text stringlengths 1 2.12k | source dict |
|---|---|
Let’s have a look at the kind of estimates that the different kernels produce.
def plot(x_train, y_train, x_val, y_val, kernels, names, attention=False):
fig, axes = d2l.plt.subplots(1, 4, sharey=True, figsize=(12, 3))
for kernel, name, ax in zip(kernels, names, axes):
y_hat, attention_w = nadaraya_watson(x_train, y_t... | {
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... |
plot(x_train, y_train, x_val, y_val, kernels, names)
def plot(x_train, y_train, x_val, y_val, kernels, names, attention=False):
fig, axes = d2l.plt.subplots(1, 4, sharey=True, figsize=(12, 3))
for kernel, name, ax in zip(kernels, names, axes):
y_hat, attention_w = nadaraya_watson(x_train, y_train, x_val, kernel)
if at... | {
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... |
sigmas = (0.1, 0.2, 0.5, 1)
names = ['Sigma ' + str(sigma) for sigma in sigmas]
def gaussian_with_width(sigma):
return (lambda x: torch.exp(-x**2 / (2*sigma**2)))
kernels = [gaussian_with_width(sigma) for sigma in sigmas]
plot(x_train, y_train, x_val, y_val, kernels, names)
sigmas = (0.1, 0.2, 0.5, 1)
names = ['Sigm... | {
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... |
plot(x_train, y_train, x_val, y_val, kernels, names, attention=True)
As we would expect, the narrower the kernel, the narrower the range of large attention weights. It is also clear that picking the same width might not be ideal. In fact, Silverman (1986) proposed a heuristic that depends on the local density. Many m... | {
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1. What happens if you just use the above estimates to minimize $$(f(\mathbf{x_i}) - y_i)^2$$ directly? Hint: $$y_i$$ is part of the terms used to compute $$f$$.
2. Remove $$(\mathbf{x}_i, y_i)$$ from the estimate for $$f(\mathbf{x}_i)$$ and optimize over the kernel widths. Do you still observe overfitting?
3. Assume... | {
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# Moderate Probability Solved QuestionAptitude Discussion
Q. A man can hit a target once in 4 shots. If he fires 4 shots in succession, what is the probability that he will hit his target?
✖ A. 1 ✖ B. 1/256 ✖ C. 81/256 ✔ D. 175/256
Solution:
Option(D) is correct
The man will hit the target even if he hits it once ... | {
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VeeraraghavanK
()
It looks apparently that the answer is 1/4 since he fires 4 shots. The question could have been : "probability that the target is fired at least once"
Is there any other alternate method | {
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# Time complexity of euler totient function
Code for finding $\phi$(n) is
int phi(int n)
{
int result = n; // Initialize result as n
// Consider all prime factors of n and subtract their
// multiples from result
for (int p=2; p*p<=n; ++p)
{
// Check if p is a prime factor.
if (n % p == 0)
{
// If yes, then update ... | {
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# IMO 2016 Day 1
I'm surprised nobody else has posted this. Yes, there are plenty of discussions in Art of Problem Solving, but why not here? And of course there are more problems.
## Problem 1
Triangle $$BCF$$ has a right angle at $$B$$. Let $$A$$ be the point on line $$CF$$ such that $$FA = FB$$ and $$F$$ lies bet... | {
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Note by Ivan Koswara
2 years, 9 months ago
MarkdownAppears as
*italics* or _italics_ italics
**bold** or __bold__ bold
- bulleted- list
• bulleted
• list
1. numbered2. list
1. numbered
2. list
Note: you must add a full line of space before and after lists for them to show up correctly
paragraph 1paragraph 2
paragraph... | {
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- 2 years, 9 months ago
Here's my synthetic solution. Let $$XF$$ intersect $$BD$$ at $$O$$ and $$EM$$ at $$O^{'}$$. We need to show that $$O$$ coincides with $$O^{'}$$. A clear observation shows that $$B,C,F,D,X$$ are concyclic because $$\angle\{DCF\}$$=$$\angle\{DBF\}$$. Moreover $$D$$ is the circumcenter of triangle... | {
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# In how many ways can they rearrange such that each person is in a new seat and no 2 persons swap?
Beforehand, each person is assigned a seat (n seats). How many ways can they rearrange such that each person is in a new seat and no 2 people swap?
I'm guessing $$(n-1)!$$, but I have no idea on how to get the answer.
... | {
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I think there is a way to write a recurrence relation by the following reasoning: we are counting the permutations of $$\left\{1 , \ldots , n\right\}$$ which decomposition in cycles contains only cycles having at least $$3$$ elements. Let $${X}_{n}$$ be that number. We can start with $${X}_{1} = {X}_{2} = 0$$ and $${X}... | {
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Clearly, the same reasoning shows that if we generalize and we count instead the number $$Y_n$$ of permutations of $$\{1,\cdots, n\}$$ where all the cycles have a length $$\ge p$$, then we have the relation
$$$$Y_n = (n-1) Y_{n-1} + (n-1)\cdots (n-p+1) Y_{n-p}$$$$
• I see how you can use the summation equation to pro... | {
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# Heat Transfer Pipe Problem dilemma
Tags:
1. Aug 8, 2016
### williamcarter
1. The problem statement, all variables and given/known data
Problem data:
http://imgur.com/a/hEz16
2. Relevant equations
Rconduction=ln(rout/rin)/2*pi*Length*thermal conduct
Rconvection=1/2*pi*Length*routside*convective heat transfer outs... | {
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Last edited: Aug 8, 2016
4. Aug 8, 2016
### benny_91
You are missing an important point here. Note that the heat that is conducted through the pipe is then convected from the surface of the pipe to the atmosphere. Heat transfer passes through different stages: first conduction and then convection.
As long as there is... | {
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10. Aug 8, 2016
### williamcarter
Heat is transferred firstly by conduction within the pipe , then by convection from pipe surface to the air.
To answer the questions I would say , yes Qtotal passes through tube wall, yes Qtotal passes through boundary layer.
There is no insulation at the moment, furthermore no gener... | {
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16. Aug 20, 2016
### Staff: Mentor
Do you see any Cp in the exercise?
17. Aug 21, 2016
### williamcarter
No , I don't.
I cannot understand why they did in this problem ΔT=Tin-Tout=300-5.Why they did like that?
Usually we know ΔT=Tout-Tin=Tfinal-Tinitial not Tinitial-Tfinal
18. Aug 21, 2016
### Staff: Mentor
Don... | {
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# Proving that $3|x^2 \implies 3|x$.
Given a positive integer $x$, I would like to show that $3|x^2 \implies 3|x$. This is my missing step in the proof that $\sqrt{3}$ is irrational. My thoughts so far:
Assuming $3|x^2,$ then since $3$ is a prime number, we have that the prime factorization of $x^2$ is $m3^n$, where ... | {
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• You are implcitly using uniqueness of prime factorizations. For the proof to be complete/correct you need to make that explicit. – Bill Dubuque Sep 4 '14 at 21:47
• Can you suggest how that can done? Thanks. – Sheheryar Zaidi Sep 4 '14 at 21:55
• I imagine he is asking you to state that "Due to unique prime factoriza... | {
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# Area of circle (double integral and cartesian coordinates)?
I know that the area of a circle, $x^2+y^2=a^2$, in cylindrical coordinates is $$\int\limits_{0}^{2\pi} \int\limits_{0}^{a} r \, dr \, d\theta = \pi a^2$$
But how can find the same result with a double integral and only cartesian coordinates?
## 5 Answers... | {
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Just as an alternative solution to the qbert answer.
Note that $$|x|$$, $$|y|$$ are not strictly less than $$r$$, but instead $$|x| \leq r$$, $$|y| \leq r$$. This can still insure that $$r^2 \geq y^2$$ and so $$\sqrt{r^2-y^2}$$ is real.
After the first integral
$$\int_{-r}^r\int_{-\sqrt{r^2-y^2}}^{\sqrt{r^2-y^2}}\ma... | {
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Do you mean something like this?
$$\int_{-a}^{a} \int_{-\sqrt{a^2-x^2}}^{+\sqrt{a^2-x^2}} 1 \, dy \, dx$$
• Yes, how do you evaluate that without cylindrical coordinates? – JDoeDoe Jul 18 '16 at 17:53
• I don't understand your question. These are Cartesian coordinates. – Martin Kochanski Jul 19 '16 at 8:28 | {
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# Is it true that if $\limsup\limits_{n \to \infty}\left|\frac{a_{n+1}}{a_n}\right| > 1$, then $\sum a_n$ diverges?
I am reading "A Course in Analysis vol.2" by Kazuo Matsuzaka.
There is the following theorem ("ratio test") in this book.
Let $$a_n \neq 0$$ for all $$n$$.
(a) If $$\limsup\limits_{n \to \infty}\left|... | {
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# problem with joint probability function
The joint probability density function of $X$ and $Y$ is given by $$f(x,y) = \begin{cases}c(y^2-x^2) e^{-y}, & -y \le x < +y, \ 0 \le y < \infty,\\ 0, & \text{otherwise}. \end{cases}$$
(a) Find $c$.
(b) Find the marginal densities of $X$ and $Y$.
(c) Find $\operatorname{E}[... | {
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So, the marginal density of $Y$ is $\frac{1}{6} y^3 e^{-y}$ as you obtained.
The marginal density of $X$ is $$\int_{|x|}^\infty c(y^2-x^2) e^{-y} \mathop{dy} .$$ The tricky part is the limits of integration, which comes from the condition $|x| \le y$. First, integration by parts twice gives \begin{align} \int_{|x|}^\i... | {
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$$f_X(x) = \int_{y=|x|}^\infty \frac18 (y^2-x^2)e^{-y}dy =\frac14 (1+|x|) e^{-|x|}$$
They are marginal distributions for two different variables and the joint distribution is in no way symmetric so in general they would be two distinct distributions unless some staggering coincidence occurred.
Your confusion was prob... | {
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CLRS Chapter 3. Growth of Functions
3.1. Asymptotic notation
Exercises
3.1-1. Let $f(n)$ and $g(n)$ be asymptotically nonnegative functions. Using the basic definition of $\Theta$-notation, prove that $\max(f(n), g(n)) = \Theta(f(n) + g(n)).$
Since
for all $n \in \mathbb{N}$, we see that
for all $n \in \mathbb{N}... | {
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for all $n \geq n_0$. Since
for all $n \geq n_0$, we have that $f(n) = O(g(n))$. Similarly,
for all $n \geq n_0$, and so $f(n) = \Omega(g(n))$.
$(\Leftarrow)$ Suppose that $f(n) = O(g(n))$ and $f(n) = \Omega(g(n))$. The first assumption furnishes positive constants $c_1$ and $n_1$ such that
for all $n \geq n_1$. Th... | {
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Let $n \leq m$. Since $f(n) \leq f(m)$ and $g(n) \leq g(m)$, we see that
If, in addition, $f$ and $g$ are nonnegative, then
Finally, $g(n) \leq g(m)$ and the monotonicity of $f$ implies that
as was to be shown. $\square$
3.2-2. Prove equation (3.16).
3.2-3. Prove equation (3.19). Also prove that $n! = \omega(2^n)$... | {
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It follows that $n! = o(n^n)$.
Observe also that
whenever $n \geq 1$. Since $(n/2e)^n \to \infty$ as $n \to \infty$, we can find, for each constant $c>0$, a positive integer $N_c$ such that $n \geq N_c$ implies
Therefore,
whenever $n \geq \max(n_c, 1)$. Since the choice of $c$ was arbitrary, it follows that $n! = \... | {
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Since $k \log k \geq k$ for all $k \geq 2$, we see that
for all $k \geq 2$. It follows that
for all $n \geq n_0$ and $k \geq 2$. Now, if $% $, then $% $. This cannot hold if $n \geq \max(n_0, 1/c_1)$, and so we conclude that
whenever $n \geq \max(n_0,1/c_1)$.
Now, $k \geq \log k$ for all $k \geq 1$, and so
for all... | {
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for all $n \geq N_{C,p}$.
We fix a positive constant $c$. Observe that, for each $1 \leq i \leq d$,
whenever $n \geq N_{2(d+1)\vert a_i \vert/c, k-i}$. Moreover,
whenever $n \geq \vert 2(d+1) a_0 / c \vert^{1/k}$.
Setting
we see that, for each $n \geq N$,
Since the choice of $c > 0$ was arbitrary, it follows that... | {
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for each $n \geq 1$ and every $\epsilon > 0$.
An argument analogous to the above shows that
for all integers $n \geq 1$. Therefore,
for all $i \geq 1$ and $n \geq 1$, and so it suffices to show that
for an arbitrary $\epsilon > 0$.
To this end, we fix a positive constant $C$. Since $n^{\epsilon/2} \to \infty$ as $... | {
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3-3. Ordering by asymptotic growth rates
a. Rank the following functions by order of growth; that is, find an arrangement $g_1,g_2,\ldots,g_{30}$ of the functions satisfying $g_1 = \Omega(g_2)$, $g_2 = \Omega(g_3)$, $\ldots$, $g_{29} = \Omega(g_{30})$. Partition your list into equivalence classes such that functions $... | {
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Problem 3-4e
$f(n) = \frac{1}{n}$ is a counterexample. $\square$
Problem 3-4f
This follows at once from the definitions of $O$ and $\Omega$. $\square$
Problem 3-4g
$f(n) = \sin n$ is a counterexample. To see this, we observe that
which is not asymptotically equivalent to $\sin \frac{n}{2}$. $\square$
Problem 3-4... | {
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Some authors define $\tilde{O}$ (read “soft-oh”) to mean $O$ with logarithmic factors ignored:
d. Define $\tilde{\Omega}$ and $\tilde{\Theta}$ in a similar manner. Prove the corresponding analog to Theorem 3.1.
Problem 3-5a
We assume that $f(n) \notin O(g(n))$. This means that each $C > 0$ and every $N > 0$ admits a... | {
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Problem 3-5d
We define
% and
If $f(n) = \tilde{\Theta}(g(n))$, then we can find positive constants $c_1$, $c_2$, $k_2$, $k_2$, and $n_0$ such that
for all $n \geq n_0$. This, in particular, implies that
for all $n \geq n_0$, and so $f(n) = \tilde{\Omega}(g(n))$. Similarly,
for all $n \geq n_0$, and so $f(n) = \ti... | {
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Since $g$ and $g^{-1}$ are monotonically increasing, we see that
and that
It follows that $f_c^*(n) = i$. $\square$
Problem 3-6c
$f^{(i)}(n) = n/2^i$, and so $f_c^*(n) = \lceil \log n \rceil$. $\square$
Problem 3-6d
$f^{(i)}(n) = n^{1/2^i}$, and so $f_c^*(n) = \lceil \log n \rceil - 1$. $\square$
Problem 3-6e
$... | {
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To generalize the discrete Euler–Mclaurin summation formula, we introduce the Bernoulli polynomials
which constitute the coefficients of the Taylor series
Differentiating the Taylor series term-by-term, we see that
for each $m > 1$ and every $x \in \mathbb{R}$.
We now observe that integrating a function by parts wi... | {
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Knuth (TAOCP vol.1, sec.1.2.11.3) gives a refined estimate:
Ramanujan distributions
Recall from the previous section that the $Q$-distribution is
The term-by-term asymptotic estimate (Sedgewick/Flajolet, Theorem 4.4) is as follows:
Closely related to the $Q$-distribution is the $R$-distribution
which is also asymp... | {
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Under what circumstances can one divide by a variable?
In light of recent responses to my other questions, I would like to know when it is mathematically acceptable to undergo division by a variable or a function of a variable, i.e., $x$ or $\cos x$. From what I sort of understand, it is only acceptable to divide by $... | {
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• So, if you know that $x \neq (2n-1)\frac{\pi}{2}$ then you can safely divide by $\cos x$, etc... if you know that $x \neq 1$ then you can safely divide by $\ln x$, and so on. – Zain Patel Aug 17 '16 at 0:38
• See this previous question: Why should you never divide both sides by a variable when solving an equation? In... | {
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$x^2 + 4 = 4x$ ... and so on....
2) Okay... second example. Suppose you are asked to solve:
$x^4 -3x^2 -4 = 4x^3 - 4x$. And you decide to factor both sides:
$x^4 -3x^2 - 4 = 4x^3 - 4x$
$(x^2 + 4)(x^2 - 1) = 4x(x^2 - 1)$
And now at this point you realllllly want to divide both sides by $x^2 - 1$. Can you? Well, you... | {
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$(x^2 + 4)(x^2 + 1) = 4x(x^2 + 1)$
$(x^2+4)\frac{x^2 + 1}{x^2 + 1}=4x\frac{x^2 + 1}{x^2 + 1}$
$x^2 + 4 = 4x$
$x^2 - 4x + 4 = 0$
$x = \frac {2 \pm \sqrt{2^2 - 4*1}}{2*1} = 2$.
• I upvoted this because it covers the essential cases as I see them. I would like the answer even better if it reached the same conclusions... | {
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First things first: I think you meant $n \in \mathbb{Z}$ instead of $n \in \mathbb{R}$ for $x= \frac{(2n-1)\pi}{2}$. $\mathbb{R}$ stands for real numbers. This are numbers like 1 and 2, but also $\frac{2}{3}$ and $\pi$. $\mathbb{Z}$ stands for integers. So $1 \in \mathbb{Z}$ and $1 \in \mathbb{R}$, but $\pi \notin \mat... | {
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So we can calculate the value of $h(x)$ except for $x = -1$ or $x = -5$, where it is undefined.
Hope it helped you a bit. Please not that I'm not a professional mathematician, so please ask someone else too. The last thing I want is you getting a bad grade because of my answer (I'm not responsible for that. $:)$). If ... | {
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## Moment Of Inertia Of T Section Problems | {
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Then equations (4. Since the section modulus depends on the value of the moment of inertia, an efficient beam must have most of its material located as far from. Centroid and Moment of Inertia/ Centre of Gravity/ Problems. Introduction: Problem 1 on Centroid & Moment of Inertia. I've already calculated the centroid wit... | {
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Problem 819 Determine the moment of inertia of the T-section shown in Fig. The SI unit for polar moment of inertia, like the area moment of inertia, is metre to the fourth power (m 4) Application. Moment of Inertia About Y-axis. Then this moment of inertia is transferred about the axis passing through the centroid of t... | {
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axis. (yield moment is when steel is starting. The separation points on that area moments. Calculate Yield Moment for the Beam Assume concrete accepts no tension. Where, I = second moment of inertia about the line k. Where: J M = Polar Mass Moment of Inertia (in-lbs-sec 2, Kg-m-sec 2). This means that the portion of li... | {
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The moment of inertia of a rod of mass and length about an axis, perpendicular to its length, which passes through its midpoint is. Then equations (4. W - T = m*a W-T = m*T*L^2/(16*I) The Attempt at a Solution Well, in my attempting of solving the problem, I considered that the moment of inertia utilized in these equat... | {
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and Rotational Inertia 2 Torque Torque is the rotational equivalence of force. dA = the area of the elementary portion. 7(4): 193-197 (Apr. In general, moment of inertia is just a measure of how hard it is to get something rotating. – The cylinder is cut into infinitesimally thin rings centered at the middle. Dt i th t... | {
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rho from the center, and it's going to have a thickness or width of D rho. In general, moment of inertia is just a measure of how hard it is to get something rotating. I need to calculate the change in moment of intertia due to modifing a simple angled beam from 120 x 120 x 10 to 120 x 112 x 10. The moment of inertia i... | {
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moment of inertia of a 12-cylinder V-engine having a piston–crank mechanism with main and auxiliary connecting rods. Simply Supported Beam|calculate support reaction|Engineering. Dear Friends, this session helps you to to find the centroid of composite areas, especially a T Section ,understand from basics, step by step... | {
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Friends, this session helps you to to find the centroid of composite areas, especially a T Section ,understand from basics, step by step tutorial on Moment of Inertia. Centroid, Area, Moments of Inertia, Polar Moments of Inertia, & Radius of Gyration of a Complex Cross-Sectional Area. See next page for section properti... | {
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"A body maintains the current state of motion unless acted upon some external force". Inventor seems to handle. This ratio of moment of inertia was also obtained by Graham and Rodriguez (1952) and cited in Roberts et al. The moment of inertia, otherwise known as the mass moment of inertia, angular mass or rotational in... | {
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f t is the tensile strength of concre te. Angular momentum. Moment of inertia can be defined by the equation The moment of inertia is the sum of the masses of the particles making up the object multiplied by their respective distances squared from the axis of rotation. The Inertia object has a density property that is ... | {
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has a function for moments but it rotates the center plane to something like VxV in the link below. Problem on Moment of Inertia of T Section - Moment of Inertia - Strength of Materials - Duration: 17:24. Torsional stress: where T is the torque, r is the radius and J x is the polar moment of area. Wallace Torque or Tor... | {
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the moment of inertia and radius of gyration with respect to a polar centroidal axis of the cross section of a hollow tube whose outside diameter is 6 in. The object in the diagram below consists of five thin cylinders arranged in a circle. c) The moment of inertia of the bar will be higher about the axis at the end of... | {
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9. Then this moment of inertia is transferred about the axis passing through the centroid of the given section, using theorem of parallel axis. Problem on Moment of Inertia of T Section - Moment of Inertia - Strength of Materials - Duration: 17:24. It will help in deciding whether the failure will be on the compression... | {
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from an axis: I = ∫ Q r 2 d m {\displaystyle I=\textstyle \int _ {Q}r^ {2. The ability to resist bending = I/y. 1) Today s Objectives: Students will be able to Determine the mass moment of inertia of a rigid body or a system of rigid bodies. Moment of Inertia Formula for Plane Figures - Moment of. It's trivial to find:... | {
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also have high moments of inertia and can be just as hard to turn. The bending moment M applied to a cross-section is related with its moment of inertia with the following equation:. Sample Learning Goals. Centroid and Moment of Inertia/ Centre of Gravity/ Problems. Moment of inertia represents the body's tendency to r... | {
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me work through it. The formula for moment of inertia is the "sum of the product of mass" of each particle. However, it represents the bending stiffness of the structural member, i. The equations that they give for the rotational component give the shear STRESS in the bolts. I just can't get the moment of inertia for t... | {
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problem solving. Lecture Notes: Area-Moment. The moment of inertia is the quantitative measure of rotational inertia, just as in translational motion, and mass is the quantitative measure of linear inertia—that is, the more massive an object is, the more inertia it has, and the greater is its resistance to change in li... | {
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linear and rotational motion. (3) x is the distance from the y axis to an infinetsimal area dA. The method has been developed to evaluate the variable mass moment of inertia of a 12-cylinder V-engine having a piston–crank mechanism with main and auxiliary connecting rods. For example: The Second Rectangular Area Moment... | {
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needed in these problems. I = I ¯ + A d 2. Note: If you are lost at any point, please visit the beginner’s lesson or comment below. Hi, I'm in Year 12 and my teacher gave me a problem about moment of inertia, and it wants me to find the moment of inertia of a truck about its centre of mass and I'm given the following: ... | {
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on composite sections. That measurement is calculated based upon the distribution of mass within the object and the position of the axis, meaning that the same object can have very. In the same way that the larger the mass of an object the smaller the acceleration for a given force, the larger the moment of inertia of ... | {
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Determine the distance 'd' between the sections such that the centroidal moment of inertia about the x- and y-axis are equal. If we compare Figure to the way we wrote kinetic energy in Work and Kinetic Energy, $(\frac{1}{2}m{v}^{2})$, this suggests we have a new rotational variable to add to our list of our relations b... | {
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r = Distance from neutral axis to the outer most fibre = D/2. Polar Moment of Inertia of a Circle about its Center: Letting dA 2 d, the area of the dark-shaded ring in Fig. Physically, the moment of inertia gives the ability of a body to resist angular acceleration (analogous to how mass resists linear acceleration) wh... | {
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PowerPoint PPT presentation (displayed as a Flash slide show) on PowerShow. Problem (10-110) Page 57 Determine the mass moment of inertia of the thin plate about an axis perpendicular to the page and passing through point O. Sample Problem 9. 5 2 3 A 4-0. Look up or calculate the moment of inertia and section modulus (... | {
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ΣM G = I Gα). Moment of Inertia Formula for Plane Figures - Moment of. 14 ratings • 1 review. Moment of Inertia. A good example of this is an I-Beam. Moment of Inertia. Ekeeda 174,723 views. The differential element, dA, is usually broken into two parts, dx and dy (dA = dx dy), which makes integration easier. This is a... | {
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Now consider a compound object such as that in , which depicts a thin disk at the end of a thin rod. Calculate the moment of inertia. The moment of inertia of a particle of mass m rotating about a particular point is given by: M o m e n t o f i n e r t i a = m d 2 \displaystyle\text {Moment of inertia}= {m} {d}^ {2} Mo... | {
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have a moment of inertia. Z = Elastic Section Modulus, in 3 or mm 3 Online Rectangular Angle Property Calculator Using the structural engineering calculator located at the top of the page (simply click on the the "show/hide calculator" button) the following properties can be calculated:. Properties of Sections. Find th... | {
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104 mm4. Moment of inertia, denoted by I, measures the extent to which an object resists rotational acceleration about a particular axis, and is the rotational analogue to mass. If we compare Figure to the way we wrote kinetic energy in Work and Kinetic Energy, $(\frac{1}{2}m{v}^{2})$, this suggests we have a new rotat... | {
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of its material ρ. Gross Moment Of Inertia T Beam April 12, 2018 - by Arfan - Leave a Comment Area moment of inertia typical cross sections i get consider the t beam shown in figure 1 get consider the t beam shown in figure 1 ing moment exle problems of design methods for beams and columns ering feed. Calculator for Mo... | {
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the greater is its resistance to change in linear velocity. Problem 821 Find the moment of inertia about the indicated x-axis for the shaded area shown in Fig. RE: Moment of Inertia - Tube with Slot GBor (Mechanical) 5 Aug 08 08:39 If this slot is sufficiently long, you also need to take a look at whether the edges of ... | {
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Therefore, the moment of inertia I x of the tee section, relative to non-centroidal x0-x0 axis, is determined like this:. The more far away from the axis, the more moment of inertia the object has. Calculating the moment of inertia for compound objects. which have solved the classification problem for 3 manifolds. Area... | {
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mast. Inertia can be thought of as another word for mass. A wagon wheel is constructed. 5 in c c A I k x x k xc 5. 5 2 3 A 4-0. 6 Review Recall from previous lesson the first moment about y-axis The moment of inertia (or second moment) is the measure of the. Problem on Moment of Inertia of T Section - Moment of Inertia... | {
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circle and basically divide it by two to get the result for a semicircle. So based on that I have solved problems on composite sections. The objects resistance for being rotated is measured by area moment of inertia calculator. 2, Figure 7. Learn exactly what happened in this chapter, scene, or section of Rotational Dy... | {
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# Is $\mathbb R^3$ the square of some topological space?
The other day, I was idly considering when a topological space has a square root. That is, what spaces are homeomorphic to $X \times X$ for some space $X$. $\mathbb{R}$ is not such a space: If $X \times X$ were homeomorphic to $\mathbb{R}$, then $X$ would be pat... | {
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So, is there a space $X$ so that $X \times X$ is homeomorphic to $\mathbb{R}^3$?
• I'm wondering to what extent there is unique factorization of topological spaces relative to $\times$. $\mathbb{Q}$ is an idempotent (as is its complement in $\mathbb{R}$), but are there more interesting failures of UF involving connect... | {
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• I hope this fine illustration of the power of relative homology will find its way in a textbook or, meanwhile, in algebraic topology courses. Apr 2, 2011 at 19:40
• I have a question regarding the top answer given by Tyler Lawson. As far as I know you can only apply the relative version of the Kunneth formula to cofi... | {
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A linear map $$\Bbb R^n \to \Bbb R^n$$ can be understood to preserve or reverse orientation, depending on whether its determinant is $$+1$$ or $$-1$$. This notion of orientation can be generalized to arbitrary homeomorphisms, giving a "degree" $$\deg(m)$$ for every homeomorphism which is $$+1$$ if it is orientation-pre... | {
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• I hope no one misses this nice alternative proof because it's behind a link. Apr 4, 2011 at 2:24
• Quoting from the link: "The paper also refers to an earlier paper ("The cartesian product of a certain nonmanifold and a line is E4", R.H. Bing, Annals of Mathematics series 2 vol 70 1959 pp. 399–412) which constructs a... | {
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• Would you care to elaborate? Jan 19, 2013 at 5:09
• All things considered, perhaps "S" is not the best name for the topological space for this assertion. Jan 19, 2013 at 5:52
• @Terry Tao True enough, but in all honesty it's precisely the notational perversity that brought this to mind to begin with. Jan 19, 2013 at ... | {
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• You say that $\chi_c$ is "available for any reasonable space" - does the argument above genuinely show that $\mathbb{R}^{2k+1}$ is not a topological square, or just that it isn't the square of a "reasonable" space in the appropriate sense? Oct 4, 2020 at 18:41
• The answer assumes that the square root has a well defi... | {
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# Contradiction between integration by partial fractions and substitution
Integration by substitution:
$$\int \frac {dx}{x^2-1}$$ Let $x=\sec\theta$ and $dx=\sec\theta\tan\theta \,d\theta$ $$\int \frac {dx}{x^2-1} = \int \frac{\sec\theta\tan\theta \,d\theta}{\sec^2\theta-1} = \int \frac {\sec\theta\tan\theta\, d\thet... | {
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Let's try to clear out some of the confusion on what is going on here.
First of all, $x\mapsto \frac 1 {x^2 - 1}$ is continuous function everywhere except for $x=\pm 1$, so it is Riemann integrable on any segment not containing $\pm 1$. That said, we would very much like to find primitive function defined on $\mathbb ... | {
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• Very well explained! Jul 20, 2015 at 21:56
• Well. Your simplification which equated the result of the two integrations depends on the assumption of $|x|>1$ and that's fine, since for $|x|<1$ this equation can't hold. I found many problems when using trigonometric substitution in integration and don't know exactly ho... | {
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• Without absolute value, what you wrote is right. but when you deal with absolute value you can't do this way. Jul 20, 2015 at 20:51
• Thank you Michael for editing. Now it looks so beautiful. Please give me a link to the tutorial where i can learn to write equations and mathematical symbols on internet. Jul 20, 2015 ... | {
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• Without absolute value, what you wrote is right. but when you deal with absolute value you can't do this way. Jul 20, 2015 at 20:52
• Correct .... This is not a complete answer. It works when $x>1$. But with this information, the original poster should be able to figure everything out. Jul 20, 2015 at 20:55
• Exactly... | {
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# A prime number problem.
If $n$ is a positive integer and $(p_1,p_2,p_3,p_4,\ldots, p_n)$ are distinct positive primes, show that the integer $(p_1\cdot p_2\cdot p_3\cdot p_4\cdots p_n)+1$ is divisible by none of these primes. How do I figure out that none of the primes divide that new integer?
• Start with a simple... | {
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Proof. If $x < y$ then we must have $q = 0$ and $r = x$, so in this case the $q, r$ exist and are unique. Assume that there is a natural number $x \geq y$ such that no such $q, r$ exist, then without loss of generality we may assume that $x$ is the smallest such number. However, then $x - y$ cannot be written in the de... | {
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"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9852713874477227,
"lm_q1q2_score": 0.8432456397760394,
"lm_q2_score": 0.8558511396138366,
"openwebmath_perplexity": 199.20457504441737,
"openwebmath_score": 0.9677867293357849,
"ta... |
$\Bbb {R^R}$ equinumerous to $\{f\in\Bbb{R^R}\mid f\text{ surjective}\}$
In an introductory course on axiomatic set theory, I am asked whether $\Bbb {R^R}$ is equinumerous to the set of all surjective functions in $\Bbb{R^R}$ (say $\mathcal S$). Because of the evocative nature of the question, I assume this to be true... | {
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"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9852713848528318,
"lm_q1q2_score": 0.8432456393664784,
"lm_q2_score": 0.8558511414521923,
"openwebmath_perplexity": 102.4125726070227,
"openwebmath_score": 0.9540027379989624,
"tag... |
It suffices to divide $$\mathbb R$$ into two sets $$A$$ and $$B$$ such that $$|A|=|B|=|\mathbb R|$$.
This means you have some bijection $$\varphi$$ from $$A$$ to $$B$$. I.e., you have pairs $$a\in A$$ and $$\varphi(a)\in B$$
Using these two subsets you can construct bijections $$\mathbb R\to\mathbb R$$ by choosing so... | {
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"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9852713848528318,
"lm_q1q2_score": 0.8432456393664784,
"lm_q2_score": 0.8558511414521923,
"openwebmath_perplexity": 102.4125726070227,
"openwebmath_score": 0.9540027379989624,
"tag... |
And you could find other similar questions.
Remark on use of Axiom of Choice. In connection with another question on this site, it might be also interesting whether we can do this in ZF. It's not difficult to see that we might replace $$\mathbb R$$ with $$\mathbb R\setminus\{0\}$$. (Since we can write down an explicit... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9852713848528318,
"lm_q1q2_score": 0.8432456393664784,
"lm_q2_score": 0.8558511414521923,
"openwebmath_perplexity": 102.4125726070227,
"openwebmath_score": 0.9540027379989624,
"tag... |
prove that for all x : $ae^{\frac{x}{a}} + be^{-\frac{x}{b}}\leq e^{\frac{x^2}{8a^2b^2}}$ if a + b = 1
I am trying to solve the following question, but I did not reach to any answer, I would be sol glad if anyone could help me on that.
If a, b are positive numbers such that a + b = 1 prove that for all x,
$ae^{\frac... | {
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"openwebmath_perplexity": 117.82262669871962,
"openwebmath_score": 0.909011960029602,
... |
It looks to be neat but true. Denote $x=abt$, we rewrite inequality as $ae^{bt}+be^{-at}\leqslant e^{t^2/8}$. We may suppose $t>0$, else replace $t$ to $-t$ and $b$ to $a$. Multiply by $e^{at}$ and rewrite as $ae^t+1-a\leqslant e^{at+t^2/8}$. Fix $t$ and vary $a\in [0,1]$. We should consider the minimal value of $H(a)=... | {
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"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9852713831229043,
"lm_q1q2_score": 0.8432456378859178,
"lm_q2_score": 0.8558511414521922,
"openwebmath_perplexity": 117.82262669871962,
"openwebmath_score": 0.909011960029602,
... |
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