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As user1952500 says, most (in a quite precise sense) numbers are normal and hence as you expect. However, it is very simple to create exceptions: numbers which are irrational but are not normal. The rational numbers have decimal expansions which, after a while, terminate or are periodic so just create a sequence that d... | {
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• "very hard" is an understatement, I think. Noone has the slightest idea, how a proof of the normality of , lets say , $\pi$ could look like. It cannot even be ruled out that eventually, only digits $0$ and $1$ appear. It is very likely that such a proof is completely out of reach. Apr 21, 2019 at 8:27
• Good answer (... | {
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# Convergence of $\sum\limits_{n = 1}^{\infty} \sin^2(\pi/n)$
I am trying to determine the convergence of $$\sum\limits_{n = 1}^{\infty} \sin^2(\pi/n)$$
After some time I found out that $$sin^2(\pi x) \leq (\pi x)^2$$ holds true for all $$x$$ using a graphing calculator. Which means I can substitute $$x={1\over n}$$ ... | {
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Directly related to series convergence is the Leibniz test for series $$\sum_{k=0}^\infty(-1)^ka_k$$, $$a_k>a_{k+1}>0$$ converging to $$0$$. One result of that test is that the value of that series is bounded by its partial sums $$s_n=\sum_{k=0}^n(-1)^ka_k$$, from below by the odd index sums $$s_{2m+1}$$ and from above... | {
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# Calculate $I(t,n) = \int_{-\infty}^{\infty} \big( \frac{1}{1-jq} \big)^{n} e^{-jqt} dq$
I am trying to calculate integrals of the form:
$$I(t, n) = \int_{-\infty}^{\infty} \Big(\frac{1}{1-jq}\big)^{n} e^{-jqt} dq$$
where $$j = \sqrt{-1}$$. In the case when $$n=1$$, I have:
$$I(t, 1) = \int_{-\infty}^{\infty} \fra... | {
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• This seems to solve it. Thank you very much. I need to revisit my contour integration! So what happens if t is a nonpositive real number? Also, if q = -z, then shouldnt dq = -dz ? But it doesnt matter because of the limits of integration, right? Sorry, for the basic questions, but the only way to get better is to ask... | {
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This gives us $$I(t, 1)=2\pi e^{-t}H(t)$$ and you can verify that the $$n$$-th self-convolution is given by $$I(t,n)=2\pi\frac{t^{n-1}}{(n-1)!}e^{-t}H(t)$$
• Where did you find this fourier transform pair? After looking this up, I only found the first result. I'm not sure still how the (n-1)! comes out of this ,,Fouri... | {
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# Stating the induction hypothesis
I would like to ask about the best way to state the induction hypothesis in a proof by induction.
Just to use a concrete example, suppose I wanted to prove that $n!\ge 2^{n-1}$ for every positive integer $n$.
Assuming that I have already verified the case $n=1$, which of the follow... | {
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The Principle of Mathematical Induction says that for all "properties" $P$, $$\left(P(0)\land\forall k\in \mathbb N\left(P(k) \implies P(k+1)\right)\right)\implies \forall n\in \mathbb N(P(n)).$$
So you're basically asking how to write the $\forall k\in \mathbb N\left(P(k)\implies P(k+1)\right)$ bit.
It's a universal s... | {
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# Can I still use Linear Regression assumptions test on a linear model with a Polynomial variable
I have a multivariate linear model (y=x1+x2) which gives me the following results when using R's plot() function:
I can clearly see that the Normality and Linearity assumptions are not the best. Thus, I decided to add a ... | {
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Is the model still considered linear
Given a dataset composed of a vector $\mathbf{x} = \{ x_1, x_2,...,x_n\}$ of $n$ explanatory variables and one dependent variable $y$ we assume in this model that the relationship between $\mathbf{x}$ and $y$ is linear
$$y = \beta_0 1 + \beta_1 x_1 + \beta_2 x_2 + ... + \beta_n x_... | {
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88 views
Let G be a graph with 10 vertices and 31 edges. If G has 3 vertices of degree 10, 1 vertex of degree 8 and 2 vertices of degree 5 and the other four vertices of degree at least 3, how many vertices are of degree 3________?
my solution:
Σ deg(v) = 2|E|
3*10 + 1*8 + 2*5 + 4*(>=3) = 2*31
4*(>=3) = 62- (30+8+... | {
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ContourPlot and 3Dplot of $f(x,y,z)=x z + y z - x y z$
I have a function of 3 variables: $x$, $y$, and $z$. This is the function: $$f(x,y,z)=x z + y z - x y z$$
1. Is there a way for me to graph this function? (3D graph)
2. Can you sketch several representative contour plots from the family of equations for various c... | {
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f[x_, y_, z_] := x z + y z - x y z
cp[c_, z0_] :=
ContourPlot[f[x, y, z0] == c, {x, -4, 4}, {y, -4, 4},
FrameLabel -> {Row[{"c=", c, ", z=", z0}], None}, BaseStyle -> 12]
Grid[Table[cp[i, j], {j, Range[-2, 2]}, {i, Range[-2, 2]}],
Frame -> All, Spacings -> {2, 0}]
g[x_, y_, z_, c_] := f[x, y, z] - c
scp[c_, z0_] :=
Sli... | {
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# Newton Method To Find Roots | {
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Remark 1 The new ninth order method requires six function evaluations and has the order of convergence nine. And let's say that x is the cube root of 3. This program is not a generalised one. GRAPHICAL INTERPRETATION :Let the given equation be f(x) = 0 and the initial approximation for the root is x 0. ' The function. ... | {
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Durand-Kerner, Ostrowski or Eigenvalue method. Below is the syntax highlighted version of Newton. Newton's Method in Matlab. Newton Search for a Minimum Newton's Method The quadratic approximation method for finding a minimum of a function of one variable generated a sequence of second degree Lagrange polynomials, and ... | {
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methods: approximating the Hessian on the fly ¶ BFGS : BFGS (Broyden-Fletcher-Goldfarb-Shanno algorithm) refines at each step an approximation of the Hessian. Bisection method is one of the many root finding methods. Newton's method, also known as Newton-Raphson, is an approach for finding the roots of nonlinear equati... | {
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single initial guess. A brief overview of the Newton-Raphson method can be found in 8. We will find root by this method in mathematica here. Atul Roy 4,273. The find_zerofunction provides the primary interface. It is based on the simple idea of linear approximation. 3 Newton's method Newton's method is an algorithm to ... | {
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or root of the equation f(x) = 0 or an x intercept of the graph of f. This tutorial explores a numerical method for finding the root of an equation: Newton's method. Newton's Method In this section we will explore a method for estimating the solutions of an equation f(x) = 0 by a sequence of approximations that approac... | {
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To find the roots of a function using graphing and Newton's method. Also, this method is not 100% in finding roots. Newton's Method in Matlab. We start with this case, where we already have the quadratic formula, so we can check it works. I want generate R code to determine the real root of the polynomial x^3-2*x^2+3*x... | {
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line. Use Newton's method to find the absolute maximum value of the function f(x) = 2x sin x, 0 ≤ x ≤ π correct to six decimal places. 1–3) • introducing the problem • bisection method • Newton-Raphson method • secant method • fixed-point iteration method x 2 x 1 x 0. This method uses the derivative of f(x) at x to esti... | {
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on the simple idea of linear approximation. Newton's method calculator or Newton-Raphson Method calculator is an essential free online tool to calculate the root for any given function for the desired number of decimal places. Kite is a free autocomplete for Python developers. It's also called a zero of f. The Newton-R... | {
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point. This guess is based on the reasoning that a value of 2 will be too high since the cube of. Newton-Raphson Method is a root finding iterative algorithm for computing equations numerically. The Attempt at a Solution First I attempted to write the fifth root of 36 in exponential form as show below: Let the 5th root... | {
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the values of X for which F(X) = 0. The only tricky part about using Newton's method is picking a. Take for example the 6th degree polynomial shown below. Worksheet 25: Newton’s Method Russell Buehler b. Newton Raphson method: it is an algorithm that is used for finding the root of an equation. It includes solvers for ... | {
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rewriting this Newton iteration are shown to have excellent. Newton's method is a tool you can use to estimate the root of a function, which is the point at which the function crosses the x-axis. The iteration goes on in this way:. To remedy this, let's look at some Quasi-Newtonian methods. Beginning with the classical... | {
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times. The Newton Method, when properly used, usually comes out with a root with great efficiency. multiplicity 2 # [int] The multiplicity of the root when using the modified newton method Exercise: In the Newton's root finding algorithm, it is important to choose a reasonable initial search value. To get started with ... | {
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of finding approximate solutions. Adjust the Julia/SymPy function so it works with initial values with nonzero imaginary parts. We want to solve the equation f(x) = 0. Exercise 2: Find a root of f(x) =ex −3x. Newton's Method is an application of derivatives will allow us to approximate solutions to an equation. However... | {
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having its minimum , if a root exists). This is an iterative method invented by Isaac Newton around 1664. For example, if y = f(x), it helps you find a value of x that y = 0. Newton-Rapson’s Method Norges teknisk-naturvitenskapelige universitet Professor Jon Kleppe Institutt for petroleumsteknologi og anvendt geofysikk... | {
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it in terms of its steps. Note that for a quadratic equation ax2+bx+c = 0, we can solve for the solutions using the quadratic formula. The Newton Method, properly used, usually homes in on a root with devastating e ciency. Just decide how much of the complex plane to draw, and for each pixel in the image, iterate Newto... | {
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is a root-finding algorithm which produces successively better approximations to the roots (or zeroes) of a real-valued function. It's also called a zero of f. It's a solution or root of the equation f (x) = 0, ie, a point where the graph of f intersects the x-axis. f(x) = 0 Themethodconsistsofthe following steps: Pick... | {
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a nutshell, the former is slow but robust and the latter is fast but not robust. In 1976, my Cornell colleague John Hubbard began looking at the dynamics of Newton’s method, a powerful algorithm for finding roots of equations in the complex plane. We see that the function graph crosses the x-axis somewhere between -0. ... | {
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Tangent. I have another form to the function f(x) ,but I don't know if it's suitable to be solved by Newton's method in matlab,the other form is:. Explore complex roots or the step‐by‐step symbolic details of the calculation. Note: In Maple 2018, context-sensitive menus were incorporated into. So while Newton’s Method ... | {
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# Using Darboux integrals to prove even integration on a symmetric interval
Recall a function $f:[-a,a] \rightarrow R$ is said to be even if $f(x)=f(-x)$.
Let $f$ be an integrable, even function.
Prove that: $\int_{-a}^a f\,$ = $2 \int_0^a f\,$.
.
I'm trying to prove this using Darboux/Riemann Integrals, as stated... | {
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• I edited the last line "Hence..." because I forgot a negative sign on one of the limits. Also, I may have to include the set {0} in the union, but I am not sure. So it may have to look like this: $Q_n = P_n \cup (-P_n) \cup$ {$0$}. – Marcus Apr 2 '18 at 19:22
Since $f$ is Riemann integrable on $[-a,a]$, by the Riema... | {
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Notice that the sums appearing on the right-hand sides of the two equations in (2) are themselves upper and lower sums with respect to a partition of $[0,a]$ since $-y_j \in [0,a]$.
Consequently,
$$\tag{3}L(f,P^-) \leqslant \int_0^a f \leqslant U(f,P^-).$$
Adding (1) and (3) we get
$$\tag{4}L(f,P) \leqslant 2\int_0... | {
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### Author Topic: FE-P5 (Read 6327 times)
#### Victor Ivrii
• Administrator
• Elder Member
• Posts: 2563
• Karma: 0
##### FE-P5
« on: April 11, 2018, 08:47:26 PM »
For the system of ODEs
\begin{equation*}
\left\{\begin{aligned}
&x'_t = x (x -y+1)\, , \\
&y'_t = y (x - 2)\,.
\end{aligned}\right.
\end{equation*}
a. ... | {
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#### Victor Ivrii
• Administrator
• Elder Member
• Posts: 2563
• Karma: 0
##### Re: FE-P5--solution
« Reply #3 on: April 18, 2018, 06:48:47 AM »
a. Solving $x(x-y+1)=0$, $y(x-2)=0$ we get cases
\begin{align*}
&x=y=0 &&\implies A_1=(0,0),\\
&x=x-2=0 &&\implies \text{impossible}\\
&y=x-y+1=0 &&\implies A_2=(-1,0),\\
... | {
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#### Victor Ivrii
• Administrator
• Elder Member
• Posts: 2563
• Karma: 0
##### FE-P5 Comments
« Reply #4 on: April 19, 2018, 01:16:12 PM »
a. Some students missed some stationary points and/or reported wrong points. All further analysis in the wrong points was ignored as irrelevant. For all three correct points found... | {
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# How many binary sequences of length 7 have at least two 1's?
How many binary sequences of length 7 have at least two 1's? Can someone please explain the procedure in detail please. I tried solving it using the "count what you do not want" procedure, but I got nowhere. Thank you in advance
• What do you get if you c... | {
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# Finding eigenvalues and corresponding eigenvectors
I have the matrix:
$$A= \begin{bmatrix} 7 & -2 \\ 15 & -4 \\ \end{bmatrix}$$
and I am asked to find the eigenvalues and eigenvectors. I found the eigenvalues to be $\lambda = 1,2$. Now I need to find the eigenvectors:
$$(A-\lambda I)\mathbf u=\mathbf 0$$ $$\begin... | {
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# Arranging numbers 1 to 1000 such that the difference of two adjacent numbers is not a square nor a prime number
I've been working on the following problem for a while: Prove that it's possible to arrange numbers 1 to 1000 an order such that each number appears once and |$$x_j - x_{j+1}$$| is not a perfect square nor... | {
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You can also construct it:
Start with $$1$$ and keep adding $$6$$ i.e $$1,7,13$$ until you hit $$997$$ then go back to $$3$$ and keep adding $$6$$ until you get to $$999$$ and go back to $$5$$ repeat until $$995$$ then go back to $$2$$ repeat until $$998$$ and go back to $$4$$ repeat until $$1000$$ and go back to $$6$$... | {
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# A Triangle out of Three Broken Sticks
### Solution, Part 1
Assume the sticks are of length $1$ and consider the cube with vertices $A(0,0,0),$ $B(1,0,0),$ $D(0,1,0),$ $E(0,0,1),$ $C(1,1,0),$ $F(1,0,1),$ $H(0,1,1),$ and $G(1,1,1).$
The left pieces of the sticks are defined by their right points and I shall describe... | {
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### Solution, Part 3
If now $x,y,z$ stand for the lengths of shortest pieces then $0\lt x,y,z\lt\frac{1}{2}$ and each of the three is drawn uniformly randomly from the interval $\displaystyle\left(0,\frac{1}{2}\right).$ The situation is similar to Part 1 so that the probability in this case is $\displaystyle \frac{1}{... | {
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Now, there is a harder road to arrive at the same conclusion. Assume $\displaystyle 0\le x,y,z\le\frac{1}{2},$ what is the probability of $P(\Delta(x,y,z)=1)?$ In other words, what is $\displaystyle P\left(\Delta(x,y,z)=1\text{ and }0\le x,y,z\le\frac{1}{2}\right)?$ The formula for conditional probability gives
\displ... | {
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### SLL
In this case we need to satisfy the following inequalities:
\displaystyle \begin{align} x&\lt\frac{1}{2}\\ y&\gt\frac{1}{2}\\ z&\gt\frac{1}{2}\\ x+y&\gt z\\ x+z&\gt y. \end{align}
The points $(x,y,z)$ that satisfy all five inequalities lie outside two pyramids in the small cube:
The relative volume of that ... | {
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The probability of triangulation is given by
\displaystyle \begin{align} P&=\frac{\int_{z=0}^{1/2}\int_{y=z}^{1-z}\int_{x=y}^{y+z}dxdydz + \int_{y=1/2}^{1}\int_{z=1-y}^{y}\int_{x=y}^{1} dxdzdy} {\int_{z=0}^{1/2}\int_{y=z}^{1-z}\int_{x=y}^{1}dxdydz + \int_{y=1/2}^{1}\int_{z=1-y}^{y}\int_{x=y}^{1} dxdzdy} \\ &=\frac{1/2... | {
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# How to evaluate this integral? (relating to binomial)
I saw some result that some article used, (without proving) that stated:$$\int_0^1 p^k (1-p)^{n-k} \mathrm{d}p = \frac{k!(n-k)!}{(n+1)!}$$
But I was wondering, how would you integrate it? How did this integral come about? Is it something to do with the binomial ... | {
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• Hmm interesting way, cheers! – Heijden Mar 20 '12 at 16:11
• $p^k(1-p)^{n-k}$ is the probability you described. But what about the integral? Integral makes us talk not about single $p$ but about whole bunch of them. – Yola Jan 22 '18 at 13:55
• @Yola $p^k(1-p)^{n-k}$ is the probability of drawing a sequence of $n$ nu... | {
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Let's consider the quantity
$$I(n,k) = \int_{0}^{1} \binom{n}{k} p^k (1-p)^{n-k} \; dp.$$
Then by integration by parts, as in two former answers, we have
$$I(n, k+1) = I(n, k).$$
Let $I$ denote this common value. Thus
$$1 = \int_{0}^{1} 1 \; dp = \int_{0}^{1} \sum_{k=0}^{n} \binom{n}{k} p^k (1-p)^{n-k} \; dp = \su... | {
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# Math Help - counting
1. ## counting
arrange 12 people into 4 teams of 3 people
would u choose the team first then the players: 4C1 x 12C3 first team
3C1 x 9C3 second team and carry on for third and fourth then multiply all of ur results together?
2. Originally Posted by qwerty10
arrange 12 people into 4 teams of ... | {
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~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~
$\text{Suppose two players, }X\text{ and }Y\text{, are }not\text{ allowed on the same team.}$
$\text{Count the ways that }X\text{ and }Y\;are\text{ on the same team.}$
$\text{Place }X\text{ and }Y\text{ on team A.}$
$\text{There are 10 choices for the thir... | {
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8. Originally Posted by qwerty10
I see where I have gone wrong-ive just been doing the standard distinguishable approach and not taking into account that the teams are not distinguishable.
For part 1 for the total number of teams, I don't understand why in the distinguishable case( say we have teams 1,2,3) why we dont ... | {
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Can every true theorem that has a proof be proven by contradiction?
After reading and being inspired by, Can every proof by contradiction also be shown without contradiction? and after some thought, I still don't have an answer to this.
Does every theorem with a true proof have a proof by contradiction?
• Let $P$ be... | {
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Therefore $$\neg \neg T$$, by negation introduction.
Thus $$T$$, by double negation elimination.
One may object that this proof is essentially the same as $$P$$, and is just wrapped up. That is true, but it is a perfectly legitimate proof of $$T$$, even if it is longer than $$P$$, and it is indeed of the form of a pr... | {
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Note that intuitionistic logic plus the rule "$$\neg A \to \bot \vdash A$$" gives back classical logic, and one could say that this rule embodies the 'true principle' of proof by contradiction, in which case one can say that some true theorems require the use of a proof by contradiction somewhere.
If you can prove a s... | {
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A contradiction arising in a proof does not necessarily warrant that the method of proof being employed is what we call "proof by contradiction." It is the derivation of this contradiction which warrants the name "proof by contradiction." It is how the contradiction arises. And here, our contradiction doesn't really ar... | {
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# Reduced Row Echelon Form with a Variable
For what value of k does the system of equations not have a unique solution?
$$\left\{ \begin{array}{c} x-2y+2z=0 \\ 2x+ky-z=3 \\ x-y+3z=-5 \end{array} \right.$$
I know that this means I have to find the value(s) of k where the system of equations has either no solutions or... | {
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Any help will be greatly appreciated, thanks in advance.
• Keep going: do $4R_2+kR_3$ – Santiago Canez Aug 14 '15 at 2:27
• Now what? I'm sorry if it's a silly question - am a tenth grader who's trying to self-study :P – StopReadingThisUsername Aug 14 '15 at 2:33
• It would have been better to put the result into the ... | {
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$=\left[\begin{array}{ccc|c}1&-2&2&0\\0&4k&4k&-20k\\0&0&4k+36&-92-20k\end{array}\right]$
It will be that there are no solutions when the final row looks like $[0~0~0~|~n]$ with $n\neq 0$, or infinitely many solutions if the final row looks like $[0~0~0~|~0]$ (assuming that there are no other $[0~0~0~|~n]$ rows elsewhe... | {
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## The Locker Problem
There is a hallway of lockers with lockers numbered 1 through 100. There are also 100 students. Student 1 opens every locker, then student 2 closes every other locker, then student 3 opens or closes every 3rd locker (if it’s open then she closes it and if it’s closed then she opens it), and so on... | {
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4. Posted by Shobhit Gupta on January 25, 2011 at 5:40 am
Oh yeah, I should have put a disclaimer in my 1st comment as well.
Next time I will be careful about it.
Well let me do that here:
—————
—————
My True and False meant:
True = ‘door is closed’
False = ‘door is open’
But anyway, in better words, all those door... | {
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8. Posted by Shobhit Gupta on January 26, 2011 at 12:38 am
>> I saw a variation on this problem – where only odd-numbered students come to flip doors. Is there a nice way count the open doors in that case?
I think the answer would remain same if only the odd-numbered students come to flip doors.
I have my own simplis... | {
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11. Posted by Shobhit Gupta on February 2, 2011 at 3:12 am
I tried it the naive way. And I couldn’t match up 17. Maybe I did something wrong. | {
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# Indeterminate form $1^\infty$ vs. $0^\infty$
Why is $1^\infty$ an indeterminate form while $0^\infty = 0$? If $0\cdot0\cdot0\cdots = 0$ shouldn't $1\cdot1\cdot1\cdots = 1$?
• Check out these two links: math.stackexchange.com/questions/520795/… math.stackexchange.com/questions/10490/… – Brenton Sep 16 '17 at 23:35
•... | {
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• Sir, I'm not sure if you can count, but that's three different numbers ;-) – Simply Beautiful Art Sep 17 '17 at 1:21
• @SimplyBeautifulArt : That depends on which numbers are referred to. However, there are of course three kinds of people in the world: those who can count and those who can't. – Michael Hardy Oct 5 '1... | {
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$$0 \le \left| (x_n)^{y_n} \right| \le \left| (1/2)^{y_n} \right| \to 0$$
as $n \to \infty$. It follows that $\lim_{n \to \infty} (x_n)^{y_n} = 0$.
What you'll notice is that $1/2$ could have been any number between $0$ and $1$. That is, if $x_n$ is "close to $0$" in the sense that $|x_n| < r$ and $r < 1$ then we can... | {
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which we can take logarithms and use a Taylor series for $\log(1 + x)$to get
\begin{align} \log\left( \lim_{n \to \infty} \left( 1 + \frac{1}{q_n} \right)^{p_n} \right) &= \lim_{n \to \infty} p_n \log\left( 1 + \frac{1}{q_n} \right) \\ &= \lim_{n \to \infty} p_n \left( \frac{1}{q_n} - \frac{1}{2q_n^2} + \frac1{3q_n^3}... | {
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Without logarithms...
Here are three different $1^\infty$s:
• $\lim_{n \rightarrow \infty} (2^{1/\ln n})^n = \infty$
• $\lim_{n \rightarrow \infty} (2^{1/n})^n = 2$
• $\lim_{n \rightarrow \infty} (2^{1/n^2})^n = 1$
Only the last one is doing what you seem to expect. This is because $1/n^2$ is going to $0$ faster tha... | {
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+0
# HELP PLZ
+3
76
11
+304
In how many ways can you spell the word COOL in the grid below? You can start on any letter, then on each step, you can step one letter in any direction (up, down, left, right, or diagonal).
$$\begin{array}{ccccc} C&C&C&C&C\\ L&O&O&O&L\\ L&O&O&O&L\\ L&O&O&O&L\\ C&C&C&C&C\\ \end{array}$$
... | {
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I have been trying and trying and trying, and I am counting 96.
A "C" in the corner has 3 ways to make the word "cool."
Since there are 4 of them, 3*4=12
A "C" adjacent to a C in the corner has 13 ways to make the word "cool."
Since there are 4 of them, 4*13=52
A "C" in the center column has 16 ways to make the wo... | {
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$$(C3,R3)$$ is adjacent to 2 red and 2 blue O's. Because there are 3 possibilities for a red one and 2 possibilities for a blue one, the number of possibilities is $$3+3+2+2=10$$. However, this is only one sequence. Let's consider the next sequence of $$(C4,R4)\Rightarrow(C4,R3)\Rightarrow(C4,R2)$$. Oh look! This is a... | {
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# Bound on the derivative of a holomorphic function from right half plane to unit disc
Let $D=\{z\in\mathbb{C}:|z|<1\}$ and let $V=\{z\in\mathbb{C}:\Re(z)>0\}$.
Let $f:V\to D$ be a holomorphic function. Prove that
$$\forall z\in V:|f'(z)|\leq\frac{1-|f(z)|^2}{2\Re(z)}.$$
By taking $B_z=\{\xi\in V:|\xi-z|<\Re(z)\}$ ... | {
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\begin{align} \lvert f'(z)\rvert &\leqslant \frac{1 - \lvert f(z)\rvert^2}{(1 - \lvert w\rvert^2)\cdot \lvert\phi'(w)\rvert} \\ &= \bigl(1 - \lvert f(z)\rvert^2\bigr)\cdot \frac{\lvert 1-w\rvert^2}{2(1 - \lvert w\rvert^2)} \\ &= \frac{1 - \lvert f(z)\rvert^2}{2}\cdot \frac{\lvert 1 - \psi(z)\rvert^2}{1 - \lvert \psi(z)... | {
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Regularity of the heat kernel
Let $(M,g)$ be a compact Riemannian manifold. Let $H:M\times M\times\mathbb{R}_{>0}\to\mathbb{R}$ be the heat kernel. i.e. $H\in C^0(M\times M\times\mathbb{R}_{>0})$ is the unique continuous function such that for all $y\in M$,
(A) $H^y\in C^{2,1}(M\times\mathbb{R}_{>0})$
(B) $\left(\De... | {
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Smoothness in all the questions has a local character. And in local coordinates it's a parabolic equation $Lu=F$ with (smooth) variable coefficients. So the local theory in $\mathbb R^n$ will do.
On Question2 the answer is no. If function $F$ is continuous it does not follow that $u$ is locally from $C^{2,1}$. For the... | {
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To get continuity at $t=0$, simply note that $$|u(x,t)| \le t \cdot\left(\sup_{M \times [0,t]} |F|\right)\left( \sup_{\tau \in [0,t]} \int_M H(x,y,t-\tau) \mu_g(dy)\right)$$ But $\sup_{M \times [0,t]} |F|$ is finite if $F$ is bounded, and $\int_M H(x,y,t-\tau) \mu_g(dy) = 1$ for any $x, t, \tau$.
I guess you still wan... | {
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# Ellipse bounding rectangle
I'm trying to find the ellipse that bounds a rectangle in a way that the "distance" between the rectangle and the ellipse is the same vertically and horizontally.
Here is an image to illustrate what I mean:
It's not perfectly drawn but the two "x" dimension need to be the same.
What I t... | {
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An ellipse centred at the origin has equation $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ where $a$ is the 'horizontal' radius, and $b$ is the 'vertical' radius (here I am taking $a$, $b$ positive). Then the ellipse you are looking for must satisfy
$$a-\frac{w}{2} = b - \frac{h}{2}$$
or equivalently
$$2a - w = 2b - h.$$... | {
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The result can be seen below (made using WolframAlpha).
• I thought that the OP was looking for the largest recatngle or the smallest ellipse. If this is not the case, you are right and .... I am wrong. Happy New Year !! – Claude Leibovici Jan 1 '14 at 13:09
• So yeah... it's a quartic as I suspected :( Which means th... | {
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Writing piecewise functions
Rated 5/5 based on 28 review
# Writing piecewise functions
Yes, piecewise functions isn’t particularly exciting but it can, at least, be enjoyable we dare you to prove us wrong. Worksheet piecewise functions name: algebra 2 part i carefully graph each of the following identify whether or... | {
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Match the formula of a piecewise function to its graph. Write the piecewise functions for the graph shown solution: step 1: locate the break point here it is at x = 2 step 2: find the equation of the graph to the left. Writing equations for piecewise functions and word problems mr swartz. How we can define piecewise fu... | {
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functions are all. How to write this piecewise function using latex i tried$ \begin{array}{cc} \{ & \begin {array}{cc how to write a function (piecewise) with bracket outside. | {
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Help with piecewise function can't use learn more about piecewise, symbolic, calculus symbolic math toolbox. Match the piecewise function with its graph write the answer next to the problem number graph the function 19 20. Represents a piecewise function with values val i in the regions defined by the conditions cond i... | {
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# Largest rectangle not touching any rock in a square field
You want to build a rectangular house with a maximal area. You are offered a square field of area 1, on which you plan to build the house. The problem is, there are $n$ rocks scattered in unknown locations throughout the field. The rocks are unmovable, and yo... | {
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Is there another arrangement of rocks in which the largest rectangle is smaller?
• Just out of curiosity, where does this problem come from? (+1) – A.P. Nov 17 '15 at 21:32
• @A.P. It comes from my Ph.D. research about fair division of land. See my profile :) – Erel Segal-Halevi Nov 17 '15 at 21:37
• Is it necessary t... | {
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• Thanks! Is it the same paper here? stetson.edu/~efriedma/mathmagic/0899/ravsky.ps The problem in your paper seems very similar to my problem, with one difference: your function, $T(n)$, is a supremum (on all possible n-tuples of points), while I defined it as an infimum. Apparently, the supremum is always 1, as it is... | {
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However, even better bounds are possible if we work directly with $M(n)$. Let $n(\epsilon)$ be the size of the smallest point set $P$ such that $M(P) \le \epsilon$ (this is called an $\epsilon$-net). Then it is known that $n(\epsilon) = O(\frac{1}{\epsilon}\log \log \frac{1}{\epsilon})$, which implies that $M(n) = O(\l... | {
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# Summing (0,1) uniform random variables up to 1 [duplicate]
So I'm reading a book about simulation, and in one of the chapters about random numbers generation I found the following exercise:
For uniform $(0,1)$ random independent variables $U_1, U_2, \dots$ define
$$N = \min \bigg \{ n : \sum_{i=1}^n U_i > 1 \bigg ... | {
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## marked as duplicate by Rahul, Ross Millikan, Hans Lundmark, Arkamis, NorbertOct 15 '12 at 21:34
Here is a way to compute $\mathbb E(N)$. We begin by complicating things, namely, for every $x$ in $(0,1)$, we consider $m_x=\mathbb E(N_x)$ where $$N_x=\min\left\{n\,;\,\sum_{k=1}^nU_k\gt x\right\}.$$ Our goal is to com... | {
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Explain why the graph of $y=\frac{4x}{x^2+1}$ and $y=2\sin(2\arctan x)$ are the same.
Explain why the graph of $y=\frac{4x}{x^2+1}$ and $y=2\sin(2\arctan x)$ are the same.
The first equation is of the form of Newton's Serpentine. When you graph the second equation it appears to overlap the first equation.
I'm not su... | {
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# Thread: Simple Probability Question
1. ## Simple Probability Question
I am a bit confused by the way this question is solved:-
From a of well shuffled pack 52 cards, three cards are drawn at random. Find the probability of drawing an ace, a king and a jack.
Solution given:- There are 4 aces, 4 king and 4 jacks an... | {
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Solution given:- There are 4 aces, 4 king and 4 jacks and their selection can be made in the following ways:
12C1 X 8C1 X 4C1 = 12 X 8 X 4.
Total selections can be made = 52C3= 52 X 51 X 50.
Therefore required probability = $\frac{(12)(8)(4)}{ (52)(51)(50)}$
I don't understand why are we taking 12C1 X 8C1 X 4C1 = 12 X... | {
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4. ## Re: Simple Probability Question
Both solutions are correct, and both give the same answer. Neither is "more correct" than the other, though I would admit that my instinct is to solve it the way that Halls did. I tend to think first in terms of selecting cards in a particular order, then multiply by the number of... | {
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7. ## Re: Simple Probability Question
Hi ebaines,
But the original solution mentions: Total selections can be made in 52C3 ways, which is equivalent to $\frac{(52)(51)(50)}{3!}$, which is the denominator. So where will the 3! go then ?
8. ## Re: Simple Probability Question
This part of your first post is incorrect:... | {
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Lindelof Exercise 2
The preceding post is an exercise showing that the product of countably many $\sigma$-compact spaces is a Lindelof space. The result is an example of a situation where the Lindelof property is countably productive if each factor is a “nice” Lindelof space. In this case, “nice” means $\sigma$-compac... | {
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Exercise 2.E
Prove that the product of finitely many $\sigma$-compact spaces is a $\sigma$-compact space. Give an example of a space showing that the product of countably and infinitely many $\sigma$-compact spaces does not have to be $\sigma$-compact. For example, show that $\mathbb{R}^\omega$, the product of countab... | {
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Show that the product of uncountably many copies of the real line does not have countable extent. Specifically, focus on either one of the following two examples.
• Show that the product space $\mathbb{R}^c$ has a closed and discrete subspace of cardinality continuum where $c$ is cardinality of continuum. Hence $\math... | {
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$\text{ }$
Further Hints for Exercise 2.A
The hints here focus on the example $\mathbb{R}^c$.
Let $I=[0,1]$. Let $\omega$ be the first infinite ordinal. For convenience, consider $\omega$ the set $\{ 0,1,2,3,\cdots \}$, the set of all non-negative integers. Since $\omega^I$ is a closed subset of $\mathbb{R}^I$, any ... | {
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