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Begin with n + 1 interpolation points evenly spaced in [0; 2pi]. For example, Figure 1 shows 4 points and a polynomial which passes through them. Create a new le named Newton interpolant. Purpose Native implementation of the Lagrange interpolation algorithm over finite fields. W8V5 Python:Lagrange Interpolation 6:33. w... | {
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module to extract multiple point data. The Foundation region is where the parent Interpolation class is defined. mechtutor com 568 views. PolynomialInterpolationPolynomial Interpolation Thepolynomialinterpolationproblemistheproblemofconstructingapolynomialthatpassesthroughor interpolatesn+1datapoints(x0. 1000 loops, be... | {
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is to determine a polynomial of degree n which interpolates f at the points in question. There is a unique straight line passing through these points. Nominators and denominators fo the base-polynomials are calculated and used to build ab the interpolation polynomial. lagrange taken from open source projects. 93 KB #!/... | {
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interpolation scheme in three dimensions that is both C1 and isotropic. 4) x k+1 = x k 1 1 2 (x k 1 x. y array of data: ydata = 1 1 3 LINEAR INTERPOLATION: x data chosen: x1 = 1, x2 = 0 , x3 = -2. InterpolatingFunction works like Function. where is the barycentric weight, and the Lagrange interpolation can be written a... | {
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and Spline Interpolation using 5 data points. interpolate packages wraps the netlib FITPACK routines (Dierckx) for calculating smoothing splines for various kinds of data and geometries. In that sense, in Section 2 we consider the construction of the unique Lagrange interpolating polynomial on a set of interpolating no... | {
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and, with another variable, the Lagrange polynomials can be written by using the given data. In this situation, g(x, y, z) = 2x + 3y - 5z. lagrange's inverse interpolation method Basic GAUSS ELIMINATION METHOD, GAUSS ELIMINATION WITH PIVOTING, GAUSS JACOBI METHOD, GAUSS SEIDEL METHOD Program to construct Newton's Forwa... | {
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text box. First, Lagrange interpolation is O(n2) where other. The algorithms use their respective interpolation/basis functions, so are capable of producing curves of any order. %It is a matrix and is filled with 1s for multiplication purposes. We discuss the remedies for this, including: optimal distribution of. The L... | {
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is a useful mathematical and statistical tool used to estimate values between two points. Testing You can test the code by cloning the directory, entering it, and typing make test. When graphical data contains a gap, but data is available on either side of the gap or at a few specific points within the gap, an estimate... | {
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Le but pour. This piece of code is a Matlab/GNU Octave function to perform Lagrange interpolation. Looking for the full power of Python 3? Check out our Python 3 Trinket. Survey: Interpolation Methods in Medical Image Processing Thomas M. The derivative of a lineal function is a constant function. Learning a. (2020) Ba... | {
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specifies the domain of the data from which the InterpolatingFunction was constructed. where are the data-points. This implies that $$\displaystyle p(x) = \sum_{i=0}^n y_i \cdot L_i(x)$$ is an interpolation of our data points. The function utilizes the rSymPy library to build the interpolating polynomial and approximat... | {
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is the estimation of an unknown quantity between two known quantities. • It is also possible to set up specialized Hermite interpolation functions which do not include all functional and/or derivative values at all nodes • There may be some missing functional or derivative values at certain nodes. Mathematical interpol... | {
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Volterra integral equations of the second kind. The Lagrange polynomial, displayed in red, has been calculated using this class. You must implement a interpolation what you do by hand when interpolate. abedkime 13 août 2013 à 3:49:33. There are several approaches to polynomial interpolation, of which one of the most we... | {
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this code. The double prime notation in the summation indicates that the first and last terms are halved. In this method, one of the variables is forced to be constant and, with another variable, the Lagrange polynomials can be written by using the given data. For instance, if you. And in another article Linear Interpo... | {
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less than or equal to , the CGL quadrature formula is exact. He is the author of the asciitable, cosmocalc, and deproject packages. But let us explain both of them to appreciate the method later. a guest May 15th, 2014 2,132 Never Not a member of Pastebin yet? it unlocks many cool features! raw download clone embed rep... | {
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we need to exactly fit all the data points whereas it's not the case in regression. format (), string. Regarding number of lines we have: 34 in Python and 37 in Julia. 60 gx f o xx– 1 xx– 2 xx– 3 x o – x 1 x o – x. Let's have a look how to implement Lagrange polynomials and interpolation with Lagrange polynomials on th... | {
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'knot_points. We see that they indeed pass through all node points at , , and. By construction, on. Lagrange I n terpolat io. I don't think you can say splines are always better, but for a lot of data sets it can be beneficial. the latter only guarantees continuity of the zeroeth derivative (the interpolated function i... | {
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entering it, and typing make test. The problem is to find f in a continuum that includes N. They are of degree n−1. The purpose of this paper is to give a local tricubic interpolation scheme in three dimensions that is both C1 and isotropic. Find the Lagrange Interpolation Formula given below, Solved Examples. This is ... | {
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was constructed. i think the bicubic interpolation is more likely 3rd-order Hermite polynomial than 3rd-order Lagrange polynomial interpolation. The function values and sample points , etc. derive Lagrangian method of interpolation, 2. Returns the same object type as the caller, interpolated at some or all NaN values. ... | {
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# Does a 15-puzzle always have a solution
For those that are not familiar with (this type of) sliding puzzles, basically you have a number of tiles on a board equal to (n * m) - 1 (possibly more holes if you want). The goal is to re-arrange the tiles in such a way that solves the puzzle.
The puzzle could be anything,... | {
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Martin Gardner had a very good writeup on the 14-15 puzzle in one of his Mathematical Games books.
Sam Loyd invented the puzzle. He periodically posted rewards for solutions to certain starting configurations. None of those rewards were claimed.
Much analysis was expended, and it was finally determined, through a par... | {
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For the math behind this, please see http://mathworld.wolfram.com/15Puzzle.html
• While this link may answer the question, it is better to include the essential parts of the answer here and provide the link for reference. Link-only answers can become invalid if the linked page changes. - From Review Oct 3, 2016 at 3:1... | {
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# Number of ways to group students with certain conditions
So i have this specific question and i really can't seem to wrap my head around it. The question is as below;
There are 4 girls in a class of 16 students where 2 of the girls are twins. If we want to split them into two equal groups how many different ways ca... | {
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Solution to c) can start by allocating the twins one to each group (which are now therefore distinct eg. "Brenda's team" and "Glenda's team") and picking the other $$7$$ children to be on one of these.
There are $$4$$ girls in a class of $$16$$ students where $$2$$ of the girls are twins. If we want to split them into... | {
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# Fermat's Little Theorem 2
• October 7th 2008, 05:06 PM
bigb
Fermat's Little Theorem 2
Use Fermat's Little Theorem to compute 29^202 mod 13
• October 7th 2008, 06:53 PM
o_O
By Fermat's theorem: $a^{12} \equiv 1 \ (\text{mod } 13)$
Note that $29 \equiv 3 \ (\text{mod } 13)$.
So: $\left(3^{12}\right)^{17} \equiv 1^{1... | {
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or
3^4 = 1 (mod 5) by Fermat theorems.
So (raise to the 75) both sides,
3^300 = 1 (mod 5)
Multiply 3^2 both sides,
3^302 = 3^2 = 9 = 4 (mod 5)
• October 7th 2008, 07:40 PM
o_O
(Yes) I would do it the second way
• October 7th 2008, 07:43 PM
bigb
Quote:
Originally Posted by o_O
(Yes) I would do it the second way
It ca... | {
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###### Example4.3.4
We will try to find a diagonalization of $$A = \left[\begin{array}{rr} -5 \amp 6 \\ -3 \amp 4 \\ \end{array}\right] \text{.}$$
First, we find the eigenvalues of $$A$$ by solving the characteristic equation
\begin{equation*} \det(A-\lambda I) = (-5-\lambda)(4-\lambda)+18 = (-2-\lambda)(1-\lambda) ... | {
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If we choose a different basis for the eigenspaces, we will also find a different matrix $$P$$ that diagonalizes $$A\text{.}$$ The point is that there are many ways in which $$A$$ can be written in the form $$A=PDP^{-1}\text{.}$$
in-context | {
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Hi,
I'm reviewing some basic calculus concepts, and I have the following questions that will probably be very easy for you guys to answer, although I'm stuck since a long time on them.
Given the usual limit that is used in the definition of "e": lim (n->infinity) (1+1/n), how do we prove that the limit exists and it ... | {
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We should find a way to prove it without using the natural logarithm...
Alternatively, we could define ln(x) as the integral of 1/t from 1 to x, and then napier's inequality should be simple to infer, but we need to use a integral definition...
there should be an elementary method to prove it...
4. There are so many... | {
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Now if we let $h = \frac{1}{n}$, then as $h \to 0, n \to \infty$.
Thus $e = \lim_{n \to \infty}\left(1 + \frac{1}{n}\right)^n$.
9. Another useful way to define ln(x) is simply:
$ln(x)= \int_1^x \frac{1}{t}dt$
Since $\frac{1}{t}$ is defined for all non-zero t, this defines ln(x) for all positive x. It is clearly cont... | {
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# Is $ax^2 + bx = 0$ considered a quadratic equation? Or is it linear, since it simplifies to $ax+b=0$?
I know that a quadratic equation can be represented in the form $$ax^2 + bx + c = 0$$ where $$a$$ is not equal to $$0$$, and $$a$$, $$b$$, and $$c$$ are real numbers. However, if there is an equation in the form $$a... | {
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$$ax^2+bx=0$$ is not equivalent to $$ax+b=0$$. Because, $$x=0$$ is not always a root of $$ax+b=0.$$
• I see. But upon multiplying x to both sides of ax + b = 0, we get ax^2 + bx = 0. Would this be a quadratic equation (since a is not equal to 0)? Or would it be linear since it is the same as ax + b = 0? Apr 19, 2021 a... | {
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What is the difference between collinear vectors and parallel vectors?
| 241 views
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Suppose that $\mathbf{v}$ and $\mathbf{w}$ are vectors in 2 -space or 3 -space with a common initial point. If one of the vectors is a scalar multiple of the other, then the vectors lie on a ... | {
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## Kainui one year ago Here's a fun problem I came across. During an hour two independent events can happen at any time. What's the probability that the events are at least 10 minutes apart?
1. rational
|dw:1432999492312:dw|
2. Kainui
Hahaha yeah you got it.
3. rational
P(|E1-E2| > 10 minutes) = (Area of shaded r... | {
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19. dan815
i solved it by looking at a number line
20. dan815
|dw:1433020285038:dw|
21. dan815
so i thought about summing all these individual events up (kind of like an integral) however its constant here its always 5/6 of the smaller case, that event 1 happens at the one single point
22. dan815
Hence 5/6 for t... | {
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40. dan815
i dont think i understand the graph
41. Kainui
Honestly I don't think it gets much better than this general case: $\frac{(60-t)^2}{60^2}$ this tells you the probability that two random events happen with at least a separation of time t within an hour. Like if t=10 in the original then we get: $\frac{(60-1... | {
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52. Kainui
|dw:1433021690423:dw|
53. dan815
ok wat u drew there looks like its as long all 3 dont happen within 10, so u can have 2 with in 10 and other out
54. Kainui
So one example of an event can be like (10, 20, 30) since they are all separated by 10 min.
55. dan815
|dw:1433021882772:dw|
56. Kainui
yeah wh... | {
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Question 1
Consider an undirected random graph of eight vertices. The probability that there is an edge between a pair of vertices is 1/2. What is the expected number of unordered cycles of length three?
A 1/8 B 1 C 7 D 8
GATE CS 2013 Graph Theory
Discuss it
Question 1 Explanation:
Question 2
Which of the followi... | {
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Question 5 Explanation:
Question 6
Let G be a complete undirected graph on 6 vertices. If vertices of G are labeled, then the number of distinct cycles of length 4 in G is equal to
A 15 B 30 C 45 D 360
GATE CS 2012 Graph Theory
Discuss it
Question 6 Explanation:
There can be total 6C4 ways to pick 4 vertices from... | {
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Question 8 Explanation:
The expression ξ(G) is basically sum of all degrees in a tree. For example, in the following tree, the sum is 3 + 1 + 1 + 1.
a
/ | \
b c d
Now the questions is, if sum of degrees in trees are same, then what is the relationship between number of vertices present in both trees? The answer... | {
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Question 9 Explanation: | {
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A generic algorithm or method to solve this question is 1: procedure isV alidDegreeSequence(L) 2: for n in list L do 3: if L doesn’t have n elements next to the current one then return false 4: decrement next n elements of the list by 1 5: arrange it back as a degree sequence, i.e. in descending order 6: if any element... | {
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is non negative so its not a graphical. Option III) 7,6,6,4,4,3,2,2 → 5,5,3,3,2,1,1 → 4,2,2,1,1,0 → 1,1,0,0,0 → 0,0,0,0 so its graphical. Option IV) 8,7,7,6,4,2,1,1 , here degree of a vertex is 8 and total number of vertices are 8 , so it’s impossible, hence it’s not graphical. Hence only option I) and III) are graphic... | {
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Question 10
What is the chromatic number of an n-vertex simple connected graph which does not contain any odd length cycle? Assume n >= 2.
A 2 B 3 C n-1 D n
GATE-CS-2009 Graph Theory
Discuss it | {
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Question 10 Explanation:
The chromatic number of a graph is the smallest number of colours needed to colour the vertices of so that no two adjacent vertices share the same colour. These types of questions can be solved by substitution with different values of n. 1) n = 2 This simple graph can be coloured with 2 colours... | {
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Question 12 Explanation:
A planar graph is a graph which can drawn on a plan without any pair of edges crossing each other. A) FALSE: A disconnected graph can be planar as it can be drawn on a plane without crossing edges. B) FALSE: An Eulerian Graph may or may not be planar. An undirected graph is eulerian if all vert... | {
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Question 14 Explanation:
For a simple, connected, planar graph with v vertices and e edges, the following simple conditions hold: If v ≥ 3 then e ≤ 3v − 6; Note that the question is about non-planar graph G. Only option C doesn't follow above. Alternate Explanation: We know that K5K5 (which has 10 edges and 5 vertice... | {
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Question 16 Explanation:
If we reverse directions of all arcs in a graph, the new graph has same set of strongly connected components as the original graph. Se http://www.geeksforgeeks.org/strongly-connected-components/ for more details.
Question 17
Consider an undirected graph G where self-loops are not allowed. The ... | {
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Since graph is undirected, two edges from v1 to v2 and v2 to v1
should be counted as one.
So total number of undirected edges = 1012/2 = 506.
Question 18
An ordered n-tuple (d1, d2, … , dn) with d1 >= d2 >= ⋯ >= dn is called graphic if there exists a simple undirected graph with n vertices having degrees d1, d2, … ,... | {
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Question 20 Explanation:
Below is a cyclic graph with 5 vertices and its complement graph. The complement graph is also isomorphic (same number of vertices connected in same way) to given graph.
Question 21
If G is a forest with n vertices and k connected components, how many edges does G have?
A floor(n/k) B ceil(n/... | {
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Question 23 Explanation:
There are n nodes which are single and 1 node which belong to empty set. And since they are not having 2 or more elements so they won’t be connected to anyone hence total number of nodes with degree 0 are n+1 hence answer should be none. Thanks to roger for the explanation.
Question 24
The 2n ... | {
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Question 25 Explanation:
An undirected graph is called a planar graph if it can be drawn on a paper without having two edges cross and such a drawing is called Planar Embedding. We say that a graph can be embedded in the plane, if it planar. A planar graph divides the plane into regions (bounded by the edges), called f... | {
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Question 26 Explanation:
Background Explanation: Vertex cover is a set S of vertices of a graph such that each edge of the graph is incident to at least one vertex of S. Independent set of a graph is a set of vertices such that none of the vertices in this set have an edge connecting them i.e. no two are adjacent. A si... | {
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Question 27 Explanation:
A graph is planar if it can be redrawn in a plane without any crossing edges. G1 is a typical example of nonplanar graphs.
Question 28
Let s and t be two vertices in a undirected graph G + (V, E) having distinct positive edge weights. Let [X, Y] be a partition of V such that s ∈ X and t ∈ Y. C... | {
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Question 30 Explanation:
Minimum: The removed vertex itself is a separate connected component. So removal of a vertex creates k-1 components. Maximum: It may be possible that the removed vertex disconnects all components. For example the removed vertex is center of a star. So removal creates n-1 components.
Question 3... | {
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Question 34 Explanation:
Background required - Basic Combinatorics Since the given graph is undirected, that means the order of edges doesn't matter. Since we have to insert an edge between all possible pair of vertices, therefore problem reduces to finding the count of the number of subsets of size 2 chosen from the s... | {
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Question 36 Explanation:
An n-vertex self-complementary graph has exactly half number of edges of the complete graph, i.e., n(n − 1)/4 edges, and (if there is more than one vertex) it must have diameter either 2 or 3. Since n(n −1) must be divisible by 4, n must be congruent to 0 or 1 mod 4; for instance, a 6-vertex gr... | {
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Question 39 Explanation:
A planar graph is a graph on a plane where no two edges are crossing each other. The set of regions of a map can be represented more abstractly as an undirected graph that has a vertex for each region and an edge for every pair of regions that share a boundary segment. Hence the four color theo... | {
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Question 41 Explanation:
The minimum weight happens when (S,U) + (U,V) = (S,V) Else (S,U) + (U,V) >= (S,V) Given (S,U) = 53, (S,V) = 65 53 + (U,V) >= 63 (U,V) >= 12. This solution is contributed by Anil Saikrishna Devarasetty
Question 42
What is the chromatic number of the following graph?
A 2 B 3 C 4 D 5
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# Domain of $f(x)=\arctan\left(\frac{x^2}{2-x^2}\right)$
I need to find the domain of the function $f(x)=\arctan\left(\frac{x^2}{2-x^2}\right)$. I used a graphing calculator and found that the domain is $(-\infty,-\sqrt 2) ∪ (-\sqrt 2, \sqrt 2)∪(\sqrt 2, \infty)$ but I'm unsure as to how to do it without a calculator.... | {
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What you have with $\arctan (x^2/(2-x^2))$ is an expression. The set of all $x$ for which an expression is defined is often called the "natural domain" of the expression.
Assuming the natural domain is what we're after here leads to two questions: For which $x$ is $x^2/(2-x^2)$ defined, and for which $y$ is $\arctan y... | {
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# number of ways to arrange books
I am solving number of arrangements for following question:
Eight books are placed on a shelf. Three of them form a 3-volume series, two form a 2-volume series, and 3 stand on their own. In how many ways can the eight books be arranged so that the books in the 3-volume series are pla... | {
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Since the number of ways in which $$5$$ books can be arranged is $$5!=120,$$ we have $$120$$ ways.
• thanks for explanation. Can you point out what is wrong in my analysis Sep 17, 2020 at 11:24
• As mentioned in the comments, you calculated the $C$ books to be next to each other, but they do not need to be next to eac... | {
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# 3.10 When are propositions truncated?
At first glance, it may seem that the truncated versions of $+$ and $\Sigma$ are actually closer to the informal mathematical meaning of “or” and “there exists” than the untruncated ones. Certainly, they are closer to the precise meaning of “or” and “there exists” in the first-o... | {
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A very common example is “$A$ is isomorphic to $B$”, which strictly speaking means only that there exists some isomorphism between $A$ and $B$. But almost invariably, when proving such a statement, one exhibits a specific isomorphism or proves that some previously known map is an isomorphism, and it often matters later... | {
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A universal consensus may not be possible. Perhaps depending on the sort of mathematics being done, one convention or the other may be more useful — or, perhaps, the choice of convention may be irrelevant. In this case, a remark at the beginning of a mathematical paper may suffice to inform the reader of the linguistic... | {
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“there merely exists an $x:A$ such that $P(x)$
indicates the type $\mathopen{}\left\|\mathchoice{\sum_{x:A}\,}{\mathchoice{{\textstyle\sum_{(x:A)% }}}{\sum_{(x:A)}}{\sum_{(x:A)}}{\sum_{(x:A)}}}{\mathchoice{{\textstyle\sum_{(x% :A)}}}{\sum_{(x:A)}}{\sum_{(x:A)}}{\sum_{(x:A)}}}{\mathchoice{{\textstyle\sum_% {(x:A)}}}{\s... | {
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2. 2.
Using truncated logic as the default in type theory suffers from the same sort of “abuse of language” problems as set-theoretic foundations, which untruncated logic avoids. For instance, our definition of “$A\simeq B$” as the type of equivalences between $A$ and $B$, rather than its propositional truncation, mea... | {
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# Thread: Enough conditions for this question? smallest number if exactly 93.6% answered
1. ## Enough conditions for this question? smallest number if exactly 93.6% answered
For the following question, do we have enough conditions to get the solution?
-----
What is the fewest number of people surveyed if exactly 93.... | {
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5. Originally Posted by mathdaughter
0.936x = y;
The fewest x should be 105 to make y a whole number. Because 6*5 is ended with 0, so 93.6*5 will a whole number, then 105 must be the smallest. Am I correct?
If 105 people were surveyed, then 0.936*105 =98.28 people completed the survey?
That's a fractional number of p... | {
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# Math Help - Deduce to inequality
1. ## Deduce to inequality
By using the formulae expressing $sin\theta$ and $cos\theta$ in terms of t, where $(t\equiv tan\frac{\theta}{2})$ or otherwise, show that $\frac{1+sin\theta}{5+4cos\theta}\equiv\frac{(1+t)^ 2}{9+t^2}$.
Deduce that $0\leq\frac{1+sin\theta}{5+4cos\theta}\leq... | {
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$= \frac{ (1+\tan{\theta/2})^2}{ 9 + \tan^2{\theta/2}}$
for part 2 , besides using $b^2 - 4ac \geq 0$
here is another method :
$\frac{(1+t)^2 }{t^2 + 9}$
$= \frac{ t^2 + 9 + 2t - 8}{(t-4)^2 + 8(t-4) + 25}$
$= 1 + \frac{2}{ (t-4) + 8 + \frac{25}{t-4}}$
$\leq 1 + \frac{2}{2\sqrt{25} + 8}$
$= \frac{ 10}{9}$ | {
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# A collection of sets that cover all edges in Kn?
The problem is the following:
Let $$\mathcal{F}$$ be a family of distinct proper subsets of {1,2,...,n}. Suppose that for every $$1\leq i\neq j\leq n$$ there is a unique member of $$\mathcal{F}$$ that contains both $$i$$ and $$j$$. Prove that $$\mathcal{F}$$ has at l... | {
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• A set of objects called points.
• A collection of sets of points called lines. If a line $$\ell$$ contains a point $$P$$, we also say that $$P$$ lies on $$\ell$$, or $$\ell$$ passes through $$P$$.
We require that for any two points, there is a unique line passing through both.
We also make some nontriviality assump... | {
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Case 1: $$t\ge \frac{n}{2}$$:
For each $$i \in S$$, we must have a set containing the pair $$(t+1,i)$$, and as we can't have $$2$$ elements from $$S$$ together in another set, we must have $$t$$ new sets.
As $$t+2$$ can be in at most $$1$$ of the above sets, we need at least $$t-1$$ new sets for $$t+2$$ (to have the ed... | {
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# Douglas-Peucker line simplification algorithm time complexity
I am analyzing the time complexity of the Douglas-Peucker line simplification algorithm. Reading online I've found that it has a worst-case running time of $$O(n^2)$$ where $$n$$ is the number of points on the line. However, I haven't been able to provide... | {
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## Best case scenario
We assume the input in the recursion will always be cut down by half. That is, the recursion is $$T(n)=2T(n/2)+2n$$.
Define $$b=a=2$$ and $$f(n)=2n$$. Then, $$T(n)=aT(n/b)+f(n)$$.
We can calculate $$c_{crit}=\log_{b}(a)=\log_2(2)=1$$ and thus $$f(n)=2n=\Theta(n)=\Theta(n^{c_{crit}})$$.
Now we ... | {
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# How does subtracting an exponent from an exponent return a greater value?
I'm just stuck on a problem where I have to simplify an expression. This is the expression:
$$\sqrt{x^2\:-\:\left(\frac{x}{2}\right)^2}$$
The textbook has the answer as:
$$\frac{\sqrt{3}x}{2}$$
I have no idea how to get to that. I've tried... | {
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Hello, I just wanna clear up a confusion.
Is f(x,y) = a + bx + cy + dxy
where x and y are variables and a,b,c,d are constants.
Mark44
Mentor
Hello, I just wanna clear up a confusion.
Is f(x,y) = a + bx + cy + dxy
where x and y are variables and a,b,c,d are constants.
No. A quadratic polynomial in two variables loo... | {
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chiro
Hey n0ya and welcome to the forums.
As a guiding rule for further questions like this, look at the highest term (in terms of the order) and use that as a basis to figure out what kind of equation/polynomial/blah something is on top of other things that need to be considered (for example polynomials only have int... | {
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Anyway, I think both quadratic and bilinear are both correct terms here. However, it might be more intuative to call it a bilinear mapping, instead of a quadratic polynomial, since we have no x^2 or y^2 terms.
coolul007
Gold Member
Thanks for the replies.
Then, as far as I understand it, if you for example have: x = ... | {
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coolul007
Gold Member
f(x,y) IS the 3rd dimension variable. It's basically equivalent to z=xy. For any fixed value z, then yes, you're dealing with a hyperbola, but if you consider the graph intersected by the plane y=mx, then you're dealing with a parabola z=mx2, hence why it's called a hyperbolic paraboloid.
It must ... | {
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# Find $\int_0^{2\pi}\frac1{5-4\cos x}\ dx$
$$\int_0^{2\pi}\frac1{5-4\cos x}\ dx$$ How do I compute this integral? An online integral calculator gives an antiderivative as $$\frac{2\arctan(3\tan\frac x2)}3$$ but then gives the definite integral as $\frac{2\pi}3$. Obviously this doesn't make sense as the antiderivative... | {
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$$\int \frac{\mathrm dx}{5-4\cos x}= \int \frac{1}{5-4\dfrac{1-u^2}{1+u^2}}\frac{2\mathrm du}{1+u^2} =\int \frac{2\mathrm du}{1+(3u)^2}=\frac23\arctan(3u)\\=\frac23\arctan(3\tan \frac x2)$$
But you can do that only on an interval where $u=\tan \frac x2$ is a continuous and differentiable bijection, so not on $[0,2\pi]... | {
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• Nice answer! Here's an interesting article by D. J. Jeffrey about such matters: jstor.org/stable/2690852 – Hans Lundmark Sep 27 '16 at 8:21
• @HansLundmark Very interesting article! Actually, I was wrong when I stated that the antiderivative is not given by a single expression: this article gives $$\frac13x+\frac23\a... | {
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# Can I get a Cholesky decomposition from the inverse of a matrix?
I have the inverse of a giant covariance matrix from which I'd like to draw random instances.
The way I know how to do this is to do a Cholesky decomposition on the covariance matrix and use it to transform a vector of independent Gaussians. So the st... | {
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A <- rbind(c(1,0.8,-0.4), c(0.8,2,0.3), c(-0.4,0.3,3))
e1 <- eigen(A, symmetric=TRUE)
set.seed(1)
X <- matrix(rnorm(5000*ncol(A)), ncol=ncol(A))
draws1 <- t(e1$vectors %*% sqrt(diag(e1$values)) %*% t(X))
draws1.cov <- cov(draws1)
draws1.cov
# [,1] [,2] [,3]
#[1,] 0.9765023 0.8030752 -0.3970233
#[2... | {
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# Why is the dot product of a vector with itself not a linear function? [closed]
If an inner product is linear by definition, i.e., $$\langle\mathbf{v+w},\mathbf{u}\rangle=\langle\mathbf{v},\mathbf{u}\rangle+\langle\mathbf{w},\mathbf{u}\rangle$$ and $$\langle a\mathbf{v},\mathbf{w}\rangle=a\langle\mathbf{v},\mathbf{w}... | {
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A linear map is a function between two vector spaces that preserves vector addition and scalar multiplication. Given a vector space $$V$$ over $$K$$ (where $$K$$ equals $$\mathbb R$$ or $$\mathbb C$$), an inner product on $$V$$ is a function $$V\times V\to K$$ that satisfies certain axioms: conjugate symmetry, linearit... | {
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Yes, the function \begin{align} A_v:V&\to F\\ w&\mapsto\langle w,v\rangle \end{align} is linear for all $$v\in V$$. This does not imply that \begin{align} B:V&\to F\\ w&\mapsto\langle w,w\rangle \end{align} is linear since there is no $$v\in V$$ s.t. $$B=A_v$$, so there is no contradiction.
The definition of a linear ... | {
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# Flirting Sequences (Real Analysis)
I need help with a homework problem and am pretty sure my real analysis teacher made the following definition up:
In a metric space, a sequence $$\{P_n\}_n$$ $$\$$ flirts with $$p$$ iff for each $$\epsilon > 0$$, there is a $$n \in \mathbb{N}$$ and $$m > n$$ such that $$0 < d(p_n,... | {
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• It looks like you just need the sequence to come near $1$ a lot, but then move away. If the definition didn't have the requirement $d(p_n,1)>0$ you could just take the sequence $\{p_n\}$ with $p_{2n}=1,p_{2n+1}=2$. Can you modify that sequence to meet the requirements? – lulu Dec 13 '18 at 21:50
• Hi lulu, thanks for... | {
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• a sequence can flirt with $$p$$ and converge to $$p$$: $$p_n = \begin{cases} \frac{1}{2^n} & \text{ if } n \text{ is even}\\ \frac{1}{n} &\text{ if } n \text{ is odd} \end{cases}$$
• Hello and thank you. I can see why the sequence flirts with 0! I was starting to believe no sequences flirt with anything, but you hav... | {
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# Dot product notation
Let $\mathbf{A=(a_1,a_2,\ldots, a_n)}$ and $\mathbf{B=(b_1,b_2,\ldots,b_n)}$. Many linear algebra books and texts define the dot product as $$\mathbf{A\cdot B^T=a_1b_1+a_2b_2+\cdots+a_nb_n}$$ where $\mathbf{B^T}$ is the transpose of row vector $\mathbf{B}$. But Serge Lang in Linear Algebra defin... | {
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The notation $\mathbf{A \cdot B}$ doesn't sugest any of these things, and you can think directly of the termwise multiplication, then sum.
In Linear Algebra, we often talk about inner products in arbitrary vector spaces, a sort of generalization of the dot product. Given vectors $\mathbf{A}$ and $\mathbf{B}$, a widely... | {
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Incidentally, in terms of notation, I think it is nicer to reserve bold type ($\mathbf a$) for matrices and vectors, and italic type ($a$) for their entries. This provides a visual advantage to the reader, so that it is easy to see at a glance that, for example, $\mathbf A = [\mathbf a_1, \ldots, \mathbf a_n]$ is a mat... | {
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# Factorial
• Sep 27th 2013, 01:45 PM
Natasha1
Factorial
So here goes my problem. I am not allowed a calculator through this problem...
4! = 4 x 3 x 2 x 1
5! = 5 x 4 x 3 x 2 x 1 etc
How many zeros are there at the end of 50! ?
It would take me hours to do 50! = 50 x 49 x....x 1 by hand so how can I show the number ... | {
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So what is the answer for $50!$?
Here is a routine to do what you want: $126!$ as thirty-one trailing zeros.
You can 'play with this change 126 to 119 to 121, see what happens.
• Sep 27th 2013, 02:14 PM
Natasha1
Re: Factorial
Is this correct?
• Sep 27th 2013, 02:17 PM
Natasha1
Re: Factorial
50! has 10 factors of five ... | {
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Just one thing... 50! has 10 factors of five right. What does the 50/25 mean exactly in your sum when you say 50/5 + 50/25? That 25 is a factor of 50, right? But isn't it already accounted for in the 10 factors of five?
• Sep 28th 2013, 02:04 PM
Plato
Re: Factorial
Quote:
Originally Posted by Natasha1
I have a questio... | {
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5.10.15.20.25.30.35.40.45.50.55.60.65.70.75.80.85. 90.95.100.105.110.120.125.126.127.128 =>
5.10.15.20.5.5.30.35.40.45.2.5.5.55.60.65.70.3.5.5.80.85.90.95.4.5.5.105.110.120.5.5.5.126.127.128
In general, you need to n/5 + n/25 + n/125 + n/625 + .... n/(5^kmax).
I hope this help. Let me know if you need more explanati... | {
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