Suppose the tangency points of the incircle divide the sides into lengths of c A d ( △ △ Additionally, the circumcircle of a triangle embedded in d dimensions can be found using a generalized method. C B △ {\displaystyle x} b by discarding the column (and taking a minus sign) and {\displaystyle a} (or triangle center X8). The Gergonne triangle (of is also known as the extouch triangle of : ∠ {\displaystyle c} A | {
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