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the x-value 11 has two y-values pair with it. Yes, because the vertical line test shows there are no repeating input values. No. Identify the mapping diagram that represents the relation and determine whether the relation is a function {(-5,-4), (-1,5),(-5,3),(7,8)} Mathematics. Let's ask ourselves: Is each input paire... | {
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One way is to analyze the ordered pairs, and the other way is to use If any vertical line intersects the graph more than once, then the graph does not represent a function. If a relation is not a function, list two ordered pairs that show … Recognizing functions from verbal description word problem. by momanomany. The ... | {
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is paired with more than 1 element in the range. test is used when you graph the ordered pairs. The vertical line test can be used to determine whether a graph represents a function. The equation of a vertical line in the graph, that is parallel to the y-axis is x = a. If a vertical line intersects a curve on an xy -pl... | {
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\) We're asked: Do the points on the graph below represent a function? use the vertical line test to determine whether the relation graphed below is a function. Homework. Relations and Functions (Vertical Line Test) Notes. Vertical Line Test Strategy Try to draw a vertical line on the graph so it intersects the graph i... | {
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for it. However if no vertical line exists that will intersect the graph in more than one location, then the relation is a function. Your LinkedIn profile and activity data to personalize ads and to show you more relevant.... Value of x coordinate represents a function if there is a function can have. C '' graph would ... | {
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only a function ordered-pair numbers can represent relations or functions ( { x } {! Function or not the relation graphed below is a function given a set ordered... By convention, graphs are typically constructed with the input values recap a few things that will intersect graph... 4, 7 ) because of the different amoun... | {
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only touches the graph not represent a function x-value! Ways to determine if the graph represents a function then the relation at only one output the relation by. Check out the Algebra Class e-courses provided by DepEd, p.151 function always the... Determine if a relation is a function because of the relation is a pai... | {
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input values also try a few things will..., the graph more than once, the vertical line test is used you! Is any such line, determine that the graph of a relation and function vertical line test is a line and passes the horizontal will. How each element of the vertical line test proves that a relation is a function mor... | {
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on Algebra Class what constraints are there on the domain the! The points of intersection ; Share ; Edit ; Delete ; Host a game y=f\left ( x\right?... Text … the vertical line test is used to determine whether each relation is a function if input... By a comma curve is a line and passes the vertical line,... No repeati... | {
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Adrian College Hockey D3, Clanked Meaning In Urdu, Reflexis Qr Code Cvs, Planta Guaco En Ingles, Rockford Fosgate P3 12 Box Specs, | {
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# Comparing Areas under Curves
I remembered back in high school AP Calculus class, we're taught that for a series: $$\int^\infty_1\frac{1}{x^n}dx:n\in\mathbb{R}_{\geq2}\implies\text{The integral converges.}$$
Now, let's compare $$\int^\infty_1\frac{1}{x^2}dx\text{ and }\int^\infty_1\frac{1}{x^3}dx\text{.}$$
Of cours... | {
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$$\int \limits _a^b \frac{1}{x^p}\text{ }dx = \int \limits _a^1 \frac{1}{x^p}\text{ }dx + \int \limits _1^b \frac{1}{x^p}\text{ }dx$$
Vary $p$ and look at what happens to each piece individually...
So for your examples, you can examine the four integrals
$\displaystyle \int \limits _{1/2}^1 \frac{1}{x^2}\text{ }dx$,... | {
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# Brachistochrone Problem | {
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Particular attention will be given to the description and analysis of methods that can be used to solve practical problems. Take two points in space, A and B. The brachistochrone problem is to find the curve of the roller coaster's track that will yield the shortest possible time for the ride. Given two points Aand B, ... | {
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Derivation of Lagrangian mechanics We begin with the idea of Fermat’s principle, the idea that light moves in a way to minimize the time it takes to travel along its path. one dimensional scalar problems) and in Chapter 4 (for the general case). com - id: 69e0cb-NWIzO. A famous instance of this is found in what is know... | {
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for Chapter 2. The time for a body to move along a curve y(x) under gravity is given by f = 1 + y ' 2 2 g y ,. 3 the eycloid's geometrical properties, while the rnechanical ones in Sect. 이 문서는 28,420번 읽혔습니다. With that in mind, we consider a related and equally classic problem. This is the way to use Solver Add-in to so... | {
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und einem gleich hoch oder tiefer gelegenen Endpunkt, auf der ein sich reibungsfrei bewegender Massenpunkt unter dem Einfluss der Gravitationskraft am schnellsten zum Endpunkt gleitet. j'ai voulu utiliser le tournebroche pour cuire un poulet , celui ci à tourner environ 5 mn puis plus rien. The brachistochrone problem ... | {
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exhaust velocity by tenfold, to 3000 km/s, and mission delta v to 1800 km/s, while keeping the same comfortable 0. The spline collocation method is considered to find the approximate solution of the brachistochrone problem. The earliest discovery about the nature of light Light II How to rescue drowning people, and how ... | {
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a vital role in determining the brachistochrone of a fluid-filled cylinder. The AP Calculus BC exam is one of the longest AP exams, clocking in at three hours and 15 minutes. every point of the rope has a distan. With that in mind, we consider a related and equally classic problem. The word Brachistochrone cames from A... | {
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carry a bead from rest along the curve, without friction, under constant gravity, to an end point in the shortest amount of time. In this article, we will propose and solve a new problem of the General Brachistochrone Curve (GBC),. An upside down cycloid is the solution to the famous Brachistochrone problem (curve of f... | {
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a generalization of the Fermat’s principle, which was used to derive the Snell’s law in optics, he considers a light travelling in a medium with nonuniform refractive index such that the speed of light increases at the rate of g (to simulate the gravitational acceleration),. The solution, a segment of a cycloid, was fo... | {
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from point A to point B in the least time. Although this problem might seem simple it offers a counter-intuitive result and thus is fascinating to watch. Throughout, the theory is set in the context of examples from applied mathematics and mathematical physics such as the brachistochrone problem, Fraunhofer diffraction... | {
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way to accelerate initially), whereas the tautochrone always includes the minimum point (it is not isochronous to any other point, as can be seen by examining the integral for the descent time given on MathWorld with a more general angle than $\pi$). Once you have already guessed that the brachistochrone is a cycloid, ... | {
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Johann Bernoulli proposed the brachistochrone problem, which asks what shape a wire must be for a bead to slide from one end to the other in the shortest possible time, as a challenge to other mathematicians in June 1696. Question: We Have Shown That Brachistochrone Problem Is Solved By Cycloid. Brachistochrone Problem... | {
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in all textbooks dealing with the calculus of variations. Posts about Brachistochrone written by soyoungsocurious. This problem was formulated by Johann Bernoulli, in Acta Eruditorum, June 1696 14. Solve the Tautochrone Problem. The word brachistochrone, coming from the root words brachistos ( chrone, meaning shortest,... | {
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C, a useful LCP is the Brachistochrone Problem: the. La braquistocrona, Whistler Alley Mathematics. See examples in github repository. There is an optimal solution to this problem, and the path that describes this curve of fastest descent is given the name Brachistochrone curve (after the Greek for shortest 'brachistos... | {
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of the longest AP exams, clocking in at three hours and 15 minutes. In mathematics and physics, a brachistochrone curve, or curve of fastest descent, is the one lying on the plane between a point A and a lower point B, where B is not directly below A, on which a bead slides frictionlessly under the influence of a unifo... | {
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at point A. This book covers the major topics typically addressed in an introductory undergraduate course in real analysis in their historical order. 1 How can one choose the shape of the wire so that the time of descent under gravity (from rest) is smallest possible? (One can also phrase this in terms of designing the... | {
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Jun 26 at 5:16. H Nguyen Eastern Oregon University June 3, 2014 Abstract This paper consists of some detailed analysis of the classic mathematical. Many problems are entirely straightforward,but many others. As Bernoulli noted in the challenge, the fastest curve is in fact not a straight line, even if it is the shortes... | {
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the curve between two points through which a body moves under the force of gravity in a shorter time than for any other curve; the path of quickest. In the work a systematic method is developed for the dynamic analysis of structures with sliding isolation, which is a highly non-linear dynamic problem. Given two points ... | {
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problem (Johann Bernoulli, 1696) served as the starting point for the development of the calculus of variations. References. We constrain the particle to follow a path (r; ;’) = (r( ); ( );’( )), where is an arbitrary parameter. In this article, we will propose and solve a new problem of the General Brachistochrone Cur... | {
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finish with point C. Download The Brachistochrone Curve: The Problem of Quickest Descent book pdf free download link or read online here in PDF. 3 : note1: Aug. In this article, we will propose and solve a new problem of the General Brachistochrone Curve (GBC),. The Brachistochrone problem, which describes the curve th... | {
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everywhere on the internet and is the one remembered with the explanation of the brachistochrone problem. In this paper we consider several gen-eralizations of the classical brachistochrone problem in which friction is considered. The solution of the brachistochrone problem (Johann Bernoulli, 1696) served as the starti... | {
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and cleverness, but also are worthwhile beeause oí their. Posts about Brachistochrone written by soyoungsocurious. Thismeansthatanyforceexertedby. On the brachistochrone In a resistant medium while a body is attracted to a centre of forces in one way or another By the author L. The solution of the Brachistochrone probl... | {
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# Let $A=\pi^2\int_{0}^{1}\frac{\sin(\pi x)}{1 + \sin(\pi x)}dx$ and $B=\int_{0}^{\pi}\frac{x\sin( x)}{1 + \sin( x)}dx$. Find $\frac{A}{B}$.
I am trying to solve the following problem:
Let $$A=\pi^2\int_{0}^{1}\frac{\sin(\pi x)}{1 + \sin(\pi x)}dx \quad \text{and}\quad B=\int_{0}^{\pi}\frac{x\sin( x)}{1 + \sin( x)}dx... | {
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"openwebmath_perplexity": 255.79358853938626,
"openwebmath_score": 0.9999876022338867,
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Can you give me some advice to find the desired value?
Lemma: $$\int_{0}^{\pi} xf(\sin(x))\, dx = \frac{\pi}{2}\int_{0}^{\pi} f(\sin(x)) \, dx$$ Proof: Taking the substitution $$u = \pi -x$$ gives us $$\int_{0}^{\pi} xf(\sin(x))\, dx = \int_{\color{blue}{\pi}}^{\color{blue}{0}}(\pi - u)f(\sin(\pi - u)) (\color{blue}{-... | {
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"openwebmath_score": 0.9999876022338867,
"tag... |
# How to prove that $\sum_{k=0}^{n-1}\cos\left(\frac{2\pi k}{n}+\phi\right)=0$ [duplicate]
Prove that $\displaystyle\sum_{k=0}^{n-1}\cos\left(\frac{2\pi k}{n}+\phi\right)=0$ for $n\in\mathbb{N},n>1$
I'm thinking at a demonstration by induction, as base case $n=2$
$$\sum_{k=0}^{n-1}\cos\left(\frac{2\pi k}{n}+\phi\rig... | {
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"openwebmath_score": 0.7669275999069214,
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• and so simple that I do not know how I did not think about it :c – cand Apr 20 '16 at 22:35
\begin{align} \sum_{k=0}^{n-1}\cos\left(\frac{2\pi k}{n}+\phi\right) &=\mathrm{Re}\left(\sum_{k=0}^{n-1}e^{\frac{2\pi ik}n+i\phi}\right)\\ &=\mathrm{Re}\left(e^{i\phi}\sum_{k=0}^{n-1}e^{\frac{2\pi ik}n}\right)\\ &=\mathrm{Re}... | {
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"openwebmath_score": 0.7669275999069214,
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# Degree of the eighth vertex given the other degrees
Consider a graph with $8$ vertices. If the degrees of seven of the vertices are $1,2,3,4,5,6,7$, find the degree of the eighth vertex. I also have to check the graph's planarity and its chromatic number.
I know that the sum of degrees of vertices is twice the numb... | {
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If $d$ is the missing degree, the Handshaking Lemma implies that $1+2+3+4+5+6+7+d=28+d$ is even, so $d$ is even. Since the degree-$7$ vertex is adjacent to it, $d>0$ and thus $d \in \{2,4,6\}$.
If $d=6$, then the vertex of degree $7$ (which is adjacent to all other vertices) and the two vertices of degree $6$ (which a... | {
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Therefore the degree 5 vertex must be connected to every vertex apart from the degree 1 and 2 vertices. So the degree 3 vertex is connected to the degree 5, degree 6 and 7 vertices only.
Therefore the degree 4 vertex is connected to the degree 5, 6 and 7 vertices but not to the degree 1, 2 and 3 vertices. To have degr... | {
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"openwebmath_score": 0.8016057014465332,
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# I'm not sure I'm correct: $\|\sum_{n=1}^\infty x_n\| \leq \sum_{n=1}^\infty \|x_n\|$
I was solving this Functional Analysis problem, but I'm not sure I'm correct on this one, the problem is:
Let $\big(E,\|\bullet\|\big)$ be a normed vector space, and $(x_n)_n$ a sequence in $E$ such that $\sum_{n=1}^\infty x_n$ con... | {
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"openwebmath_score": 0.9800519347190857,
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If $\sum_{n=1}^\infty \|x_n\|=\infty$ then it is true. So let us assume that $\sum_{n=1}^\infty \|x_n\|<\infty$. So, $$\left\|\sum_{n=1}^m x_n\right\| \leq \sum_{n=1}^m \|x_n\| \le \left\|\sum_{n=1}^m x_n\right\| \leq \sum_{n=1}^m \|x_n\|=l$$ Since, norm is a continuous function so $$\lim_{m\to \infty}\left\|\sum_{n=1}... | {
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# Goldbach Partition
I want to check the Goldbach conjecture for a big number of $n$, but I don't know how to define this in Mathematica.
These are my questions:
1. Find a pair of primes $(p,q)$ for every even integer $n$, such $p+q=n$, using Mathematica.
and
1. How to calculate number of ways to write an even num... | {
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Row[{"Number of prime partitions: ", Length[res]}]
(* "Number of prime partitions: ", 1668 *)
Row[{"Sample: ", Take[SortBy[res, Abs[Subtract @@ #] &], 4]}]
(* "Sample: ", {{59981, 60041}, {60041, 59981}, {59921,
60101}, {60101, 59921}} *)
Here is another computation with a larger number:
In[9]:= AbsoluteTiming[
res... | {
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Given a list $p$ of the first $k$ primes, there is a smallest even $n$ which cannot be represented as $n=p+q$, with prime $q$. The sequence begins $\{6,12,30,98,98,98,98,220,308,...\}$, which is Sloane's A152522. This page links to a paper by Granville, Van de Lune, and te Riele, where they conjecture that the smallest... | {
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f5[n_] := Select[FrobeniusSolve[{1, 1}, n], And @@ Map[PrimeQ, #] &]
f6[n_] := Pick[#, And @@@ PrimeQ[#]] &[FrobeniusSolve[{1, 1}, n]]
Pick is faster than Select, but the fastest formulation is f4[n] with Cases. The number of solutions is twice as large because both solutions {p,q} and {q,p} are returned. Since half... | {
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# Plane and span Vector
I have the following problem: the plane equation is given by $$x_1-2x_2+4x_3=0$$ I need to come up with the two vectors that spans the plane. So the normal from the equation can be written as $$(1,-2,4)$$ So $$(1,-2,4)$$ has to be equal to the cross product of two vectors:
$$1=a_2b_3-a_3b_2$$ ... | {
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"openwebmath_score": 0.9132746458053589,
"ta... |
# How many $n\times m$ binary matrices are there, up to row and column permutations?
I'm interested in the number of binary matrices of a given size that are distinct with regard to row and column permutations.
If $\sim$ is the equivalence relation on $n\times m$ binary matrices such that $A \sim B$ iff one can obtai... | {
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Here is a computational contribution that treats the case of a square matrix. As pointed out this problem can be solved using the Polya Enumeration Theorem. In fact if we are interested only in counting these matrices, then the Burnside lemma will suffice. We just need to compute the cycle index of the group acting on ... | {
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and
$$Z(Q_5) = {\frac {{a_{{1}}}^{25}}{14400}}+{\frac {{a_{{1}}}^{15}{a_{{2} }}^{5}}{720}}+{\frac {{a_{{1}}}^{9}{a_{{2}}}^{8}}{144}}+{ \frac {{a_{{1}}}^{10}{a_{{3}}}^{5}}{360}}+{\frac {{a_{{1}}}^{ 5}{a_{{2}}}^{10}}{480}}\\+1/48\,{a_{{1}}}^{3}{a_{{2}}}^{11}+{ \frac {a_{{1}}{a_{{2}}}^{12}}{64}}+1/36\,{a_{{1}}}^{6}{a_{{2... | {
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"lm_q1_score": 0.9845754479181589,
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"tag... |
$$Z(Q_3)(A+B+C) = 1/36\, \left( A+B+C \right) ^{9}+1/6\, \left( A+B+C \right) ^{3} \left( {A}^{2}+{B}^{2}+{C}^{2} \right) ^{3}\\+2/9\, \left( {A}^{3}+{B}^{3}+{C}^{3} \right)^{3} +1/4\, \left( A+B+C \right) \left( {A}^{2}+{B}^{2}+{C}^{2 } \right) ^{4}\\+1/3\, \left( {A}^{3}+{B}^{3}+{C}^{3} \right) \left( {A}^{6}+{B}^{6}... | {
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pet_cycleind_symm :=
proc(n)
option remember;
if n=0 then return 1; fi;
end;
pet_varinto_cind :=
proc(poly, ind)
local subs1, subs2, polyvars, indvars, v, pot, res;
res := ind;
polyvars := indets(poly);
indvars := indets(ind);
for v in indvars do
pot := op(1, v);
subs1 :=
[seq(polyvars[k]=polyvars[k]^pot,
k=1..n... | {
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"openwebmath_score": 0.6676205992698669,
"tag... |
# Is a matrix multiplied with its transpose something special?
In my math lectures, we talked about the Gram-Determinant where a matrix times its transpose are multiplied together.
Is $A A^\mathrm T$ something special for any matrix $A$?
• The matrix $A^TA^{-1}$ is generally self similar... – DVD Jun 12 '15 at 3:42
... | {
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The determinant is just the product of these eigenvalues.
• Then you can write $\mathbb R^n\cong A\bot V$. What is $AA^TA?$ and what is $AA^Tv$ for $v\in V$? How does $AA^T$ hence look like? – Simon Markett Jul 25 '12 at 7:57
• What "if $A$ is regular" exactly mean? There seem to be several interpretations on Wikipedi... | {
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$AA^TAu_i=\lambda_iAu_i$.
Now, the originally searched eigenvectors $v_i$ of $AA^T$ can easily be obtained by $v_i:=Au_i$. Note, that the resulted eigenvectors are not yet normalized.
One could name some properties, like if $B=AA^T$ then
$$B^T=(AA^T)^T=(A^T)^TA^T=AA^T=B,$$
so
$$\langle v,Bw\rangle=\langle Bv,w\ran... | {
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• What is this key role? – Matthew C Apr 16 '16 at 0:07
• @lhf, can you extend the properties of this key role? – PlagTag May 6 '17 at 20:54
• From my guessing its because we get a rectangular matrix --> R with the least squares problem beeing Rx = b. In short: The residuals are orthogonal to the fit line. We aim for f... | {
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Software demo (Delphi Pascal):
program Demo;
type
matrix = array of array of double;
procedure Inverteren(var b : matrix);
{
Matrix inversion
pivots on diagonal
}
var
pe : double;
NDM,i,j,k : integer;
begin
NDM := Length(b);
for k := 0 to NDM-1 do
begin
pe := b[k,k];
b[k,k] := 1;
for i := 0 to NDM-1 do
begin
b[i,k] ... | {
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If you have a real vector space equipped with a scalar product, and an Orthogonal matrix $$A$$ then $$AA^T=I$$ holds. A matrix is orthogonal if for the scalar product $$\langle v,w \rangle = \langle Av, Aw \rangle$$ holds for any $$v,w \in V$$
However I don't see a direct link to the Gram-Determinant.
The matrix $$A$... | {
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# Evaluate $\int e^x \sin^2 x \mathrm{d}x$
Is the following evaluation of correct?
\begin{align*} \int e^x \sin^2 x \mathrm{d}x &= e^x \sin^2 x -2\int e^x \sin x \cos x \mathrm{d}x \\ &= e^x \sin^2 x -2e^x \sin x \cos x + 2 \int e^x (\cos^2 x - \sin^2x) \mathrm{d}x \\ &= e^x \sin^2 x -2e^x \sin x \cos x + 2 \int e^x ... | {
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"id": null,
"lm_label": "1. YES\n2. YES",
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"ta... |
$$=\frac{1}{2}e^x-\frac{1}{2}\frac{e^x}{1^2+2^2}(\cos 2x+2\sin 2x)+C$$ $$=\frac{1}{2}e^x-\frac{1}{10}e^x(\cos 2x+2\sin 2x)+C$$ $$=-\frac{e^x(2\sin 2x+\cos 2x-5)}{10}+C$$
We have $$\int e^{x}\sin^{2}\left(x\right)dx=\frac{1}{2}\int e^{x}dx-\frac{1}{2}\int e^{x}\cos\left(2x\right)dx$$ and $$\int e^{x}\cos\left(2x\right)... | {
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"openwebmath_score": 0.9999685287475586,
"ta... |
Your solution is correct. To reach to required form See here,You allready Got this
$$\int e^x \sin^2 x \mathrm{d}x = \frac{e^x \sin^2 x -2e^x \sin x \cos x + 2 e^x}{5}+C$$
Now multiply and divide your result by $2$, you will get $$\frac{2e^x \sin^2 x -4e^x \sin x \cos x +4e^x}{10}+C\\ =\frac{2e^x \sin^2 x -4e^x \sin ... | {
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"id": null,
"lm_label": "1. YES\n2. YES",
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"lm_q2_score": 0.8596637505099168,
"openwebmath_perplexity": 133.12129674819235,
"openwebmath_score": 0.9999685287475586,
"ta... |
# A subgroup of a cyclic group is cyclic - Understanding Proof
I'm having some trouble understanding the proof of the following theorem
A subgroup of a cyclic group is cyclic
I will list each step of the proof in my textbook and indicate the places that I'm confused and hopefully somewhere out there can clear some t... | {
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so $$b$$ is a power of $$c.$$
This final step is confusing as well, but I think its just because of the previous parts I was confused about. Any help in understanding this proof would be greatly appreciated
• "Where does the division algorithm come from?" is a very vague question. In this cases, there is a deeper the... | {
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First, the use of the division algorithm is for the sake of utility, as it provides a basis for the structure of the proof. Here, we want to relate $n$ with $m$, that is, we want to show that $n$ must be a multiple of $m$, but we start with the fact that what we know of any $n$, then given any positive integer $m$, by ... | {
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and since $K$ is proper no generating set contains $h$
for $K$ choose a generating set $\mathfrak{K}$ which contains an element $h^m$ where $m \gt 1$ is minimal amongst all the powers of $h$ occurring in generating sets for K.
without loss of generality we may assume that for any $p \gt 1$ we have $h^{pm} \notin \mat... | {
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# Solving for $\frac 13 +\frac 29 +\frac 3{27}+\cdots$
Evaluate the sum $\frac{1}{3^1} + \frac{2}{3^2} + \frac{3}{3^3} + \cdots + \frac{k}{3^k} + \cdots$
How would I go about solving this problem? I'm thinking of setting the sum to $S$, multiplying by $3$, and then subtracting from the original equation. HELP!
I hav... | {
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$$2S=\frac1{3^0}+\frac1{3^1}+\frac1{3^2}+\cdots=\frac32$$
So that we get $S=\frac 34$.
• So I was correct! Thank you. – A Piercing Arrow Mar 10 '18 at 23:05
• Yup, you were! Be sure to accept one of the answers so that this question doesn't end up in the unanswered section ;) – vrugtehagel Mar 10 '18 at 23:06
Nice a... | {
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## If $$2.00X$$ and $$3.00Y$$ are $$2$$ numbers in decimal form with thousandths digits $$X$$ and $$Y,$$ is
##### This topic has expert replies
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### If $$2.00X$$ and $$3.00Y$$ are $$2$$ numbers in decimal form with thousandths digits $$X$$ and $$Y,... | {
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Aside: the video below has tips on rephrasing the target question
Statement 1:3X < 2Y
PERFECT!!
The answer to the REPHRASED target question is NO, 3X is NOT greater than 2Y
Since we can answer the REPHRASED target question with certainty, statement 1 is SUFFICIENT
Statement 2: X < Y − 3
Add 3 to both sides to get: X ... | {
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# For what pair of integers $(a,b)$ is $3^a + 7^b$ a perfect square.
Problem: For what pair of positive integers $(a,b)$ is $3^a + 7^b$ a perfect square.
First obviously $(1,0)$ works since $4$ is a perfect square, $(0,0)$ does not work, and $(0,1)$ does not work, so we can exclude cases where $a$ or $b$ are zero for... | {
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I would guess that an elementary method shown in an answer by Gyumin Roh to http://math.stackexchange.com/questions/1551324/exponential-diophantine-equation-7y-2-3x can be modified for this task. My way of working with this takes a while...
http://math.stackexchange.com/questions/1941354/elementary-solution-of-exponen... | {
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In the other parity case, if $a=2m$ is even and $b$ is odd, then, along similar lines, we must have $2\cdot3^m=7^b-1$, which has $(m,b)=(1,1)$ as one solution. After checking that there is no solution with $m=0$, it remains to show there are no solutions with $m\gt1$. If there were, then we would have $7^b\equiv1$ mod ... | {
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$$7^x \equiv 1 \pmod {13},$$ $$x \equiv 0 \pmod {12}.$$ Next, $7^x - 1$ is divisible by $7^{12} - 1 = 13841287200 = 32 \cdot 9 \cdot 25 \cdot 13 \cdot 19 \cdot 43 \cdot 181.$
$$3^y \equiv 1 \pmod {43},$$ $$y \equiv 0 \pmod {42}.$$ Next, $3^y - 1$ is divisible by $3^{42} - 1 = 109418989131512359208 = 8 \cdot 7^2 \cdot ... | {
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# probability of picking 4 numbers out of 100
I have $100$ balls and on each ball is labelled a number between $1$ and $100$. No two balls can have the same number, so the interval $1\ldots 100$ is represented by the balls.
I now pick 20 balls at random, one by one without putting them back at any point. I am trying ... | {
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Your solution is not correct. But, here's a hint on a different way to proceed:
Hint: You want to count all permutations (of length $20$) of distinct elements in $\{1,2,\ldots,100\}$, such that four fixed numbers (say $1,2,3,4$) are all contained in the permutation.
To do this: pick the $16$ numbers OTHER than the on... | {
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# Find the probability that no kid will have his own pair of shoes
In the morning, $n$ children come to the kindergarten and leave their shoes in the locker room. Leaving the kindergarten one by one, each child takes one left and one right shoe, accidentally equiprobably choosing them from among the remaining ones. Fi... | {
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$3$) Choose one left included his/her and one right excluded his/her own: $$P(\text{left})P(\text{right})=\left(\frac{n}{\binom{2n}{2}}\cdot\frac{n-1}{\binom{2n-1}{2}}\right)^n$$
$4$) Choose one right inlcuded his her and one left excluded his/her own $$P(\text{right})P(\text{left})=\left(\frac{n}{\binom{2n}{2}}\cdot\... | {
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rperm := nextperm(rperm);
od;
lperm := nextperm(lperm);
od;
res/n!^2;
end;
EX := n -> add(binomial(n,q)*(-1)^q*(n-q)!^2, q=0..n)/n!^2;
Remark. Apparently the asymptotics here follow by inspection, i.e. by writing the probability as
$$\frac{1}{n!} \sum_{q=0}^n \frac{1}{q!} (-1)^q (n-q)! \\ = 1 - \frac{1}{n} + \fra... | {
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# Proof that $C\exp(x)$ is the only set of functions for which $f(x) = f'(x)$
I was wondering the following. And I probably know the answer already: NO.
Is there another number with similar properties as $e$? So that the derivative of $\exp(x)$ is the same as the function itself.
I can guess that it’s probably not, ... | {
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$$T_f(x) = \sum_{k=0}^\infty c_k x^k.$$
Notice that the fact that $f(x)=f^{(k)}(x)$, for all $k\geq 0$, implies that the Taylor series $T_f(x_0)$ converges to $f(x_0)$ for every $x_0\in \mathbb{R}$ (more on this later), so we may write $f(x)=T_f(x)$. Since $f'(x) = \sum_{k=0} (k+1)c_{k+1}x^k = f(x)$, we conclude that ... | {
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If you are worried about $f(x)$ being zero, the above shows $f(x)$ is of the form $C''e^x$ on any interval for which $f(x)$ is nonzero. Since $f(x)$ is continuous, this implies $f(x)$ is always of that form, unless $f(x)$ is identically zero (in which case we can just take $C'' = 0$ anyhow).
Hint \rm\displaystyle\:\ \... | {
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Proceed by induction on $\rm\:n\:.\:$ The Theorem is clearly true if $\rm\:n = 1\:.\:$ Suppose that $\rm\: n > 1\:$ and $\rm\:W(f_1,\ldots,f_n) = 0\:$ for all $\rm\:x\in I.\:$
If $\rm\:f_1 = 0\:$ throughout $\rm\:I\:$ then $\rm\: f_1,\ldots,f_n\:$ are dependent on $\rm\:I.\:$ Else $\rm\:f_1\:$ is nonzero at some point ... | {
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Pick some $t>0$. Consider the equation $$f(x)=\frac{f(x+t)-f(x)}{t}$$
It is not hard to show by induction that there is a function $C_t:[0,t)\to \mathbb{R}$ so that $$f(x)=C_t(\{\frac{x}{t}\})(1+t)^{\lfloor\frac{x}{t}\rfloor}$$
where $\{\cdot\}$ and $\lfloor\cdot\rfloor$ denote fractional and integer part, respectively... | {
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Note that $e$ is defined by the following Limit: $e=\lim_{n \rightarrow \infty}(1+ \frac{1}{n})^n$. Then: $e^x=\lim_{n \rightarrow \infty}(1+ \frac{1}{n})^{nx}$. Applying the Definition of the derivative $f'(x) = \lim_{h \rightarrow 0} \frac{f(x+h)-f(x)}{h}$ one obtains: $(e^x)’=\lim_{h \rightarrow 0} \frac{ \lim_{n \r... | {
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# How to simplify a square root
How can the following:
$$\sqrt{27-10\sqrt{2}}$$
Be simplified to:
$$5 - \sqrt{2}$$
Thanks
-
If you're faced with a question that says "Prove that $\sqrt{27-10\sqrt{2}}$ $=5 - \sqrt{2}$", then it's just a matter of squaring $5 - \sqrt{2}$ and seeing that you get $27-10\sqrt{2}$. Bu... | {
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and this has $\rm\ \sqrt{trace}\: =\: 10,\ \ hence\ \ \ \color{brown}{dividing\ it\ out}\$ of this yields $\rm\ 5 - \sqrt{2} =\:$ sought sqrt.
Remark $\$ The sign of the norm sqrt was chosen to make the trace sqrt rational. The same answer would arise using the opposite sign, but with slightly more work (rationalizing... | {
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I don't know any way to notice this except to just get lucky and notice it; I didn't realize it could happen until sometime in high school when I was astounded to discover that $\sqrt{7+4\sqrt3} = 2+\sqrt3$.
-
Set the nested radical as the difference of two square roots so that $$\sqrt{27-10\sqrt{2}}=(\sqrt{a}-\sqrt{... | {
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# Find a + b.
#### anemone
##### MHB POTW Director
Staff member
If $$\displaystyle f(x)=x^3-6x^2+17x$$ and $$\displaystyle f(a)=16$$ and $$\displaystyle f(b)=20$$, find $$\displaystyle a+b$$.
#### Opalg
##### MHB Oldtimer
Staff member
Re: Find a+b.
If $$\displaystyle f(x)=x^3-6x^2+17x$$ and $$\displaystyle f(a)=16... | {
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$$\displaystyle k^3-6k^2+32+(3ab-17)(4-k)=0$$
$$\displaystyle (k+2)(k-4)^2+(3ab-17)(4-k)=0$$
Therefore, we can conclude that $$\displaystyle k=4$$ must be true.
This implies $$\displaystyle a+b=4$$.
But hey, it is very obvious that my method is the least impressive/worst one...(bh)...
#### Klaas van Aarsen
##### ... | {
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##### Well-known member
MHB Math Helper
Re: Find a+b.
The problem should say:
If $$\displaystyle f(x)=x^3-6x^2+17x$$ and $$\displaystyle f(a)=16$$ and $$\displaystyle f(b)=20$$ with $a,b\in\mathbb{R}$, find $$\displaystyle a+b$$.
Oherwise, we can't guarantee $a+b=4$. I solved the question with that additional hypoth... | {
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# Additional information in conditional probability problem gives a different answer but this does not seem correct to me?
Q1.) I tell you that I have two children and that one of them is a girl. What is the probability that I have two girls? Assume girls and boys equally likely to be born and that the gender of one c... | {
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Baz
• You say for the first scenario that "the probability that I have two girls" yet the probability of $\frac{2}{3}$ seems to be for "the elder child is a girl." Shouldn't the probability of two girls in that scenario instead be $\frac{1}{3}$? – JMoravitz Oct 16 '15 at 18:21
• As for defining the difference between ... | {
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I would say then that the problem is poorly worded and there is not yet enough information to conclude that the probability is indeed $\frac{1}{2}$ without additional assumptions (even if the assumption seems valid).
• Thanks, have to agree that the question is too loosely worded but I understand the difference better... | {
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• OK but this is where I was/am confused. If a girl answers the door and you are not making any assumptions that she is the eldest (or indeed youngest) then surely you can still have {GB, BG, GG}? The observation that a girl answered the door is consistent with any of those three combinations? – Bazman Oct 16 '15 at 18... | {
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1. Suppose that the elder child opens the door with probability $p$ and the younger with probability $1-p$. Let G@D denote the event that you see a girl at the door. We have: $$P\left(GG|G@D\right)=\frac{P\left(G@D|GG\right)P(GG)}{P\left(G@D\right)}=\frac{1/3}{1/3+p/3+\left(1-p\right)/3}=\frac{1}{2}.$$
2. Suppose that... | {
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# Trouble with meeting design specifications of a second order system using Matlab
I am trying to solve the following question.
Consider the transfer function $$G(s) = \frac{1.247}{s^2+9.76s+23.8}$$ is in the forward path of a unity feedback control loop. Assume that it is compensated using a static gain K in the for... | {
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• The formula for settling time is approximate. In this case, $t_s \approx \frac{4}{4.88}=0.82$, and you cannot change this value. You can, however, select K to achieve the required 10%-90% rise time (once again, the formula for this is approximate). So, yes, this can all be done via the root locus, but don't expect th... | {
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# Methods for Finding Asymptotic Lower Bounds
I've found in many exercises where I'm asked to show that $f(n)=\Theta(g(n))$ where the two functions are of the same order of magnitude I have difficulty finding a constant $c$ and a value $n_0$ for the lower bound. I'm using Corman's definition of $\Theta$:
$$\exists c_... | {
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Thus we have $0 < (\frac{2}{3})^bn^b \leq (n+a)^b \leq 2^bn^b$ for any $n \geq 3|a|$.
Is there an easier way to do this?
Normally when looking for upper limit constants where the two functions are of the same magnitude I simply eliminate negative lower order terms and change positive ones into multiples of the highes... | {
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